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Iterative Solution Methods (iterative + solution_methods)
Selected AbstractsOptimal Control of Iterative Solution Methods for Linear Systems of EquationsPROCEEDINGS IN APPLIED MATHEMATICS & MECHANICS, Issue 1 2005Uwe Helmke Iterative solution methods for linear systems of equations can be regarded as discrete-time control systems, for which a stabilizing feedback control has to be found. Well known algorithms such as GMRES(m) may exhibit unstable dynamics or sensitive dependence on initial conditions, thus preventing the algorithm to converge to the desired solution. Based on linear system feedback design techniques a new algorithm is proposed that does not suffer under such shortcomings. Global convergence to the desired solution is shown for any initial state. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source] Algebraic preconditioning versus direct solvers for dense linear systems as arising in crack propagation problemsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING, Issue 2 2005Erik Bängtsson Abstract Preconditioned iterative solution methods are compared with the direct Gaussian elimination method to solve dense linear systems Ax=b which originate from problems, discretized by boundary element method (BEM) techniques. Numerical experiments are presented and compared with the direct solution method available in a commercial BEM package, which show that the preconditioned iterative schemes are highly competitive with respect to both arithmetic operations required and memory demands. Copyright © 2004 John Wiley & Sons, Ltd. [source] Accelerating iterative solution methods using reduced-order models as solution predictorsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 5 2006R. Markovinovi Abstract We propose the use of reduced-order models to accelerate the solution of systems of equations using iterative solvers in time stepping schemes for large-scale numerical simulation. The acceleration is achieved by determining an improved initial guess for the iterative process based on information in the solution vectors from previous time steps. The algorithm basically consists of two projection steps: (1) projecting the governing equations onto a subspace spanned by a low number of global empirical basis functions extracted from previous time step solutions, and (2) solving the governing equations in this reduced space and projecting the solution back on the original, high dimensional one. We applied the algorithm to numerical models for simulation of two-phase flow through heterogeneous porous media. In particular we considered implicit-pressure explicit-saturation (IMPES) schemes and investigated the scope to accelerate the iterative solution of the pressure equation, which is by far the most time-consuming part of any IMPES scheme. We achieved a substantial reduction in the number of iterations and an associated acceleration of the solution. Our largest test problem involved 93 500 variables, in which case we obtained a maximum reduction in computing time of 67%. The method is particularly attractive for problems with time-varying parameters or source terms. Copyright © 2006 John Wiley & Sons, Ltd. [source] An inner product lemmaNUMERICAL LINEAR ALGEBRA WITH APPLICATIONS, Issue 7 2004Karl Gustafson Abstract Given the operator product BA in which both A and B are symmetric positive-definite operators, for which symmetric positive-definite operators C is BA symmetric positive-definite in the C inner product ,x, y,C? This question arises naturally in preconditioned iterative solution methods, and will be answered completely here. Copyright © 2004 John Wiley & Sons, Ltd. [source] | ||||||||||