Volume 55, Number 3, May-June 2021
|Page(s)||735 - 761|
|Published online||05 May 2021|
Probabilistic error estimation for non-intrusive reduced models learned from data of systems governed by linear parabolic partial differential equations
Courant Institute of Mathematical Sciences, New York, NY 10012, USA
* Corresponding author: firstname.lastname@example.org
Accepted: 22 February 2021
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.
Mathematics Subject Classification: 65M60 / 65P99 / 65Y99 / 65F99 / 93C05 / 62G15 / 65C05 / 65N15
Key words: Model reduction / error estimation / non-intrusive model reduction / small sample statistical estimates
© EDP Sciences, SMAI 2021
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