FFT-based Kronecker product approximation to micromagnetic long-range interactions

TL;DR

This paper introduces an FFT-based Kronecker product approximation method for micromagnetic long-range interactions, achieving efficient, scalable computations with proven convergence properties and validated through numerical experiments.

Contribution

It develops a novel Kronecker product approximation using sinc quadrature and FFT, enabling fast, scalable evaluation of micromagnetic interactions in structured tensor formats.

Findings

Evaluation scales below linear in volume size

Achieves quasi linear complexity in collocation points

Proves quadratic and exponential convergence

Abstract

We derive a Kronecker product approximation for the micromagnetic long range interactions in a collocation framework by means of separable sinc quadrature. Evaluation of this operator for structured tensors (Canonical format, Tucker format, Tensor Trains) scales below linear in the volume size. Based on efficient usage of FFT for structured tensors, we are able to accelerate computations to quasi linear complexity in the number of collocation points used in one dimension. Quadratic convergence of the underlying collocation scheme as well as exponential convergence in the separation rank of the approximations is proved. Numerical experiments on accuracy and complexity confirm the theoretical results.