In this thesis, a new smoothing penalty algorithm is introduced to solve a mathematical program with equilibrium constraints (MPEC). By smoothing the exact penalty function, an MPEC is reformulated as a series of subprograms which belong to a class of MPECs with simple linear complementarity constraints. To deal with the subproblems, a hybrid algorithm is proposed, which combines the active set algorithm, the 6-active search algorithm and the PSQP algorithm. It is shown that the smoothing penalty algorithm converges globally to a M-stationary point of MPEC under weak conditions. Supervisor: Dr. Jane Ye (Department of Mathematics and Statistics) Co-Supervisor: Dr. Wu-Sheng Lu (Department of Electrical and Computer Engineering)

Modeling sediment erosion is important in a wide range of environmental problems. The effects of various environmental factors on erosion rates have been studied, but the effects of surface slope on erosion rates of a wide range of sediments have not been quantified. The effects of surface slope, both in the direction of flow (pitch) and perpendicular to the flow (roll), on erosion rates of quartz particles were investigated using the Sediment Erosion at Depth Flume (Sedflume). / US Navy (USN) author.

<div>This PhD thesis is an important development in the theories, methods, and applications of eigenvalue algorithms for structured matrices. Though eigenvalue problems have been well-studied, the class of matrices that admit very fast (near-linear time) algorithms was quite small until very recently. We developed and implemented a generalization of the famous symmetric tridiagonal divide-and-conquer algorithm to a much larger class of rank structured matrices (symmetric hierarchically semisperable, or HSS) that appear frequently in applications. Altogether, this thesis makes valuable contributions to three different major areas of scientific computing: algorithmic development, numerical analysis, and applications. In addition to the previously stated divide-and-conquer algorithm, we generalize to larger classes of eigenvalue problems and provide several key new low-rank update algorithms. A major contribution the analysis of the structured eigenvalue problem. In addition to standard perturbation analysis, we elucidate some subtle and previously under-examined issues in structured matrix eigenvalue problems such as subspace contributions and secular equation conditioning. Finally, several applications are studied.</div>

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Deng, Xiaomao. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2010. / Includes bibliographical references (leaves 98-104). / Chapter 1 --- Introduction to Inverse Scattering Problems --- p.6 / Chapter 1.1 --- Direct Problems --- p.6 / Chapter 1.1.1 --- Far-field Patterns --- p.10 / Chapter 1.2 --- Inverse Problems --- p.16 / Chapter 1.2.1 --- Introduction --- p.16 / Chapter 2 --- Numerical Methods in Inverse Obstacle Scattering --- p.19 / Chapter 2.1 --- Linear Sampling Method --- p.19 / Chapter 2.1.1 --- History Review --- p.19 / Chapter 2.1.2 --- Numerical Scheme of LSM --- p.21 / Chapter 2.1.3 --- Theoretic Justification --- p.25 / Chapter 2.1.4 --- Summarize --- p.38 / Chapter 2.2 --- Point Source Method --- p.38 / Chapter 2.2.1 --- Historical Review --- p.38 / Chapter 2.2.2 --- Superposition of Plane Waves --- p.40 / Chapter 2.2.3 --- Approximation of Domains --- p.42 / Chapter 2.2.4 --- Algorithm --- p.44 / Chapter 2.2.5 --- Summarize --- p.49 / Chapter 2.3 --- Singular Source Method --- p.49 / Chapter 2.3.1 --- Historical Review --- p.49 / Chapter 2.3.2 --- Algorithm --- p.51 / Chapter 2.3.3 --- Far-field Data --- p.54 / Chapter 2.3.4 --- Summarize --- p.55 / Chapter 2.4 --- Probe Method --- p.57 / Chapter 2.4.1 --- Historical Review --- p.57 / Chapter 2.4.2 --- Needle --- p.58 / Chapter 2.4.3 --- Algorithm --- p.59 / Chapter 3 --- Numerical Experiments --- p.61 / Chapter 3.1 --- Discussions on Linear Sampling Method --- p.61 / Chapter 3.1.1 --- Regularization Strategy --- p.61 / Chapter 3.1.2 --- Cut off Value --- p.70 / Chapter 3.1.3 --- Far-field data --- p.76 / Chapter 3.2 --- Numerical Verification of PSM and SSM --- p.80 / Chapter 3.3 --- Inverse Medium Scattering --- p.83 / Bibliography --- p.98

Thesis (M. Phil.)--University of Hong Kong, 2006. / Title proper from title frame. Also available in printed format.

Thesis (Ph. D.)--Ohio State University, 2006. / Title from first page of PDF file. Includes bibliographical references (p. 137-143).

