# Poster Session and Reception

Thursday, June 9, 2016 - 4:15pm - 6:00pm

Lind 400

**Efficient PDE-Constrained Optimization using Adaptive Model Reduction**

Matthew Zahr (Stanford University)

Optimization problems constrained by partial differential equations are ubiquitous in modern science and engineering. They play a central role in optimal design and control of multiphysics systems, as well as nondestructive evaluation and execution, and inverse problems. Methods to solve these optimization problems rely on, potentially many, numerical solutions of the underlying equations. For complicated physical interactions taking place on complex domains, these solutions will be computationally expensive — in terms of both time and resources — to obtain, rending the optimization procedure difficult or intractable. This situation is exacerbated when a risk-averse solution is sought that accounts for the uncertainties inherent in any physical system.

This work introduces a framework for accelerating optimization problems governed by stochastic partial differential equations by leveraging adaptive sparse grids and model reduction. Adaptive sparse grids perform efficient integration approximation in a high-dimensional stochastic space and reduced-order models reduce the cost of objective and gradient evaluations by decreasing the complexity of primal and dual PDE solves. A globally convergent trust-region framework accounts for these two levels of inexactness in the objective and gradient.**Impact of Using Hypothetical Tracer Data on Reservoir Parameter Estimation Problems**

Dimitar Trenev (ExxonMobil)

This poster summarizes work assessing the impact of tracer data – the flow of a hypothetical tracer, which could be tracked remotely from the earth surface - on oil reservoir parameter estimation. Numerical examples are presented, indicating that the additional information provided by the tracer data is largely localized and propagates with the tracer front. It is observed that, even with the addition of the tracer data, the reservoir permeability scalar field may not be uniquely inverted.**Accelerating Newton's Method for Large-Scale Time-Dependent Optimal Control via Reduced- Order Modeling**

Caleb Magruder (Rice University)

We use projection based reduced order models (ROMs) to substantially decrease the computational cost of Newton's method for large-scale time-dependent optimal control problems. Previous approaches of using projection based ROMs in optimization solve sequences of computationally less expensive optimization problems, which are obtained by replacing the high fidelity discretization of the governing partial differential equation (PDE) by a ROM. For several nonlinear PDEs this approach may not be feasible or efficient for the following reasons. 1) Projection based ROMs may not be well-posed because the ROM solution does not satisfy properties obeyed by the solution of the high-fidelity PDE discretization, such as conservation properties, or non-negativity. 2) ROMs can become untrustworthy when the controls differ slightly from those at which the ROM was generated. 3) Validation that a ROM is still sufficiently accurate at a new control can be very expensive.

Therefore, we use the high-fidelity PDE discretization for objective function and gradient evaluation, and use state PDE and adjoint PDE solves to generate a ROM to compute approximate Hessian information. The ROM generation is computationally inexpensive, since it reuses information that is already computed for objective function and gradient evaluation, and the ROM-Hessian approximations are inexpensive to evaluate. The resulting method often enjoys convergence rates similar to those of Newton's method, but at the computational cost per iteration of the gradient method. We demonstrate our approach on a semilinear parabolic optimal control problem.**A Space-time Fractional Optimal Control Problem: Analysis and Discretization**

Abner Salgado (University of Tennessee)

We study a linear-quadratic optimal control problem involving a parabolic equation with fractional diffusion and Caputo fractional time derivative of orders $s \in (0,1)$ and $\gamma \in (0,1]$, respectively. The spatial fractional diffusion is realized as the Dirichlet-to-Neumann map for a nonuniformly elliptic operator. Thus, we consider an equivalent formulation with a quasi-stationary elliptic problem with a dynamic boundary condition as state equation. The rapid decay of the solution to this problem suggests a truncation that is suitable for numerical approximation. We consider a fully-discrete scheme: piecewise constant functions for the control and, for the state, first-degree tensor product finite elements in space and a finite difference discretization in time. We show convergence of this scheme and, for $s\in(0,1)$ and $\gamma = 1$, we derive a priori error estimates.**Efficient Multigrid Methods for Large-scale Optimization Problems Constrained by PDEs**

