Computational linear algebra over finite fields

We present here algorithms for efficient computation of linear algebra problems over finite fields

A fast and well-conditioned spectral method for singular integral equations

We develop a spectral method for solving univariate singular integral equations over unions of intervals by utilizing Chebyshev and ultraspherical polynomials to reformulate the equations as almost-banded infinite-dimensional systems. This is accomplished by utilizing low rank approximations for sparse representations of the bivariate kernels. The resulting system can be solved in ${\cal O}(m^2n)$ operations using an adaptive QR factorization, where $m$ is the bandwidth and $n$ is the optimal number of unknowns needed to resolve the true solution. The complexity is reduced to ${\cal O}(m n)$ operations by pre-caching the QR factorization when the same operator is used for multiple right-hand sides. Stability is proved by showing that the resulting linear operator can be diagonally preconditioned to be a compact perturbation of the identity. Applications considered include the Faraday cage, and acoustic scattering for the Helmholtz and gravity Helmholtz equations, including spectrally accurate numerical evaluation of the far- and near-field solution. The Julia software package SingularIntegralEquations.jl implements our method with a convenient, user-friendly interface

B-splines for sparse grids : algorithms and application to higher-dimensional optimization

In simulation technology, computationally expensive objective functions are often replaced by cheap surrogates, which can be obtained by interpolation. Full grid interpolation methods suffer from the so-called curse of dimensionality, rendering them infeasible if the parameter domain of the function is higher-dimensional (four or more parameters). Sparse grids constitute a discretization method that drastically eases the curse, while the approximation quality deteriorates only insignificantly. However, conventional basis functions such as piecewise linear functions are not smooth (continuously differentiable). Hence, these basis functions are unsuitable for applications in which gradients are required. One example for such an application is gradient-based optimization, in which the availability of gradients greatly improves the speed of convergence and the accuracy of the results. This thesis demonstrates that hierarchical B-splines on sparse grids are well-suited for obtaining smooth interpolants for higher dimensionalities. The thesis is organized in two main parts: In the first part, we derive new B-spline bases on sparse grids and study their implications on theory and algorithms. In the second part, we consider three real-world applications in optimization: topology optimization, biomechanical continuum-mechanics, and dynamic portfolio choice models in finance. The results reveal that the optimization problems of these applications can be solved accurately and efficiently with hierarchical B-splines on sparse grids.In der Simulationstechnik werden zeitaufwendige Zielfunktionen oft durch einfache Surrogate ersetzt, die durch Interpolation gewonnen werden können. Vollgitter-Interpolationsmethoden leiden unter dem sogenannten Fluch der Dimensionalität, der sie unbrauchbar macht, falls der Parameterbereich der Funktion höherdimensional ist (vier oder mehr Parameter). Dünne Gitter sind eine Diskretisierungsmethode, die den Fluch drastisch lindert und die Approximationsqualität nur leicht verschlechtert. Leider sind konventionelle Basisfunktionen wie die stückweise linearen Funktionen nicht glatt (stetig differenzierbar). Daher sind sie für Anwendungen ungeeignet, in denen Gradienten benötigt werden. Ein Beispiel für eine solche Anwendung ist gradientenbasierte Optimierung, in der die Verfügbarkeit von Gradienten die Konvergenzgeschwindigkeit und die Ergebnisgenauigkeit deutlich verbessert. Diese Dissertation demonstriert, dass hierarchische B-Splines auf dünnen Gittern hervorragend geeignet sind, um glatte Interpolierende für höhere Dimensionalitäten zu erhalten. Die Dissertation ist in zwei Hauptteile gegliedert: Der erste Teil leitet neue B-Spline-Basen auf dünnen Gittern her und untersucht ihre Implikationen bezüglich Theorie und Algorithmen. Der zweite Teil behandelt drei Realwelt-Anwendungen aus der Optimierung: Topologieoptimierung, biomechanische Kontinuumsmechanik und Modelle der dynamischen Portfolio-Wahl in der Finanzmathematik. Die Ergebnisse zeigen, dass die Optimierungsprobleme dieser Anwendungen durch hierarchische B-Splines auf dünnen Gittern genau und effizient gelöst werden können

