On the purely algebraic data‐sparse approximation of the inverse and the triangular factors of sparse matrices

The approximation of the inverse and the factors of the LU decomposition of general sparse matrices by hierarchical matrices is investigated. In this first approach, we present and motivate a new matrix partitioning algorithm which is based on the matrix graph by proving logarithmic-linear complexity of the approximant in the case of bounded condition numbers. In contrast to the usual partitioning, the new algorithm allows to treat general grids if the origin of the sparse matrix is the finite element discretization of differential operators. Numerical examples indicate that the restriction to bounded condition numbers has only technical reasons. Copyright © 2010 John Wiley & Sons, Ltd.

[1]  Mario Bebendorf,et al.  Hierarchical Matrices: A Means to Efficiently Solve Elliptic Boundary Value Problems , 2008 .

[2]  M. Benzi,et al.  A comparative study of sparse approximate inverse preconditioners , 1999 .

[3]  Edmond Chow,et al.  A Priori Sparsity Patterns for Parallel Sparse Approximate Inverse Preconditioners , 1999, SIAM J. Sci. Comput..

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

[5]  G. Meinardus.  Approximation of Functions: Theory and Numerical Methods , 1967 .

[6]  Wolfgang Hackbusch,et al.  A Sparse Matrix Arithmetic Based on H-Matrices. Part I: Introduction to H-Matrices , 1999, Computing.

[7]  Edmond Chow,et al.  Approximate Inverse Preconditioners via Sparse-Sparse Iterations , 1998, SIAM J. Sci. Comput..

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

[9]  Michele Benzi,et al.  A Sparse Approximate Inverse Preconditioner for Nonsymmetric Linear Systems , 1998, SIAM J. Sci. Comput..

[10]  Wolfgang Hackbusch,et al.  Construction and Arithmetics of H-Matrices , 2003, Computing.

[11]  Ronald Kriemann,et al.  Hierarchical Matrices Based on a Weak Admissibility Criterion , 2004, Computing.

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

[13]  Marcus J. Grote,et al.  Parallel Preconditioning with Sparse Approximate Inverses , 1997, SIAM J. Sci. Comput..

[14]  Mario Bebendorf,et al.  Why Finite Element Discretizations Can Be Factored by Triangular Hierarchical Matrices , 2007, SIAM J. Numer. Anal..

[15]  Vipin Kumar,et al.  A Fast and High Quality Multilevel Scheme for Partitioning Irregular Graphs , 1998, SIAM J. Sci. Comput..

[16]  Vladimir Rokhlin,et al.  A Fast Direct Algorithm for the Solution of the Laplace Equation on Regions with Fractal Boundaries , 1994 .

[17]  W. Barrett,et al.  Inverses of banded matrices , 1981 .

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

[19]  Robert Sedgewick,et al.  Algorithms in C++ - part 5: graph algorithms (3. ed.) , 2014 .