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Multigrid methods for fractional diffusion equations

Staff - Faculty of Informatics

Date: 15 December 2021 / 10:00 - 12:00

You are cordially invited to attend the PhD Dissertation Defence of Ken Trotti on Wednesday 15 December 2021 at 10:00 on MS Teams.

Abstract:
Recent years have seen the rapid growth in interest towards fractional calculus. Fractional calculus plays an important role in modelling anomalous diffusion phenomena, however closed-form analytical solutions of such equations are rarely available, hence numerical estimates are needed.

In this thesis we consider various fractional diffusion equations (FDEs), where different fractional derivative definitions and related discretizations are involved and we focus on multigrid-based approaches for solving the associated linear systems. Precisely, we will leverage the spectral properties of the coefficient matrix, retrieved by exploiting its structure, to design ad-hoc (tailored) multigrid solvers or preconditioners for Krylov methods.

We develop a new approach to compute the Jacobi weight, which is versatile enough to work with various FDEs and allows to build a parameter free multigrid method. Moreover, in the case of uniform meshes, we exploit the knowledge about anisotropic integer-order partial diffusion equations to deal with anisotropic FDEs, by building a robust multigrid-based solver. Furthermore, we study the behavior of multigrid methods as parallel-in-time solvers and, then, we provide a new second-order accurate finite volume approximation and related ad-hoc multigrid solver. Finally, we extend our multigrid strategies to deal with a singular one-dimensional space-FDE discretized over non-uniform meshes. strategies to deal with a singular one-dimensional space-FDE discretized over non-uniform meshes.

Dissertation Committee:
- Prof. Rolf Krause, Università della Svizzera italiana, Switzerland (Research Advisor)
- Prof. Marco Donatelli, Università degli Studi dell’Insubria, Italy (Research co-Advisor)
- Prof. Cesare Alippi, Università della Svizzera italiana, Switzerland (Internal Member)
- Prof. Michael Multerer, Università della Svizzera italiana, Switzerland (Internal Member)
- Prof. Matthias Bolten, University of Wuppertal, Germany (External Member)