TY - JOUR

T1 - Probabilistic error estimation for non-intrusive reduced models learned from data of systems governed by linear parabolic partial differential equations

AU - Uy, Wayne Isaac Tan

AU - Peherstorfer, Benjamin

N1 - Funding Information:
Acknowledgements. This work was partially supported by US Department of Energy, Office of Advanced Scientific Computing Research, Applied Mathematics Program (Program Manager Dr. Steven Lee), DOE Award DESC0019334, and by the National Science Foundation under Grant No. 1901091.
Publisher Copyright:
© EDP Sciences, SMAI 2021.

PY - 2021/5/1

Y1 - 2021/5/1

N2 - 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.

AB - 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.

KW - Error estimation

KW - Model reduction

KW - Non-intrusive model reduction

KW - Small sample statistical estimates

UR - http://www.scopus.com/inward/record.url?scp=85105726375&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85105726375&partnerID=8YFLogxK

U2 - 10.1051/m2an/2021010

DO - 10.1051/m2an/2021010

M3 - Article

AN - SCOPUS:85105726375

VL - 55

SP - 735

EP - 761

JO - ESAIM: Mathematical Modelling and Numerical Analysis

JF - ESAIM: Mathematical Modelling and Numerical Analysis

SN - 0764-583X

IS - 3

ER -