A Tensor-Train accelerated solver for integral equations in complex geometries

Understand

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.