Probabilistic error estimation for non-intrusive reduced models learned from data of systems governed by linear parabolic partial differential equations

Uy, Wayne Isaac Tan; Peherstorfer, Benjamin

doi:10.1051/m2an/2021010

Probabilistic error estimation for non-intrusive reduced models learned from data of systems governed by linear parabolic partial differential equations

Journal Article · Wed May 05 00:00:00 EDT 2021 · Mathematical Modelling and Numerical Analysis

DOI:https://doi.org/10.1051/m2an/2021010· OSTI ID:1784724

^[1]; Peherstorfer, Benjamin ^[2]

Courant Inst. of Mathematical Sciences, New York, NY (United States); New York University
Courant Inst. of Mathematical Sciences, New York, NY (United States)

This work derives a residual-based a posteriori error estimator for reduced models learned with non-intrusive model reduction from data of high-dimensional systems governed by linear parabolic partial differential equations with control inputs. It is shown that quantities that are necessary for the error estimator can be either obtained exactly as the solutions of least-squares problems in a non-intrusive way from data such as initial conditions, control inputs, and high-dimensional solution trajectories or bounded in a probabilistic sense. Here, the computational procedure follows an offline/online decomposition. In the offline (training) phase, the high-dimensional system is judiciously solved in a black-box fashion to generate data and to set up the error estimator. In the online phase, the estimator is used to bound the error of the reduced-model predictions for new initial conditions and new control inputs without recourse to the high-dimensional system. Numerical results demonstrate the workflow of the proposed approach from data to reduced models to certified predictions.

View Accepted Manuscript (DOE)

Research Organization:: New York Univ., NY (United States)

Sponsoring Organization:: USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR)

Grant/Contract Number:: SC0019334

OSTI ID:: 1784724

Journal Information:: Mathematical Modelling and Numerical Analysis, Journal Name: Mathematical Modelling and Numerical Analysis Journal Issue: 3 Vol. 55; ISSN 0764-583X

Publisher:: EDP SciencesCopyright Statement

Country of Publication:: United States

Language:: English

References (41)

