Monte Carlo cubature construction

Abstract

In numerical integration, cubature methods are effective, especially when the integrands can be well-approximated by known test functions, such as polynomials. However, the construction of cubature formulas has not generally been known, and existing examples only represent the particular domains of integrands, such as hypercubes and spheres. In this study, we show that cubature formulas can be constructed for probability measures provided that we have an i.i.d. sampler from the measure and the mean values of given test functions. Moreover, the proposed method also works as a means of data compression, even if sufficient prior information of the measure is not available.

Positively Weighted Kernel Quadrature via Subsampling

“…Remark 1. Hayakawa (2021) recently proposed a simple approach for constructing a quadrature formula: randomly sample candidate points and find a solution by using a linear programming (LP) solver. Indeed, for an independent sample X 1 , .…”

“…with positive probability (e.g., Hayakawa, 2021). By considering a basic feasible solution of the following LP problem for such points X 1 , .…”

Markov Chain Approximations to Stochastic Differential Equations by Recombination on Lattice Trees

“…Remark 4. There are other recombination algorithms [20,21,22,37,43,53] besides Algorithm 2 from [6]. However, what makes Algorithm 2 especially attractive in the current situation is that it can take as input a sequence of points -in our case, points sampled uniformly on the lattice -and the moments that need to be matched.…”

“…However, the proof of Carathéodory's Theorem is not constructive and the design of algorithms that carry out such a measure construction efficiently is still the subject of recent research; e.g. just over the last ten years the articles [6,20,21,22,37,43,53] provide novel algorithms. Recombination has already been used in the context of SDE simulations: in order to make "Cubature on Wiener Space" [42] efficient, a recombination step is applied iteratively over time, similar in spirit to our construction; see [37] for a discussion on how powerful this can be for cubature methods on Wiener space.…”

InRS: implementing the indicator function of NURBS-shaped planar domains

“…NURBS-shaped domains produced by CAGD algorithms play a central role in digital design and modelling processes. The capability of locating quasi-uniform or random sample points in such domains can be useful in a vast range of applications, for example within several meshfree bivariate approximation algorithms developed in the last twenty years, among which we may quote (without any pretence of completeness) kernel-based and partition-of-unity collocation methods [5,7], construction of algebraic cubature formulas [8,18,19] potentially useful for curved FEM/VEM elements [1,17], compressed MC/QMC integration [2,9], compressed polynomial regression [4,15].…”