Fractional diffusion equations have played an increasingly important role in ex- plaining long-range interactions, nonlocal dynamics, and anomalous diffusion, pro- viding effective means of describing the memory and hereditary properties of such processes.
This dissertation explores the uncertainty propagation in space fractional diffusion equations in one and multiple dimensions with variable diffusivity and order parameters. This is achieved by: (I) deploying accurate numerical schemes of the forward problem and (ii) employing uncertainty quantification tools that accelerate the inverse problem. We begin by focusing on parameter calibration of a variable-diffusivity fractional diffusion model. Next, we address the numerical challenges when multidimensional space-fractional diffusion equations have spatially varying diffusivity and fractional order. We present a singularity-aware discretization scheme that regularizes the singular integrals through a singularity subtraction technique adapted to the spatial variability of diffusivity and fractional order. We explore the application of a Bayesian formalism to detect an anomaly in a fractional medium. Specifically, a computational method is presented for inferring the location and properties of an inclusion inside a two-dimensional domain.
Hasnaa Alzahrani is a Ph.D. candidate supervised by Prof. Omar Knio in the Applied Mathematics program at King Abdullah University of Science and Technology. She received her Master of Applied Mathematics from King Abdullah University of Science and Technology in 2016 and her Bachelor in Mathematics from King Abdulaziz University in 2014. Her research interests include fractional diffusion equations with uncertainty and Bayesian inference.