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  • Articles  (186)
  • global optimization  (110)
  • 65F10  (76)
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  • 2000-2004  (19)
  • 1995-1999  (62)
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  • Mathematics  (186)
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  • Articles  (186)
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  • 1
    Electronic Resource
    Electronic Resource
    Concave minimization via conical partitions and polyhedral outer approximation (1991)
    Springer
    Mathematical programming 50 (1991), S. 259-274 
    Details
    ISSN: 1436-4646
    Keywords: Concave minimization ; global optimization ; nonlinear programming ; nonconvex programming ; branch-and-bound ; outer approximation ; cutting planes
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract An algorithm is proposed for globally minimizing a concave function over a compact convex set. This algorithm combines typical branch-and-bound elements like partitioning, bounding and deletion with suitably introduced cuts in such a way that the computationally most expensive subroutines of previous methods are avoided. In each step, essentially only few linear programming problems have to be solved. Some preliminary computational results are reported.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01594938
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  • 2
    Electronic Resource
    Electronic Resource
    Using copositivity for global optimality criteria in concave quadratic programming problems (1993)
    Springer
    Mathematical programming 62 (1993), S. 575-580 
    Details
    ISSN: 1436-4646
    Keywords: Non-convex non-linear programming ; global optimization ; second-order optimality conditions ; copositive matrices ; quadratic programming ; convex maximization problem
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract In this note we specify a necessary and sufficient condition for global optimality in concave quadratic minimization problems. Using this condition, it follows that, from the perspective of worst-case complexity of concave quadratic problems, the difference between local and global optimality conditions is not as large as in general. As an essential ingredient, we here use theε-subdifferential calculus via an approach of Hiriart-Urruty and Lemarechal (1990).
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01585185
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  • 3
    Electronic Resource
    Electronic Resource
    Pure adaptive search in global optimization (1992)
    Springer
    Mathematical programming 53 (1992), S. 323-338 
    Details
    ISSN: 1436-4646
    Keywords: Random search ; Monte Carlo optimization ; global optimization ; complexity
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract Pure adaptive seach iteratively constructs a sequence of interior points uniformly distributed within the corresponding sequence of nested improving regions of the feasible space. That is, at any iteration, the next point in the sequence is uniformly distributed over the region of feasible space containing all points that are strictly superior in value to the previous points in the sequence. The complexity of this algorithm is measured by the expected number of iterations required to achieve a given accuracy of solution. We show that for global mathematical programs satisfying the Lipschitz condition, its complexity increases at mostlinearly in the dimension of the problem.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01585710
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  • 4
    Electronic Resource
    Electronic Resource
    Linear multiplicative programming (1992)
    Springer
    Mathematical programming 56 (1992), S. 51-64 
    Details
    ISSN: 1436-4646
    Keywords: Non-convex quadratic programming problem ; global optimization ; parametric simplex algorithm
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract An algorithm for solving a linear multiplicative programming problem (referred to as LMP) is proposed. LMP minimizes the product of two linear functions subject to general linear constraints. The product of two linear functions is a typical non-convex function, so that it can have multiple local minima. It is shown, however, that LMP can be solved efficiently by the combination of the parametric simplex method and any standard convex minimization procedure. The computational results indicate that the amount of computation is not much different from that of solving linear programs of the same size. In addition, the method proposed for LMP can be extended to a convex multiplicative programming problem (CMP), which minimizes the product of two convex functions under convex constraints.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01580893
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  • 5
    Electronic Resource
    Electronic Resource
    Deciding uniqueness in norm maximization (1992)
    Springer
    Mathematical programming 57 (1992), S. 203-214 
    Details
    ISSN: 1436-4646
    Keywords: 90C30 ; 68Q15 ; 90C20 ; 90C25 ; 52A25 ; global optimization ; quadratic programming ; unique optimum ; polytope ; parallelotope ; norm ; NP-hardness ; diameter ; width ; inradius ; circumradius
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract NP-hardness is established for the problem whose instance is a system of linear inequalities defining a polytopeP, and whose question is whether, onP, the global maximum of the Euclidean norm is attained at more than one vertex ofP. The NP-hardness persists even for the restricted problem in whichP is a full-dimensional parallelotope with one vertex at the origin. This makes it possible to establish NP-hardness for other uniqueness problems, including some from pseudoboolean programming and computational convexity.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01581081
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  • 6
    Electronic Resource
    Electronic Resource
    Global Minimization of Increasing Positively Homogeneous Functions over the Unit Simplex (2000)
    Springer
    Annals of operations research 98 (2000), S. 171-187 
    Details
    ISSN: 1572-9338
    Keywords: global optimization ; cutting angle method ; increasing positively homogeneous function
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Economics
    Notes: Abstract In this paper we study a method for global optimization of increasing positively homogeneous functions over the unit simplex, which is a version of the cutting angle method. Some properties of the auxiliary subproblem are studied and a special algorithm for its solution is proposed. A cutting angle method based on this algorithm allows one to find an approximate solution of some problems of global optimization with 50 variables. Results of numerical experiments are discussed.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1019204407420
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  • 7
    Electronic Resource
    Electronic Resource
    Minimum concave-cost network flow problems: Applications, complexity, and algorithms (1990)
    Springer
    Annals of operations research 25 (1990), S. 75-99 
    Details
    ISSN: 1572-9338
    Keywords: Concave-cost network flow ; global optimization ; complexity theory ; NP-hard transportation problems
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Economics
    Notes: Abstract We discuss a wide range of results for minimum concave-cost network flow problems, including related applications, complexity issues, and solution techniques. Applications from production and inventory planning, and transportation and communication network design are discussed. New complexity results are proved which show that this problem is NP-hard for cases with cost functions other than fixed charge. An overview of solution techniques for this problem is presented, with some new results given regarding the implementation of a particular branch-and-bound approach.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02283688
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  • 8
    Electronic Resource
    Electronic Resource
    An intelligent decision support system for assisting industrial wastewater management (1995)
    Springer
    Annals of operations research 58 (1995), S. 455-477 
    Details
    ISSN: 1572-9338
    Keywords: Environment-economy integration ; industrial waste management ; “black box” systems design ; global optimization ; decision support systems
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Economics
    Notes: Abstract Sustainable economic development requires the inclusion of environmental factors in the decision making procedure. The generic objective of the Environmentally Sensitive Investment System (ESIS) Project is to provide industry and governmental departments or agencies with a tool to assess the technical and economic implications of capital-intensive projects, in response to stated environmental policies. More specifically, the ESIS prototype helps to find wastewater management alternatives that meet given environmental regulatory standards in a technologically sound and cost-efficient manner. The use of this decision support system will enhance the ability of managers and planners to explore the quantitative implications of a wide range of options. ESIS incorporates a combination of artificial intelligence and operations research techniques, database management and visualization tools, integrated under a graphical user interface. The ESIS prototype runs on top-of-the-line personal computers.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02032381
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  • 9
    Electronic Resource
    Electronic Resource
    The continuous reactive tabu search: Blending combinatorial optimization and stochastic search for global optimization (1996)
    Springer
    Annals of operations research 63 (1996), S. 151-188 
    Details
    ISSN: 1572-9338
    Keywords: Tabu search ; local search ; approximate algorithms ; heuristics ; global optimization ; stochastic minimization ; hybrid algorithms
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Economics
    Notes: Abstract A novel algorithm for the global optimization of functions (C-RTS) is presented, in which a combinatorial optimization method cooperates with a stochastic local minimizer. The combinatorial optimization component, based on the Reactive Tabu Search recently proposed by the authors, locates the most promising “boxes”, in which starting points for the local minimizer are generated. In order to cover a wide spectrum of possible applications without user intervention, the method is designed with adaptive mechanisms: the box size is adapted to the local structure of the function to be optimized, the search parameters are adapted to obtain a proper balance of diversification and intensification. The algorithm is compared with some existing algorithms, and the experimental results are presented for a variety of benchmark tasks.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02125453
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  • 10
    Electronic Resource
    Electronic Resource
    A Continuous Genetic Algorithm Designed for the Global Optimization of Multimodal Functions (2000)
    Springer
    Journal of heuristics 6 (2000), S. 191-213 
    Details
    ISSN: 1572-9397
    Keywords: genetic algorithm ; global optimization ; continuous variables
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Genetic algorithms are stochastic search approaches based on randomized operators, such as selection, crossover and mutation, inspired by the natural reproduction and evolution of the living creatures. However, few published works deal with their application to the global optimization of functions depending on continuous variables. A new algorithm called Continuous Genetic Algorithm (CGA) is proposed for the global optimization of multiminima functions. In order to cover a wide domain of possible solutions, our algorithm first takes care over the choice of the initial population. Then it locates the most promising area of the solution space, and continues the search through an “intensification” inside this area. The selection, the crossover and the mutation are performed by using the decimal code. The efficiency of CGA is tested in detail through a set of benchmark multimodal functions, of which global and local minima are known. CGA is compared to Tabu Search and Simulated Annealing, as alternative algorithms.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1009626110229
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  • 11
    Electronic Resource
    Electronic Resource
    CHESS—Changing Horizon Efficient Set Search: A Simple Principle for Multiobjective Optimization (2000)
    Springer
    Journal of heuristics 6 (2000), S. 405-418 
    Details
    ISSN: 1572-9397
    Keywords: MOCO ; multiobjective optimization ; global optimization ; local search methods ; metaheuristics ; quad-trees
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract This paper presents a new concept for generating approximations to the non-dominated set in multiobjective optimization problems. The approximation set A is constructed by solving several single-objective minimization problems in which a particular function D(A, z) is minimized. A new algorithm to calculate D(A, z) is proposed. No general approach is available to solve the one-dimensional optimization problems, but metaheuristics based on local search procedures are used instead. Tests with multiobjective combinatorial problems whose non-dominated sets are known confirm that CHESS can be used to approximate the non-dominated set. Straightforward parallelization of the CHESS approach is illustrated with examples. The algorithm to calculate D(A, z) can be used in any other applications that need to determine Tchebycheff distances between a point and a dominant-free set.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1009638700683
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  • 12
    Electronic Resource
    Electronic Resource
    An explicit characterization of the convex envelope of a bivariate bilinear function over special polytopes (1990)
    Springer
    Annals of operations research 25 (1990), S. 197-209 
    Details
    ISSN: 1572-9338
    Keywords: Convex envelopes ; bilinear functions ; bilinear programming problems ; global optimization
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Economics
    Notes: Abstract In this paper, we constructively derive an explicit characterization of the convex envelope of a bilinear function over a special type of polytope in ℝ2. Our motivation stems from the use of such functions for deriving strengthened lower bounds within the context of a branch-and-bound algorithm for solving bilinear programming problems. For the case of polytopes with no edges having finite positive slopes, that is polytopes with “downward” sloping edges (which we call D-polytopes), we obtain a direct, explicit characterization of the convex envelope. This case subsumes the analysis of Al-Khayyal and Falk (1983) for constructing the convex envelope of a bilinear function over a rectangle in ℝ2. For non-D-polytopes, the analysis is more complex. We propose three strategies for this case based on (i) encasing the region in a D-polytope, (ii) employing a discretization technique, and (iii) providing an explicit characterization over a triangle along with a triangular decomposition approach. The analysis is illustrated using numerical examples.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02283695
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  • 13
    Electronic Resource
    Electronic Resource
