System and computer-implemented method for evaluating integrals using stratification by rank-1 lattices

Abstract

A system for numerically evaluating an integral of a function over an s-dimensional integration domain is described, the system comprising a sample point generator, a function value generator and an integral value estimate generator. The sample point generator is configured to generate a selected number of sample points over the integration domain, the sample points being generated such that there is at least one sample point in each of a plurality of strata distributed over the integration domain, the strata being defined by a rank-I lattice. The function value generator is configured to, for respective ones of the sample points, generate a value for the function at the respective sample point. The integral value estimate generator is configured to use the function values generated by the function value generator at the respective sample points in generating an estimate for the value of the integral. The system finds utility in a number of areas, including computer graphics.

Description

FIELD OF THE INVENTION [0001] The invention relates generally to the field of systems and computer-implemented methods for evaluating integrals, and more particularly to such systems and computer-implemented methods that evaluate integrals using Monte Carlo and quasi-Monte Carlo methodologies. The invention particularly provides a new and improved system and computer-implemented method for evaluating integrals using a Monte Carlo methodology in which the integration domain is stratified using a stratification methodology in which the number of strata is independent of the number of dimensions in the integration domain. Specifically, the invention provides a methodology that makes use of stratification based on rank-1 lattices. Systems and computer-implemented methods according to the invention find utility in a number of applications, including but not limited to computer graphics. BACKGROUND OF THE INVENTION [0002] In computer graphics, a computer is used to generate digital data t...

AI Technical Summary

Benefits of technology

"The invention provides a new and improved system and method for evaluating integrals using a Monte Carlo methodology. The method involves generating a sample points over an integration domain using a rank-1 lattice, and then using these sample points to generate a value for the function being integrated. This process allows for a more efficient and accurate evaluation of integrals over large domains. The invention also provides a computer program product for use with a computer to implement this method."

Problems solved by technology

Typically, previously-proposed stratification strategies, particularly stratification strategies that are based on axis-aligned grids, suffer exponential growth in the number of strata with increasing numbers of dimensions comprising the integration domain.

Since integrands ƒ that are used in fields such as, for example, computer graphics, often have very high-dimensional integration domains, stratifying domains using such stratification strategies can be computationally quite intensive.

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