An expansion method for the stationary Schrodinger equation of a particle moving freely in an arbitrary axisymmeric three dimensional region defined by an analytic function is introduced. The region is transformed into the unit ball by means of coordinate substitution. As a result the Schrodinger equation is considerably changed. The wavefunction is expanded into a series of spherical harmonics, thus, reducing the transformed partial differential equation to an infinite system of coupled ordinary differential equations. A Fourier-Bessel expansion of the solution vector in terms of Bessel functions with real orders is employed, resulting in a generalized matrix eigenvalue problem. The method is applied to two particular examples. The first example is a prolate spheroidal billiard which is also treated by using an alternative method. The numerical results obtained by using both the methods are compared. The second exampleis a billiard family depending on a parameter. Numerical results concerning the second example include the statistical analysis of the eigenvalues.

Diss. Uppsala : Univ., 2002.

Differential equations arise frequently in modeling of physical systems, often resulting in linear eigenvalue problems. However, when dealing with large physical domains, solving such problems can be computationally expensive. This thesis examines an alternative approach to solving these problems, which involves utilizing absorbing boundary conditions and a Dirichlet-to-Neumann maps to transform the large sparse linear eigenvalue problem into a smaller nonlinear eigenvalue problem (NEP). The NEP is then solved using augmented Newton’s method. The specific equation investigated in this thesis is the two-dimensional Helmholtz equation, defined on the interval (x, y) ∈ [0, 10] × [0, 1], with the absorbing boundary condition introduced at x = 1. The results show a significant reduction in computational time when using this method compared to the original linear problem, making it a valuable tool for solving large linear eigenvalue problems. Another result is that the NEP does not affect the computational error compared to solving the linear problem, which further supports the NEP as an attractive alternative method.

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Sequences of matrices arise in many applications in science and engineering. In this thesis we consider matrices that are closely related (or closely related in groups), and we take advantage of the small differences between them to efficiently solve sequences of linear systems and eigenproblems. Recycling techniques, such as recycling preconditioners or subspaces, are popular approaches for reducing computational cost. In this thesis, we introduce two novel approaches for recycling previously computed information for a subsequent system or eigenproblem, and demonstrate good results for sequences arising in several applications. Preconditioners are often essential for fast convergence of iterative methods. However, computing a good preconditioner can be very expensive, and when solving a sequence of linear systems, we want to avoid computing a new preconditioner too often. Instead, we can recycle a previously computed preconditioner, for which we have good convergence behavior of the preconditioned system. We propose an update technique we call the sparse approximate map, or SAM update, that approximately maps one matrix to another matrix in our sequence. SAM updates are very cheap to compute and apply, preserve good convergence properties of a previously computed preconditioner, and help to amortize the cost of that preconditioner over many linear solves. When solving a sequence of eigenproblems, we can reduce the computational cost of constructing the Krylov space starting with a single vector by warm-starting the eigensolver with a subspace instead. We propose an algorithm to warm-start the Krylov-Schur method using a previously computed approximate invariant subspace. We first compute the approximate Krylov decomposition for a matrix with minimal residual, and use this space to warm-start the eigensolver. We account for the residual matrix when expanding, truncating, and deflating the decomposition and show that the norm of the residual monotonically decreases. This method is effective in reducing the total number of matrix-vector products, and computes an approximate invariant subspace that is as accurate as the one computed with standard Krylov-Schur. In applications where the matrix-vector products require an implicit linear solve, we incorporate Krylov subspace recycling. Finally, in many applications, sequences of matrices take the special form of the sum of the identity matrix, a very low-rank matrix, and a small-in-norm matrix. We consider convergence rates for GMRES applied to these matrices by identifying the sources of sensitivity. / Doctor of Philosophy / Problems in science and engineering often require the solution to many linear systems, or a sequence of systems, that model the behavior of physical phenomena. In order to construct highly accurate mathematical models to describe this behavior, the resulting matrices can be very large, and therefore the linear system can be very expensive to solve. To efficiently solve a sequence of large linear systems, we often use iterative methods, which can require preconditioning techniques to achieve fast convergence. The preconditioners themselves can be very expensive to compute. So, we propose a cheap update technique that approximately maps one matrix to another in the sequence for which we already have a good preconditioner. We then combine the preconditioner and the map and use the updated preconditioner for the current system. Sequences of eigenvalue problems also arise in many scientific applications, such as those modeling disk brake squeal in a motor vehicle. To accurately represent this physical system, large eigenvalue problems must be solved. The behavior of certain eigenvalues can reveal instability in the physical system but to identify these eigenvalues, we must solve a sequence of very large eigenproblems. The eigensolvers used to solve eigenproblems generally begin with a single vector, and instead, we propose starting the method with several vectors, or a subspace. This allows us to reduce the total number of iterations required by the eigensolver while still producing an accurate solution. We demonstrate good results for both of these approaches using sequences of linear systems and eigenvalue problems arising in several real-world applications. Finally, in many applications, sequences of matrices take the special form of the sum of the identity matrix, a very low-rank matrix, and a small-in-norm matrix. We examine the convergence behavior of the iterative method GMRES when solving such a sequence of matrices.