Andrei Draganescu (University of Maryland Baltimore County)

This project is centered on developing efficient solvers for optimization problems constrained by partial differential equations (PDEs), with main focus on multigrid methods. Our strategy targets the reduced optimization problem obtained by eliminating the state variables from the PDE constraints, which we then solve using the most efficient optimization method available for the problem – preferably second-order based, superlinearly convergent algorithms such as Newton’s method or the semismooth Newton’s method. Thus, the challenge remains to devise efficient preconditioners for the linear systems arising in the optimization iteration. The focus of our work has been on developing optimal-order multigrid preconditioners for the linear systems under scrutiny. By using these preconditioners in conjunction with a Krylov-space solver, the process converges in a number of iterations that decreases with increasing resolution. Consequently, relatively few fine-level matrix-vector multiplications are required at the finest level. This is critical, since for the reduced formulation each matrix-vector multiplication is equivalent in cost to two linearized PDE solves. The strategy outlined proved efficient in a number of model problems that include distributed optimal control problems constrained by linear and semilinear elliptic or parabolic equations, and by the steady-state Navier-Stokes equations. A case study involving the latter example is presented.**Mitigating Cycle Skipping Issue in FWI: An Optimal Transport Investigation**

Jean Virieux (University Grenoble Alpes)**A Coupling Strategy for Nonlocal and Local Diffusion Models with Mixed Volume Constraints and Boundary Conditions**

Marta D'Elia (Sandia National Laboratories)

The use of nonlocal models in science and engineering applications has been steadily increasing over the past decade. The ability of nonlocal theories to accurately capture effects that are difficult or impossible to represent by PDE models motivates and drives the interest in this type of simulations. However, the improved accuracy of nonlocal models comes at the price of a significant increase in computational costs compared to, e.g., traditional PDEs. As a result, it is important to develop Local-to-Nonlocal coupling strategies, which aim to combine the accuracy of nonlocal models with the computational efficiency of PDEs. The basic idea is to use the more efficient PDE model everywhere except in those parts of the domain that require the improved accuracy of the nonlocal model.

We develop and analyze an optimization-based method for the coupling of nonlocal and local diffusion problems with mixed volume constraints and boundary conditions. The approach formulates the coupling as a control problem where the states are the solutions of the nonlocal and local equations, the objective is to minimize their mismatch on the overlap of the nonlocal and local domains, and the controls are virtual volume constraints and boundary conditions. We prove that the resulting optimization problem is well-posed and we implement it using Sandia's agile software components toolkit. Numerical results for nonlocal diffusion in three-dimensions illustrate key properties of the coupling method.**Coupling of Nonlocal and Local Diffusion Through a Divergence Formulation**

Xingjie Li (University of North Carolina)

We developed a new consistent quasinonlocal coupling diffusion model in the spirit of atomistic-to-continuum modeling for solid crystal. We proved that the coupling model is symmetric, patch-test consistent and pointwise convergent to the local diffusion limit. In addition, the coupling model is positive-definite with respect to the energy norms induced by the nonlocal diffusion kernels as well as the L^2 norm, and it satisfies the maximum principle as the local diffusion does. Besides the discussions on the continuous level, we also provided a first order finite difference approximation for the implementation purpose. This numerical approximation keeps the symmetry, consistency, positive-definiteness and maximum principle. In addition, the first order accuracy is verified in the numerical examples. Furthermore, we demonstrated a special example which shows the discrepancy between the fully nonlocal and fully local diffusion, whereas the result of the coupling diffusion kernel agrees with that of the fully nonlocal one.**On Parametrized Domain Optimization Problems**

Cristina Letona Bolivar (Virginia Polytechnic Institute and State University)

We study domain optimization problems subject to PDE constraints. Problems that involve Dirichlet constraints, or Neumann constraints, have been extensively studied; but not much is known when mixed boundary conditions are given. To solve this type of minimization problem we try to find the parametrization of the minimizing domain. The goal is to describe the conditions so that existence of the (weak) solution can be guaranteed. Efficient means to calculate the shape gradient are required so that we can develop practical numerical algorithms to approximate the solution.