Data-driven model order reduction of quadratic-bilinear systems: The Loewner framework for QB systems Gosea, Ion Victor; Antoulas, Athanasios C. Numerical Linear Algebra with Applications, Vol. 25, Issue 6 https://doi.org/10.1002/nla.2200	journal	July 2018
Dynamic data-driven reduced-order models Peherstorfer, Benjamin; Willcox, Karen Computer Methods in Applied Mechanics and Engineering, Vol. 291 https://doi.org/10.1016/j.cma.2015.03.018	journal	July 2015
Data-driven operator inference for nonintrusive projection-based model reduction Peherstorfer, Benjamin; Willcox, Karen Computer Methods in Applied Mechanics and Engineering, Vol. 306 https://doi.org/10.1016/j.cma.2016.03.025	journal	July 2016
Dynamic mode decomposition of numerical and experimental data Schmid, Peter J. Journal of Fluid Mechanics, Vol. 656 https://doi.org/10.1017/s0022112010001217	journal	July 2010
Certified reduced-basis solutions of viscous Burgers equation parametrized by initial and boundary values Janon, Alexandre; Nodet, Maëlle; Prieur, Clémentine ESAIM: Mathematical Modelling and Numerical Analysis, Vol. 47, Issue 2 https://doi.org/10.1051/m2an/2012029	journal	January 2013
A Mathematical and Computational Framework for Reliable Real-Time Solution of Parametrized Partial Differential Equations Prud'homme, Christophe; Rovas, Dimitrios V.; Veroy, Karen ESAIM: Mathematical Modelling and Numerical Analysis, Vol. 36, Issue 5 https://doi.org/10.1051/m2an:2002035	journal	September 2002
Reduced basis method for finite volume approximations of parametrized linear evolution equations Haasdonk, Bernard; Ohlberger, Mario ESAIM: Mathematical Modelling and Numerical Analysis, Vol. 42, Issue 2 https://doi.org/10.1051/m2an:2008001	journal	March 2008
Efficient reduced models and a posteriori error estimation for parametrized dynamical systems by offline/online decomposition Haasdonk, Bernard; Ohlberger, Mario Mathematical and Computer Modelling of Dynamical Systems, Vol. 17, Issue 2 https://doi.org/10.1080/13873954.2010.514703	journal	March 2011
Rational approximation of frequency domain responses by vector fitting Gustavsen, B.; Semlyen, A. IEEE Transactions on Power Delivery, Vol. 14, Issue 3 https://doi.org/10.1109/61.772353	journal	July 1999
Reliable Real-Time Solution of Parametrized Partial Differential Equations: Reduced-Basis Output Bound Methods Prud’homme, C.; Rovas, D. V.; Veroy, K. Journal of Fluids Engineering, Vol. 124, Issue 1 https://doi.org/10.1115/1.1448332	journal	November 2001
Norm and Trace Estimation with Random Rank-one Vectors Bujanovic, Zvonimir; Kressner, Daniel SIAM Journal on Matrix Analysis and Applications, Vol. 42, Issue 1 https://doi.org/10.1137/20m1331718	journal	January 2021
Certified real-time solution of the parametrized steady incompressible Navier-Stokes equations: rigorous reduced-basisa posteriori error bounds Veroy, K.; Patera, A. T. International Journal for Numerical Methods in Fluids, Vol. 47, Issue 8-9 https://doi.org/10.1002/fld.867	journal	January 2005
Certified Reduced Basis Methods for Parametrized Partial Differential Equations Hesthaven, Jan S.; Rozza, Gianluigi; Stamm, Benjamin SpringerBriefs in Mathematics https://doi.org/10.1007/978-3-319-22470-1	book	January 2016
Reduced basis approximation and a posteriori error estimation for affinely parametrized elliptic coercive partial differential equations Rozza, G.; Huynh, D. B. P.; Patera, A. T. Archives of Computational Methods in Engineering, Vol. 15, Issue 3 https://doi.org/10.1007/BF03024948	journal	September 2007
Reduced basis approximation and a posteriori error estimation for the time-dependent viscous Burgers’ equation Nguyen, Ngoc-Cuong; Rozza, Gianluigi; Patera, Anthony T. Calcolo, Vol. 46, Issue 3 https://doi.org/10.1007/s10092-009-0005-x	journal	June 2009
A successive constraint linear optimization method for lower bounds of parametric coercivity and inf–sup stability constants Huynh, D. B. P.; Rozza, G.; Sen, S. Comptes Rendus Mathematique, Vol. 345, Issue 8 https://doi.org/10.1016/j.crma.2007.09.019	journal	October 2007
A posteriori error bounds for the empirical interpolation method Eftang, Jens L.; Grepl, Martin A.; Patera, Anthony T. Comptes Rendus Mathematique, Vol. 348, Issue 9-10 https://doi.org/10.1016/j.crma.2010.03.004	journal	May 2010
Lift & Learn: Physics-informed machine learning for large-scale nonlinear dynamical systems Qian, Elizabeth; Kramer, Boris; Peherstorfer, Benjamin Physica D: Nonlinear Phenomena, Vol. 406 https://doi.org/10.1016/j.physd.2020.132401	journal	May 2020
Spectral analysis of nonlinear flows Rowley, Clarence W.; MeziĆ, Igor; Bagheri, Shervin Journal of Fluid Mechanics, Vol. 641 https://doi.org/10.1017/S0022112009992059	journal	November 2009
Dynamic mode decomposition of numerical and experimental data Schmid, Peter J. Journal of Fluid Mechanics, Vol. 656 https://doi.org/10.1017/S0022112010001217	journal	July 2010
