Search

We present a framework using the Quantized Tensor Train ( qtt ) decomposition to accurately and efficiently solve volume and boundary integral equations in three dimensions. We describe how the qtt decomposition can be used as a hierarchical compression and inversion scheme for matrices arising from the discretization of integral equations. For a broad range of problems, computational and storage costs of the inversion scheme are extremely modest O ( log ⁡ N ) and once the inverse is computed, it can be applied in O ( N log ⁡ N ) . We analyze the qtt ranks for hierarchically low rank matrices and discuss its relationship to commonly used hierarchical compression techniques such as fmm and hss . We prove that the qtt ranks are bounded for translation-invariant systems and argue that this behavior extends to non-translation invariant volume and boundary integrals. For volume integrals, the qtt decomposition provides an efficient direct solver requiring significantly less memory compared to other fast direct solvers. We present results demonstrating the remarkable performance of the qtt -based solver when applied to both translation and non-translation invariant volume integrals in 3d . For boundary integral equations, we demonstrate that using a qtt decomposition to construct preconditioners for a Krylov subspace method leads to an efficient and robust solver with a small memory footprint. We test the qtt preconditioners in the iterative solution of an exterior elliptic boundary value problem (Laplace) formulated as a boundary integral equation in complex, multiply connected geometries. [ABSTRACT FROM AUTHOR]

An efficient direct solver for volume integral equations with O ( N ) complexity for a broad range of problems is presented. The solver relies on hierarchical compression of the discretized integral operator, and exploits that off-diagonal blocks of certain dense matrices have numerically low rank. Technically, the solver is inspired by previously developed direct solvers for integral equations based on “recursive skeletonization” and “Hierarchically Semi-Separable” (HSS) matrices, but it improves on the asymptotic complexity of existing solvers by incorporating an additional level of compression. The resulting solver has optimal O ( N ) complexity for all stages of the computation, as demonstrated by both theoretical analysis and numerical examples. The computational examples further display good practical performance in terms of both speed and memory usage. In particular, it is demonstrated that even problems involving 10 7 unknowns can be solved to precision 10 − 10 using a simple Matlab implementation of the algorithm executed on a single core. [ABSTRACT FROM AUTHOR]