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An efficient block variant of robust structured multifrontal factorization method

Chinese Physics B 2013年第8期22卷 225-232页

作者：左宪禹莫则尧谷同祥 School of Computer and Information Engineering Henan University Laboratory of Computational Physics Institute of Applied Physics and Computational Mathematics

Based on the two-dimensional three-temperature （2D3T） radiation diffusion equations and its discrete system, using the block diagonal structure of the three-temperature matrix, the reordering and symbolic decomposition parts of the RSMF method are replaced with corresponding block operation in order to improve the solution efficiency. We call this block form method block RSMF （in brief, BRSMF） method. The new BRSMF method not only makes the reordering and symbolic decomposition become more effective, but also keeps the cost of numerical factorization from increasing and ensures the precision of solution very well. The theoretical analysis of the computation complexity about the new BRSMF method shows that the solution efficiency about the BRSMF method is higher than the original RSMF method. The numerical experiments also show that the new BRSMF method is more effective than the original RSMF method.

关键词： HSS structure low-rank property multifrontal method two-dimensional three-temperature radia-tive diffusion equations

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A sweeping preconditioner for Yee＇s finite difference approximation of time-harmonic Maxwell＇s equations

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Frontiers of Mathematics in China 2012年第2期7卷 347-363页

作者： Paul TSUJI Lexing YING ICES University of Texas at Austin Austin TX 78712 USA Department of Mathematics University of Texas at Austin Austin TX 78712 USA

This paper is concerned with the fast iterative solution of linear systems arising from finite difference discretizations in electromagnetics. The sweeping preconditioner with moving perfectly matched layers previously developed for the Helmholtz equation is adapted for the popular Yee grid scheme for wave propagation in inhomogeneous, anisotropic media. Preliminary numerical results are presented for typical examples.

关键词： Electromagnetic scattering Yee grid finite difference methods perfectly matched layers LDLT factorizations multifrontal method wave propagation in inhomogeneous and anisotropic media matrix preconditioners

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A High Performance multifrontal Code for Linear Solution of Structures Using Multi-Core Microprocessors

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Tsinghua Science and Technology 2008年第S1期13卷 34-39页

作者： Efe Guney Kenneth Will Computer Aided Structural Engineering Center School of Civil and Environmental Engineering Georgia Institute of Technology

A multifrontal code is introduced for the efficient solution of the linear system of equations arising from the analysis of structures. The factorization phase is reduced into a series of interleaved element assembly and dense matrix operations for which the BLAS3 kernels are used. A similar approach is generalized for the forward and back substitution phases for the efficient solution of structures having multiple load conditions. The program performs all assembly and solution steps in parallel. Examples are presented which demonstrate the code’s performance on single and dual core processor computers.

关键词： multifrontal method Cholesky decomposition high performance computing finite element method multi-core programming BLAS3 parallel computing

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Intel^(■) Math Kernel Library PARDISO* forIntel^(■) Xeon Phi^(TM) Manycore Coprocessor

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Applied Mathematics 2015年第8期6卷 1276-1281页

作者： Alexander Kalinkin Anton Anders Roman Anders Intel Corporation Software and Services Group (SSG) Novosibirsk Russia

The paper describes an efficient direct method to solve an equation Ax = b, where A is a sparse matrix, on the Intel®Xeon PhiTM coprocessor. The main challenge for such a system is how to engage all available threads (about 240) and how to reduce OpenMP* synchronization overhead, which is very expensive for hundreds of threads. The method consists of decomposing A into a product of lower-triangular, diagonal, and upper triangular matrices followed by solves of the resulting three subsystems. The main idea is based on the hybrid parallel algorithm used in the Intel®Math Kernel Library Parallel Direct Sparse Solver for Clusters [1]. Our implementation exploits a static scheduling algorithm during the factorization step to reduce OpenMP synchronization overhead. To effectively engage all available threads, a three-level approach of parallelization is used. Furthermore, we demonstrate that our implementation can perform up to 100 times better on factorization step and up to 65 times better in terms of overall performance on the 240 threads of the Intel®Xeon PhiTM coprocessor.

关键词： multifrontal method Direct method Sparse Linear System HPC OpenMP* Intel^(■) MKL Intel^(■) Xeon Phi^(TM) Coprocessor

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Schur Complement Computations in Intel^(■) Math Kernel Library PARDISO

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Applied Mathematics 2015年第2期6卷 304-311页

作者： Alexander Kalinkin Anton Anders Roman Anders Intel Corporation Software and Services Group(SSG)NovosibirskRussia

This paper describes a method of calculating the Schur complement of a sparse positive definite matrix A. The main idea of this approach is to represent matrix A in the form of an elimination tree using a reordering algorithm like METIS and putting columns/rows for which the Schur complement is needed into the top node of the elimination tree. Any problem with a degenerate part of the initial matrix can be resolved with the help of iterative refinement. The proposed approach is close to the “multifrontal” one which was implemented by Ian Duff and others in 1980s. Schur complement computations described in this paper are available in Intel®Math Kernel Library (Intel®MKL). In this paper we present the algorithm for Schur complement computations, experiments that demonstrate a negligible increase in the number of elements in the factored matrix, and comparison with existing alternatives.

关键词： multifrontal method Direct method Sparse Linear System Schur Complement HPC Intel^(■) MKL

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