Convergence of sum-up rounding schemes for cloaking problems governed by the Helmholtz equation

Leyffer, Sven; Manns, Paul; Winckler, Malte

doi:10.1007/s10589-020-00262-3

Convergence of sum-up rounding schemes for cloaking problems governed by the Helmholtz equation

Journal Article · Fri Feb 05 00:00:00 EST 2021 · Computational Optimization and Applications

DOI:https://doi.org/10.1007/s10589-020-00262-3· OSTI ID:1782731

Leyffer, Sven ^[1]; ^[1]; Winckler, Malte ^[2]

Argonne National Lab. (ANL), Argonne, IL (United States)
Universität Duisburg-Essen (Germany)

In this work, we consider the problem of designing a cloak for waves described by the Helmholtz equation from an integer programming point of view. The problem can be modeled as a PDE-constrained optimization problem with integer-valued control inputs that are distributed in the computational domain. A first-discretize-then-optimize approach results in a large-scale mixed-integer nonlinear program that is in general intractable because of the large number of integer variables that arise from the discretization of the domain. Instead, we propose an efficient algorithm that is able to approximate the local infima of the underlying nonconvex infinite-dimensional problem arbitrarily close without the need to solve the discretized finite-dimensional integer programs to optimality. We optimize only the continuous relaxations of the approximations for local minima and then apply the sum-up rounding methodology to obtain integer-valued controls. If the solutions of the discretized continuous relaxations converge to a local minimizer of the continuous relaxation, then the resulting discrete-valued control sequence converges weakly\(^*\) in \(L^\infty\) to the same local minimizer. These approximation properties follow under suitable refinements of the involved discretization grids. Our results use familiar concepts arising from the analytical properties of the underlying PDE and complement previous results, derived from a topology optimization point of view.

View Accepted Manuscript (DOE)

Research Organization:: Argonne National Laboratory (ANL), Argonne, IL (United States)

Sponsoring Organization:: German Research Foundation (DFG); USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR)

Grant/Contract Number:: AC02-06CH11357

OSTI ID:: 1782731

Journal Information:: Computational Optimization and Applications, Journal Name: Computational Optimization and Applications Journal Issue: 1 Vol. 79; ISSN 0926-6003

Publisher:: SpringerCopyright Statement

Country of Publication:: United States

Language:: English

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