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Methods for generating sequences of surrogates approximating fine scale models of two-phase random heterogeneous media are presented that are designed to adaptively control the modeling error in key quantities of interest (QoIs). For…
Multistable L\'evy motions are extensions of L\'evy motions where the stability index is allowed to vary in time. Several constructions of these processes have been introduced recently, based on Poisson and Ferguson-Klass-LePage series…
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…
The strong convergence of the semi-implicit Euler-Maruyama (EM) method for stochastic differential equations with non-linear coefficients driven by a class of L\'evy processes is investigated. The dependence of the convergence order of the…
We introduce a class of adaptive timestepping strategies for stochastic differential equations with non-Lipschitz drift coefficients. These strategies work by controlling potential unbounded growth in solutions of a numerical scheme due to…
In many real-world engineering systems, the performance or reliability of the system is characterised by a scalar parameter. The distribution of this performance parameter is important in many uncertainty quantification problems, ranging…
In this paper, a multilevel Monte Carlo theta EM scheme is provided for stochastic differential delay equations with small noise. Under a global Lipschitz condition, the variance of two coupled paths is derived. Then, the global Lipschitz…
Models of stochastic processes are widely used in almost all fields of science. Theory validation, parameter estimation, and prediction all require model calibration and statistical inference using data. However, data are almost always…
We consider a multi-step algorithm for the computation of the historical expected shortfall such as defined by the Basel Minimum Capital Requirements for Market Risk. At each step of the algorithm, we use Monte Carlo simulations to reduce…
We present a new, for plasma physics, highly efficient multilevel Monte Carlo numerical method for simulating Coulomb collisions. The method separates and optimally minimizes the finite-timestep and finite-sampling errors inherent in the…
Since Giles introduced the multilevel Monte Carlo path simulation method [18], there has been rapid development of the technique for a variety of applications in computational finance. This paper surveys the progress so far, highlights the…
We consider the numerical approximation of $\mathbb{P}[G\in \Omega]$ where the $d$-dimensional random variable $G$ cannot be sampled directly, but there is a hierarchy of increasingly accurate approximations $\{G_\ell\}_{\ell\in\mathbb{N}}$…
We show how to extend a recently proposed multi-level Monte Carlo approach to the continuous time Markov chain setting, thereby greatly lowering the computational complexity needed to compute expected values of functions of the state of the…
In recent years, tremendous progress has been made on numerical algorithms for solving partial differential equations (PDEs) in a very high dimension, using ideas from either nonlinear (multilevel) Monte Carlo or deep learning. They are…
Cr\'epey, Frikha, and Louzi (2025) introduced a multilevel stochastic approximation scheme to compute the value-at-risk of a financial loss that is only simulatable by Monte Carlo. The best complexity of the scheme is in…
We develop a novel procedure for estimating the optimizer of general convex stochastic optimization problems of the form $\min_{x\in\mathcal{X}} \mathbb{E}[F(x,\xi)]$, when the given data is a finite independent sample selected according to…
The identification of parameters in mathematical models using noisy observations is a common task in uncertainty quantification. We employ the framework of Bayesian inversion: we combine monitoring and observational data with prior…
In the nano era in integrated circuit fabrication technologies, the performance variability due to statistical process and circuit parameter variations is becoming more and more significant. Considerable effort has been expended in the EDA…
In this paper, we study the error in first order Sobolev norm in the approximation of solutions to linear parabolic PDEs. We use a Monte Carlo Euler scheme obtained from combining the Feynman--Kac representation with a Euler discretization…
Nonlinear differential equations (DEs) are used in a wide range of scientific problems to model complex dynamic systems. The differential equations often contain unknown parameters that are of scientific interest, which have to be estimated…