Hybrid Symbolic-Numeric Computing in Linear and Polynomial Algebra

In this thesis, we introduce hybrid symbolic-numeric methods for solving problems in linear and polynomial algebra. We mainly address the approximate GCD problem for polynomials, and problems related to parametric and polynomial matrices. For symbolic methods, our main concern is their complexity and for the numerical methods we are more concerned about their stability. The thesis consists of 5 articles which are presented in the following order: Chapter 1, deals with the fundamental notions of conditioning and backward error. Although our results are not novel, this chapter is a novel explication of conditioning and backward error that underpins the rest of the thesis. In Chapter 2, we adapt Victor Y. Pan\u27s root-based algorithm for finding approximate GCD to the case where the polynomials are expressed in Bernstein bases. We use the numerically stable companion pencil of G. F. Jónsson to compute the roots, and the Hopcroft-Karp bipartite matching method to find the degree of the approximate GCD. We offer some refinements to improve the process. In Chapter 3, we give an algorithm with similar idea to Chapter 2, which finds an approximate GCD for a pair of approximate polynomials given in a Lagrange basis. More precisely, we suppose that these polynomials are given by their approximate values at distinct known points. We first find each of their roots by using a Lagrange basis companion matrix for each polynomial. We introduce new clustering algorithms and use them to cluster the roots of each polynomial to identify multiple roots, and then marry the two polynomials using a Maximum Weight Matching (MWM) algorithm, to find their GCD. In Chapter 4, we define ``generalized standard triples\u27\u27 X, zC1 - C0, Y of regular matrix polynomials P(z) in order to use the representation X(zC1 - C0)-1 Y=P-1(z). This representation can be used in constructing algebraic linearizations; for example, for H(z) = z A(z)B(z) + C from linearizations for A(z) and B(z). This can be done even if A(z) and B(z) are expressed in differing polynomial bases. Our main theorem is that X can be expressed using the coefficients of 1 in terms of the relevant polynomial basis. For convenience we tabulate generalized standard triples for orthogonal polynomial bases, the monomial basis, and Newton interpolational bases; for the Bernstein basis; for Lagrange interpolational bases; and for Hermite interpolational bases. We account for the possibility of common similarity transformations. We give explicit proofs for the less familiar bases. Chapter 5 is devoted to parametric linear systems (PLS) and related problems, from a symbolic computational point of view. PLS are linear systems of equations in which some symbolic parameters, that is, symbols that are not considered to be candidates for elimination or solution in the course of analyzing the problem, appear in the coefficients of the system. We assume that the symbolic parameters appear polynomially in the coefficients and that the only variables to be solved for are those of the linear system. It is well-known that it is possible to specify a covering set of regimes, each of which is a semi-algebraic condition on the parameters together with a solution description valid under that condition.We provide a method of solution that requires time polynomial in the matrix dimension and the degrees of the polynomials when there are up to three parameters. Our approach exploits the Hermite and Smith normal forms that may be computed when the system coefficient domain is mapped to the univariate polynomial domain over suitably constructed fields. Our approach effectively identifies intrinsic singularities and ramification points where the algebraic and geometric structure of the matrix changes. Specially parametric eigenvalue problems can be addressed as well. Although we do not directly address the problem of computing the Jordan form, our approach allows the construction of the algebraic and geometric eigenvalue multiplicities revealed by the Frobenius form, which is a key step in the construction of the Jordan form of a matrix

Algorithmes rapides pour les polynômes, séries formelles et matrices

Notes d'un cours dispensé aux Journées Nationales du Calcul Formel 2010International audienceLe calcul formel calcule des objets mathématiques exacts. Ce cours explore deux directions : la calculabilité et la complexité. La calculabilité étudie les classes d'objets mathématiques sur lesquelles des réponses peuvent être obtenues algorithmiquement. La complexité donne ensuite des outils pour comparer des algorithmes du point de vue de leur efficacité. Ce cours passe en revue l'algorithmique efficace sur les objets fondamentaux que sont les entiers, les polynômes, les matrices, les séries et les solutions d'équations différentielles ou de récurrences linéaires. On y montre que de nombreuses questions portant sur ces objets admettent une réponse en complexité (quasi-)optimale, en insistant sur les principes généraux de conception d'algorithmes efficaces. Ces notes sont dérivées du cours " Algorithmes efficaces en calcul formel " du Master Parisien de Recherche en Informatique (2004-2010), co-écrit avec Frédéric Chyzak, Marc Giusti, Romain Lebreton, Bruno Salvy et Éric Schost. Le support de cours complet est disponible à l'url https://wikimpri.dptinfo.ens-cachan.fr/doku.php?id=cours:c-2-2

Spectral and High Order Methods for Partial Differential Equations ICOSAHOM 2018

This open access book features a selection of high-quality papers from the presentations at the International Conference on Spectral and High-Order Methods 2018, offering an overview of the depth and breadth of the activities within this important research area. The carefully reviewed papers provide a snapshot of the state of the art, while the extensive bibliography helps initiate new research directions