Exponential Integrators for Large Systems of Differential Equations
SIAM Journal on Scientific Computing
A sparse H -matrix arithmetic: general complexity estimates
Journal of Computational and Applied Mathematics - Special issue on numerical anaylsis 2000 Vol. VI: Ordinary differential equations and integral equations
Computing All or Some Eigenvalues of Symmetric $\mathcal{H}_{\ell}$-Matrices
SIAM Journal on Scientific Computing
Eigenvalue computations in the context of data-sparse approximations of integral operators
Journal of Computational and Applied Mathematics
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A class of matrices (H-matrices) has recently been introduced by Hackbusch for approximating large and fully populated matrices arising from FEM and BEM applications. These matrices are data-sparse and allow approximate matrix operations of almost linear complexity. In the present paper, we choose a special class of H-matrices that provides a good approximation to the inverse of the discrete 2D Laplacian. For these 2D H-matrices we study the blockwise recursive schemes for block triangular linear systems of equations and the Cholesky and LDLT factorization in an approximate arithmetic of almost linear complexity. Using the LDLT factorization we compute eigenpairs of the discrete 2D Laplacian in H-matrix arithmetic by means of a so-called simultaneous iteration for computing invariant subspaces of non-Hermitian matrices due to Stewart. We apply the H-matrix techniques to approximate the solutions of the high-frequency 2D wave equation for smooth initial data and the 2D heat equation for arbitrary initial data by spectral decomposition of the discrete 2D Laplacian in, up to logarithmic factors, optimal complexity.