A generalized Conjugate Gradient like method is used to solve the linear systems of equations formed at each time-integration step of the unsteady, two-dimensional, compressible Navier-Stokes equations of fluid flow. The Navier-Stokes equations are cast in an implicit, upwind finite-volume, flux split formulation. Preconditioning techniques are employed with the Conjugate Gradient like method to enhance the stability and convergence rate of the overall iterative method. The superiority of the new solver is established by comparisons with a conventional Line GaussSeidel Relaxation (LGSR) solver. Comparisons are based on 'number of iterations required to converge to a steady-state solution' and 'total CPU time required for convergence'. Three test cases representing widely varying flow physics are chosen to investigate the performance of the solvers. Computational test results for very low speed (incompressible flow over a backward facing step at Mach 0.1), transonic flow (trailing edge flow in a transonic turbine cascade) and hypersonic flow (shockon- shock interactions on a cylindrical leading edge at Mach 6.0) are presented. For the 1vfach 0.1 case, speed-up factors of 30 (in terms of iterations) and 20 (in terms of CPU time) are found in favor of the new solver when compared with the LGSR solver. The corresponding speed-up factors for the transonic flow case are 20 and 18, respectively. The hypersonic case shows relatively lower speed-up factors of 5 and 4, respectively. This study reveals that preconditioning can greatly enhance the range of applicability and improve the performance of Conjugate Gradient like methods. / Ph. D.

Over the last 25 years, interior-point methods (IPMs) have emerged as a viable class of algorithms for solving various forms of conic optimization problems. Most IPMs use a modified Newton method to determine the search direction at each iteration. The system of equations corresponding to the modified Newton system can often be reduced to the so-called normal equation, a system of equations whose matrix ADA' is positive definite, yet often ill-conditioned. In this thesis, we first investigate the theoretical properties of the maximum weight basis (MWB) preconditioner, and show that when applied to a matrix of the form ADA', where D is positive definite and diagonal, the MWB preconditioner yields a preconditioned matrix whose condition number is uniformly bounded by a constant depending only on A. Next, we incorporate the results regarding the MWB preconditioner into infeasible, long-step, primal-dual, path-following algorithms for linear programming (LP) and convex quadratic programming (CQP). In both LP and CQP, we show that the number of iterative solver iterations of the algorithms can be uniformly bounded by n and a condition number of A, while the algorithmic iterations of the IPMs can be polynomially bounded by n and the logarithm of the desired accuracy. We also expand the scope of the LP and CQP algorithms to incorporate a family of preconditioners, of which MWB is a member, to determine an approximate solution to the normal equation. For the remainder of the thesis, we develop a new preconditioning strategy for solving systems of equations whose associated matrix is positive definite but ill-conditioned. Our so-called adaptive preconditioning strategy allows one to change the preconditioner during the course of the conjugate gradient (CG) algorithm by post-multiplying the current preconditioner by a simple matrix, consisting of the identity matrix plus a rank-one update. Our resulting algorithm, the Adaptive Preconditioned CG (APCG) algorithm, is shown to have polynomial convergence properties. Numerical tests are conducted to compare a variant of the APCG algorithm with the CG algorithm on various matrices.

No description available.