This work addresses two primary topics related to vibrations in structures. The first topic is the use of a spatially distributed sensor network for localization of vibration events. I use a received signal strength (RSS) framework that presumes exponential energy decay with distance to the source. I derive the Cramér-Rao bound (CRB) for this parameter estimation problem, with the unknown parameters being source location, source intensity, and the energy dissipation rate. In this framework, I show that the CRB matches the variance of maximum likelihood estimators (MLEs) in more computationally expensive Monte Carlo trials. I also compare the CRB to the results of physical experiments to test the power of the CRB to predict spatial areas where MLEs show practical evidence of being ill-conditioned. Supported by this evidence, I recommend the CRB as a simple measure of localization accuracy, which may be used to optimize sensor layouts before installation. I demonstrate how this numerical optimization may be performed for some regions of interest with simple geometries. The second topic investigates modal vibrations of multi-body structures built from simple one-dimensional elements, with networks of elastic strings as the primary example. I introduce a method of using a nonlinear eigenvalue problem (NLEVP) to express boundary conditions of the vibrating elements so that the (infinitely many) eigenvalues of the full structure are the eigenvalues of the finite-dimensional NLEVP. The mode shapes of the structure can then be recovered in analytic form (not as a discretization) from the corresponding eigenvectors of the NLEVP. I show some advantages of this method over dynamic stiffness matrices, which is another NLEVP framework for modal analysis. In numerical experiments, I test several contour integration solvers for NLEVPs on sample problems generated from string networks. / Doctor of Philosophy / This work deals with two primary topics related to vibrations in structures. The first topic is the use of vibration sensors to detect movement or impact and to estimate the location of the detected event. Sensors that are close to the event will record a larger amount of energy than the sensors that are farther away, so comparing the signals of several sensors can approximately establish the event location. In this way, vibration sensors might be used to monitor activity in a building without the use of intrusive cameras. The accuracy of location estimates can be greatly affected by the relative positions of the sensors and the event. Generally, location estimates tend to be most accurate if the sensors closely surround the event, and less accurate if the event is outside of the sensor zone. These principles are useful, but not precise. Given a framework for how event energy and noise are picked up by the sensors, the Cramér-Rao bound (CRB) is a formula for the achievable accuracy of location estimates. I demonstrate that the CRB is usefully similar to the location estimate accuracy from experimental data collected from a volunteer walking through a sensor-rigged hallway. I then show how CRB computations may be used to find an optimal arrangement of sensors. The match between the CRB and the accuracy of the experiments suggests that the sensor layout that optimizes the CRB will also provide accurate location estimates in a real building. The other main topic is how the vibrations of a structure can be understood through the structure's natural vibration frequencies and corresponding vibration shapes, called the "modes" of the structure. I connect vibration modes to the abstract framework of "nonlinear eigenvalue problems" (NLEVPs). An NLEVP is a square matrix-valued function for which one wants to find the inputs that make the matrix singular. But these singular matrices are usually isolated---% distributed among the infinitely many matrices of the NLEVP in places that are difficult to predict. After discussing NLEVPs in general and some methods for solving them, I show how the vibration modes of certain structures can be represented by the solutions of NLEVPs. The structures I analyze are multi-body structures that are made of simple interconnected pieces, such as elastic strings strung together into a spider web. Once a multi-body structure has been cast into the NLEVP form, an NLEVP solver can be used to find the vibration modes. Finally, I demonstrate that this method can be computationally faster than many traditional modal analysis techniques.

