Department of Mathematics

Speaker: Mohsen Zayernouri
Title: Fractional PDE Approach in Multi-Fidelity Modeling and Simulation: Theory, Numerics, and Beyond
Affiliation: Brown University
Date: Friday, January 23, 2015
Place and Time: Room 101, Love Building, 3:35-4:30 pm
Refreshments: Room 204, Love Building, 3:00 pm

Abstract. Fractional PDE models generalize the standard (integer-order) calculus and PDEs to any differential form of fractional orders. That puts the existing PDEs into a subset of this larger family of mathematical models. Fractional PDEs open new possibilities for robust mathematical modeling of physical processes that exhibit anomalous (sub- or super-) diffusion, nonlocal interactions, self-similar structures, long memory dependence, and power-law effects. Fractional PDEs are the right tool for exploring fractal operators and for modeling sharp interfaces in multi-phase problems, wave propagation in disordered media, and multi-scale materials. Such phenomena occur in many applications, including non-Gaussian (Levy) processes in turbulent flows, non-Newtonian fluids and rheology, non-Brownian transport phenomena in porous and disordered materials, and non-Markovian processes in multi-scale complex fluids and multi-phase applications. In such applications, fractional PDEs naturally appear as the right governing equations leading to multi-fidelity modeling and predictive simulations, which otherwise cannot be achieved by employing the standard PDEs. However, the extension of existing numerical methods to fractional PDEs is not trivial because of their non-local and history-dependent nature.

To this end, we first present a new theory on Fractional Sturm-Liouville Eigen-Problems, which serves as a fundamental spectral theory providing explicit (non-polynomial) eigenfunctions, namely as "Jacobi Polyfractonomials". These eigenfunctions extend the well-known family of Jacobi polynomials to their fractional counterparts. We show that they enjoy many attractive properties such as orthogonality, recurrence relations, and exact fractional derivatives/integrations, in addition to their spectral approximation properties. Based upon this base fractional spectral theory, we develop a series of high-order (projection and collocation) numerical methods that efficiently treat fixed-order, variable-order, and distributed-order fractional PDEs. Of particular interest, we present a unified Petrov-Galerkin spectral method for solving Parabolic, Elliptic, and Hyperbolic (time- and space-) fractional PDEs in any (d+1)-dimensions in a hypercube. While the existing local numerical methods can take days to run a one-dimensional (1-D) problem, we formulate a "Unified Fast Linear Solver" for our scheme, which allows one to run up to a ten-dimensional (10-D) fractional PDE on a laptop with a reduced computational time by orders of magnitude.

Our approach brings to bear modern computational tools from applied mathematics to real-world physical problems. This framework motivates further fundamental and innovative research in the following three main and interwoven areas: I) fractional-order modeling, II) stochastic fractional PDEs and uncertainty quantification, and III) algorithm development and scientific computing.