In current economic research, there are many problems that are difficult to solve without powerful computers, unique software, or novel approaches. I wrote this thesis because I believe that a powerful solution technique known as the Riccati Iteration is such a novel approach, and can be applied to complex problems that would otherwise be infeasible to solve. This thesis will demonstrate the power of the Riccati iteration by employing the Riccati iteration with spreadsheet software to solve a difficult dynamic optimization problem - a capital replacement problem posed by Lotfi where multiple objectives have been identified. The Riccati iteration will be shown to be the most practicable method for solving this problem, especially when compared to the Lagrange and Least-Squares solution methods. It is hoped that the demonstration in this thesis is so compelling that others may consider using the Riccati approach in their own research. / Master of Arts

The use of Robin boundary conditions as interfacial transmission conditions in nonoverlapping domain decomposition iterative procedures was introduced by P.~L.~Lions and later discussed by a number of authors. In all of these discussions, the weighting of the flux and the trace of the solution were independent of the iterative step number. Recently, Douglas and Huang have considered an extension in which the weighting has depended on the iterative index and proved that an acceleration in the convergence rate similar to that occurring for alternating-direction iteration using a cycle of pseudo-time steps results. Combined methods have received a lot of attention on the study of the boundary value problems of elliptic equation with singularities domain. The objects of this paper are to extend this accelerated procedure to a singularity problem on a sectorial domain. We show that the convergence rate similar to that occurring for alternating-direction iteration results. Also, a combined method and the methodology of weighting parameter can be used to solve a singularity problem.

Inom nuklearmedicinsk diagnostik används radioaktiva läkemedel för att utvärdera olika organs metabolism och fysiologiska beteende. Genom att använda en scintillationskamera kan strålningen som erhålls när det administrerade läkemedlet sönderfaller i kroppen detekteras och en funktionell bild över aktivitetsfördelningen erhålls. En tredimensionell bildvolym kan erhållas om gammakameran får rotera runt patienten vilket kallas SPECT (Single Photon Emission Computed Tomography). Bildernas kvalitet är av stor betydelse för att kunna göra en noggrann bedömning av olika patologiska tillstånd. Kvaliteten begränsas av en mängd faktorer och en av dem är Comptonspridda fotoner. I denna studie optimerades bildkvaliteten för SPECT-undersökningar med 111In-Octreoscan för ett Infinia Hawkeye 4 (GE Healthcare, Wisconsin, USA) SPECT-system vid Norrlands universitetssjukhus (NUS). Optimeringen gjordes med avseende på detektion av små tumörer vid visuell inspektion av 111In-Octreoscan-bilder. Monte Carlo simuleringar användes för att utvärdera tre olika parallellhålskollimatorer med fyra olika fönsterinställningar. Rekonstruktion av bilder gjordes med den iterativa tekniken OSEM (Ordered Subset Expectation Maximization) med olika antal iterationer. Bilderna postfiltrerades med två olika filter med tre kritiska frekvenser vardera. Den lämpligaste inställningen för NUS visade sig vara MEGP-kollimatorn (Medium Energy General Purpose) tillsammans med en fönsterinställning med två huvudfönster centrerade kring 171 respektive 245 keV utan spridningskorrektion. Mest optimal rekonstruktion visade sig vara med två OSEM-iterationer (10 subsets) och postfiltrering med ett Butterworth-filter med kritisk frekvens 0.40 cm-1 och powerfaktor 8. I övrigt visade sig ELEGP-kollimatorn (Extended Low Energy General Purpose) vara den kollimator som optimerar bildkvaliteten mest med avseende på detektion av små tumörer, men den finns ännu inte på NUS. / Optimization of image quality for SPECT imaging with 111In-Octreoscan at the University Hospital of Umeå – a Monte Carlo study In nuclear medicine diagnostics, radiopharmaceuticals are used for evaluating metabolism and physiological behavior of various organs. By using a scintillation camera, radiation can be detected when the administered drug decays in the body, and the result is a functional image of the activity distribution within the patient. A three-dimensional image volume can be obtained by letting the gamma camera rotate around the patient. This method is called SPECT (Single Photon Emission Computed Tomography). Image quality is very important to make an accurate assessment of various pathological conditions. The quality is limited by many factors and one of them is the Compton scattered photons. In this study image quality of SPECT-examinations with 111In-Octreoscan were optimized for an Infinia Hawkeye 4 (GE Healthcare, Wisconsin, USA) SPECT-system at the University Hospital of Umeå (NUS). The optimization was made with respect to detecting small tumors for visual inspection of 111In-Octreoscan images. Monte Carlo simulations were used to evaluate three different parallel hole collimators with four different window settings. Reconstruction of images was performed with the iterative technique OSEM (Ordered Subset Expectation Maximization) with different numbers of iterations. The images were post-filtered with two different filters with three critical frequencies each. The most appropriate setting for the SPECT-system at NUS is the MEGP-collimator (Medium Energy General Purpose) with a window setting of two main windows centered around 171 and 245 keV, without scatter correction. The most optimal reconstruction is obtained by using two OSEM-iterations (10 subsets) and post-filtering with a Butterworth-filter with critical frequency 0.40 cm-1 and power factor 8. The ELEGP-collimator (Extended Low Energy General Purpose) proved however to be the most optimal collimator for detecting small tumors, but this collimator is currently not available at NUS.

