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A measure approximation for distributionally robust PDE-constrained optimization problems

SIAM Journal on Numerical Analysis

Kouri, Drew P.

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. In this paper, 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. We prove consistency of our approximated min-max problem and conclude with numerical results.

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TYPE Journal Article YEAR 2017
ScopusOSTIDOI

Data-Driven Optimization for the Design and Control of Large-Scale Systems: LDRD Final Report

Kouri, Drew P.

Engineering decisions are often formulated as optimization problems such as the optimal design or control of physical systems. In these applications, the resulting optimization problems are con- strained by large-scale simulations involving systems of partial differential equations (PDEs), or- dinary differential equations (ODEs), and differential algebraic equations (DAEs). In addition, critical components of these systems are fraught with uncertainty, including unverifiable model- ing assumptions, unknown boundary and initial conditions, and uncertain coefficients. Typically, these components are estimated using noisy and incomplete data from a variety of sources (e.g., physical experiments). The lack of knowledge of the true underlying probabilistic characterization of model inputs motivates the need for optimal solutions that are robust to this uncertainty. In this report, we introduce a framework for handling "distributional" uncertainties in the context of simulation-based optimization. This includes a novel measure discretization technique that will lead to an adaptive optimization algorithm tailored to exploit the structures inherent to simulation- based optimization.

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TYPE SAND Report YEAR 2016
OSTIDOI

Risk-averse PDE-constrained optimization using the conditional value-at-risk

SIAM Journal on Optimization

Kouri, Drew P.; Surowiec, T.M.

Uncertainty is inevitable when solving science and engineering application problems. In the face of uncertainty, it is essential to determine robust and risk-averse solutions. In this work, we consider a class of PDE-constrained optimization problems in which the PDE coefficients and inputs may be uncertain. We introduce two approximations for minimizing the conditional value-atrisk (CVaR) for such PDE-constrained optimization problems. These approximations are based on the primal and dual formulations of CVaR. For the primal problem, we introduce a smooth approximation of CVaR in order to utilize derivative-based optimization algorithms and to take advantage of the convergence properties of quadrature-based discretizations. For this smoothed CVaR, we prove differentiability as well as consistency of our approximation. For the dual problem, we regularize the inner maximization problem, rigorously derive optimality conditions, and demonstrate the consistency of our approximation. Furthermore, we propose a fixed-point iteration that takes advantage of the structure of the regularized optimality conditions and provides a means of calculating worst-case probability distributions based on the given probability level. We conclude with numerical results.

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TYPE Presentation YEAR 2016
ScopusOSTIDOI

Inversion for Eigenvalues and Modes Using Sierra-SD and ROL

Walsh, Timothy W.; Aquino, Wilkins A.; Ridzal, Denis R.; Kouri, Drew P.

In this report we formulate eigenvalue-based methods for model calibration using a PDE-constrained optimization framework. We derive the abstract optimization operators from first principles and implement these methods using Sierra-SD and the Rapid Optimization Library (ROL). To demon- strate this approach, we use experimental measurements and an inverse solution to compute the joint and elastic foam properties of a low-fidelity unit (LFU) model.

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TYPE SAND Report YEAR 2015
OSTIDOI
Results 51–75 of 89
Results 51–75 of 89