A new Nested Cross Approximation

TL;DR

This paper introduces a novel algebraic Nested Cross Approximation (NNCA) for H2 matrices, improving pivot selection and demonstrating enhanced performance in matrix-vector products, integral equations, and SVMs.

Contribution

The paper presents a new algebraic pivot selection technique for NNCA, leading to more efficient H2 matrix approximations and applications.

Findings

NNCA outperforms existing NCAs in accuracy and speed.

NNCA-based matrix-vector product accelerates solving 3D integral equations.

Implementation is publicly available for reproducibility.

Abstract

In this article, we present a new Nested Cross Approximation (NNCA) for constructing H2 matrices. It differs from the existing NCAs~\cite{bebendorf2012constructing, zhao2019fast} in the technique of choosing pivots, a key part of the approximation. Our technique of choosing pivots is purely algebraic and involves only a single tree traversal. We demonstrate its applicability by developing a fast H2 matrix-vector product, that uses NNCA for the appropriate low-rank approximations. We illustrate the timing profiles and the accuracy of NNCA based H2 matrix-vector product. We also provide a comparison of NNCA based H2 matrix-vector product with the existing NCA based H2 matrix-vector products. A key observation is that NNCA performs better than the existing NCAs. In addition, using the NNCA based H2 matrix-vector product, we accelerate i) solving an integral equation in 3D and ii) Support Vector Machine (SVM). In the spirit of reproducible computational science, the implementation of the algorithm developed in this article is made available at \url{https://github.com/SAFRAN-LAB/NNCA}.