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Parabolic partial differential equations (PDEs) and backward stochastic differential equations (BSDEs) are key ingredients in a number of models in physics and financial engineering. In particular, parabolic PDEs and BSDEs are fundamental…
In this article we present and analyse new multilevel adaptations of stochastic approximation algorithms for the computation of a zero of a function $f\colon D \to \mathbb R^d$ defined on a convex domain $D\subset \mathbb R^d$, which is…
We investigate the stability of a Sequential Monte Carlo (SMC) method applied to the problem of sampling from a target distribution on $\mathbb{R}^d$ for large $d$. It is well known that using a single importance sampling step one produces…
Nonlinear partial differential equations (PDEs) are used to model dynamical processes in a large number of scientific fields, ranging from finance to biology. In many applications standard local models are not sufficient to accurately…
We show that deliberately introducing a nested simulation stage can lead to significant variance reductions when comparing two stopping times by Monte Carlo. We derive the optimal number of nested simulations and prove that the algorithm is…
Bayesian model selection provides a powerful framework for objectively comparing models directly from observed data, without reference to ground truth data. However, Bayesian model selection requires the computation of the marginal…
In the following article we consider approximate Bayesian computation (ABC) inference. We introduce a method for numerically approximating ABC posteriors using the multilevel Monte Carlo (MLMC). A sequential Monte Carlo version of the…
Partial differential equation is a powerful tool to characterize various physics systems. In practice, measurement errors are often present and probability models are employed to account for such uncertainties. In this paper, we present a…
We introduce a novel numerical approach for a class of stochastic dynamic programs which arise as discretizations of backward stochastic differential equations or semi-linear partial differential equations. Solving such dynamic programs…
We consider the problem of estimating expectations with respect to a target distribution with an unknown normalizing constant, and where even the unnormalized target needs to be approximated at finite resolution. This setting is ubiquitous…
It is one of the most challenging problems in applied mathematics to approximatively solve high-dimensional partial differential equations (PDEs). Recently, several deep learning-based approximation algorithms for attacking this problem…
An optimal experimental set-up maximizes the value of data for statistical inferences and predictions. The efficiency of strategies for finding optimal experimental set-ups is particularly important for experiments that are time-consuming…
Stochastic Differential Equations (SDEs) are used as statistical models in many disciplines. However, intractable likelihood functions for SDEs make inference challenging, and we need to resort to simulation-based techniques to estimate and…
We introduce a general Monte Carlo method based on Nested Sampling (NS), for sampling complex probability distributions and estimating the normalising constant. The method uses one or more particles, which explore a mixture of nested…
We consider the probabilistic numerical scheme for fully nonlinear PDEs suggested in \cite{cstv}, and show that it can be introduced naturally as a combination of Monte Carlo and finite differences scheme without appealing to the theory of…
As the size of engineered systems grows, problems in reliability theory can become computationally challenging, often due to the combinatorial growth in the cut sets. In this paper we demonstrate how Multilevel Monte Carlo (MLMC) - a…
The computational complexity of naive, sampling-based uncertainty quantification for 3D partial differential equations is extremely high. Multilevel approaches, such as multilevel Monte Carlo (MLMC), can reduce the complexity significantly,…
We develop a multilevel approach to compute approximate solutions to backward differential equations (BSDEs). The fully implementable algorithm of our multilevel scheme constructs sequential martingale control variates along a sequence of…
Uncertainty estimation in deep models is essential in many real-world applications and has benefited from developments over the last several years. Recent evidence suggests that existing solutions dependent on simple Gaussian formulations…
Recombinant binomial trees are binary trees where each non-leaf node has two child nodes, but adjacent parents share a common child node. Such trees arise in finance when pricing an option. For example, valuation of a European option can be…