We consider in this work quantities that can be obtained as limits of powers of parametrized matrices, for instance the inverse matrix or the logarithm of the determinant. Under the assumption of affine dependence in the parameters, we use the Empirical Interpolation Method (EIM) to derive an approximation for powers of these matrices, from which we derive a nonintrusive approximation for the aforementioned limits. We derive upper bounds of the error made by the obtained formula. Finally, numerical comparisons with classical intrusive and nonintrusive approximation techniques are provided: in the considered test-cases, our algorithm performs well compared to the nonintrusive ones.
Accepté le :
DOI : 10.1051/m2an/2018048
Mots-clés : Empirical Interpolation Method, nonintrusive approximation, inverse matrix, logarithm of determinant
@article{M2AN_2019__53_1_219_0,
author = {Casenave, Fabien and Akkari, Nissrine and Charles, Alexandre and Rey, Christian},
title = {Nonintrusive approximation of parametrized limits of matrix power algorithms {\textendash} application to matrix inverses and log-determinants},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {219--248},
publisher = {EDP-Sciences},
volume = {53},
number = {1},
year = {2019},
doi = {10.1051/m2an/2018048},
zbl = {1416.65111},
mrnumber = {3937349},
language = {en},
url = {http://www.numdam.org/articles/10.1051/m2an/2018048/}
}
TY - JOUR AU - Casenave, Fabien AU - Akkari, Nissrine AU - Charles, Alexandre AU - Rey, Christian TI - Nonintrusive approximation of parametrized limits of matrix power algorithms – application to matrix inverses and log-determinants JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2019 SP - 219 EP - 248 VL - 53 IS - 1 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an/2018048/ DO - 10.1051/m2an/2018048 LA - en ID - M2AN_2019__53_1_219_0 ER -
%0 Journal Article %A Casenave, Fabien %A Akkari, Nissrine %A Charles, Alexandre %A Rey, Christian %T Nonintrusive approximation of parametrized limits of matrix power algorithms – application to matrix inverses and log-determinants %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2019 %P 219-248 %V 53 %N 1 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an/2018048/ %R 10.1051/m2an/2018048 %G en %F M2AN_2019__53_1_219_0
Casenave, Fabien; Akkari, Nissrine; Charles, Alexandre; Rey, Christian. Nonintrusive approximation of parametrized limits of matrix power algorithms – application to matrix inverses and log-determinants. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 53 (2019) no. 1, pp. 219-248. doi : 10.1051/m2an/2018048. http://www.numdam.org/articles/10.1051/m2an/2018048/
and , Regression models for ordinal responses: a review of methods and applications, Int. J. Epidemiol. 26 (1997) 1323–1333. | DOI
and , Maximum-likelihood estimation of the parameters of a multivariate normal distribution, Linear Algebra Appl. 70 (1985) 147–171. | DOI | MR | Zbl
, , and , An ‘empirical interpolation’ method: application to efficient reduced-basis discretization of partial differential equations, C.R. Math. 339 (2004) 667–672. | DOI | MR | Zbl
, Optimisation aérodynamique par réduction de modèle POD et contrôle optimal: application au sillage laminaire d’un cylindre circulaire, Ph.D. thesis, Vandoeuvre-les-Nancy, INPL, (2004).
, , , and , A randomized algorithm for approximating the log determinant of a symmetric positive definite matrix, Linear Algebra Appl. 533 (2017) 95–117. | DOI | MR | Zbl
, , , and , A priori convergence of the greedy algorithm for the parametrized reduced basis method, ESAIM: M2AN 46 (2012) 595–603. | DOI | Numdam | MR | Zbl
, and , A nonintrusive reduced basis method applied to aeroacoustic simulations, Adv. Comput. Math. 41 (2015) 961–986. | DOI | MR | Zbl
, and , Variants of the Empirical Interpolation Method: Symmetric formulation, choice of norms and rectangular extension, Appl. Math. Lett. 56 (2016) 23–28. | DOI | MR | Zbl
, , and , A Non intrusive reduced basis method: application to computational fluid dynamics, In: 2nd ECCOMAS Young Investigators Conference (YIC 2013), Bordeaux, France (2013).
