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.

A Better Place to Be: Republicanism as an Alternative to the Authoritarianism-Democracy Dichotomy

ABSTRACT Title of Dissertation: A BETTER PLACE TO BE: REPUBLICANISM AS AN ALTENATIVE TO THE AUTHORITARIANISM-DEMOCRACY DICHOTOMY Christopher Ronald Binetti, Doctor of Philosophy, and 2016 Dissertation directed by: Dr. Charled Frederick Alford, Department of Government and Politics In this dissertation, I argue that in modern or ancient regimes, the simple dichotomy between democracies and autocracies/dictatorships is both factually wrong and problematic for policy purposes. It is factually wrong because regimes between the two opposite regime types exist and it is problematic because the either/or dichotomy leads to extreme thinking in terms of nation-building in places like Afghanistan. In planning for Afghanistan, the argument is that either we can quickly nation-build it into a liberal democracy or else we must leave it in the hands of a despotic dictator. This is a false choice created by both a faulty categorization of regime types and most importantly, a failure to understand history. History shows us that the republic is a regime type that defies the authoritarian-democracy dichotomy. A republic by my definition is a non-dominating regime, characterized by a (relative) lack of domination by any one interest group or actor, mostly non-violent competition for power among various interest groups/factions, the ability of factions/interest groups/individual actors to continue to legitimately play the political game even after electoral or issue-area defeat and some measure of effectiveness. Thus, a republic is a system of government that has institutions, laws, norms, attitudes, and beliefs that minimize the violation of the rule of law and monopolization of power by one individual or group as much as possible. These norms, laws, attitudes, and beliefs ae essential to the republican system in that they make those institutions that check and balance power work. My four cases are Assyria, Persia, Venice and Florence. Assyria and Persia are ancient regimes, the first was a republic and then became the frightening opposite of a republic, while the latter was a good republic for a long time, but had effectiveness issues towards the end. Venice is a classical example of a medieval or early modern republic, which was very inspirational to Madison and others in building republican America. Florence is the example of a medieval republic that fell to despotism, as immortalized by Machiavelli’s writings. In all of these examples, I test certain alternative hypotheses as well as my own.

Development of a Real-Time Hierarchical 3D Path Planning Algorithm for Unmanned Aerial Vehicles

Unmanned aerial vehicles (UAVs) frequently operate in partially or entirely unknown environments. As the vehicle traverses the environment and detects new obstacles, rapid path replanning is essential to avoid collisions. This thesis presents a new algorithm called Hierarchical D* Lite (HD*), which combines the incremental algorithm D* Lite with a novel hierarchical path planning approach to replan paths sufficiently fast for real-time operation. Unlike current hierarchical planning algorithms, HD* does not require map corrections before planning a new path. Directional cost scale factors, path smoothing, and Catmull-Rom splines are used to ensure the resulting paths are feasible. HD* sacrifices optimality for real-time performance. Its computation time and path quality are dependent on the map size, obstacle density, sensor range, and any restrictions on planning time. For the most complex scenarios tested, HD* found paths within 10% of optimal in under 35 milliseconds.