A Symbolic-Numeric Validation Algorithm for Linear ODEs with Newton–Picard Method
A symbolic-numeric validation algorithm is developed to compute rigorous and tight uniform error bounds for polynomial approximate solutions to linear ordinary differential equations, and in particular D-finite functions. It relies on an a posteriori validation scheme, where such an error bound is c...
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| Published in | Mathematics in computer science Vol. 15; no. 3; pp. 373 - 405 |
|---|---|
| Main Author | |
| Format | Journal Article |
| Language | English |
| Published |
Cham
Springer International Publishing
01.09.2021
Springer Nature B.V Springer |
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| Online Access | Get full text |
| ISSN | 1661-8270 1661-8289 1661-8289 |
| DOI | 10.1007/s11786-021-00510-7 |
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| Abstract | A symbolic-numeric validation algorithm is developed to compute rigorous and tight uniform error bounds for polynomial approximate solutions to linear ordinary differential equations, and in particular D-finite functions. It relies on an a posteriori validation scheme, where such an error bound is computed afterwards, independently from how the approximation was built. Contrary to Newton–Galerkin validation methods, widely used in the mathematical community of computer-assisted proofs, our algorithm does not rely on finite-dimensional truncations of differential or integral operators, but on an efficient approximation of the resolvent kernel using a Chebyshev spectral method. The result is a much better complexity of the validation process, carefully investigated throughout this article. Indeed, the approximation degree for the resolvent kernel depends linearly on the magnitude of the input equation, while the truncation order used in Newton–Galerkin may be exponential in the same quantity. Numerical experiments based on an implementation in C corroborate this complexity advantage over other a posteriori validation methods, including Newton–Galerkin. |
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| AbstractList | A symbolic-numeric validation algorithm is developed to compute rigorous and tight uniform error bounds for polynomial approximate solutions to linear ordinary differential equations, and in particular D-finite functions. It relies on an a posteriori validation scheme, where such an error bound is computed afterwards, independently from how the approximation was built. Contrary to Newton-Galerkin validation methods, widely used in the mathematical community of computer-assisted proofs, our algorithm does not rely on finite-dimensional truncations of differential or integral operators, but on an efficient approximation of the resolvent kernel using a Chebyshev spectral method. The result is a much better complexity of the validation process, carefully investigated throughout this article. Indeed, the approximation degree for the resolvent kernel depends linearly on the magnitude of the input equation, while the truncation order used in Newton-Galerkin may be exponential in the same quantity. Numerical experiments based on an implementation in C corroborate this complexity advantage over other a posteriori validation methods, including Newton-Galerkin. |
| Author | Bréhard, Florent |
| Author_xml | – sequence: 1 givenname: Florent surname: Bréhard fullname: Bréhard, Florent email: florent.brehard@univ-lille.fr organization: Department of Mathematics, Uppsala University |
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| Cites_doi | 10.1016/j.aam.2020.102027 10.1145/2442829.2442849 10.1145/3208103 10.1515/9781400838974 10.1137/1.9781611970425 10.1023/A:1024467732637 10.1007/s10208-018-09411-x 10.1016/B978-0-12-505630-4.50028-6 10.1016/0378-4754(82)90046-5 10.1137/13090883X 10.1137/0728057 10.1145/178365.178368 10.1137/S0036142996304498 10.1137/120865458 10.1007/s102080010025 10.1109/SYNASC.2007.49 10.4230/LIPIcs.ITP.2019.8 10.1016/S0195-6698(80)80051-5 10.1137/0718028 10.1007/BF00933182 10.1023/A:1023061927787 10.1090/mcom/3135 10.1081/NFA-100105107 10.1090/mcom/3046 10.1115/1.3625776 10.1016/0378-4754(82)90045-3 10.1007/BF02239009 10.1090/noti1276 10.1145/1837934.1837966 10.1006/jsco.2000.0474 10.1145/1073884.1073886 |
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| Keywords | A posteriori validation Chebyshev spectral methods 65G20 65L70 65L60 D-finite functions 65Y20 Linear differential equations Validated numerics validated numerics a posteriori validation linear differential equations |
