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.

LIONEL TERTIS AND THE FLOWERING OF EARLY TWENTIETH CENTURY ENGLISH VIOLA REPERTOIRE

This performance project focused on English viola literature written in the first half of the twentieth century. During this time, numerous English composers were influenced by Lionel Tertis' unprecedented approach to the viola as a virtuosic and solo instrument. In addition to being an inspiration to composers of whom he was not in direct contact, Tertis' innovative vision for the viola led to numerous collaborations with prominent English composers of his generation. Ralph Vaughan Williams, Arnold Bax, York Bowen, Frank Bridge, Benjamin Britten, and Rebecca Clarke -his own protégé - composed some of the most important works for viola thus directly shaping the impression of the instrument as we know it today. Tertis' artistry as a performing violist was unmatched at the beginning of the twentieth century. He had a unique approach to the instrument which focused on concept of sound, tone color, concentrated listening, continuous vibrato, discreet portamento, and expressive interpretation. His convincing musical and technical ideas led him to write a treatise about how to achieve a beautiful tone. His passion for teaching and concern for the viola's posterity greatly enhanced the development of the viola. Tertis transcribed, edited, and premiered many works during his career. The music that Lionel Tertis influenced can be seen as a microcosm for a musical resurgence in England during the first half of the twentieth-century. The catalyst for this was artistic influences in the form of nationalism, folk music, romanticism, modernism, and impressionism, among others. Before this, England was widely referred to as ''the land without music" but in a very real sense, .Lionel Tertis was one of the pioneers who, through his artistry of the viola, led the way to the renaissance of music in England in the twentieth century.

IMPROVISATION AND THE CONCERT SAXOPHONIST: A SURVEY OF COMPOSITIONAL AND PERFORMANCE TECHNIQUES

Much of the contemporary concert (i.e. “classical”) saxophone literature has connections to compositional styles found in other genres like jazz, rock, or pop. Although improvisation exists as a dominant compositional device in jazz, improvisation as a performance technique is not confined to a single genre. This study looks at twelve concert saxophone pieces that are grouped into three primary categories of compositional techniques: 1) those containing unmeasured phrases, 2) those containing limited relation to improvisation but a close relationship to jazz styles, and 3) those containing jazz improvisation. In concert saxophone music, specific crossover pieces use the compositional technique of jazz improvisation. Four examples of such jazz works were composed by Dexter Morrill, Phil Woods, Bill Dobbins, and Ramon Ricker, all of which provide a foundation for this study. In addition, pieces containing varying degrees of unmeasured phrases are highlighted. As this dissertation project is based in performance, the twelve pieces were divided into three recitals that summarize a pedagogical sequence. Any concert saxophonist interested in developing jazz improvisational skills can use the pieces in this study as a method to progress toward the performance of pieces that merge jazz improvisation with the concert format. The three compositional techniques examined here will provide the performer with the necessary material to develop this individualized approach to improvisation. Specific compositional and performance techniques vary depending on the stylistic content: this study examines improvisation in the context of concert saxophone repertoire.