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  • Articles  (89)
  • Articles: DFG German National Licenses  (89)
  • CR: 5.15  (37)
  • Stability  (27)
  • AMS(MOS): 65N30  (25)
  • 2010-2014
  • 1980-1984  (89)
  • 1950-1954
  • Mathematics  (89)
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  • Articles  (89)
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  • Articles: DFG German National Licenses  (89)
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  • 1
    Electronic Resource
    Electronic Resource
    Some perturbation results for the Linear Complementarity Problem (1982)
    Springer
    Mathematical programming 23 (1982), S. 181-192 
    Details
    ISSN: 1436-4646
    Keywords: Linear Complementarity Problem ; Stability ; Classes of Matrices
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract It has been shown previously that the Linear Complementarity Problem is stable when the defining matrix is positive semidefinite and when (locally) the set of solutions is nonempty and bounded. We enlarge the class of matrices for which this is true and also demonstrate how the boundedness condition leads to other stability type questions.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01583787
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  • 2
    Electronic Resource
    Electronic Resource
    Sublinear convergence of the Chord method at singular points (1983)
    Springer
    Numerische Mathematik 42 (1983), S. 147-154 
    Details
    ISSN: 0945-3245
    Keywords: AMS (MOS): 65H10 ; CR: 5.15
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary We give sufficient conditions for convergence of the chord method for a class of singular problems. The rate of convergence is sublinear.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01395307
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  • 3
    Electronic Resource
    Electronic Resource
    A method of solving nonlinear variational problems by nonlinear transformation of the objective functional. Part I (1984)
    Springer
    Numerische Mathematik 43 (1984), S. 59-82 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65K10 ; CR: 5.15 ; 5.41
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary A general globally convergent iterative method for solving nonlinear variational problems is introduced. The method is applied to a temperature control problem and to the minimal surface problem. Several aspects of finite element implementation of the method are discussed.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01389638
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  • 4
    Electronic Resource
    Electronic Resource
    Das modifizierte Newton-Verfahren in verallgemeinerten Banach-Räumen (1984)
    Springer
    Numerische Mathematik 43 (1984), S. 91-104 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65J15 ; CR: 5.15
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary A Newton-Kantorowitsch-analysis of the modified Newton's method in generalized Banach-spaces is given. The application of generalized norms — mappings from a linear space into an ordered Banach-space — improves convergence- and existence results as well as error estimates compared with the real-norm theory.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01389640
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  • 5
    Electronic Resource
    Electronic Resource
    A general convergence theorem for the reduced gradient method (1984)
    Springer
    Numerische Mathematik 43 (1984), S. 241-247 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 90C30 ; CR: 5.15
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary Consider the problem of minimizing a real function subject to linear equality constraints and to nonnegativity bounds on the variables. A convergence theorem is established for a general algorithm model based on the reduced gradient method. The most meaningful assumptions that are made, concern two crucial points: the choice of the independent variables and that of the search direction.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01390125
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  • 6
    Electronic Resource
    Electronic Resource
    Error bounds of Newton type process on Banach spaces (1982)
    Springer
    Numerische Mathematik 39 (1982), S. 175-193 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS) (1980) ; Primary 65J05, 65J15 ; Secondary 47H10 ; CR: 5.15
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary For solving a nonlinear operator equationF(x)=0 in Banach spaces, the Newton's method or Newton type methods are important numerical techniques. We use the properties of real equationt=Φ(t) majorizing an operator equationx=Gx to find a fixed point ofG as a solution of equationF(x)=0. Various type of operatorsG are considered in this paper. For a nonlinear operatorG, we would find a real function Φ majorizing the operatorG and it will be related to a rate of convergence $$\omega (r) = \frac{{r^2 }}{{2(r^2 + d)^{1/2} }}.$$ It follows thatG has a fixed point as a solution ofF(x)=0. Practical limitations of error bounds like as in Potra and Pták [5] are discribed.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01408692
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  • 7
    Electronic Resource
    Electronic Resource
    Some convergence results for asynchronous algorithms (1982)
    Springer
    Numerische Mathematik 39 (1982), S. 325-340 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65H10 ; CR: 5.15
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Description / Table of Contents: Summary In this paper we present convergence results for the asynchronous algorithms based essentially on the notion of classical contraction. We generalize, in particular, all convergence results for those algorithms which are based on the vectorial norm hypothesis, in wide spread use recently. Certain problems, for which the vectorial norm hypothesis can be difficult or even impossible to verify, can nontheless be tackled within the scope of the classical contraction that we adopte.
    Notes: Resumé Nous présentons dans cet article des résultats de convergence des algorithmes asynchrones basés essentiellement sur la notion classique de contraction. Nous généralisons, en particulier, tous les résultats de convergence de ces algorithmes qui font l'hypothèse de contraction en norme vectorielle qui récemment a été très souvant utilisée. Par ailleurs, l'hypothèse de contraction en norme vectorielle peut se trouver difficile, voire impossible à vérifier pour certains problèmes qui peuvent être cependant abordés dans le cadre de la contraction classique que nous adoptons.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01407866
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  • 8
    Electronic Resource
    Electronic Resource
    Local convergence analysis for partitioned quasi-Newton updates (1982)
    Springer
    Numerische Mathematik 39 (1982), S. 429-448 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65K05 ; CR: 5.15
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary This paper considers local convergence properties of inexact partitioned quasi-Newton algorithms for the solution of certain non-linear equations and, in particular, the optimization of partially separable objective functions. Using the bounded deterioration principle, one obtains local and linear convergence, which impliesQ-superlinear convergence under the usual conditions on the quasi-Newton updates. For the optimization case, these conditions are shown to be satisfied by any sequence of updates within the convex Broyden class, even if some Hessians are singular at the minimizer. Finally, local andQ-superlinear convergence is established for an inexact partitioned variable metric method under mild assumptions on the initial Hessian approximations.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01407874
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  • 9
    Electronic Resource
    Electronic Resource
    Bisection is optimal (1982)
    Springer
    Numerische Mathematik 40 (1982), S. 111-117 
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    ISSN: 0945-3245
    Keywords: AMS(MOS): 65H10 ; CR: 5.15
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary We seek an approximation to a zero of a continuous functionf:[a,b]→ℝ such thatf(a)≦0 andf(b)≧0. It is known that the bisection algorithm makes optimal use ofn function evaluations, i.e., yields the minimal error which is (b−a)/2 n+1, see e.g. Kung [2]. Traub and Wozniakowski [5] proposed using more general information onf by permitting the adaptive evaluations ofn arbitrary linear functionals. They conjectured [5, p. 170] that the bisection algorithm remains optimal even if these general evaluations are permitted. This paper affirmatively proves this conjecture. In fact we prove optimality of the bisection algorithm even assuming thatf is infinitely many times differentiable on [a, b] and has exactly one simple zero.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01459080
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  • 10
    Electronic Resource
    Electronic Resource
    Maximum-norm stability and error estimates in Galerkin methods for parabolic equations in one space variable (1983)
    Springer
    Numerische Mathematik 41 (1983), S. 345-371 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65N30 ; CR: 5.17
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary Maximum-norm stability and error estimates of best approximation and nonsmooth data types are derived for the approximate solution of a parabolic equation in one space variable, using the continuous in time Galerkin method based on piecewise polynomial approximating functions on a quasi-uniform mesh.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01418330
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  • 11
    Electronic Resource
    Electronic Resource
    On a numerical treatment for the curve-tracing of the homotopy method (1983)
    Springer
    Numerische Mathematik 42 (1983), S. 323-329 
    Details
    ISSN: 0945-3245
    Keywords: AMS (MOS): 65K05 ; CR: 5.15
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary A Newton-like iteration scheme is proposed for the tracing of an implicitly defined smooth curve. This scheme originates from the study of the continuous Newton-Raphson method for underdetermined systems and, hence, inherits the characteristic property of orthogonality. Its domain of attraction is formed and makes it possible to trace this curve more efficiently.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01389577
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  • 12
    Electronic Resource
    Electronic Resource
    A generalized conjugate gradient algorithm for minimization (1980)
    Springer
    Numerische Mathematik 35 (1980), S. 223-230 
    Details
    ISSN: 0945-3245
    Keywords: AMS (MOS): 65K05 ; CR: 5.15
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary A generalized conjugate gradient algorithm which is invariant to a nonlinear scaling of a strictly convex quadratic function is described, which terminates after at mostn steps when applied to scaled quadratic functionsf: R n →R1 of the formf(x)=h(F(x)) withF(x) strictly convex quadratic andh∈C 1 (R1) an arbitrary strictly monotone functionh. The algorithm does not suppose the knowledge ofh orF but only off(x) and its gradientg(x).
