Krylov methods for large-scale modern problems in numerical linear algebra

Large-scale problems have attracted much attention in the last decades since they arise from different applications in several fields. Moreover, the matrices that are involved in those problems are often sparse, this is, the majority of their entries are zero. Around 40 years ago, the most common problems related to large-scale and sparse matrices consisted in solving linear systems, finding eigenvalues and/or eigenvectors, solving least square problems or computing singular value decompositions. However, in the last years, large-scale and sparse problems of different natures have appeared, motivating and challenging numerical linear algebra to develop effective and efficient algorithms to solve them. Common difficulties that appear during the development of algorithms for solving modern large-scale problems are related to computational costs, storage issues and CPU time, given the large size of the matrices, which indicate that direct methods can not be used. This suggests that projection methods based on Krylov subspaces are a good option to develop procedures for solving large-scale and sparse modern problems. In this PhD Thesis we develop novel and original algorithms for solving two large-scale modern problems in numerical linear algebra: first, we introduce the R-CORK method for solving rational eigenvalue problems and, second, we present projection methods to compute the solution of T-Sylvester matrix equations, both based on Krylov subspaces. The R-CORK method is an extension of the compact rational Krylov method (CORK) [104] introduced to solve a family of nonlinear eigenvalue problems that can be expressed and linearized in certain particular ways and which include arbitrary polynomial eigenvalue problems, but not arbitrary rational eigenvalue problems. The R-CORK method exploits the structure of the linearized problem by representing the Krylov vectors in a compact form in order to reduce the cost of storage, resulting in a method with two levels of orthogonalization. The first level of orthogonalization works with vectors of the same size as the original problem, and the second level works with vectors of size much smaller than the original problem. Since vectors of the size of the linearization are never stored or orthogonalized, R-CORK is more efficient from the point of view of memory and orthogonalization costs than the classical rational Krylov method applied to the linearization. Moreover, since the R-CORK method is based on a classical rational Krylov method, the implementation of implicit restarting is possible and we present an efficient way to do it, that preserves the compact representation of the Krylov vectors. We also introduce in this dissertation projection methods for solving the TSylvester equation, which has recently attracted considerable attention as a consequence of its close relation to palindromic eigenvalue problems and other applications. The theory concerning T-Sylvester equations is rather well understood, and before the work in this thesis, there were stable and efficient numerical algorithms to solve these matrix equations for small- to medium- sized matrices. However, developing numerical algorithms for solving large-scale T-Sylvester equations was a completely open problem. In this thesis, we introduce several projection methods based on block Krylov subspaces and extended block Krylov subspaces for solving the T-Sylvester equation when the right-hand side is a low-rank matrix. We also offer an intuition on the expected convergence of the algorithm based on block Krylov subspaces and a clear guidance on which algorithm is the most convenient to use in each situation. All the algorithms presented in this thesis have been extensively tested, and the reported numerical results show that they perform satisfactorily in practice.Adicionalmente se recibió ayuda parcial de los proyectos de investigación: “Structured Numerical Linear Algebra: Matrix Polynomials, Special Matrices, and Conditioning” (Ministerio de Economía y Competitividad de España, Número de proyecto: MTM2012-32542) y “Structured Numerical Linear Algebra for Constant, Polynomial and Rational Matrices” (Ministerio de Economía y Competitividad de España, Número de proyecto: MTM2015-65798-P), donde el investigador principal de ambos proyectos fue Froilán Martínez Dopico.Programa Oficial de Doctorado en Ingeniería MatemáticaPresidente: José Mas Marí.- Secretario: Fernando de Terán Vergara.- Vocal: José Enrique Román Molt

Experimental Investigation of Supersonic Jets Using Optical Diagnostics

The complexity of many fluid flows and phenomena is a well-known characteristic driven primarily by turbulence, which has been a focal point of study for decades. Most engineering applications in fluids will encounter turbulence, and hence the need to understand how turbulence might influence the problem at hand is omnipresent. In many turbulent flows, there are large-scale coherent structures which directly influence macro-scale processes of engineering relevance, such as noise production. Over decades of study, it has been demonstrated that similar structures are often observed across many flowfields, despite differences in characteristic parameters, and this has led to the pursuit of simplified models through the use of these dominant, shared structures. Large-scale, coherent structures are of particular importance in turbulent jets, as they represent efficient sources of sound. Noise reduction of subsonic and supersonic fluid jets represents a large interest in the study of acoustic production in jets, and much of it is viewed in the context of controlling these large-scale structures. Supersonic jets in particular may emit an intense sound known as jet screech as a consequence of these structures. This noise source easily has the potential to be damaging to both structures and humans in close proximity, and is a particular target of noise reduction efforts. Turbulent flowfields from two supersonic, underexpanded, screeching jets are analyzed by means of three non-intrusive, high-speed, optical diagnostics. The first technique is high-speed schlieren. The second technique is pulse-burst particle image velocimetry (PB-PIV). The third technique is known as focused laser differential interferometry (FLDI). Extensive spectral, statistical, and modal decomposition analyses are used in this work to identify, extract, and characterize the most energetic features and coherent structures associated with jet screech. The large field of view of the image-based datasets is fully taken advantage of by creating spatial maps of spectral and statistical quantities, which highlight regions of increased fluctuations or activity. These are shown to agree with, or demonstrate additional features that could not be reproduced by the modal analyses. Modal analyses are used to evaluate the structure of the most energetic components in the flow of both screeching jets

MS FT-2-2 7 Orthogonal polynomials and quadrature: Theory, computation, and applications

Quadrature rules find many applications in science and engineering. Their analysis is a classical area of applied mathematics and continues to attract considerable attention. This seminar brings together speakers with expertise in a large variety of quadrature rules. It is the aim of the seminar to provide an overview of recent developments in the analysis of quadrature rules. The computation of error estimates and novel applications also are described