    Interval Newton/generalized bisection when there are singularities near roots (1990)
    Springer
    Annals of operations research 25 (1990), S. 181-196 
    Details
    ISSN: 1572-9338
    Keywords: Nonlinear algebraic systems ; Newton's method ; interval arithmetic ; Gauss-Seidel method ; global optimization ; singularities
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Economics
    Notes: Abstract Interval Newton methods in conjunction with generalized bisection are important elemetns of algorithms which find theglobal optimum within a specified box X ⊂ ℝn of an objective function ϕ whose critical points are solutions to the system of nonlinear equationsF(X)=0with mathematical certainty, even in finite presision arithmetic. The overall efficiency of such a scheme depends on the power of the interval Newton method to reduce the widths of the coordinate intervals of the box. Thus, though the generalized bisection method will still converge in a box which contains a critical point at which the Jacobian matrix is singular, the process is much more costly in that case. Here, we propose modifications which make the generalized bisection method isolate singular solutions more efficiently. These modifications are based on an observation about the verification property of interval Newton methods and on techniques for detecting the singularity and removing the region containing it. The modifications assume no special structure forF. Additionally, one of the observations should also make the algorithm more efficient when finding nonsingular solutions. We present results of computational experiments.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02283694
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  • 14
    Electronic Resource
    Electronic Resource
    Automatic domain decomposition on unstructured grids (DOUG) (1998)
    Springer
    Advances in computational mathematics 9 (1998), S. 281-310 
    Details
    ISSN: 1572-9044
    Keywords: partial differential equations ; finite elements ; unstructured grids ; parallel solver ; domain decomposition ; additive Schwarz ; MPI ; general purpose software ; 65F10 ; 65N22 ; 65N55
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract This paper describes a parallel iterative solver for finite element discretisations of elliptic partial differential equations on 2D and 3D domains using unstructured grids. The discretisation of the PDE is assumed to be given in the form of element stiffness matrices and the solver is automatic in the sense that it requires minimal additional information about the PDE and the geometry of the domain. The solver parallelises matrix–vector operations required by iterative methods and provides parallel additive Schwarz preconditioners. Parallelisation is implemented through MPI. The paper contains numerical experiments showing almost optimal speedup on unstructured mesh problems on a range of four platforms and in addition gives illustrations of the use of the package to investigate several questions of current interest in the analysis of Schwarz methods. The package is available in public domain from the home page http://www.maths.bath.ac.uk/∼mjh/.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1018997725374
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  • 15
    Electronic Resource
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    A quasi-wavelet algorithm for second kind boundary integral equations (1999)
    Springer
    Advances in computational mathematics 11 (1999), S. 355-375 
    Details
    ISSN: 1572-9044
    Keywords: periodic quasi-wavelet ; integral equation ; multiscale ; 45G10 ; 65F10 ; 65J15 ; 65N30
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract In solving integral equations with a logarithmic kernel, we combine the Galerkin approximation with periodic quasi-wavelet (PQW) [4]. We develop an algorithm for solving the integral equations with only O(N log N) arithmetic operations, where N is the number of knots. We also prove that the Galerkin approximation has a polynomial rate of convergence.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1018992413504
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  • 16
    Electronic Resource
    Electronic Resource
    A note on best conditioned preconditioners (1994)
    Springer
    BIT 34 (1994), S. 313-317 
    Details
    ISSN: 1572-9125
    Keywords: 65F10 ; 65F15 ; 65F35 ; Toeplitz matrix ; circulant matrix ; best conditioned preconditioner ; preconditioned conjugate gradient method
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract We discuss the solution of Hermitian positive definite systemsAx=b by the preconditioned conjugate gradient method with a preconditionerM. In general, the smaller the condition numberκ(M −1/2 AM −1/2 ) is, the faster the convergence rate will be. For a given unitary matrixQ, letM Q = {Q*Λ N Q | Λ n is ann-by-n complex diagonal matrix} andM Q + ={Q*Λ n Q | Λ n is ann-by-n positive definite diagonal matrix}. The preconditionerM b that minimizesκ(M −1/2 AM −1/2 ) overM Q + is called the best conditioned preconditioner for the matrixA overM Q + . We prove that ifQAQ* has Young's Property A, thenM b is nothing new but the minimizer of ‖M −A‖ F overM Q . Here ‖ · ‖ F denotes the Frobenius norm. Some applications are also given here.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01955879
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  • 17
    Electronic Resource
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    Towards a cost-effective ILU preconditioner with high level fill (1992)
    Springer
    BIT 32 (1992), S. 442-463 
    Details
    ISSN: 1572-9125
    Keywords: 65F10 ; 76S05 ; Minimum discarded fill (MDF) ; thresholdMDF ; minimum updating matrix ; matrix ordering ; preconditioned conjugate gradient
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract There has been increased interest in the effect of the ordering of the unknowns on Preconditioned Conjugate Gradient (PCG) methods. A recently proposed Minimum Discarded Fill (MDF) ordering technique is effective in finding goodILU(l) preconditioners, especially for problems arising from unstructured finite element grids. This algorithm can identify anisotropy in complicated physical structures and orders the unknowns in an appropriate direction. TheMDF technique may be viewed as an analogue of the minimum deficiency algorithm in sparse matrix technology, and hence is expensive to compute for high levelILU(l) preconditioners. In this work, several less expensive variants of theMDF technique are explored to produce cost-effectiveILU preconditioners. The ThresholdMDF ordering combinesMDF ideas with drop tolerance techniques to identify the sparsity pattern in theILU preconditioners. The Minimum Update Matrix (MUM) ordering technique is a simplification of theMDF ordering and is an analogue of the minimum degree algorithm. TheMUM ordering method is especially effective for large matrices arising from Navier-Stokes problems.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02074880
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  • 18
    Electronic Resource
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    Circulant preconditioners for second order hyperbolic equations (1992)
    Springer
    BIT 32 (1992), S. 650-664 
    Details
    ISSN: 1572-9125
    Keywords: 65F10 ; 65N22 ; Hyperbolic equation ; circulant matrix ; condition number ; preconditioned conjugate gradient method
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Linear systems arising from implicit time discretizations and finite difference space discretizations of second-order hyperbolic equations in two dimensions are considered. We propose and analyze the use of circulant preconditioners for the solution of linear systems via preconditioned iterative methods such as the conjugate gradient method. Our motivation is to exploit the fast inversion of circulant systems with the Fast Fourier Transform (FFT). For second-order hyperbolic equations with initial and Dirichlet boundary conditions, we prove that the condition number of the preconditioned system is ofO(α) orO(m), where α is the quotient between the time and space steps andm is the number of interior gridpoints in each direction. The results are extended to parabolic equations. Numerical experiments also indicate that the preconditioned systems exhibit favorable clustering of eigenvalues that leads to a fast convergence rate.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01994848
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  • 19
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    Fourier analysis of schwarz alternating methods for piecewise hermite bicubic orthogonal spline collocation (1993)
    Springer
    BIT 33 (1993), S. 634-646 
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    ISSN: 1572-9125
    Keywords: 65F10 ; 65N22 ; 65N35 ; 65N55 ; Dirichlet problem ; Poisson's equation ; piecewise Hermite cubics ; Gauss points ; eigenvalues ; eigenfunctions ; H 1-norm
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The rates of convergence of two Schwarz alternating methods are analyzed for the iterative solution of a discrete problem which arises when orthogonal spline collocation with piecewise Hermite bicubics is applied to the Dirichlet problem for Poisson's equation on a rectangle. In the first method, the rectangle is divided into two overlapping subrectangles, while three overlapping subrectangles are used in the second method. Fourier analysis is used to obtain explicit formulas for the convergence factors by which theH 1-norm of the errors is reduced in one iteration of the Schwarz methods. It is shown numerically that while these factors depend on the size of overlap, they are independent of the partition stepsize. Results of numerical experiments are presented which confirm the established rates of convergence of the Schwarz methods.
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    URL: http://dx.doi.org/10.1007/BF01990539
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  • 20
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    A numerical study of optimized sparse preconditioners (1994)
    Springer
    BIT 34 (1994), S. 177-204 
    Details
    ISSN: 1572-9125
    Keywords: 65F10 ; 15A06 ; 65F90 ; 65K10 ; Conjugate gradient method ; preconditioning ; incomplete factorization ; polynomial preconditioner ; matrix-free method ; Fourier analysis
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Preconditioning strategies based on incomplete factorizations and polynomial approximations are studied through extensive numerical experiments. We are concerned with the question of the optimal rate of convergence that can be achieved for these classes of preconditioners. Our conclusion is that the well-known Modified Incomplete Cholesky factorization (MIC), cf. e.g., Gustafsson [20], and the polynomial preconditioning based on the Chebyshev polynomials, cf. Johnson, Micchelli and Paul [22], have optimal order of convergence as applied to matrix systems derived by discretization of the Poisson equation. Thus for the discrete two-dimensional Poisson equation withn unknowns,O(n 1/4) andO(n 1/2) seem to be the optimal rates of convergence for the Conjugate Gradient (CG) method using incomplete factorizations and polynomial preconditioners, respectively. The results obtained for polynomial preconditioners are in agreement with the basic theory of CG, which implies that such preconditioners can not lead to improvement of the asymptotic convergence rate. By optimizing the preconditioners with respect to certain criteria, we observe a reduction of the number of CG iterations, but the rates of convergence remain unchanged.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01955867
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  • 21
    Electronic Resource
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    Block elimination with one refinement solves bordered linear systems accurately (1990)
    Springer
    BIT 30 (1990), S. 490-507 
    Details
    ISSN: 1572-9125
    Keywords: 15A06 ; 65F05 ; 65F10
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Linear systems with a fairly well-conditioned matrixM of the form $$\begin{array}{*{20}c} n \\ 1 \\ \end{array} \mathop {\left( {\begin{array}{*{20}c} A & b \\ c & d \\ \end{array} } \right)}\limits^{\begin{array}{*{20}c} n & 1 \\ \end{array} } $$ , for which a ‘black box’ solver forA is available, can be accurately solved by the standard process of Block Elimination, followed by just one step of Iterative Refinement, no matter how singularA may be — provided the ‘black box’ has a property that is possessed by LU- and QR-based solvers with very high probability. The resulting Algorithm BE + 1 is simpler and slightly faster than T.F. Chan's Deflation Method, and just as accurate. We analyse the case where the ‘black box’ is a solver not forA but for a matrix close toA. This is of interest for numerical continuation methods.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01931663
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  • 22
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    A class of modified block SSOR preconditioners for symmetric positive definite systems of linear equations (1999)
    Springer
    Advances in computational mathematics 10 (1999), S. 169-186 
    Details
    ISSN: 1572-9044
    Keywords: symmetric positive definite linear system ; block SSOR iteration ; preconditioner ; hierarchical basis discretization ; 65F10 ; 65N20 ; CR: G1.8
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract A class of modified block SSOR preconditioners is presented for solving symmetric positive definite systems of linear equations, which arise in the hierarchical basis finite element discretizations of the second order self‐adjoint elliptic boundary value problems. This class of methods is strongly related to two level methods, standard multigrid methods, and Jacobi‐like hierarchical basis methods. The optimal relaxation factors and optimal condition numbers are estimated in detail. Theoretical analyses show that these methods are very robust, and especially well suited to difficult problems with rough solutions, discretized using highly nonuniform, adaptively refined meshes.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1018974514896
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  • 23
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    Circulant and skew-circulant preconditioners for skew-hermitian type Toeplitz systems (1991)
    Springer
    BIT 31 (1991), S. 632-646 
    Details
    ISSN: 1572-9125
    Keywords: 65F10 ; 65F15 ; Skew-Hermitian type Toeplitz matrix ; circulant matrix ; skew-circulant matrix ; preconditioned conjugate gradient method
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract We study the solutions of Toeplitz systemsA n x=b by the preconditioned conjugate gradient method. Then ×n matrixA n is of the forma 0 I+H n wherea 0 is a real number,I is the identity matrix andH n is a skew-Hermitian Toeplitz matrix. Such matrices often appear in solving discretized hyperbolic differential equations. The preconditioners we considered here are the circulant matrixC n and the skew-circulant matrixS n whereA n =1/2(C n +S n ). The convergence rate of the iterative method depends on the distribution of the singular values of the matricesC −1 n An andS −1 n A n . For Toeplitz matricesA n with entries which are Fourier coefficients of functions in the Wiener class, we show the invertibility ofC n andS n and prove that the singular values ofC −1 n A n andS −1 n A n are clustered around 1 for largen. Hence, if the conjugate gradient method is applied to solve the preconditioned systems, we expect fast convergence.