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01396318
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  • 13
    Electronic Resource
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    On the rate of superlinear convergence of a class of variable metric methods (1980)
    Springer
    Numerische Mathematik 35 (1980), S. 293-313 
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    ISSN: 0945-3245
    Keywords: AMS (MOS) ; 90C30 ; CR: 5.15
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary This paper considers a class of variable metric methods for unconstrained minimization. Without requiring exact line searches each algorithm in this class converges globally and superlinearly on convex functions. Various results on the rate of the superlinear convergence are obtained.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01396414
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  • 14
    Electronic Resource
    Electronic Resource
    A mixed finite element approximation of the Navier-Stokes equations (1980)
    Springer
    Numerische Mathematik 35 (1980), S. 381-404 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65N30 ; CR: 5.17
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary We study in this paper the convergence of a new mixed finite element approximation of the Navier-Stokes equations. This approximation uses low order Lagrange elements, leads to optimal order of convergence for the velocity and the pressure, and induces an efficient numerical algorithm for the solution of this problem.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01399007
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  • 15
    Electronic Resource
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    A convergence condition for the quadrilateral Wilson element (1984)
    Springer
    Numerische Mathematik 44 (1984), S. 349-361 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65N30 ; CR: G 1.8
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary The paper deals with the convergence properties of the nonconforming quadrilateral wilson element which violates the patch test. The convergence of the element is proved under a certain condition on mesh subdivisions without any modifications of the variational formulation. This result extends the range of applicability of Wilson's element. The necessity of the proposed condition is also discussed.
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    URL: http://dx.doi.org/10.1007/BF01405567
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  • 16
    Electronic Resource
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    Approximation de quelques problèmes de type Stokes par une méthode d'éléments finis mixtes (1984)
    Springer
    Numerische Mathematik 45 (1984), S. 75-91 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65N30 ; CR: 5-17
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Description / Table of Contents: Summary We present in this paper a mixed-finite element approximation of Stokes equations with boundary conditions of Fourier's or Neumann's type. For this approximation we prove that the error estimates for the velocity-vector and for the pressure are optimal. These results of convergence generalize to this kind of boundary conditions the Glowinski-Pironneau's approximation of Stokes problem with Dirichlet's boundary conditions [9, 10].
    Notes: Résumé Nous présentons dans cet article une méthode d'éléments finis mixtes qui permet la résolution des équations de Stokes avec des conditions aux limites de type Fourier ou Neumann. Pour cette méthode nous démontrons que les estimations de l'erreur d'approximation sont optimales; en vitesse et en pression. Ces résultats de convergences généralisent à ce type non standard de conditions aux limites les travaux de Glowinski-Pironneau [9, 10] pour le probleme de Stokes avec des conditions aux limites de Dirichlet.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01379662
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  • 17
    Electronic Resource
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    Estimations des erreurs de meilleure approximation polynomiale et d'interpolation de Lagrange dans les espaces de Sobolev d'ordre non entier (1984)
    Springer
    Numerische Mathematik 45 (1984), S. 301-321 
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    ISSN: 0945-3245
    Keywords: AMS(MOS): 65N30 ; CR: G18
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Description / Table of Contents: Summary Explicit bounds for the best polynomial approximation error, explicit and non-explicit bounds for the Lagrange interpolation error are derived for functions belonging to fractional order Sobolev spaces defined over a bounded open set in ℝ n . Thus the classical results of the integer order Sobolev spaces are extended.
    Notes: Résumé On établit des majorations explicites de I'erreur de meilleure approximation polynomiale ainsi que des majorations explicites et nonexplicites de I'erreur d'interpolation de Lagrange, lorsque la fonction considérée appartient à un espace de Sobolev d'ordre non entier défini sur un ouvert borné de ℝ n . Les résultats obtenus généralisent les résultats connus dans le cas des espaces de Sobolev d'ordre entier.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01389473
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  • 18
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    Error estimates for the combinedh andp versions of the finite element method (1981)
    Springer
    Numerische Mathematik 37 (1981), S. 257-277 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65N30 ; CR: 5.17
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary In theh-version of the finite element method, convergence is achieved by refining the mesh while keeping the degree of the elements fixed. On the other hand, thep-version keeps the mesh fixed and increases the degree of the elements. In this paper, we prove estimates showing the simultaneous dependence of the order of approximation on both the element degrees and the mesh. In addition, it is shown that a proper design of the mesh and distribution of element degrees lead to a better than polynomial rate of convergence with respect to the number of degrees of freedom, even in the presence of corner singularities. Numerical results comparing theh-version,p-version, and combinedh-p-version for a one dimensional problem are presented.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01398256
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  • 19
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    Quotienten-Differenzen-Algorithmus: Beweis der Regeln von Rutishauser (1982)
    Springer
    Numerische Mathematik 40 (1982), S. 93-98 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65D15 ; CR: 5.15
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Description / Table of Contents: Zusammenfassung Der Quotienten-Differenzen-Algorithmus nach Rutishauser ist geeignet zur Bestimmung von Polen meromorpher Funktionen, gegeben durch eine Taylorreihe. Sind mehrere Pole betragsgleich, so kann eine Polynomfolge bestimmt werden, deren Grenzpolynom diese Pole als Nullstellen hat. Die Konvergenz wurde von Rutishauser jedoch nicht bewiesen. Ein Beweis wird in der vorliegenden Arbeit präsentiert.
    Notes: Summary The Quotient-Difference Algorithm of Rutishauser can be used for the determination of poles of a meromorphic function given by its power series. If some of the poles have same modulus, a sequence of polynomials can be determined such that the limiting polynomial has exactly these poles as zeros. The convergence has not been proved by Rutishauser, however. A proof is presented in this paper.
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    URL: http://dx.doi.org/10.1007/BF01459078
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  • 20
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    The finite element method for parabolic equations (1982)
    Springer
    Numerische Mathematik 40 (1982), S. 339-371 
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    ISSN: 0945-3245
    Keywords: AMS(MOS): 65N30 ; 65M20 ; CR: 5.17
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary In this first of two papers, computable a posteriori estimates of the space discretization error in the finite element method of lines solution of parabolic equations are analyzed for time-independent space meshes. The effectiveness of the error estimator is related to conditions on the solution regularity, mesh family type, and asymptotic range for the mesh size. For clarity the results are limited to a model problem in which piecewise linear elements in one space dimension are used. The results extend straight-forwardly to systems of equations and higher order elements in one space dimension, while the higher dimensional case requires additional considerations. The theory presented here provides the basis for the analysis and adaptive construction of time-dependent space meshes, which is the subject of the second paper. Computational results show that the approach is practically very effective and suggest that it can be used for solving more general problems.
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    URL: http://dx.doi.org/10.1007/BF01396451
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  • 21
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    Error analysis for Laplace transform —Finite element solution of hyperbolic equations (1983)
    Springer
    Numerische Mathematik 41 (1983), S. 55-62 
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    ISSN: 0945-3245
    Keywords: AMS(MOS): 65N30 ; CR: 5.17
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary The finite element method with Laplace transform of time variable is proposed for the solution of hyperbolic equations. Error estimates in Hardy spaces of functions with values in Sobolev spaces are derived. Due to the isometric isomorphism of Hardy spaces with weighted Hilbert spaces these estimates are valid also for original formulations of hyperbolic equations.
