Randomized residual‐based error estimators for the proper generalized decomposition approximation of parametrized problems

This paper introduces a novel error estimator for the Proper Generalized Decomposition (PGD) approximation of parametrized equations. The estimator is intrinsically random: It builds on concentration inequalities of Gaussian maps and an adjoint problem with random right-hand side, which we approximate using the PGD. The effectivity of this randomized error estimator can be arbitrarily close to unity with high probability, allowing the estimation of the error with respect to any user-defined norm as well as the error in some quantity of interest. The performance of the error estimator is demonstrated and compared with some existing error estimators for the PGD for a parametrized time-harmonic elastodynamics problem and the parametrized equations of linear elasticity with a high-dimensional parameter space.

[1]  Pedro Díez,et al.  An error estimator for separated representations of highly multidimensional models , 2010 .

[2]  Walter L. Smith.  Probability and Statistics , 1959, Nature.

[3]  P ? ? ? ? ? ? ? % ? ? ? ? , 1991 .

[4]  J. Hesthaven,et al.  Certified Reduced Basis Methods for Parametrized Partial Differential Equations , 2015 .

[5]  W. Hackbusch.  Tensor Spaces and Numerical Tensor Calculus , 2012, Springer Series in Computational Mathematics.

[6]  Francisco Chinesta,et al.  Recent Advances and New Challenges in the Use of the Proper Generalized Decomposition for Solving Multidimensional Models , 2010 .

[7]  Karen Willcox,et al.  Multifidelity Dimension Reduction via Active Subspaces , 2018, SIAM J. Sci. Comput..

[8]  A. Quarteroni,et al.  Reduced Basis Methods for Partial Differential Equations: An Introduction , 2015 .

[9]  V. Koltchinskii,et al.  High Dimensional Probability , 2006, math/0612726.

[10]  Anthony Nouy,et al.  Low-rank methods for high-dimensional approximation and model order reduction , 2015, 1511.01554.

[11]  Linda R. Petzold,et al.  Approved for public release; further dissemination unlimited Error Estimation for Reduced Order Models of Dynamical Systems ∗ , 2003 .

[12]  Mario Ohlberger,et al.  Hierarchical model reduction of nonlinear partial differential equations based on the adaptive empirical projection method and reduced basis techniques , 2014, 1401.0851.

[13]  Qiang Du,et al.  Centroidal Voronoi Tessellations: Applications and Algorithms , 1999, SIAM Rev..

[14]  Anthony T. Patera,et al.  Randomized Residual-Based Error Estimators for Parametrized Equations , 2018, SIAM J. Sci. Comput..

[16]  Anthony Nouy,et al.  Randomized linear algebra for model reduction. Part I: Galerkin methods and error estimation , 2018, Advances in Computational Mathematics.

[17]  Bernhard Wieland,et al.  Reduced basis methods for partial differential equations with stochastic influences , 2013 .

[18]  Roman Vershynin,et al.  High-Dimensional Probability , 2018 .

[19]  Anthony Nouy,et al.  Interpolation of Inverse Operators for Preconditioning Parameter-Dependent Equations , 2015, SIAM J. Sci. Comput..

[20]  Michael B. Giles,et al.  Adjoint Recovery of Superconvergent Functionals from PDE Approximations , 2000, SIAM Rev..

[21]  Mario Ohlberger,et al.  A Dimensional Reduction Approach Based on the Application of Reduced Basis Methods in the Framework of Hierarchical Model Reduction , 2014, SIAM J. Sci. Comput..

[22]  Francisco Chinesta,et al.  A new family of solvers for some classes of multidimensional partial differential equations encountered in kinetic theory modeling of complex fluids , 2006 .

[23]  Nicolás Montés,et al.  Numerical strategies for the Galerkin-proper generalized decomposition method , 2013, Math. Comput. Model..

[24]  Marie Billaud-Friess,et al.  A tensor approximation method based on ideal minimal residual formulations for the solution of high-dimensional problems ∗ , 2013, 1304.6126.

[25]  W. B. Johnson,et al.  Extensions of Lipschitz mappings into Hilbert space , 1984 .

[26]  I. Babuska,et al.  On a dimensional reduction method. I. The optimal selection of basis functions , 1981 .

[27]  Kathrin Smetana,et al.  Randomized Local Model Order Reduction , 2017, SIAM J. Sci. Comput..

[28]  D. Rovas,et al.  A Posteriori Error Bounds for Reduced-Basis Approximation of Parametrized Noncoercive and Nonlinear Elliptic Partial Differential Equations , 2003 .

