Recursively Preconditioned Hierarchical Interpolative Factorization for Elliptic Partial Differential Equations
For researchers and practitioners solving elliptic PDEs, this work provides a more robust and efficient preconditioner for ill-conditioned problems, though it is an incremental improvement to an existing method.
The paper proposes a modification to the hierarchical interpolative factorization that applies a block Jacobi preconditioner before each level of skeletonization, significantly improving accuracy for ill-conditioned elliptic PDEs at no additional asymptotic cost. Numerical examples demonstrate robust performance even at modest compression tolerances.
The hierarchical interpolative factorization for elliptic partial differential equations is a fast algorithm for approximate sparse matrix inversion in linear or quasilinear time. Its accuracy can degrade, however, when applied to strongly ill-conditioned problems. Here, we propose a simple modification that can significantly improve the accuracy at no additional asymptotic cost: applying a block Jacobi preconditioner before each level of skeletonization. This dramatically limits the impact of the underlying system conditioning and enables the construction of robust and highly efficient preconditioners even at quite modest compression tolerances. Numerical examples demonstrate the performance of the new approach.