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    URL: http://dx.doi.org/10.1007/BF01396305
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  • 22
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    An analysis of thep-version of the finite element method for nearly incompressible materials (1983)
    Springer
    Numerische Mathematik 41 (1983), S. 39-53 
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    ISSN: 0945-3245
    Keywords: AMS(MOS): 65N30 ; 73K25 CR: 5.17
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary In this paper we analyze the behavior of the so-calledp-version of the finite element method when applied to the equations of plane strain linear elasticity. We establish optimal rate error estimates that are uniformly valid, independent of the value of the Poisson ratio,v, in the interval ]0, 1/2[. This shows that thep-versiondoes not exhibit the degeneracy phenomenon which has led to the use of various, only partially justified techniques of reduced integration or mixed formulations for more standard finite element schemes and the case of a nearly incompressible material.
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    URL: http://dx.doi.org/10.1007/BF01396304
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  • 23
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    On the steps of convergence of approximate eigenvectors in the Rayleigh-Schrödinger series (1983)
    Springer
    Numerische Mathematik 42 (1983), S. 31-50 
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    ISSN: 0945-3245
    Keywords: AMS(MOS): 65J10 ; 41A25 ; 41A65 ; CR: 5.15
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary Error bounds are given for the iterative computation of the eigenvectors in the Rayleigh-Schrödinger series. These bounds remove the discrepancy in the theoretical behaviour and numerical results, noted by Redont, under the assumption of collectively compact convergence. As a particular case, it follows that the eigenvector in the iterative Galerkin method proposed by Sloan improves upon the eigenvector in the Galerkin method. This is illustrated by numerical experiments.
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    URL: http://dx.doi.org/10.1007/BF01400916
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  • 24
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    Imperfect conjugate gradient algorithms for extended quadratic functions (1983)
    Springer
    Numerische Mathematik 42 (1983), S. 97-105 
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    ISSN: 0945-3245
    Keywords: AMS (MOS): 65K05 ; CR: 5.15
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary Generalized conjugate gradient algorithms which are invariant to a nonlinear scaling of a strictly convex quadratic function are described. The algorithms when applied to scaled quadratic functionsf∶R n →R 1 of the formf(x)=h(F(x)) withF(x) strictly convex quadratic andh∈C 1(R 1) an arbitrary strictly monotone functionh generate the same direction vectors as for the functionF without perfect steps.
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    URL: http://dx.doi.org/10.1007/BF01400920
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  • 25
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    On conforming finite element methods for the inhomogeneous stationary Navier-Stokes equations (1983)
    Springer
    Numerische Mathematik 42 (1983), S. 173-194 
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    ISSN: 0945-3245
    Keywords: AMS(MOS): 65N30 ; CR: 5.17
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary We consider the stationary Navier-Stokes equations, written in terms of the primitive variables, in the case where both the partial differential equations and boundary conditions are inhomogeneous. Under certain conditions on the data, the existence and uniqueness of the solution of a weak formulation of the equations can be guaranteed. A conforming finite element method is presented and optimal estimates for the error of the approximate solution are proved. In addition, the convergence properties of iterative methods for the solution of the discrete nonlinear algebraic systems resulting from the finite element algorithm are given. Numerical examples, using an efficient choice of finite element spaces, are also provided.
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    URL: http://dx.doi.org/10.1007/BF01395310
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  • 26
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    Sharp error bounds for Newton's process (1980)
    Springer
    Numerische Mathematik 34 (1980), S. 63-72 
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    ISSN: 0945-3245
    Keywords: AMS(MOS): 65H10 ; CR: 5.15
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary The method of nondiscrete mathematical induction is applied to the Newton process. The method yields a very simple proof of the convergence and sharp apriori estimates; it also gives aposteriori bounds which are, in general, better than those given in [1].
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01463998
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  • 27
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    Sur la convergence théorique de la méthode du gradient réduit généralisé (1980)
    Springer
    Numerische Mathematik 34 (1980), S. 73-85 
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    ISSN: 0945-3245
    Keywords: AMS(MOS): 65K05 ; CR: 5.15
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary The theoretical convergence of Generalized Reduced Gradient Method (GRG) has not been proved; the purpose of this paper is to propose two theoritical and general variants of the original method and a proof of their convergence.
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    URL: http://dx.doi.org/10.1007/BF01463999
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  • 28
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    The numerical solution of the inverse Stefan problem (1980)
    Springer
    Numerische Mathematik 34 (1980), S. 411-429 
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    ISSN: 0945-3245
    Keywords: AMS(MOS): 65N10 ; CR: 5.15
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    Notes: Summary The inverse Stefan problem can be understood as a problem of nonlinear approximation theory which we solved numerically by a generalized Gauss-Newton method introduced by Osborne and Watson [19]. Under some assumptions on the parameter space we prove its quadratic convergence and demonstrate its high efficiency by three numerical examples.
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    Starlike domains of convergence for Newton's method at singularities (1980)
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    Numerische Mathematik 35 (1980), S. 95-111 
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    Keywords: AMS(MOS): 65H10 ; CR: 5.15
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    Topics: Mathematics
    Notes: Summary Given a solutionx * of a system of nonlinear equationsf with singular Jacobian ∇f(x *) we construct an open starlike domainR of initial points, from which Newton's method converges linearly tox *. Under certain conditions the union of those straight lines throughx *, that do not intersect withR is shown to form a closed set of measure zero, which is necessarily disjoint from any starlike domain of convergence. The results apply to first and higher order singularities.
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    Algorithmes mixtes asynchrones. Etude de convergence monotone (1984)
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    Numerische Mathematik 44 (1984), S. 363-369 
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    Keywords: AMS(MOS): 65H10 ; CR: 5.15
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    Topics: Mathematics
    Description / Table of Contents: Summary In this paper we introduce two classes of iterative algorithms which we call “Asynchronous mixed algorithms” and we study their convergence under partial ordering. These algorithms can be implemented just as well on monoprocessors as on multiprocessors, and, along with their convergence study, constitute a generalization of the mixed classical “Newton-relaxation” algorithms.
    Notes: Résumé Nous introduisons dans cet article deux classes d'algorithmes itératifs que nous appelons «Algorithmes mixtes asynchrones» et nous en étudierons la convergence selon un ordre partiel. Ces algorithmes sont implémentables aussi bien sur les monoprocesseurs que sur les multiprocesseurs, et avec leur étude de convergence constituent une généralisation des algorithmes mixtes classiques «Newton-relaxation».
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    Eine Aussage zurL ∞-Stabilität und zur genauen Konvergenzordnung derH 0 1 -Projektionen (1984)
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    Numerische Mathematik 44 (1984), S. 393-405 
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    Keywords: AMS(MOS): 65N30 ; CR: G.1.8
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    Topics: Mathematics
    Notes: Summary In this paper we settle a question concerning the necessity of the logh-factor appearing in error estimates of linear finite element solutions. Let Ω⊂ℝ2 be a bounded convex polygon and {T h } a regular family of subdivisions of Ω into triangles. Be furtherS 0 h ⊂H 0 1 (Ω) the corresponding linear finite element spaces andP h :H 0 1 (Ω)→S 0 h theH 0 1 -projections. With respect to the uniform norm asymptotically estimates of the form $$\parallel u - P_h u\parallel _\infty \leqq c|logh|dist(u,S_0^h ), u \in C(\bar \Omega ) \cap H_0^1 (\Omega )$$ hold, as pointed out by Schatz [1980]. For functionsu with bounded second derivatives this implies $$\parallel u - P_h u\parallel _\infty \leqq ch^2 |\log h||u|_{2,\infty } ,$$ a result which was proved by Nitsche [1975] and Scott [1976] with different methods. It was an open question if the factor |logh| is necessary or only due to the methods of proof. In this paper it is shown by an example that the cited estimates are sharp in the very strong sense that in general they are no longer valid if |logh| is replaced by a termo(|logh|).
