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  • Articles  (30)
  • 65L05  (19)
  • Optimization
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  • 1980  (3)
  • Mathematics  (26)
  • Computer Science  (6)
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  • Articles  (30)
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  • Springer  (30)
  • American Chemical Society
  • American Institute of Physics
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  • Blackwell Publishing Ltd
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  • 2005-2009
  • 1990-1994  (15)
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  • 1
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    Mathematical programming 23 (1982), S. 326-340 
    ISSN: 1436-4646
    Keywords: Optimization ; Quasi-Newton ; Conjugate Gradient
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract In this paper we study conjugate gradient algorithms for large optimization problems. These methods accelerate (or precondition) the conjugate gradient method by means of quasi-Newton matrices, and are designed to utilize a variable amount of storage, depending on how much information is retained in the quasi-Newton matrices. We are concerned with the behaviour of such methods on the underlying quadratic model, and in particular, with finite termination properties.
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  • 2
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    Mathematical programming 19 (1980), S. 61-77 
    ISSN: 1436-4646
    Keywords: Optimization ; Nonlinear Programming ; Unconstrained Optimization ; Nondifferentiable Optimization ; Min—Max Problems
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract In this paper, we suggest approximations for smoothing out the kinks caused by the presence of “max” or “min” operators in many non-smooth optimization problems. We concentrate on the continuous-discrete min—max optimization problem. The new approximations replace the original problem in some neighborhoods of the kink points. These neighborhoods can be made arbitrarily small, thus leaving the original objective function unchanged at almost every point ofR n . Furthermore, the maximal possible difference between the optimal values of the approximate problem and the original one, is determined a priori by fixing the value of a single parameter. The approximations introduced preserve properties such as convexity and continuous differentiability provided that each function composing the original problem has the same properties. This enables the use of efficient gradient techniques in the solution process. Some numerical examples are presented.
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  • 3
    ISSN: 1436-5057
    Keywords: 65L05 ; CR: 5.17 ; Numer. Analysis ; verallgemeinerte Runge-Kutta-Methoden ; steife Probleme
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science
    Description / Table of Contents: Abstract Adaptive Runge-Kutta-methods are considered. Investigations of stability for these linear implicit methods are studied. For the application a LS-stable method of order four with an adaptive stepsize control is proposed. Test results for 25 stiff initial value problems for different tolerances are discussed.
    Notes: Zusammenfassung Es werden adaptive Runge-Kutta-Verfahren betrachtet und Stabilitätsuntersuchungen für diese linear impliziten Methoden durchgeführt. Für die Anwendung wird ein LS-stabiles Verfahren vierter Ordnung mit einer angepaßten Schrittweitenkontrolle vorgeschlagen. Testergebnisse von 25 stiff Anfangswertproblemen für verschiedenen Toleranzen werden diskutiert.
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  • 4
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    BIT 23 (1983), S. 262-266 
    ISSN: 1572-9125
    Keywords: 65L05 ; Implicit R-K methods ; Classification ; Nonlinear stability
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract By means of a reflection and a transposition principle one can show interesting interrelations between several classes of implicit Runge-Kutta methods. These relations are very useful in connection with the study of stability properties and the construction of implicit Runge-Kutta methods.
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  • 5
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    BIT 31 (1991), S. 647-663 
    ISSN: 1572-9125
    Keywords: 65L05 ; Runge-Kutta ; parallelism ; defect ; interpolation
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract How can small-scale parallelism best be exploited in the solution of nonstiff initial value problems? It is generally accepted that only modest gains inefficiency are possible, and it is often the case that “fast” parallel algorithms have quite crude error control and stepsize selection components. In this paper we consider the possibility of using parallelism to improvereliability andfunctionality rather than efficiency. We present an algorithm that can be used with any explicit Runge-Kutta formula. The basic idea is to take several smaller substeps in parallel with the main step. The substeps provide an interpolation facility that is essentially free, and the error control strategy can then be based on a defect (residual) sample. If the number of processors exceeds (p − 1)/2, wherep is the order of the Runge-Kutta formula, then the interpolant and the error control scheme satisfy very strong reliability conditions. Further, for a given orderp, the asymptotically optimal values for the substep lengths are independent of the problem and formula and hence can be computed a priori. Theoretical comparisons between the parallel algorithm and optimal sequential algorithms at various orders are given. We also report on numerical tests of the reliability and efficiency of the new algorithm, and give some parallel timing statistics from a 4-processor machine.