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| References_xml | – reference: WilczakDZgliczyńskiPCr\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C^r$$\end{document}-Lohner algorithmSchedae Informaticae201120942 – reference: Berinde, V.: Iterative Approximation of Fixed Points. Lecture Notes in Mathematics, vol. 1912. Springer, Berlin (2007) – reference: NeumaierATaylor forms—use and limitsReliable Comput.2003914379195212810.1023/A:10230619277871071.65070 – reference: StanleyRPDifferentiably finite power seriesEur. J. Combinat.1980117518858753010.1016/S0195-6698(80)80051-50445.05012 – reference: PlumMComputer-assisted existence proofs for two-point boundary value problemsComputing19914611934110058210.1007/BF022390090782.65107 – reference: Lalescu, T.: Introduction à la Théorie des Équations Intégrales (Introduction to the Theory of Integral Equations). Hermann, Librairie Scientifique A (1911) – reference: Gottlieb, D., Orszag, S.A.: Numerical Analysis of Spectral Methods: Theory and Applications, vol. 26. SIAM (1977) – reference: BenoitAJoldesMMezzarobbaMRigorous uniform approximation of D-finite functions using Chebyshev expansionsMath. Comput.20178630513031341361401910.1090/mcom/31351361.65045 – reference: EpsteinCMirankerWLRivlinTJUltra-arithmetic. I. Function data typesMath. Comput. Simul.19822411865465010.1016/0378-4754(82)90045-30493.42005 – reference: Bournez, O., Graça, D.S., Pouly, A.: On the complexity of solving initial value problems. In: Proceedings of the 37th International Symposium on Symbolic and Algebraic Computation, pp. 115–121 (2012). https://doi.org/10.1145/2442829.2442849 – reference: Boyd, J.P.: Chebyshev and Fourier spectral methods. Dover Publications (2001) – reference: Chyzak, F.: The ABC of Creative Telescoping: Algorithms, Bounds, Complexity. Memoir of accreditation to supervise research (HDR), Université d’Orsay (2014). https://tel.archives-ouvertes.fr/tel-01069831 – reference: BréhardFBrisebarreNJoldesMValidated and numerically efficient Chebyshev spectral methods for linear ordinary differential equationsACM Trans. Math. Softw.201844444:144:42386583210.1145/320810307003069 – reference: Bréhard, F., Mahboubi, A., Pous, D.: A certificate-based approach to formally verified approximations. In: 10th International Conference on Interactive Theorem Proving (ITP 2019), vol. 141, pp. 8:1–8:19 (2019). https://doi.org/10.4230/LIPIcs.ITP.2019.8 – reference: OlverSTownsendAA fast and well-conditioned spectral methodSIAM Rev.2013553462489308941010.1137/1208654581273.65182 – reference: SalvyBZimmermannPGfun: a Maple package for the manipulation of generating and holonomic functions in one variableACM Trans. Math. Softw. (TOMS)199420216317710.1145/178365.1783680888.65010 – reference: HungriaALessardJ-PMireles JamesJDRigorous numerics for analytic solutions of differential equations: the radii polynomial approachMath. Comput.20168529914271459345437010.1090/mcom/30461332.65114 – reference: RallLBResolvent kernels of Green’s function kernels and other finite-rank modifications of Fredholm and Volterra kernelsJ. Optim. Theory Appl.197824598848891210.1007/BF009331820348.45002 – reference: ZgliczyńskiPC1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C^1$$\end{document} Lohner algorithmFound. Comput. Math.200224429465193094610.1007/s1020800100251049.65038 – reference: BerzMMakinoKVerified integration of ODEs and flows using differential algebraic methods on high-order Taylor modelsReliable Comput.199844361369165214710.1023/A:10244677326370976.65061 – reference: Joldes, M.: Rigorous Polynomial Approximations and Applications. PhD thesis, École normale supérieure de Lyon – Université de Lyon, Lyon, France (2011). https://tel.archives-ouvertes.fr/tel-00657843 – reference: Kaucher, E., Miranker, W.L.: Validating computation in a function space. 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| SubjectTerms | A posteriori validation Algorithms Approximation Chebyshev approximation Chebyshev spectral methods Complexity Computer Science D-finite functions Differential equations Galerkin method Kernels Linear differential equations Mathematics Mathematics and Statistics Numerical Analysis Operators (mathematics) Ordinary differential equations Polynomials Spectral methods Symbolic Computation Validated numerics |
| Title | A Symbolic-Numeric Validation Algorithm for Linear ODEs with Newton–Picard Method |
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