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    Approximation of tricomi problem with Neumann boundary condition (1984)
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    Numerische Mathematik 44 (1984), S. 371-391 
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    Keywords: AMS(MOS): 65N30 ; CR: G 1.8
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    Topics: Mathematics
    Notes: Summary The Tricomi problem with Neumann boundary condition is reduced to a degenerate problem in the elliptic region with a non-local boundary condition and to a Cauchy problem in the hyperbolic region. A variational formulation is given to the elliptic problem and a finite element approximation is studied. Also some regularity results in weighted Sobolev spaces are discussed.
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    A quadratic approximation method for minimizing a class of quasidifferentiable functions (1984)
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    Numerische Mathematik 45 (1984), S. 411-430 
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    Keywords: AMS MOS ; 90C30 ; 65K05 ; CR: 5.15
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    Notes: Summary An unconstrained nonlinear programming problem with nondifferentiabilities is considered. The nondifferentiabilities arise from terms of the form max [f 1(x), ...,f n (x)], which may enter nonlinearly in the objective function. Local convex polyhedral upper approximations to the objective function are introduced. These approximations are used in an iterative method for solving the problem. The algorithm proceeds by solving quadratic programming subproblems to generate search directions. Approximate line searches ensure global convergence of the method to stationary points. The algorithm is conceptually simple and easy to implement. It generalizes efficient variable metric methods for minimax calculations.
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    Root finding by divided differences (1981)
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    Numerische Mathematik 37 (1981), S. 93-104 
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    Topics: Mathematics
    Notes: Summary A recursive method is presented for computing a simple zero of an analytic functionf from information contained in a table of divided differences of its reciprocalh=1/f. A good deal of flexibility is permitted in the choice of ordinate and derivative values, and in the choice of the number of previous points upon which to base the next estimate of the required zero. The method is shown to be equivalent to a process of fitting rational functions with linear numerators to data sampled fromf. Asymptotic and regional convergence properties of such a process have already been studied; in particular, asymptotically quadratic convergence is easily obtained, at the expense of only one function evaluation and a moderate amount of “overhead” computation per step. In these respects the method is comparable with the Newton form of iterated polynomial inverse interpolation, while its regional convergence characteristics may be superior in certain circum-stances. It is also shown that the method is not unduly sensitive to round-off errors.
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    Convergence of a penalty-finite element approximation for an obstacle problem (1981)
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    Numerische Mathematik 37 (1981), S. 105-120 
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    Keywords: AMS(MOS): 65N30 ; CR: 5.17
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    Topics: Mathematics
    Notes: Summary This study establishes an error estimate for a penalty-finite element approximation of the variational inequality obtained by a class of obstacle problems. By special identification of the penalty term, we first show that the penalty solution converges to the solution of a mixed formulation of the variational inequality. The rate of convergence of the penalization is ɛ where ɛ is the penalty parameter. To obtain the error of finite element approximation, we apply the results obtained by Brezzi, Hager and Raviart for the mixed finite element method to the variational inequality.
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    Superlinear convergence of symmetric Huang's class of methods (1981)
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    Numerische Mathematik 38 (1981), S. 209-218 
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    Keywords: AMS(MOS): 65K05 ; CR: 5.15
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    Topics: Mathematics
    Notes: Summary In this paper the problem of minimizing the functionalf:D∈R n →R is considered. Typical assumptions onf are assumed. A class of Quasi-Newton methods, namely Huang's class of methods is used for finding an optimal solution of this problem. A new theorem connected with this class is presented. By means of this theorem some convergence results known up till now only for the methods which satisfy Quasi-Newton condition are extended, that is the results of superlinear convergence of variable metric methods in the cases of exact and asymptotically exact minimization and the so-called direct-prediction case. This theorem allows to interpretate one of the parameters as the scaling parameter.
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    Quasi-Newton-Verfahren vom Rang-Eins-Typ zur Lösung unrestringierter Minimierungsprobleme (1981)
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    Numerische Mathematik 38 (1981), S. 219-228 
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    Keywords: AMS(MOS): 65K05 ; CR: 5.15
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    Topics: Mathematics
    Notes: Summary In this paper new Quasi-Newton methods of rank-one type for solving the unconstrained minimization problem are developed. The methods belong to the Oren-Luenberger class (for negative parametersΘ k ) and they generate always positive definite updating matrices. Moreover it is shown that these methods are invariant by scaling of the objective function.
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    Quasi-Newton-Verfahren vom Rang-Eins-Typ zur Lösung unrestringierter Minimierungsprobleme (1981)
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    Numerische Mathematik 38 (1981), S. 229-244 
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    Topics: Mathematics
    Notes: Summary It is shown that the methods developed in part I of this paper [6] converge at leastn-step quadratically to the global minimum point of the objective functionf, if the Hessian off is uniformly positive definite, uniformly bounded and Lipschitz continuous and if the method is used with restart and asymptotically exact line search.
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    Existence and approximation results for nonlinear mixed problems: Application to incompressible finite elasticity (1982)
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    Numerische Mathematik 38 (1982), S. 365-382 
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    Keywords: AMS(MOS): 65N30 ; CR: 5.17
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    Topics: Mathematics
    Notes: Summary This paper considers the problems of minimizing Gateaux-differentiable functionals over subsets of real Banach spaces defined by a non-linear equality constraint. The existence of a Lagrange multiplier is proved, together with approximation results on the constrained subset, provided a nonlinear compatibility condition, generalizing the classical inf-sup condition, is satisfied. These ideas are applied to equilibrium problems in incompressible finite elasticity and lead to convergence results for these problems.
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    Some supplementary results on the 1+ $$\sqrt 2 $$ order method for the solution of nonlinear equations (1982)
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    Numerische Mathematik 38 (1982), S. 383-392 
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    Keywords: AMS(MOS): 65H10 ; CR: 5.15
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    Topics: Mathematics
    Notes: Summary Recently an iterative method for the solution of systems of nonlinear equations having at leastR-order 1+ $$\sqrt 2 $$ for simple roots has been investigated by the author [7]; this method uses as many function evaluations per step as the classical Newton method. In the present note we deal with several properties of the method such as monotone convergence, asymptotic inclusion of the solution and convergence in the case of multiple roots.
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    Approximation of optimal distributed control problems governed by variational inequalities (1982)
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    Numerische Mathematik 38 (1982), S. 393-416 
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    Keywords: AMS(MOS): 49D05, 49D07, 65K10 ; CR: 5.15
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    Topics: Mathematics
    Notes: Summary A method for approximating the optimal control and the optimal state for a class of distributed control problems governed by variational inequalities is given. It uses a Rayleigh-Ritz-Galerkin scheme, regularising techniques and a gradient algorithm. A numerical example is given.
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    An error analysis for the secant method (1982)
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    Numerische Mathematik 38 (1982), S. 427-445 
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    Keywords: AMS(MOS) 65J15, 65G99 ; CR: 5.15
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    Topics: Mathematics
    Notes: Summary Sharp a priori and a posteriori error bounds are given for the secant method for solving non-linear equations in Banach spaces. The numerical stability of this method is also investigated. The stability results are analogous to those obtained by Lancaster for Newton's method.
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    Ein lokal überlinear konvergentes Verfahren zur Bestimmung von Rückkehrpunkten implizit definierter Raumkurven (1982)
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    Numerische Mathematik 38 (1982), S. 455-465 
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    Keywords: AMS(MOS): 65H10, 47H17 ; CR: 5.15
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    Topics: Mathematics
    Notes: Summary A numerical technique is described for computing turning points of a space curveL implicitly defined by a nonlinear system ofn equations inn+1 variables. The basic idea is a local parametrization ofL where the parameter that gives the next iterate is determined by applying one step of the well-known method for minimizing a real function using cubic Hermite interpolation with two nodes. The method is shown to convergeQ-super-linearly and withR-order of at least two. A numerical example concerning the analysis of nonlinear resistive circuits shows the algorithm to work effectively on real life problems.