Universiẗat, Diss.--Erlangen-Nürnberg, 2007.

Techn. Universiẗat, Diss., 2005--Berlin.

This paper introduces the hypergraph transversal problem along with thefollowing iterative solutions: naive, branch and bound, and dynamic exponentialtime (NC-D). Odometers are introduced along with the functions that manipulatethem. The traditional definitions of hyperedge, hypergraph, etc., are redefined interms of odometers and lists. All algorithms and functions necessary to implementthe solution are presented along with techniques to validate and test the results.Lastly, parallelization advanced applications, and future research directions areexamined.

The modeling of nuclear reactors involves the solution of a multi-physics problem with widely varying time and length scales. This translates mathematically to solving a system of coupled, non-linear, and stiff partial differential equations (PDEs). Multi-physics applications possess the added complexity that most of the solution fields participate in various physics components, potentially yielding spatial and/or temporal coupling errors. This dissertation deals with the verification aspects associated with such a multi-physics code, i.e., the substantiation that the mathematical description of the multi-physics equations are solved correctly (both in time and space). Conventional paradigms used in reactor analysis problems employed to couple various physics components are often non-iterative and can be inconsistent in their treatment of the non-linear terms. This leads to the usage of smaller time steps to maintain stability and accuracy requirements, thereby increasing the overall computational time for simulation. The inconsistencies of these weakly coupled solution methods can be overcome using tighter coupling strategies and yield a better approximation to the coupled non-linear operator, by resolving the dominant spatial and temporal scales involved in the multi-physics simulation. A multi-physics framework, KARMA (K(c)ode for Analysis of Reactor and other Multi-physics Applications), is presented. KARMA uses tight coupling strategies for various physical models based on a Matrix-free Nonlinear-Krylov (MFNK) framework in order to attain high-order spatio-temporal accuracy for all solution fields in amenable wall clock times, for various test problems. The framework also utilizes traditional loosely coupled methods as lower-order solvers, which serve as efficient preconditioners for the tightly coupled solution. Since the software platform employs both lower and higher-order coupling strategies, it can easily be used to test and evaluate different coupling strategies and numerical methods and to compare their efficiency for problems of interest. Multi-physics code verification efforts pertaining to reactor applications are described and associated numerical results obtained using the developed multi-physics framework are provided. The versatility of numerical methods used here for coupled problems and feasibility of general non-linear solvers with appropriate physics-based preconditioners in the KARMA framework offer significantly efficient techniques to solve multi-physics problems in reactor analysis.

No description available.

Using the idea that a quantum mechanical system drops to its ground state as its temperature goes to absolute zero several operators are devised to enable the approximation of the lowest order energy eigenstate of a given symmetry; as well as an approximation to the energy eigenvalue of the same order.

Ill-posed sets of linear equations typically arise when discretizing certain types of integral transforms. A well known example is image reconstruction, which can be modeled using the Radon transform. After expanding the solution into a finite series of basis functions a large, sparse and ill-conditioned linear system occurs. We consider the solution of such systems. In particular we study a new class of iteration methods named DROP (for Diagonal Relaxed Orthogonal Projections) constructed for solving both linear equations and linear inequalities. This class can also be viewed, when applied to linear equations, as a generalized Landweber iteration. The method is compared with other iteration methods using test data from a medical application and from electron microscopy. Our theoretical analysis include convergence proofs of the fully-simultaneous DROP algorithm for linear equations without consistency assumptions, and of block-iterative algorithms both for linear equations and linear inequalities, for the consistent case. When applying an iterative solver to an ill-posed set of linear equations the error usually initially decreases but after some iterations, depending on the amount of noise in the data, and the degree of ill-posedness, it starts to increase. This phenomenon is called semi-convergence. We study the semi-convergence performance of Landweber-type iteration, and propose new ways to specify the relaxation parameters. These are computed so as to control the propagated error. We also describe a class of stopping rules for Landweber-type iteration for solving linear inverse problems. The class includes the well known discrepancy principle, and the monotone error rule. We unify the error analysis of these two methods. The stopping rules depend critically on a certain parameter whose value needs to be specified. A training procedure is therefore introduced for securing robustness. The advantages of using trained rules are demonstrated on examples taken from image reconstruction from projections. Kaczmarz's method, also called ART (Algebraic Reconstruction Technique) is often used for solving the linear system which appears in image reconstruction. This is a fully sequential method. We examine and compare ART and its symmetric version. It is shown that the cycles of symmetric ART, unlike ART, converge to a weighted least squares solution if and only if the relaxation parameter lies between zero and two. Further we show that ART has faster asymptotic rate of convergence than symmetric ART. Also a stopping criterion is proposed and evaluated for symmetric ART. We further investigate a class of block-iterative methods used in image reconstruction. The cycles of the iterative sequences are characterized in terms of the original linear system. We define symmetric block-iteration and compare the behavior of symmetric and non-symmetric block-iteration. The results are illustrated using some well-known methods. A stopping criterion is offered and assessed for symmetric block-iteration.