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  • 6
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    BIT 31 (1991), S. 608-619 
    ISSN: 1572-9125
    Keywords: 65L05 ; error control ; Adams method ; embedded Runga-Kutta
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract We discuss error control for explicit methods when the stepsize is bounded by stability on the imaginary axis. Our main result is a formulation of a condition on the estimator of the local error which prevents the fast components to exceed the prescribed error tolerance. A PECE Adams method of 4th order accuracy is proposed for mildly stiff oscillatory systems. For comparison we also discuss embedded Runga-Kutta methods.
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  • 7
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    BIT 31 (1991), S. 89-101 
    ISSN: 1572-9125
    Keywords: 65L05 ; Numerical analysis ; stiff problems ; Rosenbrock methods
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract In this paper a new technique for avoiding exact Jacobians in ROW methods is proposed. The Jacobiansf' n are substituted by matricesA n satisfying a directional consistency conditionA n f n =f' n f n +O(h). In contrast to generalW-methods this enables us to reduce the number of order conditions and we construct a 2-stage method of order 3 and families of imbedded 4-stage methods of order 4(3). The directional approximation of the Jacobians has been realized via rank-1 updating as known from quasi-Newton methods.
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  • 8
    ISSN: 1572-9125
    Keywords: 65L05 ; stiff convergence ; linear multistep methods ; one-leg methods
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract To prove convergence of numerical methods for stiff initial value problems, stability is needed but also estimates for the local errors which are not affected by stiffness. In this paper global error bounds are derived for one-leg and linear multistep methods applied to classes of arbitrarily stiff, nonlinear initial value problems. It will be shown that under suitable stability assumptions the multistep methods are convergent for stiff problems with the same order of convergence as for nonstiff problems, provided that the stepsize variation is sufficiently regular.
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  • 9
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    BIT 31 (1991), S. 144-159 
    ISSN: 1572-9125
    Keywords: 41A20 ; 65L05 ; Rational approximation ; A-acceptability
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Rational approximations to the exponential function with real, not necessarily distinct poles are studied in this paper. The orthogonality relation is established in order to show that the zeros of the collocation polynomial of the corresponding Runge-Kutta method are all real, simple and positive. It is proven, that approximants with the smallest error constant are the Restricted Padé approximants of Nørsett. Some results concerning acceptability properties are given.
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  • 10
    ISSN: 1572-9125
    Keywords: 65L05 ; Second order periodic initial-value problem ; phase-lag
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract A four-step method with phase-lag of infinite order is developed for the numerical integration of second order initial-value problems. Extensive numerical testing indicates that this new method can be generally more accurate than other four-step methods.
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  • 11
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    BIT 31 (1991), S. 293-313 
    ISSN: 1572-9125
    Keywords: AMS(MOS) 65M10 ; 65L05 ; 47D05 ; Resolvent conditions ; evolution equations ; numerical stability
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract As many numerical processes for time discretization of evolution equations can be formulated as analytic mappings of the generator, they can be represented in terms of the resolvent. To obtain stability estimates for time discretizations, one therefore would like to carry known estimates on the resolvent back to the time domain. For different types of bounds of the resolvent of a linear operator, bounds for the norm of the powers of the operator and for their sum are given. Under similar bounds for the resolvent of the generator, some new stability bounds for one-step and multistep discretizations of evolution equations are then obtained.
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  • 12
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    BIT 31 (1991), S. 373-374 
    ISSN: 1572-9125
    Keywords: 65L05
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The polynomial associated with the largest disk of stability of anm-stage explict Runge-Kutta method of orderp is unique.
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  • 13
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    Aequationes mathematicae 24 (1982), S. 19-32 
    ISSN: 1420-8903
    Keywords: Primary 39B10 ; 22E99 ; 34A30 ; 34C35 ; 65L05
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    Topics: Mathematics
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  • 14
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    Numerische Mathematik 60 (1991), S. 465-476 
    ISSN: 0945-3245
    Keywords: 65L05 ; 65L20
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary In this paper the maximum attainable order of a special class of symmetrizers for Gauss methods is studied. In particular, it is shown that a symmetrizer of this type for thes-stage Gauss method can attain order 2s-1 only for 1 ≦s ≦ 3, and that these symmetrizers areL-stable. A classification of the maximum attainable order of symmetrizers for some higher stages is presented. AnL-stable symmetrizer is also shown to exist for each of the methods studied.
