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. 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.
Keywords: Model reduction, error estimation, non-intrusive model reduction, small sample statistical estimates
@article{M2AN_2021__55_3_735_0,
author = {Uy, Wayne Isaac Tan and Peherstorfer, Benjamin},
title = {Probabilistic error estimation for non-intrusive reduced models learned from data of systems governed by linear parabolic partial differential equations},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {735--761},
year = {2021},
publisher = {EDP-Sciences},
volume = {55},
number = {3},
doi = {10.1051/m2an/2021010},
mrnumber = {4253169},
language = {en},
url = {https://www.numdam.org/articles/10.1051/m2an/2021010/}
}
TY - JOUR AU - Uy, Wayne Isaac Tan AU - Peherstorfer, Benjamin TI - Probabilistic error estimation for non-intrusive reduced models learned from data of systems governed by linear parabolic partial differential equations JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2021 SP - 735 EP - 761 VL - 55 IS - 3 PB - EDP-Sciences UR - https://www.numdam.org/articles/10.1051/m2an/2021010/ DO - 10.1051/m2an/2021010 LA - en ID - M2AN_2021__55_3_735_0 ER -
%0 Journal Article %A Uy, Wayne Isaac Tan %A Peherstorfer, Benjamin %T Probabilistic error estimation for non-intrusive reduced models learned from data of systems governed by linear parabolic partial differential equations %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2021 %P 735-761 %V 55 %N 3 %I EDP-Sciences %U https://www.numdam.org/articles/10.1051/m2an/2021010/ %R 10.1051/m2an/2021010 %G en %F M2AN_2021__55_3_735_0
Uy, Wayne Isaac Tan; Peherstorfer, Benjamin. Probabilistic error estimation for non-intrusive reduced models learned from data of systems governed by linear parabolic partial differential equations. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 55 (2021) no. 3, pp. 735-761. doi: 10.1051/m2an/2021010
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