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Biochemical reaction networks are often modelled using discrete-state, continuous-time Markov chains. System statistics of these Markov chains usually cannot be calculated analytically and therefore estimates must be generated via…

It was recently demonstrated that a simple Monte Carlo (MC) algorithm involving the swap of particle pairs dramatically accelerates the equilibrium sampling of simulated supercooled liquids. We propose two numerical schemes integrating the…

We establish a deterministic and stochastic spherical quasi-interpolation framework featuring scaled zonal kernels derived from radial basis functions on the ambient Euclidean space. The method incorporates both quasi-Monte Carlo and Monte…

The efficient simulation of the mean value of a non-linear functional of the solution to a linear stochastic partial differential equation (SPDE) with additive Gaussian noise is considered. A Galerkin finite element method is employed along…

We introduce a Monte Carlo Virtual Element estimator based on Virtual Element discretizations for stochastic elliptic partial differential equations with random diffusion coefficients. We prove estimates for the statistical approximation…

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…

The famous least squares Monte Carlo (LSM) algorithm combines linear least square regression with Monte Carlo simulation to approximately solve problems in stochastic optimal stopping theory. In this work, we propose a quantum LSM based on…

We analyse convergence of a micro-macro acceleration method for the Monte Carlo simulation of stochastic differential equations with time-scale separation between the (fast) evolution of individual trajectories and the (slow) evolution of…

Monte Carlo simulations are widely used in many areas including particle accelerators. In this lecture, after a short introduction and reviewing of some statistical backgrounds, we will discuss methods such as direct inversion, rejection…

In this article we develop a new sequential Monte Carlo (SMC) method for multilevel (ML) Monte Carlo estimation. In particular, the method can be used to estimate expectations with respect to a target probability distribution over an…

We study one-dimensional conservation law. We develop a simple numerical method for computing the unique entropy admissible weak solution to the initial problem. The method basis on the equal-area principle and gives the solution for given…

The paper is devoted to the numerical solutions of fractional PDEs based on its probabilistic interpretation, that is, we construct approximate solutions via certain Monte Carlo simulations. The main results represent the upper bound of…

Sequential Monte Carlo methods which involve sequential importance sampling and resampling are shown to provide a versatile approach to computing probabilities of rare events. By making use of martingale representations of the sequential…

We propose an accurate data-driven numerical scheme to solve Stochastic Differential Equations (SDEs), by taking large time steps. The SDE discretization is built up by means of a polynomial chaos expansion method, on the basis of…

Solutions of initial-boundary value problems for systems of conservation laws depend on the underlying viscous mechanism, namely different viscosity operators lead to different limit solutions. Standard numerical schemes for approximating…

We introduce bounds on the finite-time performance of Markov chain Monte Carlo algorithms in approaching the global solution of stochastic optimization problems over continuous domains. A comparison with other state-of-the-art methods…

Population annealing Monte Carlo is an efficient sequential algorithm for simulating k-local Boolean Hamiltonians. Because of its structure, the algorithm is inherently parallel and therefore well suited for large-scale simulations of…

We use a Monte Carlo method to assemble finite element matrices for polynomial Chaos approximations of elliptic equations with random coefficients. In this approach, all required expectations are approximated by a Monte Carlo method. The…

Monte-Carlo simulations are routinely used for estimating the scaling exponents of complex systems. However, due to finite-size effects, determining the exponent values is often difficult and not reliable. Here we present a novel technique…

The simulation of the expectation of a stochastic quantity E[Y] by Monte Carlo methods is known to be computationally expensive especially if the stochastic quantity or its approximation Y_n is expensive to simulate, e.g., the solution of a…