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    The numerical solutions of interface problems by infinite element method (1982)
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    Numerische Mathematik 39 (1982), S. 39-50 
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    Keywords: AMS(MOS): 65N30 ; CR: 5.17
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    Topics: Mathematics
    Notes: Summary In this paper we derive error estimates for infinite element method used in the approximation of solutions of interface problems. Furthermore, approximations of stress intensity factors are given. The infinite element method may be considered as a certain scheme of mesh refinement, but it has the advantages that the refinement is easy to be constructed that the stiffness matrix can be calculated efficiently, and that an approximate solution which has a singularity at the singular point can be also obtained.
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    Analysis of finite element methods for second order boundary value problems using mesh dependent norms (1980)
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    Numerische Mathematik 34 (1980), S. 41-62 
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    Keywords: AMS(MOS): 65N30 ; CR: 5.17
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    Topics: Mathematics
    Notes: Abstract This paper presents a new approach to the analysis of finite element methods based onC 0-finite elements for the approximate solution of 2nd order boundary value problems in which error estimates are derived directly in terms of two mesh dependent norms that are closely ralated to theL 2 norm and to the 2nd order Sobolev norm, respectively, and in which there is no assumption of quasi-uniformity on the mesh family. This is in contrast to the usual analysis in which error estimates are first derived in the 1st order Sobolev norm and subsequently are derived in theL 2 norm and in the 2nd order Sobolev norm — the 2nd order Sobolev norm estimates being obtained under the assumption that the functions in the underlying approximating subspaces lie in the 2nd order Sobolev space and that the mesh family is quasi-uniform.
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    Optimal solution of nonlinear equations satisfying a Lipschitz condition (1984)
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    Numerische Mathematik 43 (1984), S. 225-240 
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    Keywords: AMS(MOS): 65 H 10 ; CR: 5.15
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    Topics: Mathematics
    Notes: Summary For a given nonnegative ε we seek a pointx * such that |f(x *)|≦ε wheref is a nonlinear transformation of the cubeB=[0,1] m into ℝ (or ℝ p ,p〉1) satisfying a Lipschitz condition with the constantK and having a zero inB. The information operator onf consists ofn values of arbitrary linear functionals which are computed adaptively. The pointx * is constructed by means of an algorithm which is a mapping depending on the information operator. We find an optimal algorithm, i.e., algorithm with the smallest error, which usesn function evaluations computed adaptively. We also exhibit nearly optimal information operators, i.e., the linear functionals for which the error of an optimal algorithm that uses them is almost minimal. Nearly optimal information operators consists ofn nonadaptive function evaluations at equispaced pointsx j in the cubeB. This result exhibits the superiority of the T. Aird and J. Rice procedure ZSRCH (IMSL library [1]) over Sobol's approach [7] for solving nonlinear equations in our class of functions. We also prove that the simple search algorithm which yields a pointx *=x k such that $$\left| {f(x_k )} \right| = \mathop {\min }\limits_{1 \leqq j \leqq n} \left| {f(x_j )} \right|$$ is nearly optimal. The complexity, i.e., the minimal cost of solving our problem is roughly equal to (K/ε) m .
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    Solving systems of polynomial equations by bounded and real homotopy (1984)
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    Numerische Mathematik 43 (1984), S. 397-418 
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    Keywords: AMS(MOS): 65H10, 58C99, 55M25 ; CR: 5.15
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    Topics: Mathematics
    Notes: Summary The homotopy method for solving systems of polynomial equations proposed by Chow, Mallet-Paret and Yorke is modified in two ways. The first modification allows to keep the homotopy solution curves bounded, the second one to work with real polynomials when solving a system of real equations. For the first method numerical results are presented.
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    Monotone explicit iterations of the finite element approximations for the nonlinear boundary value problem (1984)
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    Numerische Mathematik 43 (1984), S. 419-437 
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    Keywords: AMS(MOS): 65N30 ; CR: 5.17
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    Topics: Mathematics
    Notes: Summary In this paper, we consider monotone explicit iterations of the finite element schemes for the nonlinear equations associated with the boundary value problem Δu=bu 2, based on piecewise linear polynomials and the lumping operator. These iterations construct the monotonically decreasing and increasing sequences, and convergence proofs are given. Finally, we present some numerical examples verifying the effectiveness of the theory.
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    Free boundary problems with nonlinear source terms (1984)
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    Numerische Mathematik 43 (1984), S. 463-483 
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    Keywords: AMS(MOS): 65N30 ; CR: 5.17
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    Topics: Mathematics
    Notes: Summary The method of lines is used to semi-discretize the non-linear Poisson equation over a domain with a free boundary. The resulting multipoint free boundary problem is solved with a line Gauss-Seidel method which is shown to converge monotonically. The method of lines solution is then shown to converge to the continuous solution of the variational inequality form of the obstacle problem. Some numerical results for the diffusion-reaction equation indicate that the method is applicable to more general free boundary problems for nonlinear elliptic equations.
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    Convergence of the nonconforming Wilson element for arbitrary quadrilateral meshes (1980)
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    Numerische Mathematik 36 (1980), S. 33-52 
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    Keywords: AMS(MOS): 65N30 ; CR: 5.17
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    Topics: Mathematics
    Notes: Summary A modified variational formulation, recently introduced by Taylor, Beresford and Wilson for solving second order problems, using the nonconforming Wilson element is here analysed. It is shown that the Patch Test is satisfied and that stresses and displacements are respectively first and second order accurate for arbitrary quadrilateral meshes.
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    On the convergence order of accelerated root iterations (1984)
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    Numerische Mathematik 44 (1984), S. 463-476 
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    Keywords: AMS(MOS): 65H05 ; CR: 5.15
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    Topics: Mathematics
    Notes: Summary A Gauss-Seidel procedure for accelerating the convergence of the generalized method of the root iterations type of the (k+2)-th order (k∈N) for finding polynomial complex zeros, given in [7], is considered in this paper. It is shown that theR-order of convergence of the accelerated method is at leastk+1+σ n (k), where σ n (k)〉1 is the unique positive root of the equation σ n -σ-k-1 = 0 andn is the degree of the polynomial. The examples of algebraic equations in ordinary and circular arithmetic are given.
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    On mixed finite element methods for first order elliptic systems (1981)
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    Numerische Mathematik 37 (1981), S. 29-48 
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    Keywords: AMS(MOS): 65N30 ; CR5.17
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    Notes: Summary A physically based duality theory for first order elliptic systems is shown to be of central importance in connection with the Galerkin finite element solution of these systems. Using this theory in conjunction with a certain hypothesis concerning approximation spaces, optimal error estimates for Galerkin type approximations are demonstrated. An example of a grid which satisfies the hypothesis is given and numerical examples which illustrate the theory are provided.
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    Une méthode numérique pour le calcul des points de retournement. Application à un problème aux limites non-linéaire (1981)
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    Numerische Mathematik 37 (1981), S. 433-444 
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    Keywords: AMS(MOS): 65H10 ; CR: 5.15
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    Topics: Mathematics
    Notes: Summary An iterative method is presented for determining a turning point in a branch of solutions of an algebraic system of equations depending on a parameter. One shows that the convergence is superlinear and that this method may constitute a complement for the Newton method. With a finite element method, numerical results are given testing the process on the discrete version of a mildly nonlinear boundary value problem to determine the turning point in the positive solution branch.
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    Combined finite element and spectral approximation of the Navier-Stokes equations (1984)
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    Numerische Mathematik 44 (1984), S. 201-217 
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    Keywords: AMS(MOS): 65N30 ; CR: 5.17
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    Description / Table of Contents: Resumé On présente une méthode d'approximation numérique des équations de Navier-Stokes possédant une direction de périodicité. Dans cette direction une méthode pseudospectrale basée sur des développements en série de Fourier est utilisée, dans les deux autres on applique une méthode d'éléments finis standard. On montre que la convergence est optimale et que les deux paramètres de discrétisation peuvent être choisis de façon indépendante.
