Generalised polynomial chaos approximation for stochastic fractional partial differential equations

Abstract

Fractional diffusion equations are widely used in various fields, such as physics, chemistry, biology, finance, and economics, to model the spread of particles, diseases, and financial variables. It is characterized by the anomalous diffusion of particles, which occurs in a non-Gaussian and non-local manner. However, solving these equations is a challenging task due to the non-local nature of the fractional Laplacian operator on a bounded domain. To address this issue, this thesis studies the Caffarelli-Stinga extension problems of the fractional diffusion equations, which localises the non-local operator on a semi-infinite cylinder.

Furthermore, this thesis investigates the parametric uncertainty in the diffusion coefficient of the spectral fractional diffusion operator for the following two cases. In the first case, the coefficient is uniformly bounded and depends linearly on a countable number of random variables uniformly distributed in a compact set. In the second case, we consider the log-Gaussian coefficients where the logarithm of the coefficients follows a Gaussian distribution and depends linearly on a countable number of Gaussian random variables. The generalised polynomial chaos method is adopted to represent the random solution by an infinite series of polynomials of the random variables, which form an orthogonal basis of an infinite-dimensional Hilbert space with respect to the probability space of the random variables. We focus on studying the accuracy of the approximation when only a finite number of terms are considered.

In particular, this thesis establishes bounds for the parametric solution and the Wiener-Itˆo coefficients with respect to the norm of regularity spaces. These bounds are crucial for estimating the convergence rate of continuous, piecewise linear finite element methods.

Overall, this thesis provides valuable insights into approximating solutions to the fractional diffusion equation under parametric uncertainty and lays the foundation for future research in this area.