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    URL: http://dx.doi.org/10.1007/BF01933178
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    Hartley preconditioners for Toeplitz systems generated by positive continuous functions (1994)
    Springer
    BIT 34 (1994), S. 367-371 
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    ISSN: 1572-9125
    Keywords: 65F10 ; 65F15 ; Topelitz matrix ; circulant matrix ; Hartley preconditioner ; conjugate gradient method ; generating function
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract In this paper, we consider the solution ofn-by-n symmetric positive definite Toeplitz systemsT n x=b by the preconditioned conjugate gradient (PCG) method. The preconditionerM n is defined to be the minimizer of ‖T n −B n ‖ F over allB n εH n whereH n is the Hartley algebra. We show that if the generating functionf ofT n is a positive 2π-periodic continuous even function, then the spectrum of the preconditioned systemM n −1 T n will be clustered around 1. Thus, if the PCG method is applied to solve the preconditioned system, the convergence rate will be superlinear.
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    URL: http://dx.doi.org/10.1007/BF01935646
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    Interval method for bounding level sets: Revisited and tested with global optimization problems (1990)
    Springer
    BIT 30 (1990), S. 650-657 
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    ISSN: 1572-9125
    Keywords: 90C30 ; 65K05 ; Inclusion function ; interval arithmetic ; level set ; global optimization
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract An interval method for bounding level sets, modified to increase its efficiency and to get sharper bounding boxes, is presented. The new algorithm was tested with standard global optimization test problems. The test results show that, while the modified method gives a more valuable, guaranteed reliability result set, it is competitive with non-interval methods in terms of CPU time and number of function evaluations.
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    URL: http://dx.doi.org/10.1007/BF01933213
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    Fast transforms for tridiagonal linear equations (1994)
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    BIT 34 (1994), S. 99-112 
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    ISSN: 1572-9125
    Keywords: 65F10 ; 65N06 ; Fourier Transform ; Sine Transform ; Toeplitz matrices ; elliptic differential equation ; finite difference method ; preconditioned conjugate gradient method
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract In this paper we study the use of the Fourier, Sine and Cosine Transform for solving or preconditioning linear systems, which arise from the discretization of elliptic problems. Recently, R. Chan and T. Chan considered circulant matrices for solving such systems. Instead of using circulant matrices, which are based on the Fourier Transform, we apply the Fourier and the Sine Transform directly. It is shown that tridiagonal matrices arising from the discretization of an onedimensional elliptic PDE are connected with circulant matrices by congruence transformations with the Fourier or the Sine matrix. Therefore, we can solve such linear systems directly, using only Fast Fourier Transforms and the Sherman-Morrison-Woodbury formula. The Fast Fourier Transform is highly parallelizable, and thus such an algorithm is interesting on a parallel computer. Moreover, similar relations hold between block tridiagonal matrices and Block Toeplitz-plus-Hankel matrices of ordern 2×n 2 in the 2D case. This can be used to define in some sense natural approximations to the given matrix which lead to preconditioners for solving such linear systems.
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    The Cross-Entropy Method for Combinatorial and Continuous Optimization (1999)
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    Methodology and computing in applied probability 1 (1999), S. 127-190 
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    ISSN: 1387-5841
    Keywords: combinatorial optimization ; global optimization ; importance sampling ; markov chain monte carlo ; simulated annealing ; simulation
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract We present a new and fast method, called the cross-entropy method, for finding the optimal solution of combinatorial and continuous nonconvex optimization problems with convex bounded domains. To find the optimal solution we solve a sequence of simple auxiliary smooth optimization problems based on Kullback-Leibler cross-entropy, importance sampling, Markov chain and Boltzmann distribution. We use importance sampling as an important ingredient for adaptive adjustment of the temperature in the Boltzmann distribution and use Kullback-Leibler cross-entropy to find the optimal solution. In fact, we use the mode of a unimodal importance sampling distribution, like the mode of beta distribution, as an estimate of the optimal solution for continuous optimization and Markov chains approach for combinatorial optimization. In the later case we show almost surely convergence of our algorithm to the optimal solution. Supporting numerical results for both continuous and combinatorial optimization problems are given as well. Our empirical studies suggest that the cross-entropy method has polynomial in the size of the problem running time complexity.
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    URL: http://dx.doi.org/10.1023/A:1010091220143
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    Modified block-approximate factorization strategies (1992)
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    Numerische Mathematik 61 (1992), S. 91-110 
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    ISSN: 0945-3245
    Keywords: 65F10 ; 65F35 ; 65B99
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary Two variants of modified incomplete block-matrix factorization with additive correction are proposed for the iterative solution of large linear systems of equations. Both rigorous theoretical support and numerical evidence are given displaying their efficiency when applied to discrete second order partial differential equations (PDEs), even in the case of quasi-singular problems.
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    Convergence rate estimate for a domain decomposition method (1992)
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    Numerische Mathematik 61 (1992), S. 153-169 
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    ISSN: 0945-3245
    Keywords: 65N30 ; 65F10
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary We provide a convergence rate analysis for a variant of the domain decomposition method introduced by Gropp and Keyes for solving the algebraic equations that arise from finite element discretization of nonsymmetric and indefinite elliptic problems with Dirichlet boundary conditions in ℝ2. We show that the convergence rate of the preconditioned GMRES method is nearly optimal in the sense that the rate of convergence depends only logarithmically on the mesh size and the number of substructures, if the global coarse mesh is fine enough.
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    An incomplete-factorization preconditioning using repeated red-black ordering (1992)
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    Numerische Mathematik 61 (1992), S. 433-454 
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    ISSN: 0945-3245
    Keywords: 65F10
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary The convergence of the conjugate gradient method for the iterative solution of large systems of linear equations depends on proper preconditioning matrices. We present an efficient incomplete-factorization preconditioning based on a specific, “repeated red-black” ordering scheme and cyclic reduction. For the Dirichlet model problem, we prove that the condition number increases asymptotically slower with the number of equations than for usual incomplete factorization methods. Numerical results for symmetric and non-symmetric test problems and on locally refined grids demonstrate the performance of this method, especially for large linear systems.
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    A breakdown-free Lanczos type algorithm for solving linear systems (1992)
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    Numerische Mathematik 63 (1992), S. 29-38 
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    ISSN: 0945-3245
    Keywords: 65F10 ; 65F25
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    Topics: Mathematics
    Notes: Summary Lanczos type algorithms for solving systems of linear equations have their foundations in the theory of formal orthogonal polynomials and the method of moments which leads to a determinantal formula for their iterates. The various Lanczos type algorithms mainly differ by the way of computing the coefficients entering into the recurrence formulae. If the denominator in the formula for one of these coefficients is zero, then a breakdown occurs in the algorithm, and it must be stopped. Such a breakdown is in fact due to the non-existence of some orthogonal polynomial. In this paper we show how to jump over such a singularity by computing the next existing orthogonal polynomial by the block bordering method. The resulting algorithm, called MRZ, is equivalent to the nongeneric BIODIR algorithm (which is a look-ahead Lanczos type algorithm), but our derivation is much simpler.
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    Galerkin-wavelet methods for two-point boundary value problems (1992)
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    Numerische Mathematik 63 (1992), S. 123-144 
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    ISSN: 0945-3245
    Keywords: 65N30 ; 65N13 ; 65F10
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary Anti-derivatives of wavelets are used for the numerical solution of differential equations. Optimal error estimates are obtained in the applications to two-point boundary value problems of second order. The orthogonal property of the wavelets is used to construct efficient iterative methods for the solution of the resultant linear algebraic systems. Numerical examples are given.
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    A fast solver for the first biharmonic boundary value problem (1992)
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    Numerische Mathematik 63 (1992), S. 297-313 
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    ISSN: 0945-3245
    Keywords: 65N30 ; 65F10
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    Topics: Mathematics
    Notes: Summary This paper provides a fast and storage-saving method for the solution of the first biharmonic boundary value problem (b.v.p.). The b.v.p. is approximated via a special variational finite difference technique suggested earlier by V.G. Korneev. It is shown theoretically that our method produces an approximate solution to the finite difference equations inO(NlnNlnɛ−1) arithmetical operations, whereN is the number of unknowns and ɛ (0〈ɛ〈1) denotes the relative accuracy required. The numerical results obtained by our computer code CGMFC decisively substantiate the theoretical estimates given.
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    H-Splittings and two-stage iterative methods (1992)
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    Numerische Mathematik 63 (1992), S. 345-356 
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    ISSN: 0945-3245
    Keywords: 65F10
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    Topics: Mathematics
    Notes: Summary Convergence of two-stage iterative methods for the solution of linear systems is studied. Convergence of the non-stationary method is shown if the number of inner iterations becomes sufficiently large. TheR 1-factor of the two-stage method is related to the spectral radius of the iteration matrix of the outer splitting. Convergence is further studied for splittings ofH-matrices. These matrices are not necessarily monotone. Conditions on the splittings are given so that the two-stage method is convergent for any number of inner iterations.
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    On a class of matrices which arise in the numerical solution of Euler equations (1992)
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    Numerische Mathematik 63 (1992), S. 411-431 
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    ISSN: 0945-3245
    Keywords: 65F10 ; 15A48
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    Topics: Mathematics
    Notes: Summary We study block matricesA=[Aij], where every blockA ij ∈ℂ k,k is Hermitian andA ii is positive definite. We call such a matrix a generalized H-matrix if its block comparison matrix is a generalized M-matrix. These matrices arise in the numerical solution of Euler equations in fluid flow computations and in the study of invariant tori of dynamical systems. We discuss properties of these matrices and we give some equivalent conditions for a matrix to be a generalized H-matrix.
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    Multilevel iterative methods for mixed finite element discretizations of elliptic problems (1992)
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    Numerische Mathematik 63 (1992), S. 503-520 
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    ISSN: 0945-3245
    Keywords: 65N30 ; 65F10
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    Topics: Mathematics
    Notes: Summary For solving second order elliptic problems discretized on a sequence of nested mixed finite element spaces nearly optimal iterative methods are proposed. The methods are within the general framework of the product (multiplicative) scheme for operators in a Hilbert space, proposed recently by Bramble, Pasciak, Wang, and Xu [5,6,26,27] and make use of certain multilevel decomposition of the corresponding spaces for the flux variable.
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    Multilevel Schwarz methods (1992)
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    Numerische Mathematik 63 (1992), S. 521-539 
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    ISSN: 0945-3245
    Keywords: 65F10 ; 65N30
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    Topics: Mathematics
    Notes: Summary We consider the solution of the algebraic system of equations which result from the discretization of second order elliptic equations. A class of multilevel algorithms are studied using the additive Schwarz framework. We establish that the condition number of the iteration operators are bounded independent of mesh sizes and the number of levels. This is an improvement on Dryja and Widlund's result on a multilevel additive Schwarz algorithm, as well as Bramble, Pasciak and Xu's result on the BPX algorithm. Some multiplicative variants of the multilevel methods are also considered. We establish that the energy norms of the corresponding iteration operators are bounded by a constant less than one, which is independent of the number of levels. For a proper ordering, the iteration operators correspond to the error propagation operators of certain V-cycle multigrid methods, using Gauss-Seidel and damped Jacobi methods as smoothers, respectively.
