Nearly optimal fast preconditioning of symmetric positive definite matrices

Abstract : We consider the hierarchical off-diagonal low-rank preconditioning of symmetric positive definite matrices arising from second order elliptic boundary value problems. When the scale of such problems becomes large combined with possibly complex geometry or unstable of boundary conditions, the representing matrix is large and typically ill-conditioned. Multilevel methods such as the hierarchical matrix approximation are often a necessity to obtain an efficient solution. We propose a novel hierarchical preconditioner that attempts to minimize the condition number of the preconditioned system. The method is based on approximating the low-rank off-diagonal blocks in a norm adapted to the hierarchical structure. Our analysis shows that the new preconditioner effectively maps both small and large eigenvalues of the system approximately to $1$. Finally through numerical experiments, we illustrate the effectiveness of the new designed scheme which outperforms more classical techniques based on regular SVD to approximate the off-diagonal blocks and SVD with filtering.
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Emmanuel Agullo, Eric Darve, Luc Giraud, Yuval Harness. Nearly optimal fast preconditioning of symmetric positive definite matrices. [Research Report] RR-8984, Inria Bordeaux Sud-Ouest. 2016, pp.34. ⟨hal-01403480v2⟩

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