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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
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