Validated and numerically efficient Chebyshev spectral methods for linear ordinary differential equations

Florent Bréhard 1, 2, 3 Nicolas Brisebarre 4, 2, 3 Mioara Joldes 1, 4
1 LAAS-MAC - Équipe Méthodes et Algorithmes en Commande
LAAS - Laboratoire d'analyse et d'architecture des systèmes [Toulouse]
3 ARIC - Arithmetic and Computing
Inria Grenoble - Rhône-Alpes, LIP - Laboratoire de l'Informatique du Parallélisme
Abstract : In this work we develop a validated numerics method for the solution of linear ordinary differential equations (LODEs). A wide range of algorithms (i.e., Runge-Kutta, collocation, spectral methods) exist for numerically computing approximations of the solutions. Most of these come with proofs of asymptotic convergence, but usually, provided error bounds are non-constructive. However, in some domains like critical systems and computer-aided mathematical proofs, one needs validated effective error bounds. We focus on both the theoretical and practical complexity analysis of a so-called \emph{a posteriori} quasi-Newton validation method, which mainly relies on a fixed-point argument of a contracting map. Specifically, given a polynomial approximation, obtained by some numerical algorithm and expressed in Chebyshev basis, our algorithm efficiently computes an accurate and rigorous error bound. For this, we study theoretical properties like compactness, convergence, invertibility of associated linear integral operators and their truncations in a suitable coefficient space of Chebyshev series. Then, we analyze the almost-banded matrix structure of these operators, which allows for very efficient numerical algorithms for both numerical solutions of LODEs and rigorous computation of the approximation error. Finally, several representative examples show the advantages of our algorithms as well as their theoretical and practical limits.
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Submitted on : Friday, July 28, 2017 - 4:43:46 PM
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Florent Bréhard, Nicolas Brisebarre, Mioara Joldes. Validated and numerically efficient Chebyshev spectral methods for linear ordinary differential equations. ACM Transactions on Mathematical Software, Association for Computing Machinery, In press. ⟨hal-01526272v2⟩



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