Some a posteriori error bounds for reduced-order modelling of (non-)parametrized linear systems Feng, Lihong; Antoulas, Athanasios C.; Benner, Peter ESAIM: Mathematical Modelling and Numerical Analysis, Vol. 51, Issue 6 https://doi.org/10.1051/m2an/2017014	journal	November 2017
A posteriori error bounds for reduced-basis approximations of parametrized parabolic partial differential equations Grepl, Martin A.; Patera, Anthony T. ESAIM: Mathematical Modelling and Numerical Analysis, Vol. 39, Issue 1 https://doi.org/10.1051/m2an:2005006	journal	January 2005
A Survey of Model Reduction by Balanced Truncation and Some New Results Gugercin, Serkan; Antoulas, Athanasios C. International Journal of Control, Vol. 77, Issue 8 https://doi.org/10.1080/00207170410001713448	journal	May 2004
Partial Differential Equations Evans, Lawrence Graduate Studies in Mathematics https://doi.org/10.1090/gsm/019	book	January 1998
Estimating Extremal Eigenvalues and Condition Numbers of Matrices Dixon, John D. SIAM Journal on Numerical Analysis, Vol. 20, Issue 4 https://doi.org/10.1137/0720053	journal	August 1983
Dynamic Mode Decomposition Kutz, J. Nathan; Brunton, Steven L.; Brunton, Bingni W. Society for Industrial and Applied Mathematics https://doi.org/10.1137/1.9781611974508	book	January 2016
A State Space Error Estimate for POD-DEIM Nonlinear Model Reduction Chaturantabut, Saifon; Sorensen, Danny C. SIAM Journal on Numerical Analysis, Vol. 50, Issue 1 https://doi.org/10.1137/110822724	journal	January 2012
A Posteriori Error Estimation for DEIM Reduced Nonlinear Dynamical Systems Wirtz, D.; Sorensen, D. C.; Haasdonk, B. SIAM Journal on Scientific Computing, Vol. 36, Issue 2 https://doi.org/10.1137/120899042	journal	January 2014
Data-Driven Parametrized Model Reduction in the Loewner Framework Ionita, A. C.; Antoulas, A. C. SIAM Journal on Scientific Computing, Vol. 36, Issue 3 https://doi.org/10.1137/130914619	journal	January 2014
A Survey of Projection-Based Model Reduction Methods for Parametric Dynamical Systems Benner, Peter; Gugercin, Serkan; Willcox, Karen SIAM Review, Vol. 57, Issue 4 https://doi.org/10.1137/130932715	journal	January 2015
An Efficient Output Error Estimation for Model Order Reduction of Parametrized Evolution Equations Zhang, Yongjin; Feng, Lihong; Li, Suzhou SIAM Journal on Scientific Computing, Vol. 37, Issue 6 https://doi.org/10.1137/140998603	journal	January 2015
Vector Fitting for Matrix-valued Rational Approximation Drmač, Z.; Gugercin, S.; Beattie, C. SIAM Journal on Scientific Computing, Vol. 37, Issue 5 https://doi.org/10.1137/15M1010774	journal	January 2015
Model Reduction of Bilinear Systems in the Loewner Framework Antoulas, A. C.; Gosea, I. V.; Ionita, A. C. SIAM Journal on Scientific Computing, Vol. 38, Issue 5 https://doi.org/10.1137/15M1041432	journal	January 2016
System Identification via CUR-Factored Hankel Approximation Kramer, Boris; Gorodetsky, Alex A. SIAM Journal on Scientific Computing, Vol. 40, Issue 2 https://doi.org/10.1137/17M1137632	journal	January 2018
Randomized Residual-Based Error Estimators for Parametrized Equations Smetana, Kathrin; Zahm, Olivier; Patera, Anthony T. SIAM Journal on Scientific Computing, Vol. 41, Issue 2 https://doi.org/10.1137/18M120364X	journal	January 2019
Sampling Low-Dimensional Markovian Dynamics for Preasymptotically Recovering Reduced Models from Data with Operator Inference Peherstorfer, Benjamin SIAM Journal on Scientific Computing, Vol. 42, Issue 5 https://doi.org/10.1137/19M1292448	journal	January 2020
Norm and Trace Estimation with Random Rank-one Vectors Bujanovic, Zvonimir; Kressner, Daniel SIAM Journal on Matrix Analysis and Applications, Vol. 42, Issue 1 https://doi.org/10.1137/20M1331718	journal	January 2021
Certified reduced basis approximation for parametrized partial differential equations and applications Quarteroni, Alfio; Rozza, Gianluigi; Manzoni, Andrea Journal of Mathematics in Industry, Vol. 1, Issue 1 https://doi.org/10.1186/2190-5983-1-3	journal	January 2011
An eigensystem realization algorithm for modal parameter identification and model reduction Juang, Jer-Nan; Pappa, Richard S. Journal of Guidance, Control, and Dynamics, Vol. 8, Issue 5 https://doi.org/10.2514/3.20031	journal	September 1985
A Posteriori Error Bounds for Reduced-Basis Approximation of Parametrized Noncoercive and Nonlinear Elliptic Partial Differential Equations Veroy, Karen; Prud'homme, Christophe; Rovas, Dimitrios 16th AIAA Computational Fluid Dynamics Conference https://doi.org/10.2514/6.2003-3847	conference	June 2003
On dynamic mode decomposition: Theory and applications Kutz, J. Nathan; Brunton, Steven L.; Luchtenburg, Dirk M. Journal of Computational Dynamics, Vol. 1, Issue 2 https://doi.org/10.3934/jcd.2014.1.391	journal	December 2014

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