Efficient Multilevel and Multi-index Sampling Methods in Stochastic Differential Equations

Abstract

Most problems in engineering and natural sciences involve parametric equations in which the parameters are not known exactly due to measurement errors, lack of measurement data, or even intrinsic variability. In such problems, one objective is to compute point or aggregate values, called “quantities of interest”. A rapidly growing research area that tries to tackle this problem is Uncertainty Quantification (UQ). As the name suggests, UQ aims to accurately quantify the uncertainty in quantities of interest. To that end, the approach followed in this thesis is to describe the parameters using probabilistic measures and then to employ probability theory to approximate the probabilistic information of the quantities of interest. In this approach, the parametric equations must be accurately solved for multiple values of the parameters to explore the dependence of the quantities of interest on these parameters, using various so-called “sampling methods”. In almost all cases, the parametric equations cannot be solved exactly and suitable numerical discretization methods are required. The high computational complexity of these numerical methods coupled with the fact that the parametric equations must be solved for multiple values of the parameters make UQ problems computationally intensive, particularly when the dimensionality of the underlying problem and/or the parameter space is high. This thesis is concerned with optimizing existing sampling methods and developing new ones. Starting with the Multilevel Monte Carlo (MLMC) estimator, we first prove its normality using the Lindeberg-Feller CLT theorem. We then design the Continuation Multilevel Monte Carlo (CMLMC) algorithm that efficiently approximates the parameters required to run MLMC. We also optimize the hierarchies of one-dimensional discretization parameters that are used in MLMC and analyze the tolerance splitting parameter between the statistical error and the bias constraints. An important contribution of this thesis is the novel Multi-index Monte Carlo (MIMC) method which is an extension of MLMC in high dimensional problems with significant computational savings. Under reasonable assumptions on the weak and variance convergence, which are related to the mixed regularity of the underlying problem and the discretization method, the order of the computational complexity of MIMC is, at worst up to a logarithmic factor, independent of the dimensionality of the underlying parametric equation. We also apply the same multi-index methodology to another sampling method, namely the Stochastic Collocation method. Hence, the novel Multi-index Stochastic Collocation method is proposed and is shown to be more efficient in problems with sufficient mixed regularity than our novel MIMC method and other standard methods. Finally, MIMC is applied to approximate quantities of interest of stochastic particle systems in the mean-field when the number of particles tends to infinity. To approximate these quantities of interest up to an error tolerance, TOL, MIMC has a computational complexity of O(TOL-2log(TOL)2). This complexity is achieved by building a hierarchy based on two discretization parameters: the number of time steps in an Milstein scheme and the number of particles in the particle system. Moreover, we use a partitioning estimator to increase the correlation between two stochastic particle systems with different sizes. In comparison, the optimal computational complexity of MLMC in this case is O(TOL-3) and the computational complexity of Monte Carlo is O(TOL-4).