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  • 15
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    Numerische Mathematik 60 (1991), S. 145-161 
    ISSN: 0945-3245
    Keywords: 65L05 ; 65F20
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary This paper presents a family of methods for accurate solution of higher index linear variable DAE systems, $$A(t)\dot y(t) + B(t)y(t) = f(t)$$ . These methods use the DAE system and some of its first derivatives as constraints to a least squares problem that corresponds to a Taylor series ofy, or an approximative equality derived from a Pade' approximation of the exponential function. Accuracy results for systems transformable to standard canonical form are given. Advantages, disadvantages, stability properties and implementation of these methods are discussed and two numerical examples are given, where we compare our results with results from more traditional methods.
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  • 16
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    Numerische Mathematik 59 (1991), S. 321-348 
    ISSN: 0945-3245
    Keywords: 65D05 ; 65L05
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary Construction of optimal triangular meshes for controlling the errors in gradient estimation for piecewise linear interpolation of data functions in the plane is discussed. Using an appropriate linear coordinate transformation, rigorously optimal meshes for controlling the error in quadratic data functions are constructed. It is shown that the transformation can be generated as a curvilinear coordinate transformation for anyC data function with nonsingular Hessian matrix. Using this transformation, a construction of nearly optimal meshes for general data functions is described and the error equilibration properties of these meshes discussed. In particular, it is shown that equilibration of errors is not a sufficient condition for optimality. A comparison of meshes generated under several different criteria is made, and their equilibrating properties illustrated.
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  • 17
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    Numerische Mathematik 59 (1991), S. 723-737 
    ISSN: 0945-3245
    Keywords: 65L05
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary In this paper we give conditions for theB-convergence of Rosenbrock type methods when applied to stiff semi-linear systems. The convergence results are extended to stiff nonlinear systems in singular perturbation form. As a special case partitioned methods are considered. A third order method is constructed.
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  • 18
    ISSN: 0945-3245
    Keywords: 65F20 ; 65F25 ; 65L05
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary This paper extends the singular value decomposition to a path of matricesE(t). An analytic singular value decomposition of a path of matricesE(t) is an analytic path of factorizationsE(t)=X(t)S(t)Y(t) T whereX(t) andY(t) are orthogonal andS(t) is diagonal. To maintain differentiability the diagonal entries ofS(t) are allowed to be either positive or negative and to appear in any order. This paper investigates existence and uniqueness of analytic SVD's and develops an algorithm for computing them. We show that a real analytic pathE(t) always admits a real analytic SVD, a full-rank, smooth pathE(t) with distinct singular values admits a smooth SVD. We derive a differential equation for the left factor, develop Euler-like and extrapolated Euler-like numerical methods for approximating an analytic SVD and prove that the Euler-like method converges.
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  • 19
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    Journal of industrial microbiology and biotechnology 7 (1991), S. 27-34 
    ISSN: 1476-5535
    Keywords: Bacterium ; Fermentation ; Nutrition ; Optimization ; Liquefaction ; High-substrate
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Process Engineering, Biotechnology, Nutrition Technology
    Notes: Summary An alternative process for industrial lactic acid production was deveooped using a starch degrading lactic acid producing organism,Lactobacillus amylovorus B-4542. In this process, saccharification takes place during the fermentation, eliminating the need for complete hydrolysis of the starch to glucose prior to fermentation. The cost savings of this alternative are substantial since it eliminates the energy input, separate reactor tank, time, and enzyme associated with the typical pre-fermentation saccharification step. The only pre-treatment was gelatinization and enzyme-thinning of the starch to overcome viscosity problems associated with high starch concentrations and to make the starch more rapidly degradable. This fermentation process was optimized for temperature, substrate level, nitrogen source and level, mineral level, B-vitamins, volatile fatty acids, pH, and buffer source. The rate of the reaction and the final level of lactic acid obtained in the optimized liquefied starch process was similar to that obtained withL. delbrueckii B-445 using glucose as the substrate.
