Computational Methods in Applied Mathematics

Published by De Gruyter

Volume 10 Issue 3

Issue of Computational Methods in Applied Mathematics

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Contents
Journal Overview

Open Access January 1, 2010

Diffeomorphic Matching and Dynamic Deformable Surfaces in 3d Medical Imaging

R. Azencott, R. Glowinski, J. He, A. Jajoo, Y. Li, A. Martynenko, R.H.W. Hoppe, S. Benzekry, S.H. Little

Page range: 235-274

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Abstract

We consider optimal matching of submanifolds such as curves and surfaces by a variational approach based on Hilbert spaces of diffeomorphic transformations. In an abstract setting, the optimal matching is formulated as a minimization problem involving actions of diffeomorphisms on regular Borel measures considered as supporting measures of the reference and the target submanifolds. The objective functional consists of two parts measuring the elastic energy of the dynamically deformed surfaces and the quality of the matching. To make the problem computationally accessible, we use reproducing kernel Hilbert spaces with radial kernels and weighted sums of Dirac measures which gives rise to diffeomorphic point matching and amounts to the solution of a finite dimensional minimization problem. We present a matching algorithm based on the first order necessary optimality conditions which include an initial-value problem for a dynamical system in the trajectories describing the deformation of the surfaces and a final-time problem associated with the adjoint equations. The performance of the algorithm is illustrated by numerical results for examples from medical image analysis.

Open Access January 1, 2010

Paralel Predictor-Corrector Schemes for Parabolic Problems on Graphs

R. Čiegis, N. Tumanova

Page range: 275-282

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Abstract

We consider a predictor-corrector type finite difference scheme for solving one-dimensional parabolic problems. This algorithm decouples computations on different subdomains and thus can be efficiently implemented on parallel computers and used to solve problems on graph structures. The stability and convergence of the discrete solution is proved in the special energy and maximum norms. The results of computational experiments are presented.

Open Access January 1, 2010

On the Iterative Solution Methods for Finite-Dimensional Inclusions with Applications to Optimal Control Problems

E. Laitinen, A. Lapin, S. Lapin

Page range: 283-301

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Abstract

Iterative methods for finite-dimensional inclusions which arise in applying a finite-element or a finite-difference method to approximate state-constrained optimal control problems have been investigated. Specifically, problems of control on the right- hand side of linear elliptic boundary value problems and observation in the entire domain have been considered. The convergence and the rate of convergence for the iterative algorithms based on the finding of the control function or Lagrange multipliers are proved.

Open Access January 1, 2010

A Posteriori Error Estimates for Approximate Solutions of the Barenblatt-Biot Poroelastic Model

J.M. Nordbotten, T. Rahman, S.I. Repin, J. Valdman

Page range: 302-314

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Abstract

The paper is concerned with the Barenblatt-Biot model in the theory of poroelasticity. We derive a guaranteed estimate of the difference between exact and approximate solutions in a combined norm that encompasses errors for the pressure fields computed from the diffusion part of the model and errors related to stresses (strains) of the elastic part. Estimates do not contain generic (mesh-dependent) constants and are valid for any conforming approximation of the pressure and stress fields.

Open Access January 1, 2010

A Natural Adaptive Nonconforming FEM Of Quasi-Optimal Complexity

H. Rabus

Page range: 315-325

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Abstract

In recent years, the question on the convergence and optimality in the context of adaptive finite element methods has been the subject of intensive studies. However, for nonstandard FEMs such as mixed or nonconforming ones, the lack of Galerkin's orthogonality requires new mathematical arguments. The presented adap- tive algorithm for the Crouzeix-Raviart finite element method and the Poisson model problem is of quasi-optimal complexity. Furthermore it is natural in the sense that collective marking rather than a separate marking is applied or the estimated error and the volume term.

Open Access January 1, 2010

A Problem-Independent Slope Limiting Algorithm for the Runge-Kutta Discontinuous Galerkin Method

S.A. Tokareva

Page range: 326-342

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Abstract

This paper deals with the new algorithm of slope limiting in the Runge- Kutta discontinuous Galerkin (RKDG) method. The slope limiting is applied at each intermediate step of the Runge-Kutta process to guarantee the monotonicity of the resulting RKDG scheme. The standard formulation of the RKDG method assumes a manual prescription of the special parameter used in the limiting procedure. Such definition of the limiter makes the method problem-dependent, which is disadvantageous for practical computations. A new problem-independent way of estimating the limiting parameter is proposed and its performance in the second- and third-order RKDG methods is studied in this paper.

About this journal

Objective
The highly selective international mathematical journal Computational Methods in Applied Mathematics (CMAM) considers original mathematical contributions to computational methods and numerical analysis with applications mainly related to PDEs.

CMAM seeks to be interdisciplinary while retaining the common thread of numerical analysis, it is intended to be readily readable and meant for a wide circle of researchers in applied mathematics.

Topics

Numerical and computational mathematics
Partial differential equations
Applied mathematics

Article formats
Original research articles

Proposals for special issues of CMAM are considered.
Note that for special issue proposals not only an exciting topic within the scientific scope of the journal is required, but also the Guest Editors need to have an outstanding worldwide reputation in their field.