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Monte Carlo methods are widely used for approximating complicated, multidimensional integrals for Bayesian inference. Population Monte Carlo (PMC) is an important class of Monte Carlo methods, which utilizes a population of proposals to…

Methodology · Statistics 2022-08-30 Chaofan Huang , V. Roshan Joseph , Simon Mak

Bayesian models have become very popular over the last years in several fields such as signal processing, statistics, and machine learning. Bayesian inference requires the approximation of complicated integrals involving posterior…

Computation · Statistics 2021-07-20 Luca Martino , Víctor Elvira

In this paper we present a rigorous cost and error analysis of a multilevel estimator based on randomly shifted Quasi-Monte Carlo (QMC) lattice rules for lognormal diffusion problems. These problems are motivated by uncertainty…

Numerical Analysis · Mathematics 2016-09-05 Frances Y. Kuo , Robert Scheichl , Christoph Schwab , Ian H. Sloan , Elisabeth Ullmann

Quasi-Monte Carlo (QMC) methods are equal weight quadrature rules to approximate integrals over the unit cube with respect to the uniform measure. In this paper we discuss QMC integration with respect to general product measures defined on…

Numerical Analysis · Mathematics 2020-09-16 Josef Dick , Friedrich Pillichshammer

In this paper, we consider the numerical solution of a nonlinear Schrodinger equation with spatial random potential. The randomly shifted quasi-Monte Carlo (QMC) lattice rule combined with the time-splitting pseudospectral discretization is…

Numerical Analysis · Mathematics 2023-11-21 Zhizhang Wu , Zhiwen Zhang , Xiaofei Zhao

Antithetic coupling is a general stratification strategy for reducing Monte Carlo variance without increasing the simulation size. The use of the antithetic principle in the Monte Carlo literature typically employs two strata via antithetic…

Statistics Theory · Mathematics 2007-06-13 Radu V. Craiu , Xiao-Li Meng

Monte Carlo methods approximate integrals by sample averages of integrand values. The error of Monte Carlo methods may be expressed as a trio identity: the product of the variation of the integrand, the discrepancy of the sampling measure,…

Numerical Analysis · Mathematics 2017-08-18 Fred J. Hickernell

The following electromagnetism (EM) inverse problem is addressed. It consists in estimating local radioelectric properties of materials recovering an object from global EM scattering measurements, at various incidences and wave frequencies.…

Applications · Statistics 2015-06-12 François Giraud , Pierre Minvielle , Pierre Del Moral

We describe a number of strategies for minimizing and calculating accurately the statistical uncertainty in quantum Monte Carlo calculations. We investigate the impact of the sampling algorithm on the efficiency of the variational Monte…

Computational Physics · Physics 2012-02-14 R. M. Lee , G. J. Conduit , N. Nemec , P. Lopez Rios , N. D. Drummond

We derive and study SQMC (Sequential Quasi-Monte Carlo), a class of algorithms obtained by introducing QMC point sets in particle filtering. SQMC is related to, and may be seen as an extension of, the array-RQMC algorithm of L'Ecuyer et al.…

Computation · Statistics 2014-12-01 Mathieu Gerber , Nicolas Chopin

This paper deals with the Monte-Carlo methods for evaluating expectations of functionals of solutions to McKean-Vlasov Stochastic Differential Equations (MV-SDE) with drifts of super-linear growth. We assume that the MV-SDE is approximated…

Probability · Mathematics 2018-10-15 Goncalo dos Reis , Greig Smith , Peter Tankov

This paper concerns error bounds for recursive equations subject to Markovian disturbances. Motivating examples abound within the fields of Markov chain Monte Carlo (MCMC) and Reinforcement Learning (RL), and many of these algorithms can be…

Probability · Mathematics 2020-02-10 Shuhang Chen , Adithya M. Devraj , Ana Bušić , Sean Meyn

This article provides a high-level overview of some recent works on the application of quasi-Monte Carlo (QMC) methods to PDEs with random coefficients. It is based on an in-depth survey of a similar title by the same authors, with an…

Numerical Analysis · Mathematics 2017-10-31 Frances Y. Kuo , Dirk Nuyens

Monte Carlo (MC) sampling algorithms are an extremely widely-used technique to estimate expectations of functions f(x), especially in high dimensions. Control variates are a very powerful technique to reduce the error of such estimates, but…

Machine Learning · Statistics 2016-06-08 Brendan D. Tracey , David H. Wolpert

Monte Carlo integration with variance reduction by means of control variates can be implemented by the ordinary least squares estimator for the intercept in a multiple linear regression model with the integrand as response and the control…

Statistics Theory · Mathematics 2021-04-02 Rémi Leluc , François Portier , Johan Segers

We develop new multilevel Monte Carlo (MLMC) methods to estimate the expectation of the smallest eigenvalue of a stochastic convection-diffusion operator with random coefficients. The MLMC method is based on a sequence of finite element…

Numerical Analysis · Mathematics 2024-02-13 Tiangang Cui , Hans De Sterck , Alexander D. Gilbert , Stanislav Polishchuk , Robert Scheichl

It is well known that Monte Carlo integration with variance reduction by means of control variates can be implemented by the ordinary least squares estimator for the intercept in a multiple linear regression model. A central limit theorem…

Statistics Theory · Mathematics 2019-10-10 François Portier , Johan Segers

Variational Monte Carlo (VMC) is a powerful and fast-growing method for optimizing and evolving parameterized many-body wave functions, especially with modern neural-network quantum states. In practice, however, the stochastic estimators…

Strongly Correlated Electrons · Physics 2026-03-20 Zhou-Quan Wan , Roeland Wiersema , Shiwei Zhang

Stochastic gradient Markov Chain Monte Carlo (SG-MCMC) has been developed as a flexible family of scalable Bayesian sampling algorithms. However, there has been little theoretical analysis of the impact of minibatch size to the algorithm's…

Machine Learning · Statistics 2017-09-06 Changyou Chen , Wenlin Wang , Yizhe Zhang , Qinliang Su , Lawrence Carin

Monte Carlo sampling is a powerful toolbox of algorithmic techniques widely used for a number of applications wherein some noisy quantity, or summary statistic thereof, is sought to be estimated. In this paper, we survey the literature for…