Mathematical Reviews

Ann Arbor, MI, United States

Mathematical Reviews

Ann Arbor, MI, United States
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Adly S.,University of Limoges | Dontchev A.L.,Mathematical Reviews | Thera M.,University of Limoges
Numerical Functional Analysis and Optimization | Year: 2014

We give conditions under which the distance from a point x to the set of fixed points of the composition of the set-valued mappings F and G is bounded by a constant times the smallest distance between F -1(x) and G(x). This estimate allows us to significantly sharpen a result by T.-C. Lim [10] regarding fixed-points stability of set-valued contractions. A global version of the Lyusternik-Graves theorem is obtained from this estimate as well. We apply the generalization of Lim's result to establish one-sided Lipschitz properties of the solution mapping of a differential inclusion with a parameter. © 2014 Copyright Taylor & Francis Group, LLC.

Carlson D.A.,Mathematical Reviews
Journal of Mathematical Analysis and Applications | Year: 2017

Recently, A. Greco utilized convex rearrangements to present some new and interesting existence results for noncoercive functionals in the calculus of variations. Moreover, the integrands were not necessarily convex. In particular, using convex rearrangements permitted him to establish the existence of convex minimizers essentially considering the uniform convergence of the minimizing sequence of trajectories and the pointwise convergence of their derivatives. The desired lower semicontinuity property is now a consequence of Fatou's lemma. In this paper we point out that such an approach was considered in the late 1930's in a series of papers by E.J. McShane for problems satisfying the usual coercivity condition. In addition, we will update some hypotheses that McShane made by making use of a result due to T.S. Angell, concerning property (D) on the avoidance of the Lavrentiev phenomenon. © 2017 Elsevier Inc.

Fairweather G.,Mathematical Reviews | Karageorghis A.,University of Cyprus | Maack J.,3255 W. Chenango Ave.
Journal of Computational Physics | Year: 2011

Quadratic spline collocation methods are formulated for the numerical solution of the Helmholtz equation in the unit square subject to non-homogeneous Dirichlet, Neumann and mixed boundary conditions, and also periodic boundary conditions. The methods are constructed so that they are: (a) of optimal accuracy, and (b) compact; that is, the collocation equations can be solved using a matrix decomposition algorithm involving only tridiagonal linear systems. Using fast Fourier transforms, the computational cost of such an algorithm is O(N2logN) on an N×N uniform partition of the unit square. The results of numerical experiments demonstrate the optimal global accuracy of the methods as well as superconvergence phenomena. In particular, it is shown that the methods are fourth-order accurate at the nodes of the partition. © 2011 Elsevier Inc.

Fernandes R.I.,The Petroleum Institute | Fairweather G.,Mathematical Reviews
Journal of Computational Physics | Year: 2012

An alternating direction implicit (ADI) orthogonal spline collocation (OSC) method is described for the approximate solution of a class of nonlinear reaction-diffusion systems. Its efficacy is demonstrated on the solution of well-known examples of such systems, specifically the Brusselator, Gray-Scott, Gierer-Meinhardt and Schnakenberg models, and comparisons are made with other numerical techniques considered in the literature. The new ADI method is based on an extrapolated Crank-Nicolson OSC method and is algebraically linear. It is efficient, requiring at each time level only . O(N) operations where . N is the number of unknowns. Moreover, it is shown to produce approximations which are of optimal global accuracy in various norms, and to possess superconvergence properties. © 2012 Elsevier Inc.

Fernandes R.I.,The Petroleum Institute | Bialecki B.,Colorado School of Mines | Fairweather G.,Mathematical Reviews
Journal of Computational Physics | Year: 2015

We consider the approximate solution of nonlinear reaction-diffusion systems on evolving domains that arise in a variety of areas including biology, chemistry, ecology and physics. By mapping a fixed domain onto the evolving domain at each time level, we generalize to evolving domains the ADI extrapolated Crank-Nicolson orthogonal spline collocation technique developed in [8,9] for fixed domains. The new method is tested on the Schnakenberg model and we demonstrate numerically that it preserves the second-order accuracy in time and optimal accuracy in space for piecewise Hermite cubics in various norms. Moreover, the efficacy of the method is demonstrated on several test problems from the literature which involve various types of domain evolution but for which exact solutions are not known. © 2015 Elsevier Inc.

