NEP: A Module for the Parallel Solution of Nonlinear Eigenvalue Problems in SLEPc

[EN] SLEPc is a parallel library for the solution of various types of large-scale eigenvalue problems. Over the past few years, we have been developing a module within SLEPc, called NEP, that is intended for solving nonlinear eigenvalue problems. These problems can be defined by means of a matrix-valued function that depends nonlinearly on a single scalar parameter. We do not consider the particular case of polynomial eigenvalue problems (which are implemented in a different module in SLEPc) and focus here on rational eigenvalue problems and other general nonlinear eigenproblems involving square roots or any other nonlinear function. The article discusses how the NEP module has been designed to fit the needs of applications and provides a description of the available solvers, including some implementation details such as parallelization. Several test problems coming from real applications are used to evaluate the performance and reliability of the solvers.This work was partially funded by the Spanish Agencia Estatal de Investigacion AEI http://ciencia.gob.es under grants TIN2016-75985-P AEI and PID2019-107379RB-I00 AEI (including European Commission FEDER funds).Campos, C.; Roman, JE. (2021). NEP: A Module for the Parallel Solution of Nonlinear Eigenvalue Problems in SLEPc. ACM Transactions on Mathematical Software. 47(3):1-29. https://doi.org/10.1145/3447544S12947

The Biglobal Instability of the Bidirectional Vortex

State of the art research in hydrodynamic stability analysis has moved from classic one-dimensional methods such as the local nonparallel approach and the parabolized stability equations to two-dimensional, biglobal, methods. The paradigm shift toward two dimensional techniques with the ability to accommodate fully three-dimensional base flows is a necessary step toward modeling complex, multidimensional flowfields in modern propulsive applications. Here, we employ a two-dimensional spatial waveform with sinusoidal temporal dependence to reduce the three-dimensional linearized Navier-Stokes equations to their biglobal form. Addressing hydrodynamic stability in this way circumvents the restrictive parallel-flow assumption and admits boundary conditions in the streamwise direction. Furthermore, the following work employs a full momentum formulation, rather than the reduced streamfunction form, accounting for a nonzero tangential mean flow velocity. This approach adds significant complexity in both formulation and implementation but renders a more general methodology applicable to a broader spectrum of mean flows. Specifically, we consider the stability of three models for bidirectional vortex flow. While a complete parametric study ensues, the stabilizing effect of the swirl velocity is evident as the injection parameter, kappa, is closely examined

Das Spektrum zeitverzögerter Differentialgleichungen: numerische Methoden, Stabilität und Störungstheorie

Three types of problems related to delay-differential equations (DDEs) are treated in this thesis. We first consider the problem of numerically computing the eigenvalues of a DDE. Here, we present an application of a projection method for nonlinear eigenvalue problems (NLEPs). We compare this projection method with other methods, suggested in the literature, and used in software packages. The projection method is computationally superior to all of the other tested method for the presented large-scale examples. We give interpretations of methods based on discretizations in terms of rational approximations. Some notes regarding a special case where the spectrum can be explicitly expressed with a formula containing a matrix version of the are Lambert W function are presented. We clarify its range of applicability, and, by counter-example, show that it does not hold in general. The second part of this thesis is related to exact stability conditions of the DDE. All those combinations of the delays such that there is a purely imaginary eigenvalue (called critical delays) are parameterized. In general, an evaluation of the parameterization map consists of solving a quadratic eigenvalue problem of squared dimension. We show how the computational cost for one evaluation of the map can be reduced by exploiting a relation to a Lyapunov equation. The third and last part of this thesis is about generalizations of perturbation results for NLEPs. A sensitivity formula for the movement of the eigenvalues extends to NLEPs. We introduce a fixed point form for the NLEP, and show that some methods in the literature can be interpreted as set-valued fixed point iterations for which asymptotic convergence can be established. We also show how the Bauer-Fike theorem can be generalized to the NLEP under special conditions.In dieser Arbeit werden drei verschiedene Problemklassen im Bezug zu delay-differential equations (DDEs) behandelt. Als erstes gehen wir auf die Berechnung der Eigenwerte von DDEs ein. In dieser Arbeit wenden wir eine Projektionsmethode für nichtlineare Eigenwertprobleme (NLEPe) an. Wir vergleichen diese mit anderen bereits bekannten Verfahren, wobei die hier vorgestellte Methode bedeutend bessere numerische Eigenschaften für die verwendeten Beispiele hat. Zusätzlich treffen wir Aussagen über Diskretisierungsmethoden zur rationalen Approximation. Desweiteren betrachten wir einen Spezialfall, bei welchem das Spektrum explizit mit Hilfe einer Matrix-Version der Lambert W-Funktion dargestellt werden kann. Für diese Formel bestimmen wir einen möglichen Anwendungsbereich. Im zweiten Teil der Arbeit werden exakte Stabilitätsbedingungen von DDEs betrachtet. Die Menge der Delays, für welche die DDE einen imaginären Eigenwert hat (sogenannte kritische Delays), wird parameterisiert. Im Allgemeinen ist zur Auswertung der Parametrisierungsabbildung das Lösen eines quadratischen Eigenwertproblems nötig, dessen Größe dem Quadrat der Dimension der DDE entspricht. Wir zeigen wie der Rechenaufwand durch Ausnutzung einer Lyapunov-Gleichung reduziert werden kann. Der letzte Teil dieser Arbeit befasst sich mit der Verallgemeinerung der Störungstheorie auf NLEPe. Unter anderem lässt sich eine Sensitivitätsformel auf NLEPe erweitern. Desweiteren wird eine Fixpunktform für NLEPe vorgestellt, und gezeigt dass einige Methoden aus der Literatur als mengenwertige Fixpunktiterationen dargestellt werden können, für welche wir asymptotische Konvergenz feststellen. Wir zeigen zusätzlich, dass das Bauer-Fike Theorem unter bestimmten Bedingungen auf NLEPe verallgemeinert werden kann