Dynamical systems are a commonly used and studied tool for simulation, optimization and design. In many applications such as inverse problem, optimal control, shape optimization and uncertainty quantification, those systems typically depend on a parameter. The need for high fidelity in the modeling stage leads to large-scale parametric dynamical systems. Since these models need to be simulated for a variety of parameter values, the computational burden they incur becomes increasingly difficult. To address these issues, parametric reduced models have encountered increased popularity in recent years. We are interested in constructing parametric reduced models that represent the full-order system accurately over a range of parameters. First, we define a global joint error mea- sure in the frequency and parameter domain to assess the accuracy of the reduced model. Then, by assuming a rational form for the reduced model with poles both in the frequency and parameter domain, we derive necessary conditions for an optimal parametric reduced model in this joint error measure. Similar to the nonparametric case, Hermite interpolation conditions at the reflected images of the poles characterize the optimal parametric approxi- mant. This result extends the well-known interpolatory H2 optimality conditions by Meier and Luenberger to the parametric case. We also develop a numerical algorithm to construct locally optimal reduced models. The theory and algorithm are data-driven, in the sense that only function evaluations of the parametric transfer function are required, not access to the internal dynamics of the full model. While this first framework operates on the continuous function level, assuming repeated transfer function evaluations are available, in some cases merely frequency samples might be given without an option to re-evaluate the transfer function at desired points; in other words, the function samples in parameter and frequency are fixed. In this case, we construct a parametric reduced model that minimizes a discretized least-squares error in the finite set of measurements. Towards this goal, we extend Vector Fitting (VF) to the parametric case, solving a global least-squares problem in both frequency and parameter. The output of this approach might lead to a moderate size reduced model. In this case, we perform a post- processing step to reduce the output of the parametric VF approach using H2 optimal model reduction for a special parametrization. The final model inherits the parametric dependence of the intermediate model, but is of smaller order. A special case of a parameter in a dynamical system is a delay in the model equation, e.g., arising from a feedback loop, reaction time, delayed response and various other physical phenomena. Modeling such a delay comes with several challenges for the mathematical formulation, analysis, and solution. We address the issue of transient behavior for scalar delay equations. Besides the choice of an appropriate measure, we analyze the impact of the coefficients of the delay equation on the finite time growth, which can be arbitrary large purely by the influence of the delay. / Ph. D.

