M. Croci, M. B. Giles, P. E. Farrell, M. E. Rognes, V. Vinje, Non-nested multilevel Monte Carlo methods with applications to brain simulation, Oden Institute (2021). Download.
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 brain.