Hierarchical-Matrix Preconditioners for Parabolic Optimal Control Problems
ICCS '07 Proceedings of the 7th international conference on Computational Science, Part I: ICCS 2007
Robust Approximate Cholesky Factorization of Rank-Structured Symmetric Positive Definite Matrices
SIAM Journal on Matrix Analysis and Applications
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Hierarchical matrices provide a data-sparse way to approximate fully populated matrices. In this paper we exploit $\mathcal{H}$-matrix techniques to approximate the $LU$-decompositions of stiffness matrices as they appear in (finite element or finite difference) discretizations of convection-dominated elliptic partial differential equations. These sparse $\mathcal{H}$-matrix approximations may then be used as preconditioners in iterative methods. Whereas the approximation of the matrix inverse by an \h-matrix requires some modification in the underlying index clustering when applied to convection-dominant problems, the $\mathcal{H}$-$LU$-decomposition works well in the standard $\mathcal{H}$-matrix setting even in the convection dominant case. We will complement our theoretical analysis with some numerical examples.