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    Schwarz alternating and iterative refinement methods for mixed formulations of elliptic problems, part II: Convergence theory (1993)
    Springer
    Numerische Mathematik 65 (1993), S. 469-492 
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    ISSN: 0945-3245
    Keywords: 65F10 ; 65N30 ; 65N55
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary In this paper we discuss bounds for the convergence rates of several domain decomposition algorithms to solve symmetric, indefinite linear systems arising from mixed finite element discretizations of elliptic problems. The algorithms include Schwarz methods and iterative refinement methods on locally refined grids. The implementation of Schwarz and iterative refinement algorithms have been discussed in part I. A discussion on the stability of mixed discretizations on locally refined grids is included and quantiative estimates for the convergence rates of some iterative refinement algorithms are also derived.
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    A multidomain spectral collocation method for the Stokes problem (1990)
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    Numerische Mathematik 58 (1990), S. 441-464 
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    ISSN: 0945-3245
    Keywords: AMS(MOS): 65P30 ; 65N35 ; 65F10 ; CR: G1.8
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary We propose a multidomain spectral collocation scheme for the approximation of the two-dimensional Stokes problem. We show that the discrete velocity vector field is exactly divergence-free and we prove error estimates both for the velocity and the pressure.
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    Local refinement techniques for elliptic problems on cell-centered grids (1991)
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    Numerische Mathematik 59 (1991), S. 431-452 
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    ISSN: 0945-3245
    Keywords: 65F10 ; 65N20 ; 65N30
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    Topics: Mathematics
    Notes: Summary Algebraic multilevel analogues of the BEPS preconditioner designed for solving discrete elliptic problems on grids with local refinement are formulated, and bounds on their relative condition numbers, with respect to the composite-grid matrix, are derived. TheV-cycle and, more generally,v-foldV-cycle multilevel BEPS preconditioners are presented and studied. It is proved that for 2-D problems theV-cycle multilevel BEPS is almost optimal, whereas thev-foldV-cycle algebraic multilevel BEPS is optimal under a mild restriction on the composite cell-centered grid. For thev-fold multilevel BEPS, the variational relation between the finite difference matrix and the corresponding matrix on the next-coarser level is not necessarily required. Since they are purely algebraically derived, thev-fold (v〉1) multilevel BEPS preconditioners perform without any restrictionson the shape of subregions, unless the refinement is too fast. For theV-cycle BEPS preconditioner (2-D problem), a variational relation between the matrices on two consecutive grids is required, but there is no restriction on the method of refinement on the shape, or on the size of the subdomains.
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    A posteriori error estimators for the Stokes equations II non-conforming discretizations (1991)
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    Numerische Mathematik 60 (1991), S. 235-249 
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    ISSN: 0945-3245
    Keywords: 65F10 ; 65N22 ; 76D05
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary We present an a posteriori error estimator for the non-conforming Crouzeix-Raviart discretization of the Stokes equations which is based on the local evaluation of residuals with respect to the strong form of the differential equation. The error estimator yields global upper and local lower bounds for the error of the finite element solution. It can easily be generalized to the stationary, incompressible Navier-Stokes equations and to other non-conforming finite element methods. Numerical examples show the efficiency of the proposed error estimator.
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    A domain decomposition algorithm for elliptic problems in three dimensions (1991)
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    Numerische Mathematik 60 (1991), S. 219-234 
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    ISSN: 0945-3245
    Keywords: 65F10 ; 65N30
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    Topics: Mathematics
    Notes: Summary Most domain decomposition algorithms have been developed for problems in two dimensions. One reason for this is the difficulty in devising a satisfactory, easy-to-implement, robust method of providing global communication of information for problems in three dimensions. Several methods that work well in two dimension do not perform satisfactorily in three dimensions. A new iterative substructuring algorithm for three dimensions is proposed. It is shown that the condition number of the resulting preconditioned problem is bounded independently of the number of subdomains and that the growth is quadratic in the logarithm of the number of degrees of freedom associated with a subdomain. The condition number is also bounded independently of the jumps in the coefficients of the differential equation between subdomains. The new algorithm also has more potential parallelism than the iterative substructuring methods previously proposed for problems in three dimensions.
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    Numerical stability of the cyclic Richardson iteration (1991)
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    Numerische Mathematik 60 (1991), S. 251-290 
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    ISSN: 0945-3245
    Keywords: 65F10
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary The success of the cyclic Richardson iteration depends on the proper ordering of the acceleration parameters. We give a rigorous error analysis to show that, with the proper ordering, the relative error in the iterative method, when properly terminated, is not larger than the error incurred in stable direct methods such as Cholesky factorization. For both the computed approximation $$\tilde u$$ tou=L −1f satisfies $$\left\| {u - \tilde u} \right\|$$ ≦ cond (L)‖u‖2−t and this bound is attainable. We also show that the residual norm $$\left\| {f - L\tilde u} \right\|$$ is bounded by ‖L‖ cond $$(L)\left\| {\tilde u} \right\|2^{ - t} $$ . This bound is attainable for a small cycle lengthN. Our analysis suggests that for a larger cycle lengthN the residuals are bounded by $$\sqrt {cond(L)} \left\| L \right\|\left\| {\tilde u} \right\|2^{ - t} $$ . We construct a theoretical example in which this bound is attainable. However we observed in all numerical tests that ultimately the residual norms were of order $$\left\| L \right\|\left\| {\tilde u} \right\|2^{ - t} $$ . We explain why in practice even the factor $$\sqrt {cond(L)} $$ is never encountered. Therefore the residual stopping criterion for the Richardson iteration appears to be very reliable and the method itself appears to be stable.
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    QMR: a quasi-minimal residual method for non-Hermitian linear systems (1991)
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    Numerische Mathematik 60 (1991), S. 315-339 
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    ISSN: 0945-3245
    Keywords: 65F10
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary The biconjugate gradient (BCG) method is the “natural” generalization of the classical conjugate gradient algorithm for Hermitian positive definite matrices to general non-Hermitian linear systems. Unfortunately, the original BCG algorithm is susceptible to possible breakdowns and numerical instabilities. In this paper, we present a novel BCG-like approach, the quasi-minimal residual (QMR) method, which overcomes the problems of BCG. An implementation of QMR based on a look-ahead version of the nonsymmetric Lanczos algorithm is proposed. It is shown how BCG iterates can be recovered stably from the QMR process. Some further properties of the QMR approach are given and an error bound is presented. Finally, numerical experiments are reported.
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    Accelerated Landweber iterations for the solution of ill-posed equations (1991)
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    Numerische Mathematik 60 (1991), S. 341-373 
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    ISSN: 0945-3245
    Keywords: 65J10 ; 65F10
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary In this paper, the potentials of so-calledlinear semiiterative methods are considered for the approximate solution of linear ill-posed problems and ill conditioned matrix equations. Several efficient two-step methods are presented, most of which have been introduced earlier in the literature. Stipulating certain conditions concerning the smoothness of the solution, a notion of optimal speed of convergence may be formulated. Various direct and converse results are derived to illustrate the properties of this concept. If the problem's right hand side data are contaminated by noise, semiiterative methods may be used asregularization methods. Assuming optimal rate of convergence of the iteration for the unperturbed problem, the regularized approximations will be of order optimal accuracy. To derive these results, specific properties of polynomials are used in connection with the basic theory of solving ill-posed problems. Rather recent results onfast decreasing polynomials are applied to answer an open question of Brakhage. Numerical examples are given including a comparison to the method of conjugate gradients.
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    A hybrid Arnoldi-Faber iterative method for nonsymmetric systems of linear equations (1993)
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    Numerische Mathematik 64 (1993), S. 213-240 
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    ISSN: 0945-3245
    Keywords: 65F10
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary We present here a new hybrid method for the iterative solution of large sparse nonsymmetric systems of linear equations, say of the formAx=b, whereA ∈ ℝ N, N , withA nonsingular, andb ∈ ℝ N are given. This hybrid method begins with a limited number of steps of the Arnoldi method to obtain some information on the location of the spectrum ofA, and then switches to a Richardson iterative method based on Faber polynomials. For a polygonal domain, the Faber polynomials can be constructed recursively from the parameters in the Schwarz-Christoffel mapping function. In four specific numerical examples of non-normal matrices, we show that this hybrid algorithm converges quite well and is approximately as fast or faster than the hybrid GMRES or restarted versions of the GMRES algorithm. It is, however, sensitive (as other hybrid methods also are) to the amount of information on the spectrum ofA acquired during the first (Arnoldi) phase of this procedure.
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    Two preconditioners based on the multi-level splitting of finite element spaces (1990)
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    Numerische Mathematik 58 (1990), S. 163-184 
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    ISSN: 0945-3245
    Keywords: AMS(MOS) ; 65F10 ; 65F35 ; 65N20 ; 65N30 ; CR: G 1.8
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary The hierarchical basis preconditioner and the recent preconditioner of Bramble, Pasciak and Xu are derived and analyzed within a joint framework. This discussion elucidates the close relationship between both methods. Special care is devoted to highly nonuniform meshes; exclusively local properties like the shape regularity of the finite elements are utilized.
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    The conditioning of the stiffness matrix for certain elements approximating the incompressibility condition in fluid dynamics (1990)
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    Numerische Mathematik 58 (1990), S. 203-214 
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    ISSN: 0945-3245
    Keywords: AMS(MOS) ; 65F10 ; 65N30 ; 76D05 ; CR: G1.8
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary In order to solve the Stokes equations numerically, Crouzeix and Raviart introduced elements satisfying a discrete divergence condition. For the two dimensional case and uniform triangulations it is shown, that using the standard basis functions, the conditioning of the stiffness matrix is of orderN 2, whereN is the dimension of the corresponding finite element space. Hierarchical bases are introduced which give a condition number of orderN log(N)3.
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    An alternating direction implicit method for orthogonal spline collocation linear systems (1991)
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    Numerische Mathematik 59 (1991), S. 413-429 
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    ISSN: 0945-3245
    Keywords: 65N35 ; 65F10
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    Topics: Mathematics
    Notes: Summary An Alternating Direction Implicit method is analyzed for the solution of linear systems arising in high-order, tensor-product orthogonal spline collocation applied to some separable, second order, linear, elliptic partial differential equations in rectangles. On anNxN partition, with Jordan's selection of the acceleration parameters, the method requiresO(N 2 ln 2 N) arithmetic operations to produce an approximation whose accuracy, in theH 1-norm, is that of the collocation solution.
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    Convergence of block iterative methods for linear systems arising in the numerical solution of Euler equations (1991)
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    Numerische Mathematik 59 (1991), S. 541-559 
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    ISSN: 0945-3245
    Keywords: 65F10 ; 65N22 ; 15A48
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    Topics: Mathematics
    Notes: Summary We discuss block matrices of the formA=[A ij ], whereA ij is ak×k symmetric matrix,A ij is positive definite andA ij is negative semidefinite. These matrices are natural block-generalizations of Z-matrices and M-matrices. Matrices of this type arise in the numerical solution of Euler equations in fluid flow computations. We discuss properties of these matrices, in particular we prove convergence of block iterative methods for linear systems with such system matrices.
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    Additive Schwarz algorithms for parabolic convection-diffusion equations (1991)
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    Numerische Mathematik 60 (1991), S. 41-61 
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    ISSN: 0945-3245
    Keywords: 65N30 ; 65F10
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    Notes: Summary In this paper, we consider the solution of linear systems of algebraic equations that arise from parabolic finite element problems. We introduce three additive Schwarz type domain decomposition methods for general, not necessarily selfadjoint, linear, second order, parabolic partial differential equations and also study the convergence rates of these algorithms. The resulting preconditioned linear system of equations is solved by the generalized minimal residual method. Numerical results are also reported.