Shape optimization is an important step in many design processes. With the growing use of Computer Aided Engineering in the design chain, it has become very important to develop robust and efficient shape optimization algorithms. The field of Computer Aided Optimal Shape Design has grown substantially over the recent past. In the early days of its development, the method based on small shape perturbation to probe the parameter space and identify an optimal shape was routinely used. This method is nothing but an educated trial and error method. A key development in the pursuit of good shape optimization algorithms has been the advent of the adjoint method to compute the shape sensitivities more formally and efficiently. While undoubtedly, very attractive, this method relies on very sophisticated and advanced mathematical tools which are an impediment to its wider use in the engineering community. It that spirit, it is the purpose of this thesis to propose a new shape optimization algorithm based on more intuitive engineering principles and numerical procedures. In this thesis, the new shape optimization procedure which is proposed is based on the generation of a body-fitted mesh. This process maps the physical domain into a regular computational domain. Based on simple arguments relating to the use of the chain rule in the mapped domain, it is shown that an explicit expression for the shape sensitivity can be derived. This enables the computation of the shape sensitivity in one single solve, a performance analogous to the adjoint method, the current state-of-the art. The discretization is based on the Finite Difference method, a method chosen for its simplicity and ease of implementation. This algorithm is applied to the Laplace equation in the context of heat transfer problems and potential flows. The applicability of the proposed algorithm is demonstrated on a number of benchmark problems which clearly confirm the validity of the sensitivity analysis, the most important aspect of any shape optimization problem. This thesis also explores the relative merits of different minimization algorithms and proposes a technique to “fix” meshes when inverted element arises as part of the optimization process. While the problems treated are still elementary when compared to complex multiphysics engineering problems, the new methodology presented in this thesis could apply in principle to arbitrary Partial Differential Equations.

We present a new method for regularization of ill-conditioned problems that extends the traditional trust-region approach. Ill-conditioned problems arise, for example, in image restoration or mathematical processing of medical data, and involve matrices that are very ill-conditioned. The method makes use of the L-curve and L-curve maximum curvature criterion as a strategy recently proposed to find a good regularization parameter. We describe the method and show its application to an image restoration problem. We also provide a MATLAB code for the algorithm. Finally, a comparison to the CGLS approach is given and analyzed, and future research directions are proposed.

Kwok-po Ng. / Thesis (Ph.D.)--Chinese University of Hong Kong, 1995. / Includes bibliographical references (leaves 138-139). / Abstracts / Summary / Introduction --- p.1 / Summary of the papers A-F --- p.2 / Paper A --- p.15 / Paper B --- p.36 / Paper C --- p.63 / Paper D --- p.87 / Paper E --- p.109 / Paper F --- p.122

[pt] Muitos problemas provenientes do mundo real podem ser modelados por sistemas de equações diferenciais parciais (EDP´s). No entanto, as equações resultantes da discretização produzem matrizes grandes e freqüentementes mal condicionadas. Este trabaho implementa o método de elementos finitos mistos para resolver numericamente um sistema de EDP´s oriundo de um modelo de escoamento de fluidos em meios porosos e melhora sua performance usando precondicionadores e processamento paralelo. / [en] Many problems arising from real world can be represented by systems of partial diferential equations (PDE´s). However, the resulting discrete equations produce large and frequently bad conditioned matrices. This work implements the mixed finite element method to numerically solve a system of PDE´s coming from a multiphase flow in porous media model and improve its performance by preconditioners and parallel processing.

by Wun-Cheung Tang. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1998. / Includes bibliographical references. / Abstract also in Chinese. / Abstracts --- p.1 / Introduction --- p.3 / Toeplitz and Circulant Matrices --- p.3 / Conjugate Gradient Method --- p.6 / Cosine Transform Preconditioner --- p.7 / Regularization --- p.10 / Summary --- p.13 / Paper A --- p.19 / Paper B --- p.36