Implementación paralela de métodos iterativos para la resolución de problemas polinómicos de valores propios

The polynomial eigenvalue problem appears in many areas of scientific and technical computing. It can be seen as a generalization of the linear eigenvalue problem in which the equation P(lambda)x = 0, that defines the problem, involves a polynomial P(lambda), of degree d, in the parameter lambda (the eigenvalue), and d+1 matrix coefficients. In its turn, the polynomial eigenvalue problem is a particular case of the nonlinear eigenvalue problem, T(lambda)x = 0, in which T is a nonlinear matrix function. These problems appear in diverse areas of application such as acoustics, fluid mechanics, structure analysis, or photonics. This thesis focuses on the study of methods for the numerical resolution of the polynomial eigenvalue problem, as well as the adaptation of such methods to the most general nonlinear case. Mainly, we consider methods of projection, that are appropriate for the case of sparse matrices of large dimension, where only a small percentage of eigevalues and eigenvectors are required. The algorithms are studied from the point of view of high-performance computing, considering issues like (computational and memory) efficiency and parallel computation. SLEPc, Scalable Library for Eigenvalue Problem Computations, is a software library for the parallel solution of large-scale eigenvalue problems. It is of general purpose and can be used for standard and generalized problems, both symmetric and nonsymmetric, with real or complex arithmetic. As a result of this thesis, we have developed several solvers for polynomial an nonlinear eigenproblems, which have included in the last versions of this software. On one hand, we consider methods based on the linearization of the polynomial problem, that solves an equivalent linear eigenproblem of dimension several times the initial size. Among them, the TOAR method stands out, that repre- sents the search subspace basis in an efficient way in terms of memory, and is appropriate to handle the increase of dimension of the linear problem. The thesis also proposes specific variants for the particular case of symmetric matrices. In all these methods we consider several aspects to provide the implementations with robustness and flexibility, such as spectral transformations, scaling, and techniques of extraction. In addition to the methods of linearization, we propose methods of Newton type, such as the method of Jacobi-Davidson with deflation and the method of Newton for invariant pairs. Due to its characteristics, the latter is not usually employed as a proper method, but as a technique for refinement of the solutions obtained with another method. The previous methods can also be applied to the resolution of the nonlinear problem, using techniques like polynomial or rational interpolation, being necessary in some cases to adapt the algorithms. This thesis covers also these cases. For all the considered algorithms we have made parallel implementations in SLEPc, and have studied its numerical behaviour and its parallel performance in problems coming from real applications.El problema polinómico de valores propios aparece en muchas áreas de la computación científica y técnica. Puede verse como una generalización del problema lineal de valores propios en el que la ecuación P(lambda)x=0, que define el problema, involucra un polinomio P(lambda), de grado d, en el parámetro lambda del autovalor, y d+1 coeficientes matriciales. A su vez, el problema polinómico de valores propios es un caso particular del problema no lineal de valores propios, T(lambda)x=0, en el que T es una función matricial no lineal. Estos problemas aparecen en diversas áreas de aplicación como acústica, mecánica de fluidos, análisis de estructuras, o fotónica. Esta tesis se centra en el estudio de métodos para la resolución numérica del problema polinómico de valores propios, así como la adaptación de dichos métodos al caso más general no lineal. Principalmente, se consideran métodos de proyección, que son apropiados para el caso de matrices dispersas de gran dimensión cuando se requiere solo un pequeño porcentaje de los valores y vectores propios. Los algoritmos se estudian desde el punto de vista de la computación de altas prestaciones, teniendo en consideración aspectos como la eficiencia (computacional y de memoria) y la computación paralela. SLEPc, Scalable Library for Eigenvalue Problem Computations, es una biblioteca software para la resolución de problemas de valores propios de gran dimensión en paralelo. Es de propósito general y puede usarse para problemas estándares y generalizados, simétricos y no simétricos, con aritmética real o compleja. Como fruto de esta tesis, se han desarrollado diversos solvers para problemas polinómicos y no lineales, los cuales se han incluido