Very fine discretizations of differential operators often lead to large, sparse matrices A, where the condition number of A is large. Such ill-conditioning has well known effects on both solving linear systems and eigenvalue computations, and, in general, computing solutions with relative accuracy independent of the condition number is highly desirable. This dissertation is divided into two parts. In the first part, we discuss a method of preconditioning, developed by Ye, which allows solutions of Ax=b to be computed accurately. This, in turn, allows for accurate eigenvalue computations. We then use this method to develop discretizations that yield accurate computations of the smallest eigenvalue of the biharmonic operator across several domains. Numerical results from the various schemes are provided to demonstrate the performance of the methods. In the second part we address the role of the condition number of A in the context of multigrid algorithms. Under various assumptions, we use rigorous Fourier analysis on 2- and 3-grid iteration operators to analyze round off errors in floating point arithmetic. For better understanding of general results, we provide detailed bounds for a particular algorithm applied to the 1-dimensional Poisson equation. Numerical results are provided and compared with those obtained by the schemes discussed in part 1.

The solution of large and sparse eigenvalue problems arising from the discretization of the diffusion equation is considered. The multigroup diffusion equation is discretized by means of the Nodal expansion Method (NEM) [9, 10]. A new formulation of the higher order NEM variants revealing the true nature of the problem, that is, a generalized eigenvalue problem, is proposed. These generalized eigenvalue problems are solved using the Jacobi-Davidson (JD) method [26]. The most expensive part of the method consists of solving a linear system referred to as correction equation. It is solved using Krylov subspace methods in combination with aggregation-based Algebraic Multigrid (AMG) techniques. In that context, a particular aggregation technique used in combination with classical smoothers, referred to as oblique geometric coarsening, has been derived. Its particularity is that it aggregates unknowns that are not coupled, which has never been done to our knowledge. A modular code, combining JD with an AMG preconditioner, has been developed. The code comes with many options, that have been tested. In particular, the instability of the Rayleigh-Ritz [33] acceleration procedure in the non-symmetric case has been underlined. Our code has also been compared to an industrial code extracted from ARTEMIS.

Directional wave-field decomposition for heterogeneousanisotropic media with in-stantaneous response is establishedfor both the acoustic and the electromagnetic equations. We derive a sufficient condition for ellipticity of thesystem's matrix in the Laplace domain and show that theconstruction of the splitting matrix via a Dunford-Taylorintegral over the resolvent of the non-compact, non-normalsystem's matrix is well de ned. The splitting matrix also hasproperties that make it possible to construct the decompositionwith a generalized eigenvector procedure. The classical way ofobtaining the decomposition is equivalent to solving analgebraic Riccati operator equation. Hence the proceduredescribed above also provides a solution to the algebraicRiccati operator equation. The solution to the wave-field decomposition for theisotropic wave equation is expressed in terms of theDirichlet-to-Neumann map for a plane. The equivalence of thisDirichlet-to-Neumann map is the acoustic admittance, i.e. themapping between the pressure and the particle velocity. Theacoustic admittance, as well as the related impedance aresolutions to algebraic Riccati operator equations and are keyelements in the decomposition. In the electromagnetic case thecorresponding impedance and admittance mappings solve therespective algebraic Riccati operator equations and henceprovide solutions to the decomposition problem. The present research shows that it is advantageous toutilize the freedom implied by the generalized eigenvectorprocedure to obtain the solution to the decomposition problemin more general terms than the admittance/impedancemappings. The time-reversal approach to steer an acoustic wave eld inthe cavity and half space geometries are analyzed from aboundary control perspective. For the cavity it is shown thatwe can steer the field to a desired final configuration, withthe assumption of local energy decay. It is also shown that thetime-reversal algorithm minimizes a least square error forfinite times when the data are obtained by measurements. Forthe half space geometry, the boundary condition is expressedwith help of the wave-field decomposition. In the homogeneousmaterial case, the response of the time-reversal algorithm iscalculated analytically. This procedure uses the one-wayequations together with the decomposition operator.

No description available.