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    Schwarz alternating and iterative refinement methods for mixed formulations of elliptic problems, part I: Algorithms and numerical results (1993)
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    Numerische Mathematik 65 (1993), S. 445-468 
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    ISSN: 0945-3245
    Keywords: 65N30 ; 65F10
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    Topics: Mathematics
    Notes: Summary We describe sequential and parallel algorithms based on the Schwarz alternating method for the solution of mixed finite element discretizations of elliptic problems using the Raviart-Thomas finite element spaces. These lead to symmetric indefinite linear systems and the algorithms have some similarities with the traditional block Gauss-Seidel or block Jacobi methods with overlapping blocks. The indefiniteness requires special treatment. The sub-blocks used in the algorithm correspond to problems on a coarse grid and some overlapping subdomains and is based on a similar partition used in an algorithm of Dryja and Widlund for standard elliptic problems. If there is sufficient overlap between the subdomains, the algorithm converges with a rate independent of the mesh size, the number of subdomains and discontinuities of the coefficients. Extensions of the above algorithms to the case of local grid refinement is also described. Convergence theory for these algorithms will be presented in a subsequent paper.
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    A new technique for generating quadratic programming test problems (1993)
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    Mathematical programming 61 (1993), S. 215-231 
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    ISSN: 1436-4646
    Keywords: Test problem generation ; quadratic programming ; global optimization ; large-scale optimization
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    Topics: Computer Science , Mathematics
    Notes: Abstract This paper describes a new technique for generating convex, strictly concave and indefinite (bilinear or not) quadratic programming problems. These problems have a number of properties that make them useful for test purposes. For example, strictly concave quadratic problems with their global maximum in the interior of the feasible domain and with an exponential number of local minima with distinct function values and indefinite and jointly constrained bilinear problems with nonextreme global minima, can be generated. Unlike most existing methods our construction technique does not require the solution of any subproblems or systems of equations. In addition, the authors know of no other technique for generating jointly constrained bilinear programming problems.
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    A decomposition method using a pricing mechanism for min concave cost flow problems with a hierarchical structure (1992)
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    Mathematical programming 53 (1992), S. 339-359 
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    ISSN: 1436-4646
    Keywords: Concave cost ; hierarchical structure ; uncapacitated network ; nonconvex decomposition ; pricing mechanism ; global optimization
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract In this paper we develop a decomposition method using a pricing mechanism which has been widely applied to linear and convex programs for a class of nonconvex optimization problems that are min concave cost flow problems under directed, uncapacitated networks with a hierarchical structure.
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    A generalized Dantzig—Wolfe decomposition principle for a class of nonconvex programming problems (1993)
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    Mathematical programming 62 (1993), S. 239-260 
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    ISSN: 1436-4646
    Keywords: Decomposition ; price functions ; global optimization
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    Topics: Computer Science , Mathematics
    Notes: Abstract Since Dantzig—Wolfe's pioneering contribution, the decomposition approach using a pricing mechanism has been developed for a wide class of mathematical programs. For convex programs a linear space of Lagrangean multipliers is enough to define price functions. For general mathematical programs the price functions could be defined by using a subclass of nondecreasing functions. However the space of nondecreasing functions is no longer finite dimensional. In this paper we consider a specific nonconvex optimization problem min {f(x):h j (x)⩽g(x),j=1, ⋯,m, x∈X}, wheref(·),h j (·) andg(·) are finite convex functions andX is a closed convex set. We generalize optimal price functions for this problem in such a way that the parameters of generalized price functions are defined in a finite dimensional space. Combining convex duality and a nonconvex duality we can develop a decomposition method to find a globally optimal solution.
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    Robustness of multi-grid applied to anisotropic equations on convex domains and on domains with re-entrant corners (1993)
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    Numerische Mathematik 66 (1993), S. 373-398 
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    ISSN: 0945-3245
    Keywords: 65F10 ; 65N30
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    Topics: Mathematics
    Notes: Summary We analyse multi-grid applied to anisotropic equations within the framework of smoothing and approximation-properties developed by Hack busch. For a model anisotropic equation on a square, we give an up-till-now missing proof of an estimate concerning the approximation property which is essential to show robustness. Furthermore, we show a corresponding estimate for a model anisotropic equation on an L-shaped domain. The existing estimates for the smoothing property are not suitable to prove robustness for either 2-cyclic Gauss-Seidel smoothers or for less regular problems such as our second model equation. For both cases, we give sharper estimates. By combination of our results concerning smoothing- and approximation-properties, robustness of W-cycle multi-grid applied to both our model equations will follow for a number of smoothers.
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    On a hierarchical basis multilevel method with nonconforming P1 elements (1992)
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    Numerische Mathematik 62 (1992), S. 189-212 
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    ISSN: 0945-3245
    Keywords: 65F10 ; 65F35 ; 65N20 ; 65N30
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    Topics: Mathematics
    Notes: Summary We develop a hierarchical basis multilevel method for symmetric second order elliptic boundary value problems in two-dimensional polygonal domains based on nonconforming P1 triangular elements. The main result is that the condition number of the hierarchical discretization is bounded byO(k) wherek is the number of refinement levels. For comparison, Yserentant's result yields the sharp boundO(k 2) for the corresponding hierarchical discretization with conforming linear elements.
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    Convergence of sequential and asynchronous nonlinear paracontractions (1992)
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    Numerische Mathematik 62 (1992), S. 305-319 
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    ISSN: 0945-3245
    Keywords: MSC 1991 ; 65F10
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    Topics: Mathematics
    Notes: Summary We establish the convergence of sequential and asynchronous iteration schemes for nonlinear paracontracting operators acting in finite dimensional spaces. Applications to the solution of linear systems of equations with convex constraints are outlined. A first generalization of one of our convergence results to an infinite pool of asymptotically paracontracting operators is also presented.
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    Some comparison theorems for nonnegative splittings of matrices (1993)
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    Numerische Mathematik 65 (1993), S. 245-252 
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    ISSN: 0945-3245
    Keywords: 65F10 ; 15A06
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    Topics: Mathematics
    Notes: Summary In a recent paper the author has proposed some theorems on the comparison of the asymptotic rates of convergence of two nonnegative splittings. They extended the corresponding result of Miller and Neumann and implied the earlier theorems of Varga, Beauwens, Csordas and Varga. An open question by Miller and Neumann, which additional and appropriate conditions should be imposed to obtain strict inequality, was also answered. This article continues to investigate the comparison theorems for nonnegative splittings. The new results extend and imply the known theorems by the author, Miller and Neumann.
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    On the convergence rate of the conjugate gradients in presence of rounding errors (1993)
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    Numerische Mathematik 65 (1993), S. 301-317 
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    ISSN: 0945-3245
    Keywords: 65F10 ; 65B99
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    Topics: Mathematics
    Notes: Summary We investigate here rounding error effects on the convergence rate of the conjugate gradients. More precisely, we analyse on both theoretical and experimental basis how finite precision arithmetic affects known bounds on iteration numbers when the spectrum of the system matrix presents small or large isolated eigenvalues.
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    On additive Schwarz preconditioners for sparse grid discretizations (1993)
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    Numerische Mathematik 66 (1993), S. 449-463 
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    ISSN: 0945-3245
    Keywords: 65F10 ; 65F35 ; 65N30
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    Topics: Mathematics
    Notes: Summary Based on the framework of subspace splitting and the additive Schwarz scheme, we give bounds for the condition number of multilevel preconditioners for sparse grid discretizations of elliptic model problems. For a BXP-like preconditioner we derive an estimate of the optimal orderO(1) and for a HB-like variant we obtain an estimate of the orderO(k 2 ·2 k/2 ), wherek denotes the number of levels employed. Furthermore, we confirm these results by numerically computed condition numbers.
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    An analysis of the composite step biconjugate gradient method (1993)
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    Numerische Mathematik 66 (1993), S. 295-319 
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    ISSN: 0945-3245
    Keywords: 65N20 ; 65F10
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    Topics: Mathematics
    Notes: Summary The composite step biconjugate gradient method (CSBCG) is a simple modification of the standard biconjugate gradient algorithm (BCG) which smooths the sometimes erratic convergence of BCG by computing only a subset of the iterates. We show that 2×2 composite steps can cure breakdowns in the biconjugate gradient method caused by (near) singularity of principal submatrices of the tridiagonal matrix generated by the underlying Lanczos process. We also prove a “best approximation” result for the method. Some numerical illustrations showing the effect of roundoff error are given.
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    A generalized ADI iterative method (1993)
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    Numerische Mathematik 66 (1993), S. 215-233 
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    ISSN: 0945-3245
    Keywords: 65F10
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    Topics: Mathematics
    Notes: Summary The ADI iterative method for the solution of Sylvester's equationAX−XB=C proceeds by strictly alternating between the solution of the two equations $$\begin{gathered} \left( {A - \delta _{k + 1} I} \right)X_{2k + 1} = X_{2k} \left( {B - \delta _{k + 1} I} \right) + C, \hfill \\ X_{2k + 2} \left( {B - \tau _{k + 1} I} \right) = \left( {A - \tau _{k + 1} I} \right)X_{2k + 1} - C, \hfill \\ \end{gathered} $$ fork=0, 1, 2,... HereX o is a given initial approximate solution, and the δ k andτ k are real or complex parameters chosen so that the computed approximate solutionsX k converge rapidly to the solutionX of the Sylvester equation ask increases. This paper discusses the possibility of solving one of the equations in the ADI iterative method more often than the other one, i.e., relaxing the strict alternation requirement, in order to achieve a higher rate of convergence. Our analysis based on potential theory shows that this generalization of the ADI iterative method can give faster convergence than when strict alternation is required.
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    A finite element method for interface problems in domains with smooth boundaries and interfaces (1996)
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    Advances in computational mathematics 6 (1996), S. 109-138 
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    ISSN: 1572-9044
    Keywords: Finite element method ; interface problems ; 65N30 ; 65F10
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    Topics: Mathematics
    Notes: Abstract This paper is concerned with the analysis of a finite element method for nonhomogeneous second order elliptic interface problems on smooth domains. The method consists in approximating the domains by polygonal domains, transferring the boundary data in a natural way, and then applying a finite element method to the perturbed problem on the approximate polygonal domains. It is shown that the error in the finite element approximation is of optimal order for linear elements on a quasiuniform triangulation. As such the method is robust in the regularity of the data in the original problem.
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    The tabu search metaheuristic: How we used it (1990)
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    Annals of mathematics and artificial intelligence 1 (1990), S. 111-121 
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    ISSN: 1573-7470
    Keywords: Heuristic ; global optimization ; combinatorial programming ; artificial intelligence
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract A general description of tabu search is given and various applications to optimization problems are presented. Some guidelines for applying the tabu metaheuristic are exhibited.
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    Breakdown and near-breakdown control in the CGS algorithm using stochastic arithmetic (1996)
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    Numerical algorithms 11 (1996), S. 99-116 
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    ISSN: 1572-9265
    Keywords: Lanczos methods ; othogonal polynomials ; breakdown ; near-breakdown ; stochastic arithmetic ; accuracy ; 65F10 ; 65F25 ; 65G05
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract In the Conjugate-Gradient-Squared method, a sequence of residualsr k defined byr k=P k 2 (A)r0 is computed. Coefficients of the polynomialsP k may be computed as a ratio of scalar products from the theory of formal orthogonal polynomials. When a scalar product in a denominator is zero or very affected by round-off errors, situations of breakdown or near-breakdown appear. Using floating point arithmetic on computers, such situations are detected with the use of ∈ i in some ordering relations like |x≤∈ i . The user has to choose the ∈ i himself and these choices condition entirely the efficient detection of breakdown or near-breakdown. The subject of this paper is to show how stochastic arithmetic eliminates the problem of the ∈ i with the estimation of the accuracy of some intermediate results.