en las últimas versiones de este software. Por un lado, se abordan métodos basados en la linealización del problema polinómico, que resuelven un problema lineal equivalente de dimensión varias veces la del inicial. Entre ellos se destaca el método TOAR, que representa la base del subespacio de búsqueda de una forma eficiente en términos de memoria, y es adecuado para manejar el aumento de dimensión del problema lineal. La tesis también propone variantes específicas para el caso particular de matrices simétricas. En todos estos métodos se consideran diversos aspectos para dotar a las implementaciones de robustez y flexibilidad, tales como transformaciones espectrales, escalado, y técnicas de extracción. Además de los métodos de linealización, se proponen métodos de tipo Newton, como el método de Jacobi-Davidson con deflación y el método de Newton para pares invariantes. Por sus características, este último no suele utilizarse como un método en sí mismo sino como técnica de refinamiento de las soluciones obtenidas con otro método. Los métodos anteriores pueden aplicarse a la resolución del problema no lineal, utilizando técnicas como la interpolación polinómica o racional, siendo necesario en algunos casos adaptar los algoritmos. La tesis cubre también estos casos. Para todos los algoritmos considerados se han realizado implementaciones paralelas en SLEPc, y se ha estudiado su comportamiento numérico y sus prestaciones paralelas en problemas procedentes de aplicaciones reales.El problema polinòmic de valors propis apareix en moltes àrees de la computació científica i tècnica. Pot veure's com una generalització del problema lineal de valors propis en el qual l'equació P(lambda)x=0, que defineix el problema, involucra un polinomi P(lambda), de grau d, en el paràmetre lambda de l'autovalor, i d+1 coeficients matricials. Al seu torn, el problema polinòmic de valors propis és un cas particular del problema no lineal de valors propis, T(lambda)x=0, en el qual T és una funció matricial no lineal. Aquests problemes apareixen en diverses àrees d'aplicació com a acústica, mecànica de fluids, anàlisis d'estructures, o fotònica. Aquesta tesi se centra en l'estudi de mètodes per a la resolució numèrica del problema polinòmic de valors propis, així com l'adaptació d'aquests mètodes al cas més general no lineal. Principalment, es consideren mètodes de projecció, que són apropiats per al cas de matrius disperses de gran dimensió quan es requereix solament un reduït percentatge dels valors i vectors propis. Els algorismes s'estudien des del punt de vista de la computació d'altes prestacions, tenint en consideració aspectes com l'eficiència (computacional i de memòria) i la computació paral·lela. SLEPc, Scalable Library for Eigenvalue Problem Computations, és una biblioteca software per a la resolució de problemes de valors propis de gran dimensió en paral·lel. És de propòsit general i pot usar-se per a problemes estàndards i generalitzats, simètrics i no simètrics, amb aritmètica real o complexa. Com a fruit d'aquesta tesi, s'han desenvolupat diversos solvers per a problemes polinòmics i no lineals, els quals s'han inclòs en les últimes versions d'aquest software. D'una banda, s'aborden mètodes basats en la linealització del problema polinòmic, que resolen un problema lineal equivalent de dimensió diverses vegades la de l'inicial. Entre ells es destaca el mètode TOAR, que representa la base del subespai de cerca d'una forma eficient en termes de memòria, i és adequat per a manejar l'augment de dimensió del problema lineal. La tesi també proposa variants específiques per al cas particular de matrius simètriques. En tots aquests mètodes es consideren diversos aspectes per a dotar a les implementacions de robustesa i flexibilitat, tals com a transformacions espectrals, escalat, i tècniques d'extracció. A més dels mètodes de linealització, es proposen mètodes de tipus Newton, com el mètode de Jacobi-Davidson amb deflació i el mètode de Newton per a parells invariants. Per les seues característiques, aquest últim no sol utilitzar-se com un mètode en si mateix sinó com a tècnica de refinament de les solucions obtingudes amb un altre mètode. Els mètodes anteriors poden aplicar-se a la resolució del problema no lineal, utilitzant tècniques com la interpolació polinòmica o racional, sent necessari en alguns casos adaptar els algorismes. La tesi cobreix també aquests casos. Per a tots els algorismes considerats s'han realitzat implementacions paral·leles en SLEPc, i s'ha estudiat el seu comportament numèric i les seues prestacions paral·leles en problemes procedents d'aplicacions reals.Campos González, MC. (2017). Implementación paralela de métodos iterativos para la resolución de problemas polinómicos de valores propios [Tesis doctoral no publicada]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/86134TESI

