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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. Source

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. Source

Jones M.A.,Mathematical Reviews | Wilson J.M.,The New School
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. Source

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. Source

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. Source

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