An algorithm for estimating non-convex volumes and other integrals in n dimensions

Isaac, A and Jawlekar, A and Venkatapathi, M (2023) An algorithm for estimating non-convex volumes and other integrals in n dimensions. In: Computational and Applied Mathematics, 42 (6).

PDF com_app_mat_42-6_2023.pdf - Published Version Restricted to Registered users only Download (615kB) | Request a copy

Abstract

The computational cost in evaluation of the volume of a body using numerical integration grows exponentially with dimension of the space n. The most generally applicable algorithms for estimating n-volumes and integrals are based on Markov Chain Monte Carlo (MCMC) methods. They have well-bounded rates of convergence for convex domains, but in general, are suited for domains with smooth boundaries. We analyze a less known alternate method for estimating n-dimensional volumes, that is agnostic to the roughness and non-convexity of the boundaries of the body. It results due to the possible decomposition of an arbitrary n-volume into an integral of statistically weighted volumes of n-spheres. We establish its dimensional scaling and extend it for evaluation of arbitrary integrals over non-convex domains. Our results also show that this method is more efficient than the MCMC approach even when restricted to the typical convex domains, for n ≲ 100. An importance sampling may extend this advantage to larger n. © 2023, The Author(s) under exclusive licence to Sociedade Brasileira de Matemática Aplicada e Computacional.

Item Type:	Journal Article
Publication:	Computational and Applied Mathematics
Publisher:	Springer Nature
Additional Information:	The copyright for this article belongs to Springer Nature.
Keywords:	Estimation; High dimensions; Monte Carlo sampling; n-volumes
Department/Centre:	Division of Interdisciplinary Sciences > Computational and Data Sciences
Date Deposited:	26 Jul 2023 07:11
Last Modified:	26 Jul 2023 07:11
URI:	https://eprints.iisc.ac.in/id/eprint/82560

Actions (login required)

View Item