PDE-constrained Optimization under Uncertainty

Kouri, Drew Philip; Surowiec, Thomas Michael

Title: PDE-constrained Optimization under Uncertainty

Full Record
Other Related Research

Abstract

Uncertainty is pervasive in all science and engineering applications. Incorporating uncertainty in physical models is therefore both natural and vital. In doing so, we often arrive at parametric systems of partial differential equations (PDEs). When passing from simulation to optimization, we obtain (typically nonconvex) infinite-dimensional optimization problems that, upon discretization, result in extremely large-scale nonlinear programs. For example, consider a linear elliptic PDE on a two- dimensional domain with a single random coeffcient. If we sampled the random input with 10,000 realizations of the coeffcient, the resulting optimization problem would have 10,000 PDE constraints. Furthermore, discretizing each PDE with piecewise linear finite elements on a 100X100 uni- form quadrilateral mesh results in 100,000,000 degrees of freedom. As a result, the critical components for ensuring mesh-independent performance of numerical optimization methods in the deterministic setting, for example, solution regularity and generalized differentiability, are even more critical in the stochastic setting.

Authors:

Kouri, Drew Philip ^[1]; Surowiec, Thomas Michael ^[2]

Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)
Philipps-Univ. Marburg (Germany)

Publication Date:: 2017-11-01

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

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

OSTI Identifier:: 1429698

Report Number(s):: SAND-2017-12775J
Journal ID: ISSN 9999-0050; 659023

Grant/Contract Number:: AC04-94AL85000

Resource Type:: Accepted Manuscript

Journal Name:: SIAG/OPT Views and News

Additional Journal Information:: Journal Volume: 23; Journal Issue: 2; Journal ID: ISSN 9999-0050

Publisher:: SIAM Activity Group on Optimization

Country of Publication:: United States

Language:: English

Subject:: 42 ENGINEERING

Citation Formats


                    Kouri, Drew Philip, and Surowiec, Thomas Michael. PDE-constrained Optimization under Uncertainty.  United States: N. p., 2017. 
Web.

Copy to clipboard


                    Kouri, Drew Philip, & Surowiec, Thomas Michael. PDE-constrained Optimization under Uncertainty.  United States.

Copy to clipboard


                    Kouri, Drew Philip, and Surowiec, Thomas Michael. Wed .  
"PDE-constrained Optimization under Uncertainty".  United States.  https://www.osti.gov/servlets/purl/1429698.

Copy to clipboard


                    
@article{osti_1429698,

  title        = {PDE-constrained Optimization under Uncertainty},

  author       = {Kouri, Drew Philip and Surowiec, Thomas Michael},

  abstractNote = {Uncertainty is pervasive in all science and engineering applications. Incorporating uncertainty in physical models is therefore both natural and vital. In doing so, we often arrive at parametric systems of partial differential equations (PDEs). When passing from simulation to optimization, we obtain (typically nonconvex) infinite-dimensional optimization problems that, upon discretization, result in extremely large-scale nonlinear programs. For example, consider a linear elliptic PDE on a two- dimensional domain with a single random coeffcient. If we sampled the random input with 10,000 realizations of the coeffcient, the resulting optimization problem would have 10,000 PDE constraints. Furthermore, discretizing each PDE with piecewise linear finite elements on a 100X100 uni- form quadrilateral mesh results in 100,000,000 degrees of freedom. As a result, the critical components for ensuring mesh-independent performance of numerical optimization methods in the deterministic setting, for example, solution regularity and generalized differentiability, are even more critical in the stochastic setting.},

  doi          = {},

  journal      = {SIAG/OPT Views and News},

  number       = 2,

  volume       = 23,

  place        = {United States},

  year         = {2017},

  month        = {11}

}

Copy to clipboard

Journal Article:

Free Publicly Available Full Text

Accepted Manuscript (DOE)

Publisher's Version of Record

The DOI is not currently available

Other availability

Search WorldCat to find libraries that may hold this journal

Save / Share:

Export Metadata

Save to My Library

Similar Records in DOE PAGES and OSTI.GOV collections:

