Fractional diffusion equations have played an increasingly important role in ex plaining longrange 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 quantifications tools that accelerate the inverse problem. We begin by focusing on parameter calibration of a variable diffusivity fractional diffusion model. A random, spatiallyvarying diffusivity field is considered together with an uncertain but spatially homogeneous fractional operator order. Polynomial chaos (PC) techniques are used to express the dependence of the stochastic solution on these random variables. A nonintrusive methodology is used, and a deterministic finitedifference solver of the fractional diffusion model is utilized for this purpose. The surrogates are first used to assess the sensitivity of quantities of interest (QoIs) to uncertain inputs and to examine their statistics. In particular, the analysis indicates that the fractional order has a dominant effect on the variance of the QoIs considered. The PC surrogates are further exploited to calibrate the uncertain parameters using a Bayesian methodology. In the broad range of parameters addressed, the analysis shows that the uncertain parameters having a significant impact on the variance of the solution can be reliably inferred, even from limited observations.
Next, we address the numerical challenges when multidimensional spacefractional
diffusion equations have spatially varying diffusivity and fractional order. Significant computational challenges arise due to the kernel singularity in the fractional integral operator as well as the resulting dense discretized operators. Hence, we present a singularityaware discretization scheme that regularizes the singular integrals through a singularity subtraction technique adapted to the spatial variability of diffusivity and fractional order. This regularization strategy is conveniently formulated as a sparse matrix correction that is added to the dense operator, and is applicable to different formulations of fractional diffusion equations. Numerical results show that the singularity treatment is robust, substantially reduces discretization errors, and attains the firstorder convergence rate allowed by the regularity of the solutions.
In the last part, 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 twodimensional domain. The anomaly is assumed to have known shape, but unknown diffusivity and fractional order parameters, and is assumed to be embedded in a fractional medium of known fractional properties. To detect the presence of the anomaly, the medium is forced using a collection of localized sources, and its response is measured at the source locations. To this end, the singularityaware finitedifference scheme is applied. A nonintrusive regression approach is used to explore the dependence of the computed signals on the properties of the anomaly, and the resulting surrogates are first exploited to characterize the variability of the response, and then used to accelerate the Bayesian inference of the anomaly. In the regime of parameters considered, the computational results indicate that robust estimates of the location and fractional properties of the anomaly can be obtained, and that these estimates become sharper when high contrast ratios prevail between the anomaly and the surrounding matrix.
Date of Award  Jan 10 2023 

Original language  English (US) 

Awarding Institution   Computer, Electrical and Mathematical Sciences and Engineering


Supervisor  Omar Knio (Supervisor) 

 SpaceFractional Diffusion
 Polynomial Chaos Expansion
 Uncertainty Quantification
 Bayesian Inference
 Numerical Discretization