Certified and fast computation of supremum norms of approximation errors

The version available on HAL corresponds to the version initially submitted to the conference and slightly differs from the published version since it does not account for remarks made by the referees.International audienceIn many numerical programs there is a need for a high-quality floating-point approximation of useful functions f, such as exp, sin, erf. In the actual implementation, the function is replaced by a polynomial p, leading to an approximation error (absolute or relative) epsilon = p-f or epsilon = p/f-1. The tight yet certain bounding of this error is an important step towards safe implementations. The main difficulty of this problem is due to the fact that this approximation error is very small and the difference p-f is highly cancellating. In consequence, previous approaches for computing the supremum norm in this degenerate case, have proven to be either unsafe, not sufficiently tight or too tedious in manual work. We present a safe and fast algorithm that computes a tight lower and upper bound for the supremum norms of approximation errors. The algorithm is based on a combination of several techniques, including enhanced interval arithmetic, automatic differentiation and isolation of the roots of a polynomial. We have implemented our algorithm and timings on several examples are given

Rigorous Polynomial Approximation using Taylor Models in Coq

International audienceOne of the most common and practical ways of representing a real function on machines is by using a polynomial approximation. It is then important to properly handle the error introduced by such an approximation. The purpose of this work is to offer guaranteed error bounds for a specific kind of rigorous polynomial approximation called Taylor model. We carry out this work in the Coq proof assistant, with a special focus on genericity and efficiency for our implementation. We give an abstract interface for rigorous polynomial approximations, parameter- ized by the type of coefficients and the implementation of polynomials, and we instantiate this interface to the case of Taylor models with inter- val coefficients, while providing all the machinery for computing them. We compare the performances of our implementation in Coq with those of the Sollya tool, which contains an implementation of Taylor models written in C. This is a milestone in our long-term goal of providing fully formally proved and efficient Taylor models

Chebyshev Interpolation Polynomial-based Tools for Rigorous Computing

17 pagesInternational audiencePerforming numerical computations, yet being able to provide rigorous mathematical statements about the obtained result, is required in many domains like global optimization, ODE solving or integration. Taylor models, which associate to a function a pair made of a Taylor approximation polynomial and a rigorous remainder bound, are a widely used rigorous computation tool. This approach benefits from the advantages of numerical methods, but also gives the ability to make reliable statements about the approximated function. Despite the fact that approximation polynomials based on interpolation at Chebyshev nodes offer a quasi-optimal approximation to a function, together with several other useful features, an analogous to Taylor models, based on such polynomials, has not been yet well-established in the field of validated numerics. This paper presents a preliminary work for obtaining such interpolation polynomials together with validated interval bounds for approximating univariate functions. We propose two methods that make practical the use of this: one is based on a representation in Newton basis and the other uses Chebyshev polynomial basis. We compare the quality of the obtained remainders and the performance of the approaches to the ones provided by Taylor models

Efficient Resolution of Anisotropic Structures

We highlight some recent new delevelopments concerning the sparse representation of possibly high-dimensional functions exhibiting strong anisotropic features and low regularity in isotropic Sobolev or Besov scales. Specifically, we focus on the solution of transport equations which exhibit propagation of singularities where, additionally, high-dimensionality enters when the convection field, and hence the solutions, depend on parameters varying over some compact set. Important constituents of our approach are directionally adaptive discretization concepts motivated by compactly supported shearlet systems, and well-conditioned stable variational formulations that support trial spaces with anisotropic refinements with arbitrary directionalities. We prove that they provide tight error-residual relations which are used to contrive rigorously founded adaptive refinement schemes which converge in $L_2$. Moreover, in the context of parameter dependent problems we discuss two approaches serving different purposes and working under different regularity assumptions. For frequent query problems, making essential use of the novel well-conditioned variational formulations, a new Reduced Basis Method is outlined which exhibits a certain rate-optimal performance for indefinite, unsymmetric or singularly perturbed problems. For the radiative transfer problem with scattering a sparse tensor method is presented which mitigates or even overcomes the curse of dimensionality under suitable (so far still isotropic) regularity assumptions. Numerical examples for both methods illustrate the theoretical findings

