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Convergence rate of some hybrid multigrid methods for variational inequalities

JOURNAL OF NUMERICAL MATHEMATICS 2015年第3期23卷 195-210页

作者： Badea, Lori Acad Romana Inst Math Res Unit 6 RO-014700 Bucharest Romania

In [L. Badea, Global convergence rate of a standard multigrid method for variational inequalities, IMA J. Numer. Anal., 34 (2014), No. 1, 197-216], a global convergence rate of the standard monotone multigrid method for variational inequalities is derived. This algorithm can be also viewed as performing multiplicative iterations on each level and also multiplicative iterations over the levels. In the present paper, this algorithm together with other three algorithms, which are combinations of additive or multiplicative iterations on levels with additive or multiplicative iterations over the levels, are analyzed in a unitary manner and in a more general framework which allow us to consider problems in the Sobolev space W-1,W-sigma, 1 < sigma < infinity, not only in the usual H-1. The algorithms are given for the constrained minimization of functionals where the convex set is of two-obstacle type and have an optimal computing complexity of the iteration steps. The estimations of the global convergence rate are written in function of the number of levels.

关键词： Domain decomposition methods variational inequalities multigrid and multilevel methods nonlinear obstacle problems

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Global convergence rate of a standard multigrid method for variational inequalities

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IMA JOURNAL OF NUMERICAL ANALYSIS 2014年第1期34卷 197-216页

作者： Badea, Lori Acad Romana Inst Math RO-014700 Bucharest Romania

The main purpose of this paper is to give an estimation of the global convergence rate for the standard monotone multigrid method applied to variational inequalities whose constraints are of the two-obstacle type. The numerical experiments using this method have highlighted global upper bounds of the convergence rate, but, to our knowledge, no theoretical justification exists so far. The method was introduced by Mandel in 1984 for complementarity problems and named later by Kornhuber as the standard monotone multigrid method. First, we introduce the method as a subspace correction algorithm in a reflexive Banach space, prove its global convergence and estimate the error after making some assumptions. By introducing finite element spaces, this algorithm becomes a multilevel or multigrid method. In this case, we prove that the assumptions that we made in the general theory are satisfied and write the convergence rate as a function of the number of levels. Finally, we compare our results with the estimations of the asymptotic convergence rate existing in the literature for complementarity problems.

关键词： domain decomposition methods multigrid and multilevel methods variational inequalities nonlinear obstacle problems

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Convergence of a two-level ideal algorithm for a parametric shape inverse model problem

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INVERSE PROBLEMS IN SCIENCE AND ENGINEERING 2011年第3期19卷 363-393页

作者： Chaigne, Benoit Desideri, Jean-Antoine INRIA Sophia Antipolis Mediterranee Project Team Opale F-06902 Sophia Antipolis France

Similar to the discretization of ordinary or partial differential equations, the numerical approximation of the solution of an optimization problem is possibly subject to numerical stiffness. In the framework of parametric shape optimization, hierarchical representations of the shape can be used for preconditioning, following the idea of multigrid (MG) methods. In this article, by analogy with the Poisson equation, which is the typical example for linear MG methods, we address a parametric shape inverse problem. We describe the ideal cycle of a two-level algorithm adapted to shape optimization problems that require appropriate transfer operators. With the help of a symbolic calculus software we show that the efficiency of an optimization MG-like strategy is ensured by a small dimension-independent convergence rate. Numerical examples are worked out and corroborate the theoretical results. Applications to antenna design are realized. Finally, some connections with the direct and inverse Broyden-Fletcher-Goldfarb-Shanno preconditioning methods are shown.

关键词： parametric shape optimization inverse problem multigrid and multilevel methods convergence rate

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A precompiler to reduce the memory footprint of multiscale PDE solvers in C plus

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FUTURE GENERATION COMPUTER SYSTEMS-THE INTERNATIONAL JOURNAL OF ESCIENCE 2010年第1期26卷 175-182页

作者： Bungartz, H. -J. Eckhardt, W. Weinzierl, T. Zenger, C. Tech Univ Munich D-80748 Garching Germany

A PDE solver's value is increasingly co-determined by its memory footprint, as the increase of computational multicore power overtakes the memory access speed, and as memory restricts the maximum experiment size. Tailoring a code to require less memory is technical challenging, error-prone, and hardware-dependent. Object-oriented code typically consumes much memory, though developers favour such high-level languages offering meaningful models and good maintainability. We augment the language C++ with new keywords branding records to be memory-critical. Our precompiler DaStGen then transforms this augmented specification into plain C++ optimised for low memory requirements. Hereby, it encodes multiple attributes with fixed range within one variable, and it reduces the number of bits per floating point value. The tool also generates one user-defined MPI data type per class and, thus, facilitates the construction of parallel codes with small messages. (C) 2009 Elsevier B.V. All rights reserved.

关键词： Data compaction and compression Code generation multigrid and multilevel methods

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Efficient multilevel Eigensolvers with Applications to Data Analysis Tasks

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IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE 2010年第8期32卷 1377-1391页

作者： Kushnir, Dan Galun, Meirav Brandt, Achi Yale Univ Dept Math New Haven CT 06520 USA Weizmann Inst Sci Dept Comp Sci & Appl Math IL-76100 Rehovot Israel

multigrid solvers proved very efficient for solving massive systems of equations in various fields. These solvers are based on iterative relaxation schemes together with the approximation of the "smooth" error function on a coarser level (grid). We present two efficient multilevel eigensolvers for solving massive eigenvalue problems that emerge in data analysis tasks. The first solver, a version of classical algebraic multigrid (AMG), is applied to eigenproblems arising in clustering, image segmentation, and dimensionality reduction, demonstrating an order of magnitude speedup compared to the popular Lanczos algorithm. The second solver is based on a new, much more accurate interpolation scheme. It enables calculating a large number of eigenvectors very inexpensively.