<p>Vid analys av vattenprover tagna från t.ex. ett vattendrag betäms halten av olika ämnen. Dessa halter är ofta beroende av vattenföringen. Det är av intresse att ta reda på om observerade förändringar i halterna beror på naturliga variationer eller är orsakade av andra faktorer. För att undersöka detta har föreslagits en statistisk tidsseriemodell som innehåller okända parametrar. Modellen anpassas till uppmätta data vilket leder till ett underbestämt ekvationssystem. I avhandlingen studeras bl.a. olika sätt att säkerställa en unik och rimlig lösning. Grundidén är att införa vissa tilläggsvillkor på de sökta parametrarna. I den studerade modellen kan man t.ex. kräva att vissa parametrar inte varierar kraftigt med tiden men tillåter årstidsvariationer. Det görs genom att dessa parametrar i modellen regulariseras.</p><p>Detta ger upphov till ett minsta kvadratproblem med en eller två regulariseringsparametrar. I och med att inte alla ingående parametrar regulariseras får vi dessutom ett partiellt regulariserat minsta kvadratproblem. I allmänhet känner man inte värden på regulariseringsparametrarna utan problemet kan behöva lösas med flera olika värden på dessa för att få en rimlig lösning. I avhandlingen studeras hur detta problem kan lösas numeriskt med i huvudsak två olika metoder, en iterativ och en direkt metod. Dessutom studeras några sätt att bestämma lämpliga värden på regulariseringsparametrarna.</p><p>I en iterativ lösningsmetod förbättras stegvis en given begynnelseapproximation tills ett lämpligt valt stoppkriterium blir uppfyllt. Vi använder här konjugerade gradientmetoden med speciellt konstruerade prekonditionerare. Antalet iterationer som krävs för att lösa problemet utan prekonditionering och med prekonditionering jämförs både teoretiskt och praktiskt. Metoden undersöks här endast med samma värde på de två regulariseringsparametrarna.</p><p>I den direkta metoden används QR-faktorisering för att lösa minsta kvadratproblemet. Idén är att först utföra de beräkningar som kan göras oberoende av regulariseringsparametrarna samtidigt som hänsyn tas till problemets speciella struktur.</p><p>För att bestämma värden på regulariseringsparametrarna generaliseras Reinsch’s etod till fallet med två parametrar. Även generaliserad korsvalidering och en mindre beräkningstung Monte Carlo-metod undersöks.</p> / <p>Statistical analysis of data from rivers deals with time series which are dependent, e.g., on climatic and seasonal factors. For example, it is a well-known fact that the load of substances in rivers can be strongly dependent on the runoff. It is of interest to find out whether observed changes in riverine loads are due only to natural variation or caused by other factors. Semi-parametric models have been proposed for estimation of time-varying linear relationships between runoff and riverine loads of substances. The aim of this work is to study some numerical methods for solving the linear least squares problem which arises.</p><p>The model gives a linear system of the form <em>A</em><em>1x1</em><em> + A</em><em>2x2</em><em> + n = b</em><em>1</em>. The vector <em>n</em> consists of identically distributed random variables all with mean zero. The unknowns, <em>x,</em> are split into two groups, <em>x</em><em>1</em><em> </em>and <em>x</em><em>2</em><em>.</em> In this model, usually there are more unknowns than observations and the resulting linear system is most often consistent having an infinite number of solutions. Hence some constraint on the parameter vector x is needed. One possibility is to avoid rapid variation in, e.g., the parameters<em> x</em><em>2</em><em>.</em> This can be accomplished by regularizing using a matrix <em>A</em><em>3</em>, which is a discretization of some norm. The problem is formulated</p><p>as a partially regularized least squares problem with one or two regularization parameters. The parameter <em>x</em><em>2</em> has here a two-dimensional structure. By using two different regularization parameters it is possible to regularize separately in each dimension.</p><p>We first study (for the case of one parameter only) the conjugate gradient method for solution of the problem. To improve rate of convergence blockpreconditioners of Schur complement type are suggested, analyzed and tested. Also a direct solution method based on QR decomposition is studied. The idea is to first perform operations independent of the values of the regularization parameters. Here we utilize the special block-structure of the problem. We further discuss the choice of regularization parameters and generalize in particular Reinsch’s method to the case with two parameters. Finally the cross-validation technique is treated. Here also a Monte Carlo method is used by which an approximation to the generalized cross-validation function can be computed efficiently.</p>

