207 citations · 416 across the 27 of their papers we have counts for
20 papers · 1 filter
Multilevel lattice-based kernel approximation for elliptic PDEs with random coefficients
Alexander D. Gilbert, Michael B. Giles, Frances Y. Kuo +2
This paper introduces a multilevel kernel-based approximation method to estimate efficiently solutions to elliptic partial differential equations (PDEs) with periodic random coeffi…
Strong convergence of path sensitivities
Michael B. Giles
It is well known that the Euler-Maruyama discretisation of an autonomous SDE using a uniform timestep has a strong convergence error which is when the drift and di…
MLMC techniques for discontinuous functions
Michael B Giles
The Multilevel Monte Carlo (MLMC) approach usually works well when estimating the expected value of a quantity which is a Lipschitz function of intermediate quantities, but if it i…
Multilevel Path Branching for Digital Options
Michael B. Giles, Abdul-Lateef Haji-Ali
We propose a new Monte Carlo-based estimator for digital options with assets modelled by a stochastic differential equation (SDE). The new estimator is based on repeated path split…
Analysis of nested multilevel Monte Carlo using approximate Normal random variables
Mike Giles, Oliver Sheridan-Methven
The multilevel Monte Carlo (MLMC) method has been used for a wide variety of stochastic applications. In this paper we consider its use in situations in which input random variable…
Rounding error using low precision approximate random variables
Oliver Sheridan-Methven, Michael Giles
For numerical approximations to stochastic differential equations using the Euler-Maruyama scheme, we propose incorporating approximate random variables computed using low precisio…
Approximating inverse cumulative distribution functions to produce approximate random variables
Oliver Sheridan-Methven, Michael Giles
For random variables produced through the inverse transform method, approximate random variables are introduced, which are produced by approximations to a distribution's inverse cu…
Effects of round-to-nearest and stochastic rounding in the numerical solution of the heat equation in low precision
Matteo Croci, Michael B. Giles
Motivated by the advent of machine learning, the last few years have seen the return of hardware-supported low-precision computing. Computations with fewer digits are faster and mo…
Generalised multilevel Picard approximations
Michael B. Giles, Arnulf Jentzen, Timo Welti
It is one of the most challenging problems in applied mathematics to approximatively solve high-dimensional partial differential equations (PDEs). In particular, most of the numeri…
Multilevel quasi Monte Carlo methods for elliptic PDEs with random field coefficients via fast white noise sampling
M. Croci, M. B. Giles, P. E. Farrell
When solving partial differential equations with random fields as coefficients the efficient sampling of random field realisations can be challenging. In this paper we focus on the…
Multilevel Monte Carlo estimation of the expected value of sample information
Tomohiko Hironaka, Michael B. Giles, Takashi Goda +1
We study Monte Carlo estimation of the expected value of sample information (EVSI) which measures the expected benefit of gaining additional information for decision making under u…
An Adaptive Random Bit Multilevel Algorithm for SDEs
Michael B. Giles, Mario Hefter, Lukas Mayer +1
We study the approximation of expectations for solutions of stochastic differential equations and functionals on the path space by means of Monte C…
Random Bit Multilevel Algorithms for Stochastic Differential Equations
Michael B. Giles, Mario Hefter, Lukas Mayer +1
We study the approximation of expectations $\E(f(X))$ for solutions of SDEs and functionals by means of restricted Monte Carlo algorithms that m…
Efficient white noise sampling and coupling for multilevel Monte Carlo with non-nested meshes
Matteo Croci, Michael B. Giles, Marie E. Rognes +1
When solving stochastic partial differential equations (SPDEs) driven by additive spatial white noise, the efficient sampling of white noise realizations can be challenging. Here,…
Multilevel estimation of expected exit times and other functionals of stopped diffusions
Michael B. Giles, Francisco Bernal
This paper proposes and analyses a new multilevel Monte Carlo method for the estimation of mean exit times for multi-dimensional Brownian diffusions, and associated functionals whi…
Random Bit Quadrature and Approximation of Distributions on Hilbert Spaces
Michael B. Giles, Mario Hefter, Lukas Mayer +1
We study the approximation of expectations $\E(f(X))$ for Gaussian random elements with values in a separable Hilbert space and Lipschitz continuous functionals $f \colon H…
Adaptive Euler-Maruyama method for SDEs with non-globally Lipschitz drift: Part II, infinite time interval
Wei Fang, Michael B. Giles
This paper proposes an adaptive timestep construction for an Euler-Maruyama approximation of the ergodic SDEs with a drift which is not globally Lipschitz over an infinite time int…
Multilevel Monte Carlo methods for the approximation of invariant measures of stochastic differential equations
Michael B. Giles, Mateusz B. Majka, Lukasz Szpruch +2
We develop a framework that allows the use of the multi-level Monte Carlo (MLMC) methodology (Giles2015) to calculate expectations with respect to the invariant measure of an ergod…
A model and variance reduction method for computing statistical outputs of stochastic elliptic partial differential equations
Ferran Vidal-Codina, Ngoc-Cuong Nguyen, Mike B. Giles +1
We present a model and variance reduction method for the fast and reliable computation of statistical outputs of stochastic elliptic partial differential equations. Our method cons…
Stochastic finite differences and multilevel Monte Carlo for a class of SPDEs in finance
Michael B. Giles, Christoph Reisinger
In this article, we propose a Milstein finite difference scheme for a stochastic partial differential equation (SPDE) describing a large particle system. We show, by means of Fouri…