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    A convergence theory of multilevel additive Schwarz methods on unstructured meshes (1996)
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    Numerical algorithms 13 (1996), S. 365-398 
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    ISSN: 1572-9265
    Keywords: convergence ; multilevel additive methods ; unstructured meshes ; 65N30 ; 65F10
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    Topics: Computer Science , Mathematics
    Notes: Abstract We develop a convergence theory for two level and multilevel additive Schwarz domain decomposition methods for elliptic and parabolic problems on general unstructured meshes in two and three dimensions. The coarse and fine grids are assumed only to be shape regular, and the domains formed by the coarse and fine grids need not be identical. In this general setting, our convergence theory leads to completely local bounds for the condition numbers of two level additive Schwarz methods, which imply that these condition numbers are optimal, or independent of fine and coarse mesh sizes and subdomain sizes if the overlap amount of a subdomain with its neighbors varies proportionally to the subdomain size. In particular, we will show that additive Schwarz algorithms are still very efficient for nonselfadjoint parabolic problems with only symmetric, positive definite solvers both for local subproblems and for the global coarse problem. These conclusions for elliptic and parabolic problems improve our earlier results in [12, 15, 16]. Finally, the convergence theory is applied to multilevel additive Schwarz algorithms. Under some very weak assumptions on the fine mesh and coarser meshes, e.g., no requirements on the relation between neighboring coarse level meshes, we are able to derive a condition number bound of the orderO(ρ2 L 2), whereρ = max1≤l≤L(h l +l− 1)/δ l,h l is the element size of thelth level mesh,δ l the overlap of subdomains on thelth level mesh, andL the number of mesh levels.
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    A composite step bi-conjugate gradient algorithm for nonsymmetric linear systems (1994)
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    Numerical algorithms 7 (1994), S. 1-16 
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    ISSN: 1572-9265
    Keywords: Biconjugate gradients ; nonsymmetric linear systems ; 65N20 ; 65F10
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract The Bi-Conjugate Gradient (BCG) algorithm is the simplest and most natural generalization of the classical conjugate gradient method for solving nonsymmetric linear systems. It is well-known that the method suffers from two kinds of breakdowns. The first is due to the breakdown of the underlying Lanczos process and the second is due to the fact that some iterates are not well-defined by the Galerkin condition on the associated Krylov subspaces. In this paper, we derive a simple modification of the BCG algorithm, the Composite Step BCG (CSBCG) algorithm, which is able to compute all the well-defined BCG iterates stably, assuming that the underlying Lanczos process does not break down. The main idea is to skip over a step for which the BCG iterate is not defined.
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    Fastiterative methods for least squares estimations (1994)
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    Numerical algorithms 6 (1994), S. 353-378 
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    ISSN: 1572-9265
    Keywords: Least squares estimations ; Toeplitz matrix ; circulant matrix ; preconditioned conjugate gradient method ; signal prediction ; linear prediction ; covariance matrix ; windowing methods ; finite impulse response (FIR) system identification ; 65F10 ; 65F15 ; 43E10
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract Least squares estimations have been used extensively in many applications, e.g. system identification and signal prediction. When the stochastic process is stationary, the least squares estimators can be found by solving a Toeplitz or near-Toeplitz matrix system depending on the knowledge of the data statistics. In this paper, we employ the preconditioned conjugate gradient method with circulant preconditioners to solve such systems. Our proposed circulant preconditioners are derived from the spectral property of the given stationary process. In the case where the spectral density functions(θ) of the process is known, we prove that ifs(θ) is a positive continuous function, then the spectrum of the preconditioned system will be clustered around 1 and the method converges superlinearly. However, if the statistics of the process is unknown, then we prove that with probability 1, the spectrum of the preconditioned system is still clustered around 1 provided that large data samples are taken. For finite impulse response (FIR) system identification problems, our numerical results show that annth order least squares estimator can usually be obtained inO(n logn) operations whenO(n) data samples are used. Finally, we remark that our algorithm can be modified to suit the applications of recursive least squares computations with the proper use of sliding window method arising in signal processing applications.
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    A composite step conjugate gradients squared algorithm for solving nonsymmetric linear systems (1994)
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    Numerical algorithms 7 (1994), S. 17-32 
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    ISSN: 1572-9265
    Keywords: Lanczos method ; conjugate gradients squared algorithm ; breakdowns ; composite step ; 65F10 ; 65F25
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract We propose a new and more stable variant of the CGS method [27] for solving nonsymmetric linear systems. The method is based on squaring the Composite Step BCG method, introduced recently by Bank and Chan [1,2], which itself is a stabilized variant of BCG in that it skips over steps for which the BCG iterate is not defined and causes one kind of breakdown in BCG. By doing this, we obtain a method (Composite Step CGS or CSCGS) which not only handles the breakdowns described above, but does so with the advantages of CGS, namely, no multiplications by the transpose matrix and a faster convergence rate than BCG. Our strategy for deciding whether to skip a step does not involve any machine dependent parameters and is designed to skip near breakdowns as well as produce smoother iterates. Numerical experiments show that the new method does produce improved performance over CGS on practical problems.
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    BiCGstab(l) and other hybrid Bi-CG methods (1994)
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    Numerical algorithms 7 (1994), S. 75-109 
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    ISSN: 1572-9265
    Keywords: Bi-Conjugate gradients ; non-symmetric linear systems ; CGS ; Bi-CGSTAB ; iterative solvers ; ORTHODIR ; Krylov subspace ; 65F10
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract It is well-known that Bi-CG can be adapted so that the operations withA T can be avoided, and hybrid methods can be constructed in which it is attempted to further improve the convergence behaviour. Examples of this are CGS, Bi-CGSTAB, and the more general BiCGstab(l) method. In this paper it is shown that BiCGstab(l) can be implemented in different ways. Each of the suggested approaches has its own advantages and disadvantages. Our implementations allow for combinations of Bi-CG with arbitrary polynomial methods. The choice for a specific implementation can also be made for reasons of numerical stability. This aspect receives much attention. Various effects have been illustrated by numerical examples.
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    Treatment of near-breakdown in the CGS algorithm (1994)
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    Numerical algorithms 7 (1994), S. 33-73 
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    ISSN: 1572-9265
    Keywords: Conjugate gradient ; Lanczos' method ; systems of linear equations ; orthogonal polynomials ; 65F05 ; 65F10 ; 65F25
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract Lanczos' method for solving the system of linear equationsAx=b consists in constructing a sequence of vectors (x k ) such thatr k =b−Ax k =P k (A)r 0 wherer 0=b−Ax 0.P k is an orthogonal polynomial which is computed recursively. The conjugate gradient squared algorithm (CGS) consists in takingr k =P k 2 (A)r0. In the recurrence relation forP k , the coefficients are given as ratios of scalar products. When a scalar product in a denominator is zero, then a breakdown occurs in the algorithm. When such a scalar product is close to zero, then rounding errors can seriously affect the algorithm, a situation known as near-breakdown. In this paper it is shown how to avoid near-breakdown in the CGS algorithm in order to obtain a more stable method.
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    Maintaining convergence properties of BiCGstab methods in finite precision arithmetic (1995)
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    Numerical algorithms 10 (1995), S. 203-223 
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    ISSN: 1572-9265
    Keywords: Non-symmetric linear systems ; iterative solvers ; Bi-CG ; Bi-CGSTAB ; BiCGstab(ℓ) ; 65F10
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract It is well-known that Bi-CG can be adapted so that hybrid methods with computational complexity almost similar to Bi-CG can be constructed, in which it is attempted to further improve the convergence behavior. In this paper we will study the class of BiCGstab methods. In many applications, the speed of convergence of these methods appears to be determined mainly by the incorporated Bi-CG process, and the problem is that the Bi-CG iteration coefficients have to be determined from the BiCGstab process. We will focus our attention to the accuracy of these Bi-CG coefficients, and how rounding errors may affect the speed of convergence of the BiCGstab methods. We will propose a strategy for a more stable determination of the Bi-CG iteration coefficients and by experiments we will show that this indeed may lead to faster convergence.
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    URL: http://dx.doi.org/10.1007/BF02140769
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    A nonstandard multigrid method with flexible multiple semicoarsening for the numerical solution of the pressure equation in a Navier-Stokes solver (2000)
    Springer
    Numerical algorithms 24 (2000), S. 333-355 
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    ISSN: 1572-9265
    Keywords: multigrid method ; Krylov subspace method ; incompressible Navier-Stokes equations ; semicoarsening ; robustness ; 65F10 ; 65N55 ; 76D05 ; 76M20
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract Numerical methods for the incompressible Reynolds-averaged Navier-Stokes equations discretized by finite difference techniques on collocated cell-centered structured grids are considered in this paper. A widespread solution method to solve the pressure-velocity coupling problem is to use a segregated approach, in which the computational work is deeply controlled by the solution of the pressure problem. This pressure equation is an elliptic partial differential equation with possibly discontinuous or anisotropic coeffficients. The resulting singular linear system needs efficient solution strategies especially for 3-dimensional applications. A robust method (close to MG-S [22,34]) combining multiple cell-centered semicoarsening strategies, matrix-independent transfer operators, Galerkin coarse grid approximation is therefore designed. This strategy is both evaluated as a solver or as a preconditioner for Krylov subspace methods on various 2- or 3-dimensional fluid flow problems. The robustness of this method is shown.
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    URL: http://dx.doi.org/10.1023/A:1019113814803
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    Krylov subspace methods, biorthogonal polynomials and Padé-type approximants (1999)
    Springer
    Numerical algorithms 21 (1999), S. 97-107 
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    ISSN: 1572-9265
    Keywords: Krylov subspace methods ; biorthogonal polynomials ; Padé-type approximants ; Lanczos method ; Arnoldi method ; GMRES ; 65F10 ; 65F25 ; 41A10 ; 41A21
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract It is shown that Krylov subspace methods for solving systems of linear equations can be based on formal biorthogonal polynomials and on Padé-type and Padé approximants. New algorithms for their implementation are derived.
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    URL: http://dx.doi.org/10.1023/A:1019140909409
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    Preconditioned Barzilai-Borwein method for the numerical solution of partial differential equations (1996)
    Springer
    Numerical algorithms 13 (1996), S. 45-60 
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    ISSN: 1572-9265
    Keywords: Barzilai-Borwein method ; conjugate gradient method ; incomplete Cholesky ; symmetric successive overrelaxation ; elliptic equations ; 65F10 ; 65F15 ; 65L10 ; 65N20
    Source: Springer Online Journal Archives 1860-2000
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    Notes: Abstract The preconditioned Barzilai-Borwein method is derived and applied to the numerical solution of large, sparse, symmetric and positive definite linear systems that arise in the discretization of partial differential equations. A set of well-known preconditioning techniques are combined with this new method to take advantage of the special features of the Barzilai-Borwein method. Numerical results on some elliptic test problems are presented. These results indicate that the preconditioned Barzilai-Borwein method is competitive and sometimes preferable to the preconditioned conjugate gradient method.
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    URL: http://dx.doi.org/10.1007/BF02143126
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    Quasilinear sequence transformations (1997)
    Springer
    Numerical algorithms 15 (1997), S. 275-285 
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    ISSN: 1572-9265
    Keywords: quasilinear sequence ; transformations ; kernel ; convergence acceleration processes ; 65F10
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract We give a new characterization of quasilinear sequence transformations; we prove that any quasilinear transformation can be represented by its kernel. This approach is new and allows one to give a general result of convergence acceleration and tools for the construction of new quasilinear transformations.
