# Vico-Greengard-Ferrando quadratures in the tensor solver for integral equations

## TL;DR

This paper introduces a tensor-based method using Vico-Greengard-Ferrando quadratures for efficient, spectrally accurate convolution computations with Green's functions, applicable to integral equations like Lippmann-Schwinger.

## Contribution

It develops a low-rank tensor implementation using QTT decomposition for convolution with Green's functions, enhancing computational efficiency and accuracy.

## Key findings

- Spectral accuracy achieved with FFT-based quadratures.
- Efficient low-rank tensor representation developed.
- Applicable to various integral equations.

## Abstract

Convolution with Green's function of a differential operator appears in a lot of applications e.g. Lippmann-Schwinger integral equation. Algorithms for computing such are usually non-trivial and require non-uniform mesh. However, recently Vico, Greengard and Ferrando developed method for computing convolution with smooth functions with compact support with spectral accuracy, requiring nothing more than Fast Fourier Transform (FFT). Their approach is very suitable for the low-rank tensor implementation which we develop using Quantized Tensor Train (QTT) decomposition.