POSITIVE FILTERED P_N METHOD FOR LINEAR TRANSPORT EQUATIONS AND THE ASSOCIATED OPTIMIZATION ALGORITHM

Title of dissertation: POSITIVE FILTERED PN METHOD FOR LINEAR TRANSPORT EQUATIONS AND THE ASSOCIATED OPTIMIZATION ALGORITHM Ming Tse Paul Laiu, Doctor of Philosophy, 2016 Dissertation directed by: Professor André Tits Department of Electrical and Computer Engineering We propose a positive, accurate moment closure for linear kinetic transport equations based on a filtered spherical harmonic (FPN ) expansion in the angular variable. The FPN moment equations are accurate approximations to linear kinetic equations, but they are known to suffer from the occurrence of unphysical, negative particle concentrations. The new positive filtered PN (FP + N) closure is developed to address this issue. The FP+N closure approximates the kinetic distribution by a spherical harmonic expansion that is non-negative on a finite, predetermined set of quadrature points. With an appropriate numerical PDE solver, the FP+N closure generates particle concentrations that are guaranteed to be non-negative. Under an additional, mild regularity assumption, we prove that as the moment order tends to infinity, the FP+N approximation converges, in the L 2 sense, at the same rate as the FPN approximation; numerical tests suggest that this assumption may not be necessary. By numerical experiments on the challenging line source benchmark problem, we confirm that the FP+N method indeed produces accurate and non-negative solutions. To apply the FP+N closure on problems at large temporal-spatial scales, we develop a positive asymptotic preserving (AP) numerical PDE solver. We prove that the propose AP scheme maintains stability and accuracy with standard mesh sizes at large temporal-spatial scales, while, for generic numerical schemes, excessive refinements on temporal-spatial meshes are required. We also show that the proposed scheme preserves positivity of the particle concentration, under some time step restriction. Numerical results confirm that the proposed AP scheme is capable for solving linear transport equations at large temporal-spatial scales, for which a generic scheme could fail. Constrained optimization problems are involved in the formulation of the FP+N closure to enforce non-negativity of the FP+N approximation on the set of quadrature points. These optimization problems can be written as strictly convex quadratic programs (CQPs) with a large number of inequality constraints. To efficiently solve the CQPs, we propose a constraint-reduced variant of a Mehrotra-predictor-corrector algorithm, with a novel constraint selection rule. We prove that, under appropriate assumptions, the proposed optimization algorithm converges globally to the solution at a locally q-quadratic rate. We test the algorithm on randomly generated problems, and the numerical results indicate that the combination of the proposed algorithm and the constraint selection rule outperforms other compared constraintreduced algorithms, especially for problems with many more inequality constraints than variables. POSITIVE FILTERED PN METHOD FOR LINEAR TRANSPORT EQUATIONS AND THE ASSOCIATED OPTIMIZATION ALGORITHM by Ming Tse Paul Laiu Dissertation submitted to the Faculty of the Graduate School of the University of Maryland, College Park in partial fulfillment of the requirements for the degree of Doctor of Philosophy 2016 Advisory Committee: Professor André Tits, Chair/Advisor Dr. Cory Hauck, Co-Advisor Professor Dianne O’Leary Professor Isaak Mayergoyz Professor Konstantina Trivisa Professor Howard Elman c © Copyright by Ming Tse Paul Laiu 2016