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.
We propose and analyse randomized cubature formulae for the numerical integration of functions with respect to a given probability measure $μ$ defined on a domain $Γ\subseteq \mathbb{R}^d$, in any dimension $d$. Each cubature formula is exact on a given finite-dimensional subspace $V_n\subset L^2(Γ,μ)$ of dimension $n$, and uses pointwise evaluations of the integrand function $ϕ: Γ\to \mathbb{R}$ at $m>n$ independent random points. These points are drawn from a suitable auxiliary probability measure that depends on $V_n$. We show that, up to a logarithmic factor, a linear proportionality betwe…
Construction of quasi-Monte Carlo rules for multivariate integration in spaces of permutation-invariant functions
KU Leuven（荷语鲁汶大学）；VITO, Energy Technology（维托能源技术研究所）；Philipps-University Marburg, Faculty of Mathematics and Computer Science（马尔堡菲利普斯大学数学与计算机学院）；University of Siegen, Department of Mathematics（锡根大学数学系）
We propose and analyze a quasi-Monte Carlo (QMC) algorithm for efficient simulation of wave propagation modeled by the Helmholtz equation in a bounded region in which the refractive index is random and spatially heterogenous. Our focus is on the case in which the region can contain multiple wavelengths. We bypass the usual sign-indefiniteness of the Helmholtz problem by switching to an alternative sign-definite formulation recently developed by Ganesh and Morgenstern (Numerical Algorithms, 83, 1441-1487, 2020). The price to pay is that the regularity analysis required for QMC methods becomes m…
We propose a control variate multilevel Monte Carlo method for the kinetic BGK model of the Boltzmann equation subject to random inputs. The method combines a multilevel Monte Carlo technique with the computation of the optimal control variate multipliers derived from local or global variance minimization problems. Consistency and convergence analysis for the method equipped with a second-order positivity-preserving and asymptotic-preserving scheme in space and time is also performed. Various numerical examples confirm that the optimized multilevel Monte Carlo method outperforms the classical …
In this note, we study a concatenation of quasi-Monte Carlo and plain Monte Carlo rules for high-dimensional numerical integration in weighted function spaces. In particular, we consider approximating the integral of periodic functions defined over the $s$-dimensional unit cube by using rank-1 lattice point sets only for the first $d\, (<s)$ coordinates and random points for the remaining $s-d$ coordinates. We prove that, by exploiting a decay of the weights of function spaces, almost the optimal order of the mean squared worst-case error is achieved by such a concatenated quadrature rule as l…
Quasi-Monte Carlo methods are a way of improving the efficiency of Monte Carlo methods. Digital nets and sequences are one of the low discrepancy point sets used in quasi-Monte Carlo methods. This thesis presents the three new results pertaining to digital nets and sequences: implementing randomized digital nets, finding the distribution of the discrepancy of scrambled digital nets, and obtaining better quality of digital nets through evolutionary computation. Finally, applications of scrambled and non-scrambled digital nets are provided.
I consider the problem of integrating a function $f$ over the $d$-dimensional unit cube. I describe a multilevel Monte Carlo method that estimates the integral with variance at most $ε^{2}$ in $O(d+\ln(d)d_{t}ε^{-2})$ time, for $ε>0$, where $d_{t}$ is the truncation dimension of $f$. In contrast, the standard Monte Carlo method typically achieves such variance in $O(dε^{-2})$ time. A lower bound of order $d+d_{t}ε^{-2}$ is described for a class of multilevel Monte Carlo methods.
