Robust and Efficient Multifrontal Solver for Large Discretized PDEs

This paper presents a robust structured multifrontal factorization method for large symmetric positive definite sparse matrices arising from the discretization of partial differential equations (PDEs). For PDEs such as 2D and 3D elliptic equations, the method costs roughly O(n) and O(n 4/3) flops, respectively. The algorithm takes advantage of a low-rank property in the direct factorization of some discretized matrices. We organize the factorization with a supernodal multifrontal method after the nested dissection ordering of the matrix. Dense intermediate matrices in the factorization are approximately factorized into hierarchically semiseparable (HSS) forms, so that a data-sparse Cholesky factor is computed and is guaranteed to exist, regardless of the accuracy of the approximation. We also use an idea of rank relaxation for HSS methods so as to achieve similar performance with flexible structures in broader types of PDE. Due to the structures and the rank relaxation, the performance of the method is relatively insensitive to parameters such as frequencies and sizes of discontinuities. Our method is also much simpler than similar structured multifrontal methods, and is more generally applicable (to PDEs on irregular meshes and to general sparse matrices as a black-box direct solver). The method also has the potential to work as a robust and effective preconditioner even if the low-rank property is insignificant. We demonstrate the efficiency and effectiveness of the method with several important PDEs. Various comparisons with other similar methods are given.

[1]  Xiaoye S. Li,et al.  SuperLU Users'' Guide , 1997 .

[2]  O. Widlund,et al.  Domain Decomposition Methods in Science and Engineering XVI , 2007 .

[3]  Timothy A. Davis,et al.  The university of Florida sparse matrix collection , 2011, TOMS.

[4]  JIANLIN XIA,et al.  ROBUST STRUCTURED MULTIFRONTAL FACTORIZATION AND PRECONDITIONING FOR DISCRETIZED PDES , 2011 .

[5]  Jianlin Xia,et al.  Efficient Structured Multifrontal Factorization for General Large Sparse Matrices , 2013, SIAM J. Sci. Comput..

[6]  HackbuschW. A sparse matrix arithmetic based on H-matrices. Part I , 1999 .

[7]  Shivkumar Chandrasekaran,et al.  A Fast Solver for HSS Representations via Sparse Matrices , 2006, SIAM J. Matrix Anal. Appl..

[8]  Shivkumar Chandrasekaran,et al.  On the Numerical Rank of the Off-Diagonal Blocks of Schur Complements of Discretized Elliptic PDEs , 2010, SIAM J. Matrix Anal. Appl..

[9]  G. Golub,et al.  A bibliography on semiseparable matrices* , 2005 .

[10]  W. Hackbusch,et al.  Numerische Mathematik Existence of H-matrix approximants to the inverse FE-matrix of elliptic operators with L ∞-coefficients , 2002 .

[11]  Lexing Ying,et al.  A fast nested dissection solver for Cartesian 3D elliptic problems using hierarchical matrices , 2014, J. Comput. Phys..

[12]  Jianlin Xia,et al.  Fast algorithms for hierarchically semiseparable matrices , 2010, Numer. Linear Algebra Appl..

[13]  B. Engquist,et al.  Sweeping preconditioner for the Helmholtz equation: Hierarchical matrix representation , 2010, 1007.4290.

[14]  Ahmed H. Sameh,et al.  A parallel hybrid banded system solver: the SPIKE algorithm , 2006, Parallel Comput..

[15]  Joseph W. H. Liu,et al.  The Multifrontal Method for Sparse Matrix Solution: Theory and Practice , 1992, SIAM Rev..

[16]  R. Tewarson On the product form of inverses of sparse matrices. , 1966 .

[17]  Ananth Grama,et al.  Multipole-based preconditioners for large sparse linear systems , 2003, Parallel Comput..

[18]  Leslie Greengard,et al.  A fast algorithm for particle simulations , 1987 .

[19]  A. George Nested Dissection of a Regular Finite Element Mesh , 1973 .

[20]  L. Grasedyck,et al.  Domain-decomposition Based ℌ-LU Preconditioners , 2007 .

[21]  Mario Bebendorf,et al.  Efficient inversion of the Galerkin matrix of general second-order elliptic operators with nonsmooth coefficients , 2004, Math. Comput..

[22]  John K. Reid,et al.  The Multifrontal Solution of Indefinite Sparse Symmetric Linear , 1983, TOMS.

[23]  Shivkumar Chandrasekaran,et al.  A Fast ULV Decomposition Solver for Hierarchically Semiseparable Representations , 2006, SIAM J. Matrix Anal. Appl..

[24]  Steffen Börm,et al.  Data-sparse Approximation by Adaptive ℋ2-Matrices , 2002, Computing.

[25]  Lexing Ying,et al.  A fast direct solver for elliptic problems on general meshes in 2D , 2012, J. Comput. Phys..

[26]  Jianlin Xia,et al.  Efficient scalable algorithms for hierarchically semiseparable matrices , 2011 .

[27]  Jianlin Xia,et al.  Superfast Multifrontal Method for Large Structured Linear Systems of Equations , 2009, SIAM J. Matrix Anal. Appl..

[28]  W. Hackbusch A Sparse Matrix Arithmetic Based on $\Cal H$-Matrices. Part I: Introduction to ${\Cal H}$-Matrices , 1999, Computing.

[29]  Héctor D. Ceniceros,et al.  Fast algorithms with applications to pdes , 2005 .

[30]  Jianlin Xia,et al.  Robust Approximate Cholesky Factorization of Rank-Structured Symmetric Positive Definite Matrices , 2010, SIAM J. Matrix Anal. Appl..

[31]  W. Hackbusch,et al.  A Sparse ℋ-Matrix Arithmetic. , 2000, Computing.

[32]  I. Gohberg,et al.  On a new class of structured matrices , 1999 .

[33]  Lexing Ying,et al.  A fast direct solver for elliptic problems on Cartesian meshes in 3D , 2010 .

[34]  S. Parter The Use of Linear Graphs in Gauss Elimination , 1961 .

[35]  D. Rose,et al.  Complexity Bounds for Regular Finite Difference and Finite Element Grids , 1973 .