Globally Adaptive Control Variate for Robust Numerical Integration

Abstract

Abstract. Many methods in computer graphics require the integration of functions on lowto-middle-dimensional spaces. However, no available method can handle all the possible integrands accurately and rapidly. This paper presents a robust numerical integration method, able to handle arbitrary non-singular scalar or vector-valued functions defined on low-to-middle-dimensional spaces. Our method combines control variate, globally adaptive subdivision and Monte-Carlo estimation to achieve fast and accurate computations of any non-singular integral. The runtime is linear with respect to standard deviation while standard Monte-Carlo methods are quadratic. We additionally show through numerical tests that our method is extremely stable from a computation time and memory footprint point-of-view, assessing its robustness. We demonstrate our method on a participating media voxelization application, which requires the computation of several millions integrals for complex media.

Regression-based Monte Carlo integration

“…We show connections among control variates, least-squares regression, and MC integration to have a provable variance reduction. We will discuss more related work [Crespo et al 2021;Hickernell et al 2005;Nakatsukasa 2018;Owen and Zhou 2000;Pajot et al 2014;Rubinstein and Marcus 1985] later once we introduced our approach.…”

“…Recognizing a problem of control variates that it could perform worse than MC integration, Pajot et al [2014] proposed switching to conventional MC when the estimated variance of adaptive control variate is determined to be larger. Our approach does not need this explicit switching, as it is automatically done as a result of least-squares with theoretical guarantee.…”