A Variational Transport Theory Method for Two-Dimensional Reactor Core Calculations Scott W. Mosher 110 Pages Directed by Dr. Farzad Rahnema It seems very likely that the next generation of reactor analysis methods will be based largely on neutron transport theory, at both the assembly and core levels. Signifi-cant progress has been made in recent years toward the goal of developing a transport method that is applicable to large, heterogeneous coarse-meshes. Unfortunately, the ma-jor obstacle hindering a more widespread application of transport theory to large-scale calculations is still the computational cost. In this dissertation, a variational heterogeneous coarse-mesh transport method has been extended from one to two-dimensional Cartesian geometry in a practical fashion. A generalization of the angular flux expansion within a coarse-mesh was developed. This allows a far more efficient class of response functions (or basis functions) to be employed within the framework of the original variational principle. New finite element equations were derived that can be used to compute the expansion coefficients for an individual coarse-mesh given the incident fluxes on the boundary. In addition, the non-variational method previously used to converge the expansion coefficients was developed in a new and more thorough manner by considering the implications of the fission source treat-ment imposed by the response expansion. The new coarse-mesh method was implemented for both one and two-dimensional (2-D) problems in the finite-difference, multigroup, discrete ordinates approximation. An efficient set of response functions was generated using orthogonal boundary conditions constructed from the discrete Legendre polynomials. Several one and two-dimensional heterogeneous light water reactor benchmark problems were studied. Relatively low-order response expansions were used to generate highly accurate results using both the variational and non-variational methods. The expansion order was found to have a far more significant impact on the accuracy of the results than the type of method. The varia-tional techniques provide better accuracy, but at substantially higher computational costs. The non-variational method is extremely robust and was shown to achieve accurate re-sults in the 2-D problems, as long as the expansion order was not very low.

In spectral methods for numerical PDEs, when the solutions are periodical, the Fourier functions may be used. However, when the solutions are non-periodical, the Legendre and Chebyshev polynomials are recommended, reported in many papers and books. There seems to exist few reports for the study of non-periodical solutions by spectral Fourier methods under the Dirichlet conditions and other boundary conditions. In this paper, we will explore the spectral Fourier methods(SFM) and collocation Fourier methods(CFM) for elliptic and eigenvalue problems. The CFM is simple and easy for computation, thus for saving a great deal of the CPU time. The collocation Fourier methods (CFM) can be regarded as the spectral Fourier methods (SFM) partly with the trapezoidal rule. Furthermore, the error bounds are derived for both the CFM and the SFM. When there exist no errors for the trapezoidal rule, the accuracy of the solutions from the CFM is as accurate as the spectral method using Legendre and Chebyshev polynomials. However, once there exists the truncation errors of the trapezoidal rule, the errors of the elliptic solutions and the leading eigenvalues the CFM are reduced to O(h^2), where h is the mesh length of uniform collocation grids, which are just equivalent to those by the linear elements and the finite difference method (FDM). The O(h^2) and even the superconvergence O(h4) are found numerically. The traditional condition number of the CFM is O(N^2), which is smaller than O(N^3) and O(N^4) of the collocation spectral methods using the Legendre and Chebyshev polynomials. Also the effective condition number is only O(1). Numerical experiments are reported for 1D elliptic and eigenvalue problems, to support the analysis made. The simplicity of algorithms and the promising numerical computation with O(h^4) may grant the CFM to be competent in application in numerical physics, chemistry, engineering, etc., see [7].

In this thesis, we consider the numerical solution of a large eigenvalue problem in which the integral operator comes from a radiative transfer problem. It is considered the use of hierarchical matrices, an efficient data-sparse representation of matrices, especially useful for large dimensional problems. It consists on low-rank subblocks leading to low memory requirements as well as cheap computational costs. We discuss the use of the hierarchical matrix technique in the numerical solution of a large scale eigenvalue problem arising from a finite rank discretization of an integral operator. The operator is of convolution type, it is defined through the first exponential-integral function and hence it is weakly singular. We access HLIB (Hierarchical matrices LIBrary) that provides, among others, routines for the construction of hierarchical matrix structures and arithmetic algorithms to perform approximative matrix operations. Moreover, it is incorporated the matrix-vector multiply routines from HLIB, as well as LU factorization for preconditioning, into SLEPc (Scalable Library for Eigenvalue Problem Computations) in order to exploit the available algorithms to solve eigenvalue problems. It is also developed analytical expressions for the approximate degenerate kernels and deducted error upper bounds for these approximations. The numerical results obtained with other approaches to solve the problem are used to compare with the ones obtained with this technique, illustrating the efficiency of the techniques developed and implemented in this work