[29]  Y. Maday,et al.  Results and Questions on a Nonlinear Approximation Approach for Solving High-dimensional Partial Differential Equations , 2008, 0811.0474.

[30]  J. P. Moitinho de Almeida,et al.  A basis for bounding the errors of proper generalised decomposition solutions in solid mechanics , 2013 .

[31]  Pedro Díez,et al.  An error estimator for real-time simulators based on model order reduction , 2015, Adv. Model. Simul. Eng. Sci..

[32]  Grégory Legrain,et al.  Tensor-based methods for numerical homogenization from high-resolution images , 2013 .

[33]  Linda R. Petzold,et al.  A Posteriori Error Estimation and Global Error Control for Ordinary Differential Equations by the Adjoint Method , 2005, SIAM J. Sci. Comput..

[34]  Marc Laforest,et al.  On a Goal-Oriented Version of the Proper Generalized Decomposition Method , 2019, J. Sci. Comput..

[35]  F. Chinesta,et al.  A separated representation of an error indicator for the mesh refinement process under the proper generalized decomposition framework , 2014, Computational Mechanics.

[36]  Bernard Haasdonk,et al.  Chapter 2: Reduced Basis Methods for Parametrized PDEs—A Tutorial Introduction for Stationary and Instationary Problems , 2017 .

[37]  Anupam Gupta,et al.  An elementary proof of the Johnson-Lindenstrauss Lemma , 1999 .

[38]  Ludovic Chamoin,et al.  A posteriori error estimation and adaptive strategy for PGD model reduction applied to parametrized linear parabolic problems , 2017, 1801.07422.

[39]  Adrien Leygue,et al.  The Proper Generalized Decomposition for Advanced Numerical Simulations: A Primer , 2013 .

[40]  Pierre Ladevèze,et al.  Toward guaranteed PGD-reduced models , 2012 .

[41]  Francisco Chinesta,et al.  A new family of solvers for some classes of multidimensional partial differential equations encountered in kinetic theory modelling of complex fluids - Part II: Transient simulation using space-time separated representations , 2007 .

[42]  A. Patera,et al.  Reduced basis approximation and a posteriori error estimation for affinely parametrized elliptic coercive partial differential equations , 2007 .

[43]  F. Chinesta,et al.  A Short Review in Model Order Reduction Based on Proper Generalized Decomposition , 2018 .

[44]  Pierre Ladevèze,et al.  On the verification of model reduction methods based on the proper generalized decomposition , 2011 .

[45]  Anthony T. Patera,et al.  A natural-norm Successive Constraint Method for inf-sup lower bounds , 2010 .

[46]  Maëlle Nodet,et al.  Goal-Oriented Error Estimation for the Reduced Basis Method, with Application to Sensitivity Analysis , 2013, Journal of Scientific Computing.

[47]  N. Nguyen,et al.  An ‘empirical interpolation’ method: application to efficient reduced-basis discretization of partial differential equations , 2004 .

[48]  Anthony Nouy,et al.  Chapter 4: Low-Rank Methods for High-Dimensional Approximation and Model Order Reduction , 2017 .

[49]  J. Hesthaven,et al.  Improved successive constraint method based a posteriori error estimate for reduced basis approximation of 2D Maxwell"s problem , 2009 .

[50]  Anthony Nouy,et al.  Projection-Based Model Order Reduction Methods for the Estimation of Vector-Valued Variables of Interest , 2016, SIAM J. Sci. Comput..

[51]  T. Lelièvre,et al.  Greedy algorithms for high-dimensional non-symmetric linear problems , 2012, 1210.6688.

[52]  Alan J. Laub,et al.  Small-Sample Statistical Condition Estimates for General Matrix Functions , 1994, SIAM J. Sci. Comput..

[53]  Reinhold Schneider,et al.  Variational Monte Carlo—bridging concepts of machine learning and high-dimensional partial differential equations , 2018, Advances in Computational Mathematics.

[54]  Simona Perotto,et al.  Hierarchical Local Model Reduction for Elliptic Problems: A Domain Decomposition Approach , 2010, Multiscale Model. Simul..

[55]  A. Patera,et al.  Reduced basis approximation and a posteriori error estimation for affinely parametrized elliptic coercive partial differential equations , 2007 .

[56]  A. Nouy.  A generalized spectral decomposition technique to solve a class of linear stochastic partial differential equations , 2007 .

[57]  Mario Ohlberger,et al.  Problem adapted Hierarchical Model Reduction for the Fokker-Planck equation , 2015, 1511.02073.

[58]  A. Patera,et al.  A Successive Constraint Linear Optimization Method for Lower Bounds of Parametric Coercivity and Inf-Sup Stability Constants , 2007 .