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
SN - 2822-7840
VL - 55
SP - 735
EP - 761
JO - ESAIM: Mathematical Modelling and Numerical Analysis
JF - ESAIM: Mathematical Modelling and Numerical Analysis
IS - 3
ER -