Efficient computation of highly oscillatory integrals by using QTT tensor approximation

TL;DR

This paper introduces a QTT tensor approximation-based method for efficiently computing highly oscillatory integrals across large frequency ranges, significantly reducing computational complexity and storage requirements.

Contribution

The paper presents a novel QTT tensor approach for approximating oscillatory integrals, enabling efficient evaluation over large frequency intervals with low computational cost.

Findings

QTT approximation achieves logarithmic complexity in grid size

Method accurately computes integrals in multiple spatial dimensions

Numerical examples demonstrate high efficiency and accuracy

Abstract

We propose a new method for the efficient approximation of a class of highly oscillatory weighted integrals where the oscillatory function depends on the frequency parameter $\omega \geq 0$, typically varying in a large interval. Our approach is based, for fixed but arbitrary oscillator, on the pre-computation and low-parametric approximation of certain $\omega$-dependent prototype functions whose evaluation leads in a straightforward way to recover the target integral. The difficulty that arises is that these prototype functions consist of oscillatory integrals and are itself oscillatory which makes them both difficult to evaluate and to approximate. Here we use the quantized-tensor train (QTT) approximation method for functional $m$-vectors of logarithmic complexity in $m$ in combination with a cross-approximation scheme for TT tensors. This allows the accurate approximation and efficient storage of these functions in the wide range of grid and frequency parameters. Numerical examples illustrate the efficiency of the QTT-based numerical integration scheme on various examples in one and several spatial dimensions.