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  • 20
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    Computing 31 (1983), S. 69-82 
    ISSN: 1436-5057
    Keywords: 65L05 ; Cyclic methods ; non-smooth ordinary differential equations
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science
    Description / Table of Contents: Zusammenfassung Konvergenzsätze für zyklische Verfahren liegen vor, sofern die rechte Seite der Differentialgleichungy′=g(t, y) global Lipschitz-beschränkt ist. In der vorliegenden Arbeit wird gezeigt, daß man sich von dieser stark einschränkenden Bedingung weitgehend befreien kann. Überdies wird eine neue Definition für die Konsistenz zyklischer Verfahren eingeführt. Die hier angegebenen Konsistenzbedingungen sind im Falle echt zyklischer Verfahren (m〉1) schwächer als die klassischen Konsistenzforderungen. Neben Verfahrensordnungsargumenten ergibt sich dadurch eine zusätzliche Begründung für die Konstruktion zyklischer Verfahren, denn es werden mehr Koeffizienten zur Anpasung an andere Zwecke frei. Außerdem kann die Konvergenz weiterer Verfahren gezeigt werden. Beispiele werden angegeben.
    Notes: Abstract Convergence theorems for cyclic methods have been proved in the case of globally Lipschitz-continuous right-hand-sidesg for the differential equationy′=g(t, y). In the present paper it is shown that this strong restriction can be omitted. Moreover, a new definition for consistency of cyclic methods is pressented. The consistency conditions are weaker than the classical ones in the case of real cyclic methods (m〉1). This leadsbesides order-of-convergence-arguments — to an additional motivation for the construction of cyclic methods, since more coefficients will be available for fitting other purposes and more methods can be shown to be convergent. Examples are given.
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  • 21
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    Computing 31 (1983), S. 95-103 
    ISSN: 1436-5057
    Keywords: 65L15 ; 65L05 ; 65L10 ; 65N35 ; Eigenvalues ; differential eigenvalue problems ; differential equations ; systems of differential equations ; Tau method ; collocation ; Legendre or Chebyshev series methods
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science
    Description / Table of Contents: Zusammenfassung Es wird gezeigt, daß eine Technik zur numerischen Lösung von Eigenwertproblemen bei Differentialgleichungen, die sich auf einen operationalen Zugang zum Tau-Verfahren stützt, zu einer Methode von Chaves und Ortiz äquivalent ist. Die hier diskutierte Technik führt zu einer algorithmischen Formulierung in bemerkenswerter Einfachheit und zu numerischen Resultaten von hoher Genauigkeit. Sie erfordert keine Vorwärtsrechnung und kann auch bei komplizierten Mehrpunkt-Randbedingungen und einer nichtlinearen Abhängigkeit von Eigenwertparameter eingesetzt werden.
    Notes: Abstract A technique for the numerical solution of eigenvalue problems defined by differential equations, based on an operational approach to the Tau method recently proposed by the authors, is shown to be equivalent to a method of Chaves and Oritz. The technique discussed here leads to an algorithmic formulation of remarkable simplicity and to numerical results of high accuracy. It requires no shooting and can deal with complex multipoint boundary conditions and a nonlinear dependence on the eigenvalue parameter.
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  • 22
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    Algorithmica 6 (1991), S. 5-35 
    ISSN: 1432-0541
    Keywords: Digital circuitry ; Graph theory ; Linear programming ; Network flow ; Optimization ; Pipelining ; Propagation delay ; Retiming ; Synchronous circuitry ; Systolic circuits ; Timing analysis
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract This paper describes a circuit transformation calledretiming in which registers are added at some points in a circuit and removed from others in such a way that the functional behavior of the circuit as a whole is preserved. We show that retiming can be used to transform a given synchronous circuit into a more efficient circuit under a variety of different cost criteria. We model a circuit as a graph in which the vertex setV is a collection of combinational logic elements and the edge setE is the set of interconnections, each of which may pass through zero or more registers. We give anO(¦V∥E¦lg¦V¦) algorithm for determining an equivalent retimed circuit with the smallest possible clock period. We show that the problem of determining an equivalent retimed circuit with minimum state (total number of registers) is polynomial-time solvable. This result yields a polynomial-time optimal solution to the problem of pipelining combinational circuitry with minimum register cost. We also give a chacterization of optimal retiming based on an efficiently solvable mixed-integer linear-programming problem.
