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

We propose a positive, accurate moment closure for linear kinetic transport equations based on a filtered spherical harmonic (FP_N) expansion in the angular variable. The FP_N 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 P_N (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 L2 sense, at the same rate as the FP_N 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 constraint-reduced algorithms, especially for problems with many more inequality constraints than variables.

SELECTED CHARACTER PIECES OF FRÉDÉRIC CHOPIN (1810-1849), FRANZ LISZT (1811-1886) AND JOHANNES BRAHMS (1833-1897)

During the 19th century, Frédéric Chopin (1810-1849), Franz Liszt (1811- 1886), and Johannes Brahms (1833-1897) were among the most recognized composers of character pieces. Their compositions have been considered a significant milestone in piano literature. Frédéric Chopin (1810-1849) did not give descriptive titles to his character pieces. He grouped them into several genres such as Mazurkas, Polonaises. His Mazurkas and Polonaises are influenced by Polish dance music and inspired by the polish national idiom. Franz Liszt (1811-1886) was influenced in many ways by Chopin, and adopted Chopin’s lyricism, melodic style, and tempo rubato. However, Liszt frequently drew on non-musical subjects (e.g., art, literature) for inspiration. “Harmonies poétiques et religieuses” and “Années de pèlerinage” are especially representative of character pieces in which poetic and pictorial imagination are reflected. Johannes Brahms (1833-1897) was a conservative traditionalist, synthesizing Romantic expression and Classical tradition remarkably well. Like Chopin, Brahms avoided using programmatic titles for his works. The titles of Brahms’ short character pieces are often taken from traditional lyrical or dramatic genres such as ballade, rhapsody and scherzo. Because of his conservatism, Brahms was considered the main rival of Liszt in the Romantic Period. Brahms character pieces in his third period (e.g., Scherzo Op.4, Ballades of Op.10, and Rhapsodies of Op.79) are concise and focused. The form of Brahms’ character pieces is mostly simple ternary (ABA), and his style is introspective and lyrical. Through this recording project, I was able to get a better understanding of the styles of Chopin, Brahms and Liszt through their character pieces. This recording dissertation consists of two CDs recorded in the Dekelboum Concert Hall at the University of Maryland, College Park. These recordings are documented on compact disc recordings that are housed within the University of Maryland Library System.