    Notes: Summary We present a method for the numerical approximation of Navier-Stokes equations with one direction of periodicity. In this direction a Fourier pseudospectral method is used, in the two others a standard F.E.M. is applied. We prove optimal rate of convergence where the two parameters of discretization intervene independently.
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    On the stability of bilinear-constant velocity-pressure finite elements (1984)
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    Numerische Mathematik 44 (1984), S. 219-222 
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    ISSN: 0945-3245
    Keywords: AMS(MOS): 65N30 ; CR: 5.17
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    Topics: Mathematics
    Notes: Summary An analysis of the Babuška stability of bilinear/constant finite element pairs for viscous flow calculations is given. An unstable mode not of the checkerboard type is given for which the stability constant turns out to beO(h). Thus, the indicated spaces are not stable in general for numerical calculation.
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    On numerical asymptotic behavior of finite element solutions for parabolic equations (1984)
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    Numerische Mathematik 44 (1984), S. 285-300 
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    ISSN: 0945-3245
    Keywords: AMS(MOS): 65N30 ; CR: 5.17
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    Topics: Mathematics
    Notes: Summary In this paper, we investigate the numerical asymptotic behavior of the finite element solutions for linear parabolic equations under some appropriate conditions. We also give some results of numerical experiments in the two dimensional problems to indicate the effectiveness of our results.
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    Nonlinear successive over-relaxation (1984)
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    Numerische Mathematik 44 (1984), S. 309-315 
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    Keywords: AMS(MOS): 65H10 ; CR: 5.15
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    Topics: Mathematics
    Notes: Summary We study the convergence of Gauss-Seidel and nonlinear successive overrelaxation methods for finding the minimum of a strictly convex functional defined onR n .
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    A finite element lumped mass scheme for solving eigenvalue problems of circular arches (1980)
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    Numerische Mathematik 36 (1980), S. 267-290 
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    Keywords: AMS(MOS): 65N30 ; CR: 5.17
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    Topics: Mathematics
    Notes: Summary In this paper, we present a finite element lumped mass scheme for eigenvalue problems of circular arch structures, and give error estimates for the approximation. They assert that approximate eigenvalues and eigenfuctions converge to the exact ones. Some numerical examples are also given to illustrate our results.
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    A generalization of regula falsi (1980)
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    Numerische Mathematik 36 (1980), S. 333-346 
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    Keywords: AMS (MOS) ; 65H10 ; CR: 5.15
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    Topics: Mathematics
    Notes: Summary The method of nondiscrete mathematical induction is applied to a multistep variant of the secant method. Optimal conditions for convergence as well as error estimates, sharp in every step, are obtained.
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    Global approximate Newton methods (1981)
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    Numerische Mathematik 37 (1981), S. 279-295 
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    Keywords: AMS(MOS): 65H 10 ; CR: 5.15
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    Topics: Mathematics
    Notes: Summary We derive a class of globally and quadratically converging algorithms for a system of nonlinear equations,g(u)=0, whereg is a sufficiently smooth homeomorphism. Particular attention is directed to key parameters which control the iteration. Several examples are given that have been successful in solving the coupled nonlinear PDEs which arise in semiconductor device modelling.
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    Une méthode numérique pour le calcul des points de retournement. Application à un problème aux limites non-linéaire (1981)
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    Numerische Mathematik 37 (1981), S. 445-452 
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    ISSN: 0945-3245
    Keywords: AMS(MOS): 65H10 ; CR: 5.15
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    Topics: Mathematics
    Notes: Summary A finite element method (P1) with numerical integration for approximating the boundary value problem −Δu=λe u is considered. It is shown that the discrete problem has a solution branch (with turning point) which converges uniformely to a solution branch of the continuous problem. Error estimates are given; for example it is found that $$\left| {\lambda _0 - \lambda _h^0 } \right| = 0(h^{2 - \varepsilon } )$$ , ɛ〉0, whereλ 0 andλ h 0 are critical values of the parameter λ for continuous and discrete problems.
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    Finite dimensional approximation of nonlinear problems (1982)
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    Numerische Mathematik 38 (1982), S. 1-30 
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    Keywords: AMS(MOS): 65N30 ; CR: 5.17
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    Topics: Mathematics
    Notes: Summary In the first two papers of this series [4, 5], we have studied a general method of approximation of nonsingular solutions and simple limit points of nonlinear equations in a Banach space. We derive here general approximation results of the branches of solutions in the neighborhood of a simple bifurcation point. The abstract theory is applied to the Galerkin approximation of nonlinear variational problems and to a mixed finite element approximation of the von Kármán equations.
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    Partitioned variable metric updates for large structured optimization problems (1982)
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    Numerische Mathematik 39 (1982), S. 119-137 
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    ISSN: 0945-3245
    Keywords: AMS(MOS): 65H10 ; CR: 5.15
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    Topics: Mathematics
    Notes: Summary This paper presents a minimization method based on the idea of partitioned updating of the Hessian matrix in the case where the objective function can be decomposed in a sum of convex “element” functions. This situation occurs in a large class of practical problems including nonlinear finite elements calculations. Some theoretical and algorithmic properties of the update are discussed and encouraging numerical results are presented.
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    Dynamics of a single species in a spatially varying environment: The stabilizing role of high dispersal rates (1982)
    Springer
    Journal of mathematical biology 16 (1982), S. 49-55 
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    ISSN: 1432-1416
    Keywords: Stability ; Diffusion ; Parabolic equations
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    Topics: Biology , Mathematics
    Notes: Abstract Models for a single species that inhabits an environment that is spatially varying are presented. Simple necessary and sufficient conditions for stability, which are independent of the exact details of the dispersal process, are developed in the case of large diffusion rates. The results highlight the important stabilizing nature of diffusion in a spatially varying environment.
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    Lotka-Volterra equations: Decomposition, stability, and structure (1980)
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    Journal of mathematical biology 9 (1980), S. 65-83 
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    ISSN: 1432-1416
    Keywords: Nonnegative equilibria ; Stability ; Decompositions ; Sub-communities ; Structural perturbations ; Connective stability
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Summary The major objective of this paper is to propose a new decomposition-aggregation framework for stability analysis of Lotka-Volterra equations employing the concept of vector Liapunov functions. Both the disjoint and the overlapping decompositions are introduced to increase flexibility in constructing Liapunov functions for the overall system. Our second objective is to consider the Lotka-Volterra equations under structural perturbations, and derive conditions under which a positive equilibrium is connectively stable. Both objectives of this paper are directed towards a better understanding of the intricate interplay between stability and complexity in the context of robustness of model ecosystems represented by Lotka-Volterra equations. Only stability of equilibria in models with constant parameters is considered here. Nonequilibrium analysis of models with nonlinear time-varying parameters is the subject of a companion paper.
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    Equilibrium solutions of age-specific population dynamics of several species (1981)
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    Journal of mathematical biology 11 (1981), S. 65-84 
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    ISSN: 1432-1416
    Keywords: Population dynamics ; Age-dependent models ; Equilibrium solutions ; Stability ; Evolution equations
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    Topics: Biology , Mathematics
    Notes: Summary A mathematical model describing the dynamics of a population consisting of several species is studied. The interactions in the population are assumed to be age-specific. Using an evolution equation approach, sufficient conditions for well-posedness in L 1 of the dynamics and for existence as well as for stability of equilibrium solutions are given.
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    Stability in a class of cyclic epidemic models with delay (1981)
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    Journal of mathematical biology 11 (1981), S. 95-103 
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    Keywords: Epidemiology ; SIRS ; Deterministic models ; Distributed delays ; Stability
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    Topics: Biology , Mathematics
    Notes: Abstract A detailed analysis of a general class of SIRS epidemic models is given. Sufficient conditions are derived which guarantee the global stability of the endemic equilibrium solution. Further conditions are found which ensure instability for the equilibrium. Finally, the dependence of the stability on the contact number and the ratio of the mean length of infection to the mean removed time is considered.