Software for Exascale Computing - SPPEXA 2016-2019

This open access book summarizes the research done and results obtained in the second funding phase of the Priority Program 1648 "Software for Exascale Computing" (SPPEXA) of the German Research Foundation (DFG) presented at the SPPEXA Symposium in Dresden during October 21-23, 2019. In that respect, it both represents a continuation of Vol. 113 in Springer’s series Lecture Notes in Computational Science and Engineering, the corresponding report of SPPEXA’s first funding phase, and provides an overview of SPPEXA’s contributions towards exascale computing in today's sumpercomputer technology. The individual chapters address one or more of the research directions (1) computational algorithms, (2) system software, (3) application software, (4) data management and exploration, (5) programming, and (6) software tools. The book has an interdisciplinary appeal: scholars from computational sub-fields in computer science, mathematics, physics, or engineering will find it of particular interest

Dynamics and disorder in quantum antiferromagnets

La physique de la matière condensée, et notamment les systèmes fortement corrélés, amènent à des problèmes parmi les plus stimulants et difficiles de la physique moderne. Dans ces systèmes, les interactions à plusieurs corps et les corrélations entre les particules quantiques ne peuvent être négligées, sinon, les modèles échoueraient simplement à capturer les mécanismes physiques en jeu et les phénomènes qui en découlent. En particulier, le travail présenté dans ce manuscrit traite du magnétisme quantique et aborde plusieurs questions distinctes à l'aide d'approches computationnelles et méthodes numériques à l'état de l'art. Les effets conjoints du désordre (i.e. impuretés) et des interactions sont étudiés concernant un matériau magnétique spécifique : plutôt qu'une phase de la matière dite localisée, attendue à fort champ magnétique, une phase ordonnée induite par le désordre lui-même est mise en lumière, avec une réapparition inattendue de la cohérence quantique dans ledit composé. Par ailleurs, la réponse dynamique d'aimants quantiques à une perturbation externe, comme celle mesurée dans des expériences de résonance magnétique nucléaire ou de diffusion inélastique de neutrons est étudiée.Condensed matter physics, and especially strongly correlated systems provide some of the most challenging problems of modern physics. In these systems, the many-body interactions and correlations between quantum particles cannot be neglected; otherwise, the models would simply fail to capture the relevant physics at play and phenomena ensuing. In particular, the work presented in this manuscript deals with quantum magnetism and addresses several distinct questions through computational approaches and state-of-the-art numerical methods. The interplay between disorder (i.e. impurities) and interactions is studied regarding a specific magnetic compound, where instead of the expected many-body localized phase at high magnetic fields, a novel disorder-induced ordered state of matter is found, with a resurgence of quantum coherence. Furthermore, the dynamical response of quantum magnets to an external perturbation, such as it is accessed and measured in nuclear magnetic resonance and inelastic neutron scattering experiments is investigated