Double Greedy Algorithms: Reduced Basis Methods for Transport Dominated Problems

The central objective of this paper is to develop reduced basis methods for parameter dependent transport dominated problems that are rigorously proven to exhibit rate-optimal performance when compared with the Kolmogorov $n$-widths of the solution sets. The central ingredient is the construction of computationally feasible "tight" surrogates which in turn are based on deriving a suitable well-conditioned variational formulation for the parameter dependent problem. The theoretical results are illustrated by numerical experiments for convection-diffusion and pure transport equations. In particular, the latter example sheds some light on the smoothness of the dependence of the solutions on the parameters

Polyhedral Approximation of Multivariate Polynomials using Handelman's Theorem

International audienceConvex polyhedra are commonly used in the static analysis of programs to represent over-approximations of sets of reachable states of numerical program variables. When the analyzed programs contain nonlinear instructions, they do not directly map to standard polyhedral operations: some kind of linearization is needed. Convex polyhe-dra are also used in satisfiability modulo theory solvers which combine a propositional satisfiability solver with a fast emptiness check for polyhedra. Existing decision procedures become expensive when nonlinear constraints are involved: a fast procedure to ensure emptiness of systems of nonlinear constraints is needed. We present a new linearization algorithm based on Handelman's representation of positive polynomials. Given a polyhedron and a polynomial (in)equality, we compute a polyhedron enclosing their intersection as the solution of a parametric linear programming problem. To get a scalable algorithm, we provide several heuristics that guide the construction of the Handelman's representation. To ensure the correctness of our polyhedral approximation , our Ocaml implementation generates certificates verified by a checker certified in Coq

Méthodes de réduction de modèle pour les équations paramétrées -- Applications à la quantification d’incertitude.

Model order reduction has become an inescapable tool for the solution of high dimensional parameter-dependent equations arising in uncertainty quantification, optimization or inverse problems. In this thesis we focus on low rank approximation methods, in particular on reduced basis methods and on tensor approximation methods.The approximation obtained by Galerkin projections may be inaccurate when the operator is ill-conditioned. For projection based methods, we propose preconditioners built by interpolation of the operator inverse. We rely on randomized linear algebra for the efficient computation of these preconditioners. Adaptive interpolation strategies are proposed in order to improve either the error estimates or the projection onto reduced spaces. For tensor approximation methods, we propose a minimal residual formulation with ideal residual norms. The proposed algorithm, which can be interpreted as a gradient algorithm with an implicit preconditioner, allows obtaining a quasi-optimal approximation of the solution.Finally, we address the problem of the approximation of vector-valued or functional-valued quantities of interest. For this purpose we generalize the 'primal-dual' approaches to the non-scalar case, and we propose new methods for the projection onto reduced spaces. In the context of tensor approximation we consider a norm which depends on the error on the quantity of interest. This allows obtaining approximations of the solution that take into account the objective of the numerical simulation.Les méthodes de réduction de modèle sont incontournables pour la résolution d'équations paramétrées de grande dimension qui apparaissent dans les problèmes de quantification d'incertitude, d'optimisation ou encore les problèmes inverses. Dans cette thèse nous nous intéressons aux méthodes d'approximation de faible rang, notamment aux méthodes de bases réduites et d'approximation de tenseur.L'approximation obtenue par projection de Galerkin peut être de mauvaise qualité lorsque l'opérateur est mal conditionné. Pour les méthodes de projection sur des espaces réduits, nous proposons des préconditionneurs construits par interpolation d'inverse d'opérateur, calculés efficacement par des outils d'algèbre linéaire "randomisée". Des stratégies d'interpolation adaptatives sont proposées pour améliorer soit les estimateurs d'erreur, soit les projections sur les espaces réduits. Pour les méthodes d'approximation de tenseur, nous proposons une formulation en minimum de résidu avec utilisation de norme idéale. L'algorithme de résolution, qui s'interprète comme un algorithme de gradient avec préconditionneur implicite, permet d'obtenir une approximation quasi-optimale de la solution.Enfin nous nous intéressons à l'approximation de quantités d'intérêt à valeur fonctionnelle ou vectorielle. Nous généralisons pour cela les approches de type "primale-duale" au cas non scalaire, et nous proposons de nouvelles méthodes de projection sur espaces réduits. Dans le cadre de l'approximation de tenseur, nous considérons une norme dépendant de l'erreur en quantité d'intérêt afin d'obtenir une approximation de la solution qui tient compte de l'objectif du calcul