Publication Date

Summer 7-31-2025

Abstract

Algebraic multigrid (AMG) is a well-established and highly efficient solver for symmetric positive definite (SPD) systems arising from elliptic and parabolic PDEs, while nonsymmetric systems from hyperbolic PDEs remain a significant challenge. This dissertation develops AMG methods and theory for nonsymmetric problems. First, we develop a novel approach combining mode constraints from energy-minimization AMG with local approximations of ideal restriction in $\ell$AIR, resulting in constrained $\ell$AIR (C$\ell$AIR), which demonstrates scalable convergence across advective and diffusive problems. Second, we extend optimal AMG theory by deriving spectral radius estimates for the two-grid error transfer operator using matrix-induced orthogonality, enabling convergence predictions for nonsymmetric and indefinite systems. Third, we implement parallel $\ell$AIR for time-dependent and steady-state relativistic drift-kinetic Fokker-Planck-Boltzmann models of runaway electrons, demonstrating practical effectiveness. This work advances AMG solvers with new theoretical insights and algorithms for nonsymmetric problems across diverse applications.

Degree Name

Mathematics

Level of Degree

Doctoral

Department Name

Mathematics & Statistics

First Committee Member (Chair)

Jacob B. Schroder

Second Committee Member

James J. Brannick

Third Committee Member

Stephen R. Lau

Fourth Committee Member

Ben S. Southworth

Language

English

Keywords

algebraic multigrid, multigrid reduction, nonsymmetric, optimal interpolation, indefinite, runaway electrons

Document Type

Dissertation

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