A time-parallel method for the solution of PDE-constrained optimization problems

Technical Report Hajghassem, Mona ; Cyr, Eric C. ; Ridzal, Denis

We study a time-parallel approach to solving quadratic optimization problems with linear time-dependent partial differential equation (PDE) constraints. These problems arise in formulations of optimal control, optimal design and inverse problems that are governed by parabolic PDE models. They may also arise as subproblems in algorithms for the solution of optimization problems with nonlinear time-dependent PDE constraints, e.g., in sequential quadratic programming methods. We apply a piecewise linear finite element discretization in space to the PDE constraint, followed by the Crank-Nicolson discretization in time. The objective function is discretized using finite elements in space and the trapezoidal rule in time.more »« less
https://doi.org/10.2172/1430466

Full Text Available
Risk-averse optimal control of semilinear elliptic PDEs

Journal Article Philip Kouri, Drew ; Surowiec, Thomas - ESAIM: Control, Optimisation and Calculus of Variations

In this paper, we consider the optimal control of semilinear elliptic PDEs with random inputs. These problems are often nonconvex, infinite-dimensional stochastic optimization problems for which we employ risk measures to quantify the implicit uncertainty in the objective function. In contrast to previous works in uncertainty quantification and stochastic optimization, we provide a rigorous mathematical analysis demonstrating higher solution regularity (in stochastic state space), continuity and differentiability of the control-to-state map, and existence, regularity and continuity properties of the control-to-adjoint map. Our proofs make use of existing techniques from PDE-constrained optimization as well as concepts from the theory of measurablemore »« less
https://doi.org/10.1051/cocv/2019061

Full Text Available
Hyper-Differential Sensitivity Analysis of Uncertain Parameters in PDE-Constrained Optimization

Journal Article van Bloemen Waanders, Bart ; Hart, Joseph ; Herzog, Roland - International Journal for Uncertainty Quantification

Many problems in engineering and sciences require the solution of large scale optimization constrained by partial differential equations (PDEs). Though PDE-constrained optimization is itself challenging, most applications pose additional complexity, namely, uncertain parameters in the PDEs. Uncertainty quantification (UQ) is necessary to characterize, prioritize, and study the influence of these uncertain parameters. Sensitivity analysis, a classical tool in UQ, is frequently used to study the sensitivity of a model to uncertain parameters. In this article, we introduce "hyper-differential sensitivity analysis" which considers the sensitivity of the solution of a PDE-constrained optimization problem to uncertain parameters. Our approach is a goal-orientedmore »« less
https://doi.org/10.1615/Int.J.UncertaintyQuantification.2020032480

Full Text Available
Existence and Optimality Conditions for Risk-Averse PDE-Constrained Optimization

Journal Article Kouri, Drew Philip ; Surowiec, Thomas M. - SIAM/ASA Journal on Uncertainty Quantification

Uncertainty is ubiquitous in virtually all engineering applications, and, for such problems, it is inadequate to simulate the underlying physics without quantifying the uncertainty in unknown or random inputs, boundary and initial conditions, and modeling assumptions. Here in this paper, we introduce a general framework for analyzing risk-averse optimization problems constrained by partial differential equations (PDEs). In particular, we postulate conditions on the random variable objective function as well as the PDE solution that guarantee existence of minimizers. Furthermore, we derive optimality conditions and apply our results to the control of an environmental contaminant. Lastly, we introduce a new riskmore »« less
https://doi.org/10.1137/16M1086613

Full Text Available
A solution framework for linear PDE-constrained mixed-integer problems

Journal Article Gnegel, Fabian ; Fügenschuh, Armin ; Hagel, Michael ; ... - Mathematical Programming

Abstract We present a general numerical solution method for control problems with state variables defined by a linear PDE over a finite set of binary or continuous control variables. We show empirically that a naive approach that applies a numerical discretization scheme to the PDEs to derive constraints for a mixed-integer linear program (MILP) leads to systems that are too large to be solved with state-of-the-art solvers for MILPs, especially if we desire an accurate approximation of the state variables. Our framework comprises two techniques to mitigate the rise of computation times with increasing discretization level: First, the linear systemmore »« less
https://doi.org/10.1007/s10107-021-01626-1

Similar Records