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    URL: http://dx.doi.org/10.1023/A:1019154106316
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    Sparse approximate inverse preconditioning for dense linear systems arising in computational electromagnetics (1997)
    Springer
    Numerical algorithms 16 (1997), S. 1-15 
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    ISSN: 1572-9265
    Keywords: dense linear systems ; preconditioning ; sparse approximate inverses ; complex symmetric matrices ; scattering calculations ; Krylov subspace methods ; parallel computing ; 65F10 ; 65F50 ; 65R20 ; 65N38 ; 78-08 ; 78A50 ; 78A55
    Source: Springer Online Journal Archives 1860-2000
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    Notes: Abstract We investigate the use of sparse approximate inverse preconditioners for the iterative solution of linear systems with dense complex coefficient matrices arising in industrial electromagnetic problems. An approximate inverse is computed via a Frobenius norm approach with a prescribed nonzero pattern. Some strategies for determining the nonzero pattern of an approximate inverse are described. The results of numerical experiments suggest that sparse approximate inverse preconditioning is a viable approach for the solution of large-scale dense linear systems on parallel computers.
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    URL: http://dx.doi.org/10.1023/A:1019170609950
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    Transpose-free formulations of Lanczos-type methods for nonsymmetric linear systems (1998)
    Springer
    Numerical algorithms 17 (1998), S. 51-66 
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    ISSN: 1572-9265
    Keywords: Lanczos algorithm ; quasi-minimal residual algorithm ; bi-conjugate gradients algorithm ; nonsymmetric linear systems ; Krylov subspace methods ; 65F10 ; 65N20
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract We present a transpose-free version of the nonsymmetric scaled Lanczos procedure. It generates the same tridiagonal matrix as the classical algorithm, using two matrix–vector products per iteration without accessing AT. We apply this algorithm to obtain a transpose-free version of the Quasi-minimal residual method of Freund and Nachtigal [15] (without look-ahead), which requires three matrix–vector products per iteration. We also present a related transpose-free version of the bi-conjugate gradients algorithm.
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    URL: http://dx.doi.org/10.1023/A:1011637511962
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    Multigrid preconditioned Krylov subspace methods for three-dimensional numerical solutions of the incompressible Navier–Stokes equations (1998)
    Springer
    Numerical algorithms 17 (1998), S. 1-32 
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    ISSN: 1572-9265
    Keywords: multigrid method ; Krylov subspace method ; incompressible Navier–Stokes equations ; collocated grid ; curvilinear coordinates ; 65F10 ; 65N55 ; 76D05 ; 76M20
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract We consider numerical methods for the incompressible Reynolds averaged Navier–Stokes equations discretized by finite difference techniques on non-staggered grids in body-fitted coordinates. A segregated approach is used to solve the pressure–velocity coupling problem. Several iterative pressure linear solvers including Krylov subspace and multigrid methods and their combination have been developed to compare the efficiency of each method and to design a robust solver. Three-dimensional numerical experiments carried out on scalar and vector machines and performed on different fluid flow problems show that a combination of multigrid and Krylov subspace methods is a robust and efficient pressure solver.
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    URL: http://dx.doi.org/10.1023/A:1011689512871
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    Weighted FOM and GMRES for solving nonsymmetric linear systems (1998)
    Springer
    Numerical algorithms 18 (1998), S. 277-292 
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    Keywords: nonsymmetric linear systems ; sparse matrices ; iterative methods ; Krylov subspaces ; Arnoldi ; FOM ; GMRES ; 65F10 ; 65F50
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract This paper presents two new methods called WFOM and WGMRES, which are variants of FOM and GMRES, for solving large and sparse nonsymmetric linear systems. To accelerate the convergence, these new methods use a different inner product instead of the Euclidean one. Furthermore, at each restart, a different inner product is chosen. The weighted Arnoldi process is introduced for implementing these methods. After describing the weighted methods, we give the relations that link them to FOM and GMRES. Experimental results are presented to show the good performances of the new methods compared to FOM(m) and GMRES(m).
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    URL: http://dx.doi.org/10.1023/A:1019177600806
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    Solving stable generalized Lyapunov equations with the matrix sign function (1999)
    Springer
    Numerical algorithms 20 (1999), S. 75-100 
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    ISSN: 1572-9265
    Keywords: generalized Lyapunov equations ; mathematical software ; matrix sign function ; Newton iteration ; algebraic Riccati equations ; 65F10 ; 93B40 ; 93B51
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract We investigate the numerical solution of the stable generalized Lyapunov equation via the sign function method. This approach has already been proposed to solve standard Lyapunov equations in several publications. The extension to the generalized case is straightforward. We consider some modifications and discuss how to solve generalized Lyapunov equations with semidefinite constant term for the Cholesky factor. The basic computational tools of the method are basic linear algebra operations that can be implemented efficiently on modern computer architectures and in particular on parallel computers. Hence, a considerable speed-up as compared to the Bartels–Stewart and Hammarling methods is to be expected. We compare the algorithms by performing a variety of numerical tests.
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    URL: http://dx.doi.org/10.1023/A:1019191431273
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    Generalizations and modifications of the GMRES iterative method (1999)
    Springer
    Numerical algorithms 21 (1999), S. 119-146 
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    ISSN: 1572-9265
    Keywords: iterative methods ; Generalized Minimum Residual (GMRES) method ; minimization condition ; Galerkin condition ; Lanczos-type methods ; 65F10 ; 65H10 ; 65N20 ; 65N30
    Source: Springer Online Journal Archives 1860-2000
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    Notes: Abstract For solving nonsymmetric linear systems, the well-known GMRES method is considered to be a stable method; however, the work per iteration increases as the number of iterations increases. We consider two new iterative methods GGMRES and MGMRES, which are a generalization and a modification of the GMRES method, respectively. Instead of using a minimization condition as in the derivation of GGMRES, we use a Galerkin condition to derive the MGMRES method. We also introduce another new iterative method, LAN/MGMRES, which is designed to combine the reliability of GMRES with the reduced work of a Lanczos-type method. A computer program has been written based on the use of the LAN/MGMRES algorithm for solving nonsymmetric linear systems arising from certain elliptic problems. Numerical tests are presented comparing this algorithm with some other commonly used iterative algorithms. These preliminary tests of the LAN/MGMRES algorithm show that it is comparable in terms of both the approximate number of iterations and the overall convergence behavior.
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    URL: http://dx.doi.org/10.1023/A:1019105328973
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    Implicitly restarted and deflated GMRES (1999)
    Springer
    Numerical algorithms 21 (1999), S. 261-285 
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    ISSN: 1572-9265
    Keywords: restarted GMRES ; restarted FOM ; IRA ; deflation ; minimization with constraints ; 65F10 ; 65F15 ; 64N30
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract We introduce a deflation method that takes advantage of the IRA method, by extracting a GMRES solution from the Krylov basis computed within the Arnoldi process of the IRA method itself. The deflation is well-suited because it is done with eigenvectors associated to the eigenvalues that are closest to zero, which are approximated by IRA very quickly. By a slight modification, we adapt it to the FOM algorithm, and then to GMRES enhanced by imposing constraints within the minimization condition. The use of IRA enables us to reduce the number of matrix-vector products, while keeping a low storage.
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    URL: http://dx.doi.org/10.1023/A:1019113630790
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    Experimental comparison of three-dimensional point and line modified incomplete factorizations (2000)
    Springer
    Numerical algorithms 23 (2000), S. 51-70 
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    ISSN: 1572-9265
    Keywords: discretized partial differential equations ; large sparse linear systems ; partitionings ; modified incomplete factorizations ; preconditioned conjugate gradient ; 65F10 ; 65F35 ; 65B99 ; 65N22
    Source: Springer Online Journal Archives 1860-2000
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    Notes: Abstract We examine how the variations of the coefficients of 3-dimensional (3D) partial differential equations (PDEs) influence the convergence of the conjugate gradient method, preconditioned by standard pointwise and linewise modified incomplete factorizations. General analytical spectral bounds obtained previously are applied, which displays the conditions under which good performances could be expected. The arguments also reveal that, if the total number of unknowns is very large or the number of unknowns in one direction is much larger than in both other ones, or if there are strong jumps in the variation of the PDE coefficients or fewer Dirichlet boundary conditions, then linewise preconditionings could be significantly more efficient than the corresponding pointwise ones. We also discuss reasons to explain why in the case of constant PDE coefficients, the advantage of preferring linewise methods to pointwise ones is not as pronounced as in 2D problems. Results of numerical experiments are reported.
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    URL: http://dx.doi.org/10.1023/A:1019191715286
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    Extracting partial canonical structure for large scale eigenvalue problems (2000)
    Springer
    Numerical algorithms 24 (2000), S. 195-237 
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    ISSN: 1572-9265
    Keywords: partial Jordan canonical stucture ; eigenvalue problems ; Jordan-Schur form ; Weierstrass-Schur form ; large scale ; Krylov methods ; implicitly restarted Arnoldi ; eigenvalue clustering ; staircase algorithms ; 65F15 ; 65F10 ; 15A21 ; 15A22 ; 68N99
    Source: Springer Online Journal Archives 1860-2000
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    Notes: Abstract We present methods for computing a nearby partial Jordan-Schur form of a given matrix and a nearby partial Weierstrass-Schur form of a matrix pencil. The focus is on the use and the interplay of the algorithmic building blocks – the implicitly restarted Arnoldi method with prescribed restarts for computing an invariant subspace associated with the dominant eigenvalue, the clustering method for grouping computed eigenvalues into numerically multiple eigenvalues and the staircase algorithm for computing the structure revealing form of the projected problem. For matrix pencils, we present generalizations of these methods. We introduce a new and more accurate clustering heuristic for both matrices and matrix pencils. Particular emphasis is placed on reliability of the partial Jordan-Schur and Weierstrass-Schur methods with respect to the choice of deflation parameters connecting the steps of the algorithm such that the errors are controlled. Finally, successful results from computational experiments conducted on problems with known canonical structure and varying ill-conditioning are presented.
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    URL: http://dx.doi.org/10.1023/A:1019153528915
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    CMRH: A new method for solving nonsymmetric linear systems based on the Hessenberg reduction algorithm (1999)
    Springer
    Numerical algorithms 20 (1999), S. 303-321 
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    ISSN: 1572-9265
    Keywords: linear systems ; iterative methods ; Hessenberg’s method ; GMRES ; QMR ; 65F10
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract The Generalized Minimal Residual (GMRES) method and the Quasi-Minimal Residual (QMR) method are two Krylov methods for solving linear systems. The main difference between these methods is the generation of the basis vectors for the Krylov subspace. The GMRES method uses the Arnoldi process while QMR uses the Lanczos algorithm for constructing a basis of the Krylov subspace. In this paper we give a new method similar to QMR but based on the Hessenberg process instead of the Lanczos process. We call the new method the CMRH method. The CMRH method is less expensive and requires slightly less storage than GMRES. Numerical experiments suggest that it has behaviour similar to GMRES.
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    URL: http://dx.doi.org/10.1023/A:1019164119887
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    Adaptive Lanczos methods for recursive condition estimation (1991)
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    Numerical algorithms 1 (1991), S. 1-19 
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    ISSN: 1572-9265
    Keywords: AMS(MOS) ; 15A18 ; 65F10 ; 65F20 ; 65F35 ; Adaptive methods ; condition estimation ; control ; downdating ; eigenvalues ; Lanczos methods ; matrix modifications ; recursive least squares ; signal processing ; singular values ; updating
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract Estimates for the condition number of a matrix are useful in many areas of scientific computing, including: recursive least squares computations, optimization, eigenanalysis, and general nonlinear problems solved by linearization techniques where matrix modification techniques are used. The purpose of this paper is to propose anadaptiveLanczosestimator scheme, which we callale, for tracking the condition number of the modified matrix over time. Applications to recursive least squares (RLS) computations using the covariance method with sliding data windows are considered.ale is fast for relatively smalln-parameter problems arising in RLS methods in control and signal processing, and is adaptive over time, i.e., estimates at timet are used to produce estimates at timet+1. Comparisons are made with other adaptive and non-adaptive condition estimators for recursive least squares problems. Numerical experiments are reported indicating thatale yields a very accurate recursive condition estimator.