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    Stability in a two host epidemic model (1982)
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    Journal of mathematical biology 14 (1982), S. 71-75 
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    Keywords: Epidemiology ; Two host models ; Stability
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    Notes: Abstract An epidemic model is derived for a two host infectious disease. It is shown that if a non-trivial equilibrium solution exists, it is globally stable. This result is also proved for a similar one host model.
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    Analysis of a sterile insect release model with predation (1982)
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    Journal of mathematical biology 16 (1982), S. 33-48 
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    ISSN: 1432-1416
    Keywords: Sterile insect release ; Predation ; Stability ; Limit cycles ; Optimal control
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    Notes: Abstract A model for the sterile insect release method of pest control in which the target species is under predatory or parasitic regulation is analyzed. The equations are nondimensionalized and the rescaled parameters are interpreted. There are four types of equilibria, whose existence and stability depend on which of ten regions of parameter space contain the rescaled parameters, and in turn give minimal release rates to achieve eradication of the pest. In at least one region, Hopf bifurcation theory shows the existence of limit cycles, but they are found to be unstable. In addition, the optimal release rate to minimize a total cost functional for pest control by the sterile release method is studied. Both approaches show that when predation accounts for a large fraction of the natural deaths, the necessary release rate and total cost are higher than for weak predation. If the predators are removed without being replaced by any other source of mortality, the cost rises in all cases but rises much more dramatically for cases with strong predation. A definite danger of the sterile release method when some predatory control exists is that the predators are frequently driven extinct before the prey, so that the target species could explode to much higher levels and be more difficult to eradicate again after the sterile release is terminated.
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    Models of epidemics and endemicity in genetically variable host populations (1983)
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    Journal of mathematical biology 17 (1983), S. 289-304 
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    ISSN: 1432-1416
    Keywords: Endemicity ; Epidemics ; Genetics ; Deterministic models ; Stability
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    Topics: Biology , Mathematics
    Notes: Abstract A discrete time genetics model is developed for populations that are undergoing selection due to infectious disease. It is assumed that the generation time of the host and infectious agent are non-synchronous and that only the host population is evolving. Two classes of epidemic processes are considered. The first class is for infectious agents that confer immunity following infection, while the second class is for those that do not confer immunity. The necessary and sufficient conditions are found in order for the disease to persist in a stable polymorphic host population. These conditions are shown to depend on the density of susceptibles, the selection coefficients, and the severity and class of the disease process.
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    Constant-rate stocking of predator-prey systems (1981)
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    Journal of mathematical biology 11 (1981), S. 1-14 
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    ISSN: 1432-1416
    Keywords: Ecological modelling ; Predator-prey systems ; Ordinary non-linear differential equations ; Stability
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    Topics: Biology , Mathematics
    Notes: Abstract We examine the qualitative effects of constant-rate stocking of either or both species in a predator-prey system. The hypotheses are made as mild as possible so that several types of systems with different qualitative alternatives may be studied.
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    Coexistence properties of some predator-prey systems under constant rate harvesting and stocking (1982)
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    Journal of mathematical biology 12 (1982), S. 101-114 
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    Keywords: Predator-prey systems ; Harvesting ; Stability
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    Topics: Biology , Mathematics
    Notes: Abstract The global behaviour of a class of predator-prey systems, modelled by a pair of non-linear ordinary differential equations, under constant rate harvesting and/or stocking of both species, is presented. Theoretically possible structures and transitions are developed and validated by computer simulations. The results are presented as transition loci in the F-G (prey harvest rate-predator harvest rate) plane.
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    A predator prey model with age structure (1982)
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    Journal of mathematical biology 14 (1982), S. 231-250 
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    Keywords: Predator-prey ; Age structure ; Stability
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    Topics: Biology , Mathematics
    Notes: Abstract A general predator-prey model is considered in which the predator population is assumed to have an age structure which significantly affects its fecundity. The model equations are derived from the general McKendrick equations for age structured populations. The existence, stability and destabilization of equilibria are studied as they depend on the prey's natural carrying capacity and the maturation periodm of the predator. The main result of the paper is that for a broad class of maturation functions positive equilibria are either unstable for smallm or are destabilized asm decreases to zero. This is in contrast to the usual rule of thumb that increasing (not decreasing) delays in growth rate responses cause instabilities.
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    Stability of stationary distributions in a space-dependent population growth process (1982)
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    Journal of mathematical biology 15 (1982), S. 37-50 
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    Keywords: Reaction-diffusion system ; Stationary solution ; Stability
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    Topics: Biology , Mathematics
    Notes: Abstract We consider a spatial population growth process which is described by a reaction-diffusion equation c(x)u t = (a 2(x)u x ) x +f(u), c(x) 〉0, a(x) 〉 0, defined on an interval [0, 1] of the spatial variable x. First we study the stability of nonconstant stationary solutions of this equation under Neumann boundary conditions. It is shown that any nonconstant stationary solution (if it exists) is unstable if a xx⩽0 for all xε[0, 1], and conversely ifa xx〉0 for some xε[0, 1], there exists a stable nonconstant stationary solution. Next we study the stability of stationary solutions under Dirichlet boundary conditions. We consider two types of stationary solutions, i.e., a solution u 0(x) which satisfies u 0 x≠0 for all xε[0, 1] (type I) and a solution u 0(x) which satisfies u 0x = 0 at two or more points in [0, 1] (type II). It is shown that any stationary solution of type I [type II] is stable [unstable] if a xx ⩾0 [a xx ⩽0] for all xε[0, 1]. Conversely, there exists an unstable [a stable] stationary solution of type I [type II] if a xx 〈0 [a xx 〉0] for some xε[0, 1].
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    Effects of behavioral adaptation on a predator-prey model (1982)
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    Journal of mathematical biology 15 (1982), S. 239-247 
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    ISSN: 1432-1416
    Keywords: Predator-prey model ; Behavioral adaptation ; Stability
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    Topics: Biology , Mathematics
    Notes: Abstract The Volterra-Lotka predator-prey equations are modified so that the predator's ability to utilize the prey varies in proportion to the average number of encounters between the two species in the past. The behavior of this adaptive system is then described in terms of three parameters — the carrying capacity of the prey, the relative death rate of the predator, and the predator's memoryspan. The most stable situation is shown to occur when the carrying capacity of the prey is large, the predator's death rate is close to zero, and the predator is able to adapt quickly to changing levels of prey density.
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    On the stability of delay equation models of simple human stretch reflexes (1983)
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    Journal of mathematical biology 17 (1983), S. 331-349 
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    ISSN: 1432-1416
    Keywords: Stability ; Delay equations ; Stretch reflexes ; Mathematical studies
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    Topics: Biology , Mathematics
    Notes: Abstract Human stretch reflexes (SRs) are often too weak and ineffectual to provide adequate postural regulation or rhythmic movement boosting (e.g. in ankle pushoff at the end of stance phase in fast running). Recent improvements in methods of artificially enhancing skeletomotor responses, especially in therapeutic regimens, should not be widely employed until the clonus-resisting stability properties of SRs are better understood. We formulate an idealized linear servo model of a segmentally-mediated SR system which includes the often ignored electromechanical coupling delay. For typical closed-loop (delay/gain) ratios, the model is shown to be unstable for all values of loop gain when operating as a position servo, but maximally stable when operating as a velocity servo. We claim that the velocity servo or one of its nonlinear relatives is a better model for some well studied SRs than, e.g., Houk's stiff muscle hypothesis. We also present evidence that even feeble and quickly saturating monosynaptic postural servos are always unstable if operated as pure position regulators.
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    Existence of a globally asymptotically stable equilibrium in Volterra models with continuous time delay (1983)
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    Journal of mathematical biology 18 (1983), S. 93-102 
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    ISSN: 1432-1416
    Keywords: Stability ; Time delay ; Feasible equilibrium ; Partially feasible equilibrium
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    Topics: Biology , Mathematics
    Notes: Abstract A sufficient condition for the existence of a globally asymptotically stable equilibrium in Volterra models with continuous time delay is obtained, and some properties of the stable equilibrium are proven. Furthermore, some applications in which asymptotic stability only depends on the sign of the coefficients are considered.