Bialecki B.,Colorado School of Mines | Fairweatherm G.,Mathematical Reviews | Lopez-Marcos J.C.,University of Valladolid
Advances in Applied Mathematics and Mechanics | Year: 2013

We formulate and analyze the Crank-Nicolson Hermite cubic orthogonal spline collocation method for the solution of the heat equation in one space variable with nonlocal boundary conditions involving integrals of the unknown solution over the spatial interval. Using an extension of the analysis of Douglas and Dupont [23] for Dirichlet boundary conditions, we derive optimal order error estimates in the discrete maximum norm in time and the continuous maximum norm in space. We discuss the solution of the linear system arising at each time level via the capacitance matrix technique and the package COLROWfor solving almost block diagonal linear systems. We present numerical examples that confirm the theoretical global error estimates and exhibit superconvergence phenomena. © 2013 Global Science Press.

Dontchev A.L.,Mathematical Reviews | Rockafellar R.T.,University of Washington
Mathematical Programming | Year: 2013

For solving the generalized equation f(x)+F(x) ∩ 0, where f is a smooth function and f is a set-valued mapping acting between Banach spaces, we study the inexact Newton method described by (f(xk)+ D f(x k)(x{k+1}-xk) + F(x{k+1}) Rk(xk, x{k+1}), where Df is the derivative of f and the sequence of mappings Rk represents the inexactness. We show how regularity properties of the mappings f+F and Rk are able to guarantee that every sequence generated by the method is convergent either q-linearly, q-superlinearly, or q-quadratically, according to the particular assumptions. We also show there are circumstances in which at least one convergence sequence is sure to be generated. As a byproduct, we obtain convergence results about inexact Newton methods for solving equations, variational inequalities and nonlinear programming problems. © 2013 Springer-Verlag Berlin Heidelberg and Mathematical Optimization Society.

Jones M.A.,Mathematical Reviews | Wilson J.M.,York College
Mathematical Methods of Operations Research | Year: 2010

We define multilinear extensions for multichoice games and relate them to probabilistic values and semivalues. We apply multilinear extensions to show that the Banzhaf value for a compound multichoice game is not the product of the Banzhaf values of the component games, in contrast to the behavior in simple games. Following Owen (Manag Sci 18:64-79, 1972), we integrate the multilinear extension over a simplex to construct a version of the Shapley value for multichoice games. We compare this new Shapley value to other extensions of the Shapley value to multichoice games. We also show how the probabilistic value (resp. semivalue, Banzhaf value, Shapley value) of a multichoice game is equal to the probabilistic value (resp. semivalue, Banzhaf value, Shapley value) of an appropriately defined TU decomposition game. Finally, we explain how semivalues, probabilistic values, the Banzhaf value, and this Shapley value may be viewed as the probability that a player makes a difference to the outcome of a simple multichoice game. © Springer-Verlag 2009.

Jones M.A.,Mathematical Reviews | Wilson J.M.,York College
Mathematical Methods of Operations Research | Year: 2013

We introduce and compare several coalition values formultichoice games. Albizuri defined coalition structures and an extension of the Owen coalition value for multichoice games using the average marginal contribution of a player over a set of orderings of the player's representatives. Following an approach used for cooperative games, we introduce a set of nested or two-step coalition values on multichoice games which measure the value of each coalition and then divide this among the players in the coalition using either a Shapley or Banzhaf value at each step. We show that when a Shapley value is used in both steps, the resulting coalition value coincides with that of Albizuri. We axiomatize the three new coalition values and show that each set of axioms, including that of Albizuri, is independent. Further we show how the multilinear extension can be used to compute the coalition values. We conclude with a brief discussion about the applicability of the different values. © Springer-Verlag Berlin Heidelberg 2012.

Dontchev A.L.,Mathematical Reviews
SIAM Journal on Optimization | Year: 2012

A theorem of Dennis and Moré is generalized to characterize the q-superlinear convergence of quasi-Newton methods applied to nonsmooth equations and generalized equations under strong metric subregularity. © 2012 Society for Industrial and Applied Mathematics.

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