, An introduction to the proper orthogonal decomposition, Curr. Sci. 78 (2000) 808–817.
and , Nonlinear model reduction via discrete empirical interpolation, SIAM J. Sci. Comput. 32 (2010) 2737–2764. | DOI | MR | Zbl
and , Model order reduction based on proper generalized decomposition for the propagation of uncertainties in structural dynamics, Int. J. Numer. Methods Eng. 89 (2012) 241–268. | DOI | MR | Zbl
, , and , An overview of the Proper Generalized Decomposition with applications in computational rheology. XVIth International Workshop on Numerical Methods for Non-Newtonian Flows, J. Non-Newtonian Fluid Mech. 166 (2011) 578–592. | DOI | Zbl
, and , The Proper Generalized Decomposition for Advanced Numerical Simulations: A Primer, SpringerBriefs in Applied Sciences and Technology. Springer International Publishing, New York, 2013. | MR | Zbl
, and , A short review on model order reduction based on proper generalized decomposition, Arch. Comput. Methods Eng. 18 (2011) 395–404. | DOI
, , , and , Brief review of regression-based and machine learning methods in genetic epidemiology: the genetic analysis workshop 17 experience, Genet. Epidemiol. 35 (2011) S5–S11. | DOI
, , , Reduced basis approximation and a posteriori error estimation for the parametrized unsteady Boussinesq equations, Math. Models Methods Appl. Sci. 21 (2011) 1415–1442. | DOI | MR | Zbl
, A review of automatic selection methods for machine learning algorithms and hyper-parameter values, Network Model. Anal. Health Inf. Bioinf. 5 (2016) 18. | DOI
, , , and , Output bounds for reduced-basis approximations of symmetric positive definite eigenvalue problems, C. R. Acad. Sci. Paris Ser. I 331 (2005) 153–158. | DOI | MR | Zbl
, , , , and , Reliable real-time solution of parametrized partial differential equations: reduced-basis output bound methods, CJ Fluids Eng. 124 (2002) 70–80. | DOI
, and , Convergence analysis of the Generalized Empirical Interpolation Method, SIAM J. Numer. Anal. 54 (2016) 1713–1731. | DOI | MR | Zbl
, , and , A general multipurpose interpolation procedure: the magic points, Commun. Pure Appl, Anal. 8 (2009) 383–404. | DOI | MR | Zbl
, Approximations to the determinant term in Gaussian maximum likelihood estimation of some spatial models, Commun. Stat. Theory Methods 22 (1992) 189–205. | DOI | Zbl
, , and , Robust regression methods for computer vision: a review, Int. J. Comput. Vision 6 (1991) 59–70. | DOI
, , and , A posteriori error bounds for reduced-basis approximation of parametrized noncoercive and nonlinear elliptic partial differential equations. Proceedings of the 16th AIAA Computational Fluid Dynamics Conference, Orlando, Florida, USA, (2003).
, , , , , , , , , , , , , , and , Scikit-learn: machine learning in python, J. Mach. Learn. Res. 12 (2011) 2825–2830. | MR | Zbl
, and , Maximum projection designs for computer experiments, Biometrika 102 (2015) 371–380. | DOI | MR | Zbl
, Reduced-basis approximation and a posteriori error estimation for many-parameter heat conduction problems, Numer. Heat Transfer Part B: Fundam. 54 (2008) 369–389. | DOI
, , , , and , ``Natural norm’’ a posteriori error estimators for reduced basis approximations, J. Comput. Phys. 217 (2006) 37–62. | DOI | MR | Zbl
, Turbulence and the dynamics of coherent structures, parts I, II and III, Q. Appl. Math. XLV (1987) 561–590. | DOI | MR | Zbl
and , An overview of regression techniques for knowledge discovery, Knowl. Eng. Rev. 14 (1999) 319–340. | DOI
and , Certified real-time solution of the parametrized steady incompressible Navier-Stokes equations: rigorous reduced-basis a posteriori error bounds, Int. J. Numer. Methods Fluids 47 (2005) 773–788. | DOI | MR | Zbl
, and , Reduced-basis approximation of the viscous Burgers equation: rigorous a posteriori error bounds, C.R. Math. 337 (2003) 619–624. | DOI | MR | Zbl
, , , , , and , Non-intrusive reduced-order modeling for multiphase porous media flows using Smolyak sparse grids, Int. J. Numer. Methods Fluids 83 (2017) 205–219. | DOI | MR
, A space-time Petrov-Galerkin certified reduced basis method: Application to the Boussinesq equations, SIAM J. Sci. Comput. 36 (2014) A232–A266. | DOI | MR | Zbl
and , Interpolation of inverse operators for preconditioning parameter-dependent equations, SIAM J. Sci. Comput. 38 (2016) A1044–A1074. | DOI | MR | Zbl
, , and , Log-det approximation based on uniformly distributed seeds and its application to Gaussian process regression, J. Comput. Appl. Math. 220 (2008) 198–214. | DOI | MR | Zbl
Cité par Sources :