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    Effects of mean current and shear on stability of depth distributions of marine phytoplankton (1980)
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    Journal of mathematical biology 10 (1980), S. 33-51 
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    ISSN: 1432-1416
    Keywords: Plankton ; Stability ; Perturbations
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    Topics: Biology , Mathematics
    Notes: Abstract Linear perturbation theory is used to examine the stability of steady-state distributions of marine phytoplankton in the presence of a mean current with shear. Solutions are obtained for the general initial-value problem and it is found that all distributions are asymptotically stable so long as the rate of shear is greater than the local production. On the other hand, the early time behavior indicates that the system can be altered, if accomplished soon enough, depending upon a relative combination of diffusion, advection and production. Quantitative assessments are made where data are available.
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    Persistence of predator-prey systems in an uncertain environment (1980)
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    Journal of mathematical biology 10 (1980), S. 65-77 
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    Keywords: Predator-prey ; Persistence ; Stability ; Differential inequalities
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    Topics: Biology , Mathematics
    Notes: Summary The time derivatives of prey and predator populations are assumed to satisfy a set of inequalities, instead of a precise differential equation, reflecting an uncertain environmental and/or lack of knowledge by the modeler. A system of differential equations is found whose solution gives the boundary of a persistent set, which is positive flow invariant for any system satisfying the inequalities. Conditions are given for the persistent set to be bounded away from both axes, which show that resonance effects cannot drive either predator or prey to extinction if that does not happen for an autonomous system satisfying the inequalities. In general predator-prey systems are more persistent when there is strong asymptotic stability, when there is correlation between prey and predator dynamics, when the effect of perturbations is density dependent, and are more persistent under perturbations of the prey than of the predator.
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    One-dimensional λ-ω target patterns: Empirical stability tests (1980)
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    Journal of mathematical biology 10 (1980), S. 97-100 
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    Keywords: Reaction diffusion equations ; Wave-trains ; Stability
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    Topics: Biology , Mathematics
    Notes: Summary Stability of a class of solutions to reaction-diffusion equations is studied numerically. It is found that the solutions do not persist under a variety of boundary conditions.
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    Stability analysis for models of diseases without immunity (1981)
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    Journal of mathematical biology 13 (1981), S. 185-198 
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    ISSN: 1432-1416
    Keywords: Epidemiology ; Deterministic models ; Distributed delays ; Thresholds ; Stability
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Summary A cyclic, constant parameter epidemiological model is described for a closed population divided into susceptible, exposed and infectious classes. Distributed delays are introduced and the model is formulated as two coupled Volterra integral equations. The delays do not change the general nature of thresholds or asymptotic stability; in all cases considered the disease either dies out, or approaches an endemic steady state.
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    The community matrix in three species community models (1982)
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    Journal of mathematical biology 16 (1982), S. 25-31 
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    ISSN: 1432-1416
    Keywords: Stability ; Community matrix ; Lotka-Volterra models
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    Topics: Biology , Mathematics
    Notes: Abstract The explicit function of the community matrix of a three dimensional Lotka-Volterra model is delineated by a set of necessary and sufficient conditions for a positive equilibrium to be asymptotically stable. In the special case that the community matrix is quasi weakly diagonally dominant, it is shown that a positive determinant for the community matrix is not only necessary but is also sufficient for stability. The results are specific to three dimensional models and do not extend to communities of dimension greater than three.
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    Stability results for delayed-recruitment models in population dynamics (1984)
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    Journal of mathematical biology 19 (1984), S. 147-156 
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    ISSN: 1432-1416
    Keywords: Stability ; delay ; difference equations ; whale models
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    Topics: Biology , Mathematics
    Notes: Abstract This paper relates the stability properties of a class of delay-difference equations to those of an associated scalar difference equation. Simple but powerful conditions for testing global stability are presented which are independent of the length of the time delay involved. For models which do not have globally stable equilibria, estimates of stability regions are obtained. Some well known baleen whale models are used to illustrate the results.
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    Integral equation models for endemic infectious diseases (1980)
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    Journal of mathematical biology 9 (1980), S. 37-47 
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    ISSN: 1432-1416
    Keywords: Epidemiology ; Endemic infectious diseases ; Deterministic models ; Thresholds ; Distributed delays ; Stability
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Summary Endemic infectious diseases for which infection confers permanent immunity are described by a system of nonlinear Volterra integral equations of convolution type. These constant-parameter models include vital dynamics (births and deaths), immunization and distributed infectious period. The models are shown to be well posed, the threshold criteria are determined and the asymptotic behavior is analysed. It is concluded that distributed delays do not change the thresholds and the asymptotic behaviors of the models.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00276034
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    The existence of globally stable equilibria of ecosystems of the generalized Volterra type (1980)
    Springer
    Journal of mathematical biology 10 (1980), S. 401-415 
    Details
    ISSN: 1432-1416
    Keywords: Stability ; Volterra ecosystems ; Linear complementarity theory
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract In this paper, global asymptotic stability of ecosystems of the generalized Volterra type $$dx_i /dt = x_i \left( {b_{i - } \mathop \sum \limits_{j = 1}^n a_{ij} x_j } \right),{\text{ }}i = 1,...,n,$$ is investigated. We obtain the conditions for the existence of a nonnegative and stable equilibrium point of the system by applying a result of linear complementarity theory. The results of this paper show that there exists a class of systems that do not have multiple domains of attractions. This class is defined in terms of the species interactions alone, and does not involve carrying capacities or species net birth rates.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00276098
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    Clines in the presence of asymmetric migration (1981)
    Springer
    Journal of mathematical biology 11 (1981), S. 207-233 
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    ISSN: 1432-1416
    Keywords: Clines ; Population genetics ; Nonlinear diffusion problems ; Stability
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract If a population, which consists of individuals having genetic variation at one locus, with two alleles A and a, evolves under the influence of migration and selection, gradients in the distribution of alleles may arise. We consider the effect of asymmetry in the migration and spatial dependence of the selection process, upon the emergence and stability of such gradients.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00275443
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  • 87
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    The diffusive Lotka-Volterra system with three species can have a stable non-constant equilibrium solution (1982)
    Springer
    Journal of mathematical biology 16 (1982), S. 103-112 
    Details
    ISSN: 1432-1416
    Keywords: Bifurcation ; Competition ; Diffusive Lotka-Volterra system ; Predator-prey interaction ; Stability
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract Three examples of the diffusive 3-species Lotka-Volterra system with constant interaction parameters are given, and by bifurcation techniques shown to have stable spatially non-constant equilibrium solutions. One example is competitive; the second one predator-two-competing prey and the third involves two predators and a single prey.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00275163
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  • 88
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    Existence and stability of travelling wave solutions of a nonlinear integral operator (1983)
    Springer
    Journal of mathematical biology 16 (1983), S. 199-220 
    Details
    ISSN: 1432-1416
    Keywords: Nonlinear integral operator ; Travelling wave ; Wave speed ; Asymptotic speed of propagation ; Stability
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract In this paper, we establish the existence and stability property of travelling wave solutions of a nonlinear integral operator in the inferior case.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00276502
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    Stable spatio-temporal oscillations of diffusive Lotka-Volterra system with three or more species (1983)
    Springer
    Journal of mathematical biology 18 (1983), S. 213-221 
    Details
    ISSN: 1432-1416
    Keywords: Diffusive Lotka-Volterra system ; Hopf-bifurcation ; Spatiotemporal oscillation ; Stability
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract A stability condition for Hopf-bifurcating solutions from the uniform equilibrium of clasical Lotka-Volterra interaction-diffusion equations is presented. Using this condition, it is shown that stable spatio-temporal oscillations exist in the framework of such equations.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00276088
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