Past Event: Oden Institute Seminar

Non-nested multilevel Monte Carlo methods with applications to brain simulation

Mateo Croci, Post doctoral candidate, University of Oxford

Abstract

This talk consists of two parts. In the first, we develop two new strategies for spatial white noise and Gaussian-Matérn field sampling that work within a non-nested multilevel (quasi) Monte Carlo (ML(Q)MC) hierarchy. In the second, we apply the techniques developed to quantify the level of uncertainty in a new stochastic model for tracer transport in the brain.

The new sampling techniques are based on the stochastic partial differential equation (SPDE) approach, which recasts the sampling problem as the solution of an elliptic equation driven by spatial white noise. The efficient sampling of white noise realisations can be computationally expensive. In this talk, we present two new sampling techniques that can be used to efficiently compute white noise samples in a FEM-MLMC and FEM-MLQMC setting. The key idea is to exploit the finite element matrix assembly procedure and factorise each local mass matrix independently, hence avoiding the factorisation of a large matrix. In a multilevel framework, the white noise samples must be coupled between subsequent levels. We show how our technique can be used to enforce this coupling even in the case of non-nested mesh hierarchies.

In the MLQMC case, the QMC integrand variables must also be ordered in order of decaying importance to achieve fast convergence with respect to the number of samples. We express white noise as a Haar wavelet series whose hierarchical structure naturally exposes the leading order dimensions. We split this series in two terms which we sample via a hybrid standard Monte Carlo/QMC approach.

In the final part of the talk, we employ a combination of the methods presented to solve a PDE with random coefficients describing tracer transport within the interstitial fluid of the brai

 

Biography

Matteo Croci is a postdoctoral researcher in Computational Stochastics in the Mathematical Institute of the University of Oxford, working with Michael B. Giles. He completed is DPhil (PhD) in Mathematics at the University of Oxford under the supervision of Patrick E. Farrell and Michael B. Giles (University of Oxford, UK), and the industrial supervision of Marie E. Rognes (Simula Research Laboratory, Norway). During his PhD, Matteo developed two new state-of-the-art multilevel (quasi-) Monte Carlo techniques, which he employed to quantify the uncertainty in numerical simulations of brain fluid and solute movement in real-life geometries. Matteo's current research is in the field of uncertainty quantification and computational and industrial mathematics, with a focus on multilevel (quasi) Monte Carlo methods, reduced and mixed-precision numerical algorithms, and biomedical computing.

For more information about Matteo's research, see https://croci.github.io