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A Relaxation-based Probabilistic Approach for PDE-constrained Optimization under Uncertainty with Pointwise State Constraints

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We consider a class of convex risk-neutral PDE-constrained optimization problems subject to pointwise control and state constraints. Due to the many challenges associated with almost sure constraints on pointwise evaluations of the state, we suggest a relaxation via a smooth functional bound with similar properties to well-known probability constraints. First, we introduce and analyze the relaxed problem, discuss its asymptotic properties, and derive formulae for the gradient using the adjoint calculus. We then build on the theoretical results by extending a recently published online convex optimization algorithm (OSA) to the infinite-dimensional setting. Similar to the regret-based analysis of time-varying stochastic optimization problems, we enhance the method further by allowing for periodic restarts at pre-defined epochs. Not only does this allow for larger step sizes, it also proves to be an essential factor in obtaining high-quality solutions in practice. The behavior of the algorithm is demonstrated in a numerical example involving a linear advection-diffusion equation with random inputs. In order to judge the quality of the solution, the results are compared to those arising from a sample average approximation (SAA). This is done first by comparing the resulting cumulative distributions of the objectives at the optimal solution as a function of step numbers and epoch lengths. In addition, we conduct statistical tests to further analyze the behavior of the online algorithm and the quality of its solutions. For a sufficiently large number of steps, the solutions from OSA and SAA lead to random integrands for the objective and penalty functions that appear to be drawn from similar distributions.
Objective function empirical distribution using 104\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$10^4$$\end{document} samples for the SAA optimal control (black) and the OSA controls computed with 103\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$10^3$$\end{document} (red), 104\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$10^4$$\end{document} (blue), and 105\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$10^5$$\end{document} (green) iterations for epoch lenghts ΔN=N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varDelta _N=N$$\end{document} (left) and ΔN=500\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varDelta _N=500$$\end{document} (right) (Color figure online)
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Computational Optimization and Applications (2023) 85:441–478
https://doi.org/10.1007/s10589-023-00461-8
A relaxation-based probabilistic approach for
PDE-constrained optimization under uncertainty with
pointwise state constraints
Drew P. Kouri1·Mathias Staudigl2·Thomas M. Surowiec3
Received: 23 May 2022 / Accepted: 3 February 2023 / Published online: 27 February 2023
© The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature 2023
Abstract
We consider a class of convex risk-neutral PDE-constrained optimization problems
subject to pointwise control and state constraints. Due to the many challenges asso-
ciated with almost sure constraints on pointwise evaluations of the state, we suggest
a relaxation via a smooth functional bound with similar properties to well-known
probability constraints. First, we introduce and analyze the relaxed problem, discuss
its asymptotic properties, and derive formulae for the gradient the adjoint calculus.
We then build on the theoretical results by extending a recently published online con-
vex optimization algorithm (OSA) to the infinite-dimensional setting. Similar to the
regret-based analysis of time-varying stochastic optimization problems, we enhance
the method further by allowing for periodic restarts at pre-defined epochs. Not only
does this allow for larger step sizes, it also proves to be an essential factor in obtain-
ing high-quality solutions in practice. The behavior of the algorithm is demonstrated
in a numerical example involving a linear advection–diffusion equation with random
inputs. In order to judge the quality of the solution, the results are compared to those
arising from a sample average approximation (SAA). This is done first by comparing
the resulting cumulative distributions of the objectives at the optimal solution as a
function of step numbers and epoch lengths. In addition, we conduct statistical tests
to further analyze the behavior of the online algorithm and the quality of its solutions.
BThomas M. Surowiec
thomasms@simula.no
Drew P. Kouri
dpkouri@sandia.gov
Mathias Staudigl
m.staudigl@maastrichtuniversity.nl
1Sandia National Laboratories, P.O. Box 5800, MS-1320, Albuquerque, NM, USA
2Department of Advanced Computing Sciences (DACS), Maastricht University, Maastricht, The
Netherlands
3Department of Numerical Analysis and Scientific Computing, Simula Research Laboratory, Kristian
Augusts Gate 23, 0164 Oslo, Norway
123
Content courtesy of Springer Nature, terms of use apply. Rights reserved.
... Modern presentations can be found in [38] or [40], respectively. In recent years, there has been growing interest in considering probabilistic constraints in the framework of optimal control in general or PDE-constrained optimization in particular, e.g., [5,10,11,15,16,24,29,39]. These works include proposals for numerical approaches as well as structural investigations. ...
... Problem (24) falls into the setting of the general problem (9) with introduced before. In order to apply the results from Section 2.1 to problem (24), one has to verify first that the assumptions made there are satisfied. ...
... Problem (24) falls into the setting of the general problem (9) with introduced before. In order to apply the results from Section 2.1 to problem (24), one has to verify first that the assumptions made there are satisfied. Since by ...
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