Related papers: On the Acceleration of the Multi-Level Monte Carlo…
Accurately and efficiently estimating system performance under uncertainty is paramount in power system planning and operation. Monte Carlo simulation is often used for this purpose, but convergence may be slow, especially when detailed…
We consider the application of multilevel Monte Carlo methods to steady state Darcy flow in a random porous medium, described mathematically by elliptic partial differential equations with random coefficients. The levels in the multilevel…
Multifidelity Monte Carlo methods often rely on a preprocessing phase consisting of standard Monte Carlo sampling to estimate correlation coefficients between models of different fidelity to determine the weights and number of samples for…
Generative diffusion models have recently emerged as a powerful strategy to perform stochastic sampling in Bayesian inverse problems, delivering remarkably accurate solutions for a wide range of challenging applications. However, diffusion…
This paper considers a new approach to using Markov chain Monte Carlo (MCMC) in contexts where one may adopt multilevel (ML) Monte Carlo. The underlying problem is to approximate expectations w.r.t. an underlying probability measure that is…
We propose efficient numerical algorithms for approximating statistical solutions of scalar conservation laws. The proposed algorithms combine finite volume spatio-temporal approximations with Monte Carlo and multi-level Monte Carlo…
Statisticians often use Monte Carlo methods to approximate probability distributions, primarily with Markov chain Monte Carlo and importance sampling. Sequential Monte Carlo samplers are a class of algorithms that combine both techniques to…
We introduce a class of Monte Carlo estimators that aim to overcome the rapid growth of variance with dimension often observed for standard estimators by exploiting the target's independence structure. We identify the most basic…
We consider stochastic optimization when one only has access to biased stochastic oracles of the objective and the gradient, and obtaining stochastic gradients with low biases comes at high costs. This setting captures various optimization…
This paper proposes a new multilevel Monte Carlo (MLMC) method for the ergodic SDEs which do not satisfy the contractivity condition. By introducing the change of measure technique, we simulate the path with contractivity and add the…
Monte Carlo (MC) sampling is a popular method for estimating the statistics (e.g. expectation and variance) of a random variable. Its slow convergence has led to the emergence of advanced techniques to reduce the variance of the MC…
This paper addresses optimization problems constrained by partial differential equations with uncertain coefficients. In particular, the robust control problem and the average control problem are considered for a tracking type cost…
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…
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…
Monte Carlo algorithms are a foundational pillar of modern computational science, yet their effective application hinges on a deep understanding of their performance trade offs. This paper presents a critical analysis of the evolution of…
In this paper, we propose a new kind of numerical scheme for high-dimensional backward stochastic differential equations based on modified multi-level Picard iteration. The proposed scheme is very similar to the original multi-level Picard…
Nested integration problems arise in various scientific and engineering applications, including Bayesian experimental design, financial risk assessment, and uncertainty quantification. These nested integrals take the form $\int f\left(\int…
Monte Carlo method is a broad class of computational algorithms that rely on repeated random sampling to obtain numerical results. They are often used in physical and mathematical problems and are most useful when it is difficult or…
We propose a multilevel Markov chain Monte Carlo (MCMC) method for the Bayesian inference of random field parameters in PDEs using high-resolution data. Compared to existing multilevel MCMC methods, we additionally consider level-dependent…
We combine a generic method for finding fast orthogonal transforms for a given quasi-Monte Carlo integration problem with the multilevel Monte Carlo method. It is shown by example that this combined method can vastly improve the efficiency…