<p>Directional wave-field decomposition for heterogeneousanisotropic media with in-stantaneous response is establishedfor both the acoustic and the electromagnetic equations.</p><p>We derive a sufficient condition for ellipticity of thesystem's matrix in the Laplace domain and show that theconstruction of the splitting matrix via a Dunford-Taylorintegral over the resolvent of the non-compact, non-normalsystem's matrix is well de ned. The splitting matrix also hasproperties that make it possible to construct the decompositionwith a generalized eigenvector procedure. The classical way ofobtaining the decomposition is equivalent to solving analgebraic Riccati operator equation. Hence the proceduredescribed above also provides a solution to the algebraicRiccati operator equation.</p><p>The solution to the wave-field decomposition for theisotropic wave equation is expressed in terms of theDirichlet-to-Neumann map for a plane. The equivalence of thisDirichlet-to-Neumann map is the acoustic admittance, i.e. themapping between the pressure and the particle velocity. Theacoustic admittance, as well as the related impedance aresolutions to algebraic Riccati operator equations and are keyelements in the decomposition. In the electromagnetic case thecorresponding impedance and admittance mappings solve therespective algebraic Riccati operator equations and henceprovide solutions to the decomposition problem.</p><p>The present research shows that it is advantageous toutilize the freedom implied by the generalized eigenvectorprocedure to obtain the solution to the decomposition problemin more general terms than the admittance/impedancemappings.</p><p>The time-reversal approach to steer an acoustic wave eld inthe cavity and half space geometries are analyzed from aboundary control perspective. For the cavity it is shown thatwe can steer the field to a desired final configuration, withthe assumption of local energy decay. It is also shown that thetime-reversal algorithm minimizes a least square error forfinite times when the data are obtained by measurements. Forthe half space geometry, the boundary condition is expressedwith help of the wave-field decomposition. In the homogeneousmaterial case, the response of the time-reversal algorithm iscalculated analytically. This procedure uses the one-wayequations together with the decomposition operator.</p>

In this paper we develop new preconditioned iterative methods for solving monotone nonlinear eigenvalue problems. We investigate the convergence and derive grid-independent error estimates for these methods. Numerical experiments demonstrate the practical effectiveness of the proposed methods for a model problem.

The presence of structure in a computational problem can often be exploited and can lead to a more efficient numerical algorithm. In this dissertation, we look at structured numerical problems that arise from applications in wireless communications and machine learning that also impact other areas of scientific computing. In wireless communication system designs, certain structured matrices (frames) need to be generated. The design of such matrices is equivalent to a symmetric inverse eigenvalue problem where the values of the diagonal elements are prescribed. We present algorithms that are capable of generating a larger set of these constructions than previous algorithms. We also discuss the existence of equiangular tight frames---frames that satisfy additional structural properties. Kernel learning is an important class of problems in machine learning. It often relies on efficient numerical algorithms that solve underlying convex optimization problems. In our work, the objective functions to be minimized are the von Neumann and the LogDet Bregman matrix divergences. The algorithm that solves this optimization problem performs matrix updates based on repeated eigendecompositions of diagonal plus rank-one matrices in the case of von Neumann matrix divergence, and Cholesky updates in case of the LogDet Bregman matrix divergence. Our contribution exploits the low-rank representations and the structure of the constraint matrices, resulting in more efficient algorithms than previously known. We also present two specialized zero-finding algorithms where we exploit the structure through the shape and exact formulation of the objective function. The first zero-finding task arises during the matrix update step which is part of the above-mentioned kernel learning application. The second zero-finding problem is for the secular equation; it is equivalent to the computation of the eigenvalues of a diagonal plus rank-one matrix. The secular equation arises in various applications, the most well-known is the divide-and-conquer eigensolver. In our solutions, we build upon a somewhat forgotten zero-finding method by P. Jarratt, first described in 1966. The method employs first derivatives only and needs the same amount of evaluations as Newton's method, but converges faster. Our contributions are the more efficient specialized zero-finding algorithms. / text

No description available.

A new method is presented for the numerical computation of the generalized eigen- values of real Hamiltonian or symplectic pencils and matrices. The method is strongly backward stable, i.e., it is numerically backward stable and preserves the structure (i.e., Hamiltonian or symplectic). In the case of a Hamiltonian matrix the method is closely related to the square reduced method of Van Loan, but in contrast to that method which may suffer from a loss of accuracy of order sqrt(epsilon), where epsilon is the machine precision, the new method computes the eigenvalues to full possible accuracy.