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    URL: http://dx.doi.org/10.1007/BF02145580
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    A linear system solver based on a modified Krylov subspace method for breakdown recovery (1996)
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    Numerical algorithms 12 (1996), S. 233-251 
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    ISSN: 1572-9265
    Keywords: Lanczos breakdown ; Residual quasi-minimization ; nonsymmetric linear systems ; 65F10 ; 65N20
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract Despite its usefulness in solving eigenvalue problems and linear systems of equations, the nonsymmetric Lanczos method is known to suffer from a potential breakdown problem. Previous and recent approaches for handling the Lanczos exact and near-breakdowns include, for example, the look-ahead schemes by Parlett-Taylor-Liu [23], Freund-Gutknecht-Nachtigal [9], and Brezinski-Redivo Zaglia-Sadok [4]; the combined look-ahead and restart scheme by Joubert [18]; and the low-rank modified Lanczos scheme by Huckle [17]. In this paper, we present yet another scheme based on a modified Krylov subspace approach for the solution of nonsymmetric linear systems. When a breakdown occurs, our approach seeks a modified dual Krylov subspace, which is the sum of the original subspace and a new Krylov subspaceK m (w j ,A T ) wherew j is a newstart vector (this approach has been studied by Ye [26] for eigenvalue computations). Based on this strategy, we have developed a practical algorithm for linear systems called the MLAN/QM algorithm, which also incorporates the residual quasi-minimization as proposed in [12]. We present a few convergence bounds for the method as well as numerical results to show its effectiveness.
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    Transpose-free Lanczos-type algorithms for nonsymmetric linear systems (1998)
    Springer
    Numerical algorithms 17 (1998), S. 67-103 
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    ISSN: 1572-9265
    Keywords: Lanczos' method ; orthogonal polynomials ; CGS ; BiCGSTAB ; 65F10 ; 33A65
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract The method of Lanczos for solving systems of linear equations is implemented by using recurrence relationships between formal orthogonal polynomials. A drawback is that the computation of the coefficients of these recurrence relationships usually requires the use of the transpose of the matrix of the system. Due to the indirect addressing, this is a costly operation. In this paper, a new procedure for computing these coefficients is proposed. It is based on the recursive computation of the products of polynomials appearing in their expressions and it does not involve the transpose of the matrix. Moreover, our approach allows to implement simultaneously and at a low price a Lanczos-type product method such as the CGS or the BiCGSTAB.
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    Augmented conjugate gradient. Application in an iterative process for the solution of scattering problems (1998)
    Springer
    Numerical algorithms 18 (1998), S. 71-90 
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    ISSN: 1572-9265
    Keywords: conjugate gradient ; Krylov subspace ; least-squares ; scattering ; 65F10 ; 65C20
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract We discuss the application of an augmented conjugate gradient to the solution of a sequence of linear systems of the same matrix appearing in an iterative process for the solution of scattering problems. The conjugate gradient method applied to the first system generates a Krylov subspace, then for the following systems, a modified conjugate gradient is applied using orthogonal projections on this subspace to compute an initial guess and modified descent directions leading to a better convergence. The scattering problem is treated via an Exact Controllability formulation and a preconditioned conjugate gradient algorithm is introduced. The set of linear systems to be solved are associated to this preconditioning. The efficiency of the method is tested on different 3D acoustic problems.
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    URL: http://dx.doi.org/10.1023/A:1019109213101
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    A continuation–domain decomposition algorithm for bifurcation problems (1999)
    Springer
    Numerical algorithms 22 (1999), S. 367-383 
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    ISSN: 1572-9265
    Keywords: bifurcation problems ; continuation methods ; domain decomposition methods ; finite differences ; 65N06 ; 65F10 ; 65F15
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract We study numerical solution branches of certain parameter-dependent problems defined on compact domains with various boundary conditions. The finite differences combined with the domain decomposition method are exploited to discretize the partial differential equations. We propose efficient numerical algorithms for solving the associated linear systems and for the detection of bifurcation points. Sample numerical results are reported.
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    Fast algorithm for solving the Hankel/Toeplitz Structured Total Least Squares problem (2000)
    Springer
    Numerical algorithms 23 (2000), S. 371-392 
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    ISSN: 1572-9265
    Keywords: Hankel/Toeplitz matrix ; Structured Total Least Squares ; displacement rank ; 15A03 ; 62P30 ; 65F10
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract The Structured Total Least Squares (STLS) problem is a natural extension of the Total Least Squares (TLS) problem when constraints on the matrix structure need to be imposed. Similar to the ordinary TLS approach, the STLS approach can be used to determine the parameter vector of a linear model, given some noisy measurements. In many signal processing applications, the imposition of this matrix structure constraint is necessary for obtaining Maximum Likelihood (ML) estimates of the parameter vector. In this paper we consider the Toeplitz (Hankel) STLS problem (i.e., an STLS problem in which the Toeplitz (Hankel) structure needs to be preserved). A fast implementation of an algorithm for solving this frequently occurring STLS problem is proposed. The increased efficiency is obtained by exploiting the low displacement rank of the involved matrices and the sparsity of the associated generators. The fast implementation is compared to two other implementations of algorithms for solving the Toeplitz (Hankel) STLS problem. The comparison is carried out on a recently proposed speech compression scheme. The numerical results confirm the high efficiency of the newly proposed fast implementation: the straightforward implementations have a complexity of O((m+n)3) and O(m3) whereas the proposed implementation has a complexity of O(mn+n2).
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    Some vector sequence transformations with applications to systems of equations (1992)
    Springer
    Numerical algorithms 3 (1992), S. 75-80 
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    ISSN: 1572-9265
    Keywords: AMS (MOS) 65B05 ; 65F10 ; 65F25 ; Vector sequence ; extrapolation ; Lanczos method
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract First, recursive algorithms for implementing some vector sequence transformations are given. In a particular case, these transformations are generalizations of Shanks transformation and the G-transformation. When the sequence of vectors under transformation is generated by linear fixed point iterations, Lanczos' method and the CGS are recovered respectively. In the case of a sequence generated by nonlinear fixed point iterations, a quadratically convergent method based on the ε-algorithm is recovered and a nonlinear analog of the CGS method is obtained.
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    Robust Estimation of Rotation Angles from Image Sequences Using the Annealing M-Estimator (1998)
    Springer
    Journal of mathematical imaging and vision 8 (1998), S. 181-192 
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    ISSN: 1573-7683
    Keywords: deterministic annealing ; global optimization ; M-estimator ; motion analysis ; robust statistics
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract A robust method is presented for computing rotation angles of image sequences from a set of corresponding points containing outliers. Assuming known rotation axis, a least-squares (LS) solution are derived to compute the rotation angle from a clean data set of point correspondences. Since clean data is not guaranteed, we introduce a robust solution, based on the M-estimator, to deal with outliers. Then we present an enhanced robust algorithm, called the annealing M-estimator (AM-estimator), for reliable robust estimation. The AM-estimator has several attractive advantages over the traditional M-estimator: By definition, the AM-estimator involves neither scale estimator nor free parameters and hence avoids instabilities therein. Algorithmically, it uses a deterministic annealing technique to approximate the global solution regardless of the initialization. Experimental results are presented to compare the performance of the LS, M- and AM-estimators for the angle estimation. Experiments show that in the presence of outliers, the M-estimator outperforms the LS estimator and the AM-estimator outperforms the M-estimator.
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    Outcome-Space Cutting-Plane Algorithm for Linear Multiplicative Programming (2000)
    Springer
    Journal of optimization theory and applications 104 (2000), S. 301-322 
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    ISSN: 1573-2878
    Keywords: global optimization ; multiplicative programming ; cutting plane ; nonconvex programming ; outcome space ; extreme-point search
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract This article presents an outcome-space pure cutting-plane algorithm for globally solving the linear multiplicative programming problem. The framework of the algorithm is taken from a pure cutting-plane decision set-based method developed by Horst and Tuy for solving concave minimization problems. By adapting this method to an outcome-space reformulation of the linear multiplicative programming problem, rather than applying directly the method to the original decision-set formulation, it is expected that considerable computational savings can be obtained. Also, we show how additional computational benefits might be obtained by implementing the new algorithm appropriately. To illustrate the new algorithm, we apply it to the solution of a sample problem.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1004657629105
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    Approximation of an Analog Diffusion Network with Applications to Image Estimation (2000)
    Springer
    Journal of optimization theory and applications 107 (2000), S. 391-414 
    Details
    ISSN: 1573-2878
    Keywords: diffusion networks ; image estimation ; global optimization ; simulated annealing ; weak convergence
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract This work is concerned with a numerical procedure for approximating an analog diffusion network. The key idea is to take advantage of the separable feature of the noise for the diffusion machine and use a parallel processing method to develop recursive algorithms. The asymptotic properties are studied. The main result of this paper is to establish the convergence of a continuous-time interpolation of the discrete-time algorithm to that of the analog diffusion network via weak convergence methods. The parallel processing feature of the network makes it attractive for solving large-scale optimization problems. Applications to image estimation are considered. Not only is this algorithm useful for the image estimation problems, but it is widely applicable to many related optimization problems.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1026485504384
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  • 98
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    Duality Bound Method for the General Quadratic Programming Problem with Quadratic Constraints (2000)
    Springer
    Journal of optimization theory and applications 107 (2000), S. 331-354 
    Details
    ISSN: 1573-2878
    Keywords: general quadratic programming problem with quadratic constraints ; global optimization ; branch-and-bound algorithms ; duality bounds
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The purpose of this article is to develop a branch-and-bound algorithm using duality bounds for the general quadratically-constrained quadratic programming problem and having the following properties: (i) duality bounds are computed by solving ordinary linear programs; (ii) they are at least as good as the lower bounds obtained by solving relaxed problems, in which each nonconvex function is replaced by its convex envelope; (iii) standard convergence properties of branch-and-bound algorithms for nonconvex global optimization problems are guaranteed. Numerical results of preliminary computational experiments for the case of one quadratic constraint are reported.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1026437621223
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  • 99
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    Lagrangian duality of concave minimization subject to linear constraints and an additional facial reverse convex constraint (1996)
    Springer
    Journal of optimization theory and applications 91 (1996), S. 617-641 
    Details
    ISSN: 1573-2878
    Keywords: Lagrangian duality ; global optimization ; concave minimization ; penalty methods
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract This paper is concerned with the global optimization problem of minimizing a concave function subject to linear constraints and an additional facial reverse convex constraint. Here, the feasible set is the union of some faces of the polyhedron determined by the linear constraints. Several well-known mathematical problems can be written or transformed into the form considered. The paper addresses the Lagrangian duality of the problem. It is shown that, under slight assumptions, the duality gap can be closed with a finite dual multiplier. Finite methods based on solving concave minimization problems are also proposed. We deal with the advantages accrued when outer approximation, cutting plane, or branch-and-bound methods are used for solving these subproblems.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02190124
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    Multiplicative Programming Problems: Analysis and Efficient Point Search Heuristic (1997)
    Springer
    Journal of optimization theory and applications 94 (1997), S. 487-510 
    Details
    ISSN: 1573-2878
    Keywords: Multiplicative programming ; global optimization ; concave minimization ; efficient points ; heuristic algorithms ; multiple objectives
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Multiplicative programming problems are difficult global optimization problems known to be NP-hard. At the same time, these problems have some important applications in engineering, finance, economics, and other fields. This article has two purposes. The first is to present an analysis that shows several relationships between concave multiplicative programs and concave minimization problems, and between concave multiplicative programs and certain multiple-objective mathematical programs. The second purpose is to propose and report computational results for a heuristic efficient-point search algorithm that we have designed for use on linear multiplicative programming problems. To our knowledge, this is the first heuristic algorithm of its type. The theoretical and algorithmic results given in the article offer some potentially important new avenues for analyzing and solving multiplicative programming problems of various types.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1022600232285
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