Multilevel Monte Carlo methods for uncertainty quantification in brain simulations

This thesis consists of two parts. In the first, we develop two new strategies for spatial white noise and Gaussian-Mat\'ern 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. We present a new proof of an a priori error estimate for the finite element (FEM) solution of the white noise SPDE. The proof does not require the approximation of white noise in practice, and includes higher order elliptic operators and p-refinement. Within the SPDE approach, the efficient sampling of white noise realisations can be computationally expensive. In this thesis, 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. We demonstrate the efficacy of our sampling methods with numerical experiments. In a multilevel setting, a good coupling is enforced and the telescoping sum is respected. In the MLQMC case, the asymptotic convergence rate is the same as standard Monte Carlo, but significant computational gains are obtained in practice thanks to a pre-asymptotic QMC-like regime. In the final part of the thesis, 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. Numerical simulations support the claim that diffusion alone cannot explain the penetration of tracers within deep brain regions as observed in clinical experiments, even when uncertainties in the diffusivity have been accounted for. A convective velocity field may however increase tracer transport, provided that a directional structure is present in the interstitial fluid circulation.