# A Measure Approximation for Distributionally Robust PDE-Constrained Optimization Problems

## Abstract

In numerous applications, scientists and engineers acquire varied forms of data that partially characterize the inputs to an underlying physical system. This data is then used to inform decisions such as controls and designs. Consequently, it is critical that the resulting control or design is robust to the inherent uncertainties associated with the unknown probabilistic characterization of the model inputs. Here in this work, we consider optimal control and design problems constrained by partial differential equations with uncertain inputs. We do not assume a known probabilistic model for the inputs, but rather we formulate the problem as a distributionally robust optimization problem where the outer minimization problem determines the control or design, while the inner maximization problem determines the worst-case probability measure that matches desired characteristics of the data. We analyze the inner maximization problem in the space of measures and introduce a novel measure approximation technique, based on the approximation of continuous functions, to discretize the unknown probability measure. Finally, we prove consistency of our approximated min-max problem and conclude with numerical results.

- Authors:

- Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)

- Publication Date:

- Research Org.:
- Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)

- Sponsoring Org.:
- USDOE National Nuclear Security Administration (NNSA)

- OSTI Identifier:
- 1429682

- Report Number(s):
- SAND-2017-12112J

Journal ID: ISSN 0036-1429; 658537

- Grant/Contract Number:
- AC04-94AL85000; NA0003525

- Resource Type:
- Journal Article: Accepted Manuscript

- Journal Name:
- SIAM Journal on Numerical Analysis

- Additional Journal Information:
- Journal Volume: 55; Journal Issue: 6; Journal ID: ISSN 0036-1429

- Publisher:
- Society for Industrial and Applied Mathematics

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 97 MATHEMATICS AND COMPUTING; 54 ENVIRONMENTAL SCIENCES; PDE-constrained optimization; uncertainty; distributionally robust; data-driven

### Citation Formats

```
Kouri, Drew Philip.
```*A Measure Approximation for Distributionally Robust PDE-Constrained Optimization Problems*. United States: N. p., 2017.
Web. doi:10.1137/15M1036944.

```
Kouri, Drew Philip.
```*A Measure Approximation for Distributionally Robust PDE-Constrained Optimization Problems*. United States. doi:10.1137/15M1036944.

```
Kouri, Drew Philip. Tue .
"A Measure Approximation for Distributionally Robust PDE-Constrained Optimization Problems". United States.
doi:10.1137/15M1036944.
```

```
@article{osti_1429682,
```

title = {A Measure Approximation for Distributionally Robust PDE-Constrained Optimization Problems},

author = {Kouri, Drew Philip},

abstractNote = {In numerous applications, scientists and engineers acquire varied forms of data that partially characterize the inputs to an underlying physical system. This data is then used to inform decisions such as controls and designs. Consequently, it is critical that the resulting control or design is robust to the inherent uncertainties associated with the unknown probabilistic characterization of the model inputs. Here in this work, we consider optimal control and design problems constrained by partial differential equations with uncertain inputs. We do not assume a known probabilistic model for the inputs, but rather we formulate the problem as a distributionally robust optimization problem where the outer minimization problem determines the control or design, while the inner maximization problem determines the worst-case probability measure that matches desired characteristics of the data. We analyze the inner maximization problem in the space of measures and introduce a novel measure approximation technique, based on the approximation of continuous functions, to discretize the unknown probability measure. Finally, we prove consistency of our approximated min-max problem and conclude with numerical results.},

doi = {10.1137/15M1036944},

journal = {SIAM Journal on Numerical Analysis},

number = 6,

volume = 55,

place = {United States},

year = {Tue Dec 19 00:00:00 EST 2017},

month = {Tue Dec 19 00:00:00 EST 2017}

}