关键词： Eigenvalues and eigenvectors multigrid and multilevel methods graph algorithms segmentation clustering

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One- and Two-Level Domain Decomposition methods for Nonlinear Problems

One- and Two-Level Domain Decomposition Methods for Nonlinea...

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1st International Conference on Parallel, Distributed and Grid Computing for Engineering

作者： Badea, L. Romanian Acad Inst Math Bucharest Romania

ISBN: (纸本)9781905088294

In this paper we synthesize the results in [1] - [6] concerning the convergence rate of the one- and two-level methods for some nonlinear problems: nonlinear variational inequalities, inequalities with contraction operators, variational inequalities of the second kind and quasi-variational inequalities. Also, we verify that the convergence rates obtained by numerical tests are really in concordance with the theoretical ones. We comparatively illustrate the convergence rates of the one- and two-level methods by numerical experiments for the solution of the two-obstacle problem of a nonlinear elastic membrane.

关键词： domain decomposition methods nonlinear variational inequalities fixed-point problems quasi-variational inequalities multigrid and multilevel methods contact problems with friction nonlinear obstacle problems

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Convergence rate of a Schwarz multilevel method for the constrained minimization of nonquadratic functionals

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SIAM JOURNAL ON NUMERICAL ANALYSIS 2006年第2期44卷 449-477页

作者： Badea, L. Romanian Acad Sci Inst Math RO-70700 Bucharest Romania

In [L. Badea, Convergence Rate of a Multiplicative Schwarz Method for Strongly Nonlinear Inequalities, V. Barbu, I. Lasiecka, D. Tiba, and C. Varsan, eds., Kluwer Academic Publishers, Boston, 2003], the convergence of a subspace correction method applied to the constrained minimization of a functional in a general reflexive Banach space has been proved, provided that the convex set verifies a certain assumption. This assumption is weaker than that in which the convex set is decomposed according to the space decomposition as a sum of subsets. In the Sobolev spaces, the proposed method becomes a multiplicative Schwarz method for the solution of the variational inequalities coming from the minimization of nonquadratic functionals. We prove in this paper that this assumption holds for the one-, two- and multilevel multiplicative Schwarz methods in the finite element space, and we explicitly write the constants in the error estimations depending on the overlapping and mesh parameters. Our error estimates are similar with those obtained for the minimization of quadratic functionals in [ L. Badea, X.-C. Tai, and J. Wang, SIAM J. Numer. Anal., (2003), pp. 1052-1073], or with those obtained for the one- obstacle problem in [X.-C. Tai, Numer. Math., 93 (2003), pp. 755 - 786].

关键词： domain decomposition methods variational inequalities nonquadratic minimization multigrid and multilevel methods finite element methods nonlinear obstacle problems

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Iterative solution methods for modeling multiphase flow in porous media fully implicitly

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SIAM JOURNAL ON SCIENTIFIC COMPUTING 2003年第3期25卷 905-926页

作者： Lacroix, S Vassilevski, Y Wheeler, J Wheeler, M IFP F-92852 Rueil Malmaison France INM Moscow 117333 Russia Univ Texas TICAM Austin TX 78712 USA

We discuss several fully implicit techniques for solving the nonlinear algebraic system arising in an expanded mixed finite element or cell-centered finite difference discretization of two- and three-phase porous media flow. Every outer nonlinear Newton iteration requires solution of a nonsymmetric Jacobian linear system. Two major types of preconditioners, supercoarsening multigrid (SCMG) and two-stage, are developed for the GMRES iteration applied to the solution of the Jacobian system. The SCMG reduces the three-dimensional system to two dimensions using a vertical aggregation followed by a two-dimensional multigrid. The two-stage preconditioners are based on decoupling the systemin to a pressure and concentration equations. Several pressure preconditioners of different types are described. Extensive numerical results are presented using the integrated parallel reservoir simulator (IPARS) and indicate that these methods have low arithmetical complexity per iteration and good convergence rates.

关键词： multigrid and multilevel methods decoupling techniques reservoir simulation

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Augmented Lagrangian active set methods for obstacle problems

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JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS 2003年第3期119卷 499-533页

作者： Kärkkäinen, T Kunisch, K Tarvainen, P Univ Jyvaskyla Dept Math Informat Technol Jyvaskyla Finland Graz Univ Inst Math A-8010 Graz Austria Numerola Oy Jyvaskyla Finland

Active set strategies for two-dimensional and three-dimensional, unilateral and bilateral obstacle problems are described. Emphasis is given to algorithms resulting from the augmented Lagrangian (i.e., primal-dual formulation of the discretized obstacle problems), for which convergence and rate of convergence are considered. For the bilateral case, modifications of the basic primal-dual algorithm are also introduced and analyzed. Finally, efficient computer realizations that are based on multigrid and multilevel methods are suggested and different aspects of the proposed techniques are investigated through numerical experiments.

关键词： obstacle problems active set strategies augmented Lagrangian methods multigrid and multilevel methods

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