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Preconditioning Strategies for Linear Systems Arising in Tire Design
, 1999
"... In this paper, we consider linear systems arising in static tire equilibrium computation. The heterogeneous material properties, nonlinear constraints, and a 3D finite element formulation make the linear systems arising in tire design difficult to solve by iterative methods. An analysis of matrix ..."
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In this paper, we consider linear systems arising in static tire equilibrium computation. The heterogeneous material properties, nonlinear constraints, and a 3D finite element formulation make the linear systems arising in tire design difficult to solve by iterative methods. An analysis of matrix characteristics attempts to explain this negative effect. This paper focuses on two preconditioning techniques  a variation of an incomplete LU factorization with threshold and a multilevel recursive solver  that are able to improve the convergence of a suitable iterative accelerator. In particular, we compare these techniques and assess their applicability when the linear system difficulty varies for the same class of problems. The effect of altering the values of parameters such as number of fillin elements, block size, and number of levels is considered. 1 Introduction Static equilibrium computation routinely takes place in the tire manufacturing process. Tire stability an...
emulated and mixedprecision solvers
"... This is a preprint of an article submitted for consideration in the INTERNATIONAL JOURNAL OF PARALLEL, EMERGENT AND ..."
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This is a preprint of an article submitted for consideration in the INTERNATIONAL JOURNAL OF PARALLEL, EMERGENT AND
Fast and Accurate Finite Element Multigrid Solvers for PDE Simulations on GPU Clusters
"... [Homer] D’oh, I need to dedicate this to someone.[/Homer] vAcknowledgements It is impossible to acknowledge everyone who directly or indirectly contributed to the realisation of this thesis. I apologise to all those not mentioned by name: Your help and support is really appreciated. In particular, I ..."
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[Homer] D’oh, I need to dedicate this to someone.[/Homer] vAcknowledgements It is impossible to acknowledge everyone who directly or indirectly contributed to the realisation of this thesis. I apologise to all those not mentioned by name: Your help and support is really appreciated. In particular, I would like to express my sincere gratitude to the following people: Professors Stefan Turek and Heinrich Müller guided my scientific career from the first contact to Numerical Mathematics and Computer Graphics as an undergraduate student to the strongly interdisciplinary field of scientific computing in which I finally settled with this thesis. I am deeply indebted to them for giving me the opportunity to work in the friendly atmosphere at the Institute of Applied Mathematics and the Chair of Computer Graphics at TU Dortmund. I want to thank them for their ideas, fruitful discussions and encouraging criticisms, their belief in my work when I got lost, and last but not least for taking the risk and having the vision to support this endeavour into an emerging research area that at the beginning no one really expected to have such a big impact. I am very grateful to my fellow PhD students and friends in the FEAST group, Christian
Parallel Preconditioning with Sparse Approximate Inverses
 SIAM J. Sci. Comput
, 1996
"... A parallel preconditioner is presented for the solution of general sparse linear systems of equations. A sparse approximate inverse is computed explicitly, and then applied as a preconditioner to an iterative method. The computation of the preconditioner is inherently parallel, and its applicati ..."
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A parallel preconditioner is presented for the solution of general sparse linear systems of equations. A sparse approximate inverse is computed explicitly, and then applied as a preconditioner to an iterative method. The computation of the preconditioner is inherently parallel, and its application only requires a matrixvector product. The sparsity pattern of the approximate inverse is not imposed a priori but captured automatically. This keeps the amount of work and the number of nonzero entries in the preconditioner to a minimum. Rigorous bounds on the clustering of the eigenvalues and the singular values are derived for the preconditioned system, and the proximity of the approximate to the true inverse is estimated. An extensive set of test problems from scientific and industrial applications provides convincing evidence of the effectiveness of this approach. 1 Introduction We consider the linear system of equations Ax = b; x; b 2 IR n : (1) The work of M. Grote was...
NITSOL: A NEWTON ITERATIVE SOLVER FOR NONLINEAR SYSTEMS
"... Abstract. We introduce a welldeveloped Newton iterative (truncated Newton) algorithm for solving largescale nonlinear systems. The framework is an inexact Newton method globalized by backtracking. Trial steps are obtained using one of several Krylov subspace methods. The algorithm is implemented i ..."
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Abstract. We introduce a welldeveloped Newton iterative (truncated Newton) algorithm for solving largescale nonlinear systems. The framework is an inexact Newton method globalized by backtracking. Trial steps are obtained using one of several Krylov subspace methods. The algorithm is implemented in a Fortran solver called NITSOL that is robust yet easy to use and provides a number of useful options and features. The structure oers the user great flexibility in addressing problem specicity through preconditioning and other means and allows easy adaptation to parallel environments. Features and capabilities are illustrated in numerical experiments.
Government and no official endorsement should be inferred.
"... This article describes the design rationale, a C implementation, and conformance testing of a subset of the new Standard for the BLAS (Basic Linear Algebra Subroutines): Extended and Mixed Precision BLAS. Permitting higher internal precision and mixed input/output types and precisions allows us to i ..."
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This article describes the design rationale, a C implementation, and conformance testing of a subset of the new Standard for the BLAS (Basic Linear Algebra Subroutines): Extended and Mixed Precision BLAS. Permitting higher internal precision and mixed input/output types and precisions allows us to implement some algorithms that are simpler, more accurate, and sometimes faster than possible without these features. The new BLAS are challenging to implement and test because there
Review Jacobianfree Newton–Krylov methods: a survey of approaches and applications
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Performance and accuracy of hardwareoriented native, emulated and mixedprecision solvers in FEM simulations
"... In this survey paper, we compare native double precision solvers with emulated and mixed precision solvers of linear systems of equations as they typically arise in finite element discretisations. The emulation utilises two single float numbers to achieve higher precision, while the mixed precisio ..."
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In this survey paper, we compare native double precision solvers with emulated and mixed precision solvers of linear systems of equations as they typically arise in finite element discretisations. The emulation utilises two single float numbers to achieve higher precision, while the mixed precision iterative refinement computes residuals and updates the solution vector in double precision but solves the residual systems in single precision. Both techniques have been known since the 1960s, but little attention has been devoted to their performance aspects. Motivated by changing paradigms in processor technology and the emergence of highly parallel devices with outstanding single float performance, we adapt the emulation and mixed precision techniques to coupled hardware configurations, where the parallel devices serve as scientific coprocessors. The performance advantages are examined with respect to speedups over a native double precision implementation (time aspect) and reduced area requirements for a chip (space aspect). The paper begins with an overview of the theoretical background, algorithmic approaches and suitable hardware architectures. We then employ several conjugate gradient and multigrid solvers and study their behaviour for different parameter settings of the iterative refinement technique. Concrete speedup factors are evaluated on the coupled hardware configuration of a generalpurpose CPU and
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"... Fast iterative solvers for the discretized incompressible NavierStokes equations ..."
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Fast iterative solvers for the discretized incompressible NavierStokes equations