This thesis concerns the approximation of optimally controlled partial differential equations for inverse problems in optimal design. Important examples of such problems are optimal material design and parameter reconstruction. In optimal material design the goal is to construct a material that meets some optimality criterion, e.g. to design a beam, with fixed weight, that is as stiff as possible. Parameter reconstrucion concerns, for example, the problem to find the interior structure of a material from surface displacement measurements resulting from applied external forces. Optimal control problems, particularly for partial differential equations, are often ill-posed and need to be regularized to obtain good approximations. We here use the theory of the corresponding Hamilton-Jacobi-Bellman equations to construct regularizations and derive error estimates for optimal design problems. The constructed Pontryagin method is a simple and general method where the first, analytical, step is to regularize the Hamiltonian. Next its Hamiltonian system is computed efficiently with the Newton method using a sparse Jacobian. An error estimate for the difference between exact and approximate objective functions is derived, depending only on the difference of the Hamiltonian and its finite dimensional regularization along the solution path and its L² projection, i.e. not on the difference of the exact and approximate solutions to the Hamiltonian systems. Another treated issue is the relevance of input data for parameter reconstruction problems, where the goal is to determine a spacially distributed coefficient of a partial differential equation from partial observations of the solution. It is here shown that the choice of input data, that generates the partial observations, affects the reconstruction, and that it is possible to formulate meaningful optimality criteria for the input data that enhances the quality of the reconstructed coefficient. In the thesis we present solutions to various applications in optimal material design and reconstruction. / Denna avhandling handlar om approximation av optimalt styrda partiella differentialekvationer för inversa problem inom optimal design. Viktiga exempel på sådana problem är optimal materialdesign och parameterskattning. Inom materialdesign är målet att konstruera ett material som uppfyller vissa optimalitetsvillkor, t.ex. att konstruera en så styv balk som möjligt under en given vikt, medan ett exempel på parameterskattning är att hitta den inre strukturen hos ett material genom att applicera ytkrafter och mäta de resulterande förskjutningarna. Problem inom optimal styrning, speciellt för styrning av partiella differentialekvationer,är ofta illa ställa och måste regulariseras för att kunna lösas numeriskt. Teorin för Hamilton-Jacobi-Bellmans ekvationer används här för att konstruera regulariseringar och ge feluppskattningar till problem inom optimaldesign. Den konstruerade Pontryaginmetoden är en enkel och generell metod där det första analytiska steget är att regularisera Hamiltonianen. I nästa steg löses det Hamiltonska systemet effektivt med Newtons metod och en gles Jacobian. Vi härleder även en feluppskattning för skillnaden mellan den exakta och den approximerade målfunktionen. Denna uppskattning beror endast på skillnaden mellan den sanna och den regulariserade, ändligt dimensionella, Hamiltonianen, båda utvärderade längst lösningsbanan och dessL²-projektion. Felet beror alltså ej på skillnaden mellan den exakta och denapproximativa lösningen till det Hamiltonska systemet. Ett annat fall som behandlas är frågan hur indata ska väljas för parameterskattningsproblem. För sådana problem är målet vanligen att bestämma en rumsligt beroende koefficient till en partiell differentialekvation, givet ofullständiga mätningar av lösningen. Här visas att valet av indata, som genererarde ofullständiga mätningarna, påverkar parameterskattningen, och att det är möjligt att formulera meningsfulla optimalitetsvillkor för indata som ökar kvaliteten på parameterskattningen. I avhandlingen presenteras lösningar för diverse tillämpningar inom optimal materialdesign och parameterskattning. / QC 20100712

There are three distinct processes that are predominant in models of flowing media with interacting components: advection, reaction, and diffusion. Collectively, these processes are typically modelled with partial differential equations (PDEs) known as advection-reaction-diffusion (ARD) equations.<p> To solve most PDEs in practice, approximation methods known as numerical methods are used. The method of lines is used to approximate PDEs with systems of ordinary differential equations (ODEs) by a process known as semi-discretization. ODEs are more readily analysed and benefit from well-developed numerical methods and software. Each term of an ODE that corresponds to one of the processes of an ARD equation benefits from particular mathematical properties in a numerical method. These properties are often mutually exclusive for many basic numerical methods.<p> A limitation to the widespread use of more complex numerical methods is that the development of the appropriate software to provide comparisons to existing numerical methods is not straightforward. Scientific and numerical software is often inflexible, motivating the development of a class of software known as problem-solving environments (PSEs). Many existing PSEs such as Matlab have solvers for ODEs and PDEs but lack specific features, beyond a scripting language, to readily experiment with novel or existing solution methods. The PSE developed during the course of this thesis solves ODEs known as initial-value problems, where only the initial state is fully known. The PSE is used to assess the performance of new numerical methods for ODEs that integrate each term of a semi-discretized ARD equation. This PSE is part of the PSE pythODE that uses object-oriented and software-engineering techniques to allow implementations of many existing and novel solution methods for ODEs with minimal effort spent on code modification and integration.<p> The new numerical methods use a commutator-free exponential Runge-Kutta (CFERK) method to solve the advection term of an ARD equation. A matrix exponential is used as the exponential function, but CFERK methods can use other numerical methods that model the flowing medium. The reaction term is solved separately using an explicit Runge-Kutta method because solving it along with the diffusion term can result in stepsize restrictions and hence inefficiency. The diffusion term is solved using a Runge-Kutta-Chebyshev method that takes advantage of the spatially symmetric nature of the diffusion process to avoid stepsize restrictions from a property known as stiffness. The resulting methods, known as Integrating-factor-based 2-additive Runge-Kutta methods, are shown to be able to find higher-accuracy solutions in less computational time than competing methods for certain challenging semi-discretized ARD equations. This demonstrates the practical viability both of using CFERK methods for advection and a 3-splitting in general.