[ES] Uno de los objetivos más importantes en el análisis de la seguridad en el campo de la ingeniería nuclear es el cálculo, rápido y preciso, de la evolución de la potencia dentro del núcleo del reactor. La distribución de los neutrones se puede describir a través de la ecuación de transporte de Boltzmann. La solución de esta ecuación no puede obtenerse de manera sencilla para reactores realistas, y es por ello que se tienen que considerar aproximaciones numéricas. En primer lugar, esta tesis se centra en obtener la solución para varios problemas estáticos asociados con la ecuación de difusión neutrónica: los modos lambda, los modos gamma y los modos alpha. Para la discretización espacial se ha utilizado un método de elementos finitos de alto orden. Diversas características de cada problema espectral se analizan y se comparan en diferentes reactores. Después, se investigan varios métodos de cálculo para problemas de autovalores y estrategias para calcular los problemas algebraicos obtenidos a partir de la discretización espacial. La mayoría de los trabajos destinados a la resolución de la ecuación de difusión neutrónica están diseñados para la aproximación de dos grupos de energía, sin considerar dispersión de neutrones del grupo térmico al grupo rápido. La principal ventaja de la metodología que se propone es que no depende de la geometría del reactor, del tipo de problema de autovalores ni del número de grupos de energía del problema. Tras esto, se obtiene la solución de las ecuaciones estacionarias de armónicos esféricos. La implementación de estas ecuaciones tiene dos principales diferencias respecto a la ecuación de difusión neutrónica. Primero, la discretización espacial se realiza a nivel de pin. Por tanto, se estudian diferentes tipos de mallas. Segundo, el número de grupos de energía es, generalmente, mayor que dos. De este modo, se desarrollan estrategias a bloques para optimizar el cálculo de los problemas algebraicos asociados. Finalmente, se implementa un método modal actualizado para integrar la ecuación de difusión neutrónica dependiente del tiempo. Se presentan y comparan los métodos modales basados en desarrollos en función de los diferentes modos espaciales para varios tipos de transitorios. Además, también se desarrolla un control de paso de tiempo adaptativo, que evita la actualización de los modos de una manera fija y adapta el paso de tiempo en función de varias estimaciones del error. / [CAT] Un dels objectius més importants per a l'anàlisi de la seguretat en el camp de l'enginyeria nuclear és el càlcul, ràpid i precís, de l'evolució de la potència dins del nucli d'un reactor. La distribució dels neutrons pot modelar-se mitjançant l'equació del transport de Boltzmann. La solució d'aquesta equació per a un reactor realístic no pot obtenir's de manera senzilla. És per això que han de considerar-se aproximacions numèriques. En primer lloc, la tesi se centra en l'obtenció de la solució per a diversos problemes estàtics associats amb l'equació de difusió neutrònica: els modes lambda, els modes gamma i els modes alpha. Per a la discretització espacial s'ha utilitzat un mètode d'elements finits d'alt ordre. Algunes de les característiques dels problemes espectrals s'analitzaran i es compararan per a diferents reactors. Tanmateix, diversos solucionadors de problemes d'autovalors i estratègies es desenvolupen per a calcular els problemes obtinguts de la discretització espacial. La majoria dels treballs per a resoldre l'equació de difusió neutrònica estan dissenyats per a l'aproximació de dos grups d'energia i sense considerar dispersió de neutrons del grup tèrmic al grup ràpid. El principal avantatge de la metodologia exposada és que no depèn de la geometria del reactor, del tipus de problema d'autovalors ni del nombre de grups d'energia del problema. Seguidament, s'obté la solució de les equacions estacionàries d'harmònics esfèrics. La implementació d'aquestes equacions té dues principals diferències respecte a l'equació de difusió. Primer, la discretització espacial es realitza a nivell de pin a partir de l'estudi de diferents malles. Segon, el nombre de grups d'energia és, generalment, major que dos. D'aquesta forma, es desenvolupen estratègies a blocs per a optimitzar el càlcul dels problemes algebraics associats. Finalment, s'implementa un mètode modal amb actualitzacions dels modes per a integrar l'equació de difusió neutrònica dependent del temps. Es presenten i es comparen els mètodes modals basats en l'expansió dels diferents modes espacials per a diversos tipus de transitoris. A més a més, un control de pas de temps adaptatiu es desenvolupa, evitant l'actualització dels modes d'una manera fixa i adaptant el pas de temps en funció de vàries estimacions de l'error. / [EN] One of the most important targets in nuclear safety analyses is the fast and accurate computation of the power evolution inside of the reactor core. The distribution of neutrons can be described by the neutron transport Boltzmann equation. The solution of this equation for realistic nuclear reactors is not straightforward, and therefore, numerical approximations must be considered. First, the thesis is focused on the attainment of the solution for several steady-state problems associated with neutron diffusion problem: the $\lambda$-modes, the $\gamma$-modes and the $\alpha$-modes problems. A high order finite element method is used for the spatial discretization. Several characteristics of each type of spectral problem are compared and analyzed on different reactors. Thereafter, several eigenvalue solvers and strategies are investigated to compute efficiently the algebraic eigenvalue problems obtained from the discretization. Most works devoted to solve the neutron diffusion equation are made for the approximation of two energy groups and without considering up-scattering. The main property of the proposed methodologies is that they depend on neither the reactor geometry, the type of eigenvalue problem nor the number of energy groups. After that, the solution of the steady-state simplified spherical harmonics equations is obtained. The implementation of these equations has two main differences with respect to the neutron diffusion. First, the spatial discretization is made at level of pin. Thus, different meshes are studied. Second, the number of energy groups is commonly bigger than two. Therefore, block strategies are developed to optimize the computation of the algebraic eigenvalue problems associated. Finally, an updated modal method is implemented to integrate the time-dependent neutron diffusion equation. Modal methods based on the expansion of the different spatial modes are presented and compared in several types of transients. Moreover, an adaptive time-step control is developed that avoids setting the time-step with a fixed value and it is adapted according to several error estimations. / Carreño Sánchez, AM. (2020). Integration methods for the time dependent neutron diffusion equation and other approximations of the neutron transport equation [Tesis doctoral no publicada]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/144771 / TESIS