Drew Philip Kouri

Risk-Averse PDE-Constrained Optimization Using the Conditional Value-At-Risk

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-at-risk (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...

Inexact Objective Function Evaluations in a Trust-Region Algorithm for PDE-Constrained Optimization under Uncertainty

This paper improves the trust-region algorithm with adaptive sparse grids introduced in [SIAM J. Sci. Comput., 35 (2013), pp. A1847--A1879] for the solution of optimization problems governed by partial differential equations (PDEs) with uncertain coefficients. The previous algorithm used adaptive sparse-grid discretizations to generate models that are applied in a trust-region framework to generate a trial step. The decision whether to accept this trial step as the new iterate, however, required relatively high-fidelity adaptive discretizations of the objective function. In this paper, we extend the algorithm and convergence theory to allow the use of low-fidelity adaptive sparse-grid models in objective function evaluations. This is accomplished by extending conditions on inexact function evaluations used in previous trust-region frameworks. Our algorithm adaptively builds two separate sparse grids: one to generate optimization models for the step computation and one to approximate the objective function...

Efficient multiframe super-resolution for imagery with lateral shifts

Trade studies used to design optical imaging systems frequently result in systems being undersampled. The resolution of such systems is limited by the finite size of the detector pixels rather than the cutoff spatial frequency of the optical system. Multiframe super-resolution techniques can be used to combine a number of spatially displaced images from such systems into a single, high-resolution image. Nonlinear optimization techniques have frequently been used to solve this problem. Such techniques define an objective function and use numerical optimization methods to obtain a solution. These numerical methods are often more efficient when derivatives of the objective function with respect to the variables can be calculated analytically rather than numerically. We demonstrate for the simple motion model of pure lateral translation that the derivatives of the objective function with respect to the image lateral shifts can be calculated analytically to speed optimization calculations.

Inexact Trust-Region Methods for PDE-Constrained Optimization

Numerical solution of optimization problems with partial differential equation (PDE) constraints typically requires inexact objective function and constraint evaluations, derivative approximations, and the use of iterative linear system solvers. Over the last 30 years, trust-region methods have been extended to rigorously, robustly, and efficiently handle various sources of inexactness in the optimization process. In this chapter, we review some of the recent advances, discuss their key algorithmic contributions, and present numerical examples that demonstrate how inexact computations can be exploited to enable the solution of large-scale PDE-constrained optimization problems.

Spectral risk measures: the risk quadrangle and optimal approximation

We develop a general risk quadrangle that gives rise to a large class of spectral risk measures. The statistic of this new risk quadrangle is the average value-at-risk at a specific confidence level. As such, this risk quadrangle generates a continuum of error measures that can be used for superquantile regression. For risk-averse optimization, we introduce an optimal approximation of spectral risk measures using quadrature. We prove the consistency of this approximation and demonstrate our results through numerical examples.

Inversion for Eigenvalues and Modes Using Sierra-SD and ROL.

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.

Viscoelastic material inversion using Sierra-SD and ROL

In this report we derive frequency-domain methods for inverse characterization of the constitutive parameters of viscoelastic materials. The inverse problem is cast in a PDE-constrained optimization framework with efficient computation of gradients and Hessian vector products through matrix free operations. The abstract optimization operators for first and second derivatives are derived from first principles. Various methods from the Rapid Optimization Library (ROL) are tested on the viscoelastic inversion problem. The methods described herein are applied to compute the viscoelastic bulk and shear moduli of a foam block model, which was recently used in experimental testing for viscoelastic property characterization.