Vid analys av vattenprover tagna från t.ex. ett vattendrag betäms halten av olika ämnen. Dessa halter är ofta beroende av vattenföringen. Det är av intresse att ta reda på om observerade förändringar i halterna beror på naturliga variationer eller är orsakade av andra faktorer. För att undersöka detta har föreslagits en statistisk tidsseriemodell som innehåller okända parametrar. Modellen anpassas till uppmätta data vilket leder till ett underbestämt ekvationssystem. I avhandlingen studeras bl.a. olika sätt att säkerställa en unik och rimlig lösning. Grundidén är att införa vissa tilläggsvillkor på de sökta parametrarna. I den studerade modellen kan man t.ex. kräva att vissa parametrar inte varierar kraftigt med tiden men tillåter årstidsvariationer. Det görs genom att dessa parametrar i modellen regulariseras. Detta ger upphov till ett minsta kvadratproblem med en eller två regulariseringsparametrar. I och med att inte alla ingående parametrar regulariseras får vi dessutom ett partiellt regulariserat minsta kvadratproblem. I allmänhet känner man inte värden på regulariseringsparametrarna utan problemet kan behöva lösas med flera olika värden på dessa för att få en rimlig lösning. I avhandlingen studeras hur detta problem kan lösas numeriskt med i huvudsak två olika metoder, en iterativ och en direkt metod. Dessutom studeras några sätt att bestämma lämpliga värden på regulariseringsparametrarna. I en iterativ lösningsmetod förbättras stegvis en given begynnelseapproximation tills ett lämpligt valt stoppkriterium blir uppfyllt. Vi använder här konjugerade gradientmetoden med speciellt konstruerade prekonditionerare. Antalet iterationer som krävs för att lösa problemet utan prekonditionering och med prekonditionering jämförs både teoretiskt och praktiskt. Metoden undersöks här endast med samma värde på de två regulariseringsparametrarna. I den direkta metoden används QR-faktorisering för att lösa minsta kvadratproblemet. Idén är att först utföra de beräkningar som kan göras oberoende av regulariseringsparametrarna samtidigt som hänsyn tas till problemets speciella struktur. För att bestämma värden på regulariseringsparametrarna generaliseras Reinsch’s etod till fallet med två parametrar. Även generaliserad korsvalidering och en mindre beräkningstung Monte Carlo-metod undersöks. / Statistical analysis of data from rivers deals with time series which are dependent, e.g., on climatic and seasonal factors. For example, it is a well-known fact that the load of substances in rivers can be strongly dependent on the runoff. It is of interest to find out whether observed changes in riverine loads are due only to natural variation or caused by other factors. Semi-parametric models have been proposed for estimation of time-varying linear relationships between runoff and riverine loads of substances. The aim of this work is to study some numerical methods for solving the linear least squares problem which arises. The model gives a linear system of the form A1x1 + A2x2 + n = b1. The vector n consists of identically distributed random variables all with mean zero. The unknowns, x, are split into two groups, x1 and x2. In this model, usually there are more unknowns than observations and the resulting linear system is most often consistent having an infinite number of solutions. Hence some constraint on the parameter vector x is needed. One possibility is to avoid rapid variation in, e.g., the parameters x2. This can be accomplished by regularizing using a matrix A3, which is a discretization of some norm. The problem is formulated as a partially regularized least squares problem with one or two regularization parameters. The parameter x2 has here a two-dimensional structure. By using two different regularization parameters it is possible to regularize separately in each dimension. We first study (for the case of one parameter only) the conjugate gradient method for solution of the problem. To improve rate of convergence blockpreconditioners of Schur complement type are suggested, analyzed and tested. Also a direct solution method based on QR decomposition is studied. The idea is to first perform operations independent of the values of the regularization parameters. Here we utilize the special block-structure of the problem. We further discuss the choice of regularization parameters and generalize in particular Reinsch’s method to the case with two parameters. Finally the cross-validation technique is treated. Here also a Monte Carlo method is used by which an approximation to the generalized cross-validation function can be computed efficiently.