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The notion of periodic two-scale convergence and the method of periodic unfolding are prominent and useful tools in multiscale modeling and analysis of PDEs with rapidly oscillating periodic coefficients. In this paper we are interested in…
This paper investigates the use of stratified sampling as a variance reduction technique for approximating integrals over large dimensional spaces. The accuracy of this method critically depends on the choice of the space partition, the…
We consider a class of elasticity equations in ${\mathbb R}^d$ whose elastic moduli depend on $n$ separated microscopic scales, are random and expressed as a linear expansion of a countable sequence of random variables which are…
A new approach to defining the effective fracture toughness for heterogeneous materials is proposed. This temporal averaging approach is process-dependent, incorporating the crack velocity and material toughness. The effectiveness of the…
Improving efficiency of importance sampler is at the center of research in Monte Carlo methods. While adaptive approach is usually difficult within the Markov Chain Monte Carlo framework, the counterpart in importance sampling can be…
Due to the complexity of order statistics, the finite sample behaviour of robust statistics is generally not analytically solvable. While the Monte Carlo method can provide approximate solutions, its convergence rate is typically very slow,…
Stochastic optimization techniques are standard in variational inference algorithms. These methods estimate gradients by approximating expectations with independent Monte Carlo samples. In this paper, we explore a technique that uses…
The present paper concerns a space-time homogenization problem for nonlinear diffusion equations with periodically oscillating (in space and time) coefficients. Main results consist of corrector results (i.e., strong convergences of…
In this paper we investigate rods made of nonlinearly elastic, composite--materials that feature a micro-heterogeneous prestrain that oscillates (locally periodic) on a scale that is small compared to the length of the rod. As a main result…
The problem of sampling constrained continuous distributions has frequently appeared in many machine/statistical learning models. Many Monte Carlo Markov Chain (MCMC) sampling methods have been adapted to handle different types of…
We introduce a new method for studying stochastic homogenization of elliptic equations in nondivergence form. The main application is an algebraic error estimate, asserting that deviations from the homogenized limit are at most proportional…
The purpose of this study is to explore three numerical approaches to the elastic homogenization of disordered masonry structures with moderate meso/macro-lengthscale ratio. The methods investigated include a representative of perturbation…
In the context of stochastic homogenization, the Bourgain-Spencer conjecture states that the ensemble-averaged solution of a divergence-form linear elliptic equation with random coefficients admits an intrinsic description in terms of…
We present a way to include non local potentials in the standard Diffusion Monte Carlo method without using the locality approximation. We define a stochastic projection based on a fixed node effective Hamiltonian, whose lowest energy is an…
Traditionally, the field of computational Bayesian statistics has been divided into two main subfields: variational methods and Markov chain Monte Carlo (MCMC). In recent years, however, several methods have been proposed based on combining…
This paper interprets the stabilized finite element method via residual minimization as a variational multiscale method. We approximate the solution to the partial differential equations using two discrete spaces that we build on a…
Probabilistic prediction of sequences from images and other high-dimensional data is a key challenge, particularly in risk-sensitive applications. In these settings, it is often desirable to quantify the uncertainty associated with the…
Science and engineering problems subject to uncertainty are frequently both computationally expensive and feature nonsmooth parameter dependence, making standard Monte Carlo too slow, and excluding efficient use of accelerated uncertainty…
In this paper, we aim to compute numerical approximation integral by using an adaptive Monte Carlo algorithm. We propose a stratified sampling algorithm based on an iterative method which splits the strata following some quantities called…
We investigate corrector estimates for the solutions of a thermoelasticity problem posed in a highly heterogeneous two-phase medium and its corresponding two-scale thermoelasticity model which was derived in an earlier paper by two-scale…