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  • 23
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    Numerische Mathematik 36 (1980), S. 319-331 
    ISSN: 0945-3245
    Keywords: AMS (MOS) ; 65L05 ; CR: 5.17
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary Burrage and Butcher [1, 2] and Crouzeix [4] introduced for Runge-Kutta methods the concepts ofB-stability,BN-stability and algebraic stability. In this paper we prove that for any irreducible Runge-Kutta method these three stability concepts are equivalent.
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  • 24
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    Journal of optimization theory and applications 36 (1982), S. 495-519 
    ISSN: 1573-2878
    Keywords: Optimization ; nonlinear programming ; Numerical methods ; computational methods ; augmented Lagrangian functions
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract In this paper, a new augmented Lagrangian function is introduced for solving nonlinear programming problems with inequality constraints. The relevant feature of the proposed approach is that, under suitable assumptions, it enables one to obtain the solution of the constrained problem by a single unconstrained minimization of a continuously differentiable function, so that standard unconstrained minimization techniques can be employed. Numerical examples are reported.
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  • 25
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    Journal of optimization theory and applications 38 (1982), S. 25-31 
    ISSN: 1573-2878
    Keywords: Optimization ; Kuhn-Tucker theorem ; LegendreK-transform ; equilibrium composition
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The most common way of determining the steady states of a system is through the optimization of a concave function on a convex set. This applies only to cases where the objective function depends exclusively onextensive variables. In fields such as econometrics, physics, or chemistry, controllable quantities (and therefore constrained variables) are generally referred to asintensive parameters, and the states are described through a potential function. In the following pages, we examine how these two aspects can be connected.
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  • 26
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    Journal of optimization theory and applications 38 (1982), S. 409-422 
    ISSN: 1573-2878
    Keywords: Optimization ; scalar optimization ; vector optimization ; optimization under vector-valued criteria ; maximum principle ; dynamic programming ; optimization of dynamic systems ; multi-criteria decision problems
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The optimal control problem with vector-valued criteria is considered. A new approach to the generalization of this problem and a method of constructing the Bellman function are given.
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  • 27
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    Journal of optimization theory and applications 32 (1980), S. 463-478 
    ISSN: 1573-2878
    Keywords: Optimization ; control ; relativity ; stochastic processes ; wave mechanics
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The intention of this paper is to show how the influence of the work of Bellman has initiated a tentative new approach to quantum mechanics
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  • 28
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    Journal of optimization theory and applications 40 (1983), S. 595-623 
    ISSN: 1573-2878
    Keywords: Optimization ; partial differential equations ; elasticity ; control ; design sensitivity
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The dependence of the static response and the eigenvalues of a membrane on its shape is characterized. A transformation function is defined to determine the shape of the membrane. Differential operator properties and transformation techniques of integral calculus are employed to show that the static response and the eigenvalues of the system depend in a continuous and differentiable way on the shape of the membrane. Explicit and computable formulas are presented for the derivative (first variation) of the structural response and the eigenvalues with respect to the shape. A rigorous proof is provided, and the shape design sensitivity of a typical integral functional is determined.
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    Journal of optimization theory and applications 40 (1983), S. 321-331 
    ISSN: 1573-2878
    Keywords: Optimization ; necessary conditions ; Kuhn-Tucker theorem ; multiplier rules ; separation of convex sets ; mathematical programming ; generalized derivatives
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract We consider the following abstract mathematical programming problem: in a setD, find an element that optimizes a real function φ0, subject to inequality constraints φ1⩽0, ..., φ p ⩽0 and equality constraints φ p+1=0, ..., φ p+q =0. Necessary conditions for this problem, like the Karush-Kuhn-Tucker theorem, can be seen as a consequence of separating with a hyperplane two convex sets inR p+q+1, the image space of the map Φ=(φ0, φ1, ..., φ p+q ). This paper reviews this approach and organizes it into a coherent way of looking at necessary conditions in optimization theory.
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  • 30
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    Journal of optimization theory and applications 37 (1982), S. 315-341 
    ISSN: 1573-2878
    Keywords: Optimization ; unconstrained minimization ; updates ; line searches ; convergence ; numerical methods
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract We consider a certain generalization of the Huang family of updates and discuss, firstly, convergence, dependence on parameters, and descent property; secondly, invariance under nonlinear scaling, conjugacy of search directions, and possibility of achieving a better approximation of the inverse of the Hessian. The last three aspects are shown to be dependent on particular choices of parameters. A numerical experiment is presented comparing the performances of the usual and modified BFGS algorithms.
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