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Multilevel Monte Carlo is a key tool for approximating integrals involving expensive scientific models. The idea is to use approximations of the integrand to construct an estimator with improved accuracy over classical Monte Carlo. We…
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…
We introduce a multigrid multilevel Monte Carlo method for stochastic trace estimation in lattice QCD based on orthogonal projections. This formulation extends the previously proposed oblique decomposition and it is assessed on three…
In the stochastic gradient descent (SGD) for sequential simulations such as the neural stochastic differential equations, the Multilevel Monte Carlo (MLMC) method is known to offer better theoretical computational complexity compared to the…
It is shown that superefficient Monte Carlo computations can be carried out by using chaotic dynamical systems as non-uniform random-number generators. Here superefficiency means that the expectation value of the square of the error…
Kinetic equations model the position-velocity distribution of particles subject to transport and collision effects. Under a diffusive scaling, these combined effects converge to a diffusion equation for the position density in the limit of…
It is a well-known rule of thumb that approximations of stochastic partial differential equations have essentially twice the order of weak convergence compared to the corresponding order of strong convergence. This is already known for many…
In this paper, a nonlinear system of fractional ordinary differential equations with multiple scales in time is investigated. We are interested in the effective long-term computation of the solution. The main challenge is how to obtain the…
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 $\epsilon^{2}$ in $O(d+\ln(d)d_{t}\epsilon^{-2})$ time,…
Monte Carlo sampling of any system may be analyzed in terms of an associated glass model -- a variant of the Random Energy Model -- with, whenever there is a sign problem, complex fields. This model has three types of phases (liquid, frozen…
The stochastic-gauge representation is a method of mapping the equation of motion for the quantum mechanical density operator onto a set of equivalent stochastic differential equations. One of the stochastic variables is termed the…
We propose novel scale-invariant error estimators for the Monte Carlo and multilevel Monte Carlo estimation of mean and variance. For any linear transformation of the distribution of the quantity of interest, the computation cost across…
Since its formulation in the late 1940s, the Feynman-Kac formula has proven to be an effective tool for both theoretical reformulations and practical simulations of differential equations. The link it establishes between such equations and…
Motivated mainly by applications to partial differential equations with random coefficients, we introduce a new class of Monte Carlo estimators, called Toeplitz Monte Carlo (TMC) estimator for approximating the integral of a multivariate…
Langevin Monte Carlo (LMC) is a popular Bayesian sampling method. For the log-concave distribution function, the method converges exponentially fast, up to a controllable discretization error. However, the method requires the evaluation of…
We discuss the application of multilevel Monte Carlo methods to elliptic partial differential equations with random coefficients. Such problems arise, for example, in uncertainty quantification in subsurface flow modeling. We give a brief…
Large particle systems are often described by high-dimensional (linear) kinetic equations that are simulated using Monte Carlo methods for which the asymptotic convergence rate is independent of the dimensionality. Even though the…
Monte Carlo is a simple and flexible tool that is widely used in computational finance. In this context, it is common for the quantity of interest to be the expected value of a random variable defined via a stochastic differential equation.…
The Richards' equation is a model for flow of water in unsaturated soils. The coefficients of this (nonlinear) partial differential equation describe the permeability of the medium. Insufficient or uncertain measurements are commonly…
The order of convergence of the Monte Carlo method is 1/2 which means that we need quadruple samples to decrease the error in half in the numerical simulation. Multilevel Monte Carlo methods reach the same order of error by spending less…