Numerical integration over implicitly defined domains with topological guarantee

TL;DR

This paper introduces a method for accurate numerical integration over implicitly defined domains by using interval arithmetic to precisely identify boundaries and a geometry-based error estimate to optimize computations.

Contribution

The paper presents a novel approach combining interval arithmetic and local error estimation to improve the accuracy and efficiency of integration over complex implicit domains.

Findings

Accurate boundary detection using interval arithmetic.

Effective hierarchical subdivision guided by local error estimates.

Numerical experiments demonstrate high accuracy and efficiency.

Abstract

Numerical integration over the implicitly defined domains is challenging due to topological variances of implicit functions. In this paper, we use interval arithmetic to identify the boundary of the integration domain exactly, thus getting the correct topology of the domain. Furthermore, a geometry-based local error estimate is explored to guide the hierarchical subdivision and save the computation cost. Numerical experiments are presented to demonstrate the accuracy and the potential of the proposed method.