Mathematical analysis of transmission properties of electromagnetic meta-materials

Ohlberger, Mario; Schweizer, Ben; Urban, Maik; Verfürth, Barbara

doi:10.3934/nhm.2020002

Cited by 4 publications

(1 citation statement)

References 31 publications

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“…Parameterized multiscale problems where the parameters are optimized with respect to some userdefined quality criteria are of general interest in many physical, chemical, biomedical, or engineering applications. Examples, e.g., include optimal design of devises built from composed materials [8,7,19], optimization of reactive flow processes in porous media [37,25,58], or the design of meta-materials [17,54,50]. As a mathematical model for such constrained parameter optimization problems, we consider linear-quadratic parameter optimization, subject to the solution of a parameter-dependent elliptic variational multiscale problem.…”

Section: Introductionmentioning

confidence: 99%

A relaxed localized trust-region reduced basis approach for optimization of multiscale problems

Keil¹,

Ohlberger²

2022

Preprint

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In this contribution, we introduce and analyze a new relaxed and localized version of the trust-region method for PDE-constrained parameter optimization in the context of multiscale problems. As an underlying efficient discretization framework, we rely on the Petrov-Galerkin localized orthogonal decomposition method and its recently introduced two-scale reduced basis approximation. We derive efficient localizable a posteriori error estimates for the primal and dual equations of the optimality system, as well as for the two-scale reduced objective functional. While the relaxation of the outer trust-region optimization loop still allows for a rigorous convergence result, the resulting method converges much faster due to larger step sizes in the initial phase of the iterative algorithms. The resulting algorithm is parallelized in order to take advantage of the localization. Numerical experiments are given for a multiscale thermal block benchmark problem. The experiments demonstrate the efficiency of the approach, particularly for large scale problems, where methods based on traditional finite element approximation schemes are prohibitive or fail entirely.

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Section: Introductionmentioning

confidence: 99%

A relaxed localized trust-region reduced basis approach for optimization of multiscale problems

Keil¹,

Ohlberger²

2022

Preprint

Self Cite

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A relaxed localized trust-region reduced basis approach for optimization of multiscale problems

Keil,

Ohlberger

2024

ESAIM: M2AN

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In this contribution, we are concerned with parameter optimization problems that are constrained by multiscale PDE state equations. As an efficient numerical solution approach for such problems, we introduce and analyze a new relaxed and localized trust-region reduced basis method. Localization is obtained based on a Petrov-Galerkin localized orthogonal decomposition method and its recently introduced two-scale reduced basis approximation. We derive efficient localizable a posteriori error estimates for the optimality system, as well as for the two-scale reduced objective functional. While the relaxation of the outer trust-region optimization loop still allows for a rigorous convergence result, the resulting method converges much faster due to larger step sizes in the initial phase of the iterative algorithms. The resulting algorithm is parallelized in order to take advantage of the localization. Numerical experiments are given for a multiscale thermal block benchmark problem. The experiments demonstrate the efficiency of the approach, particularly for large scale problems, where methods based on traditional finite element approximation schemes are prohibitive or fail entirely. 1991 Mathematics Subject Classification. 49M20, 35J20, 65N15, 65N30, 90C06.

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Numerical Multiscale Methods for Waves in High-Contrast Media

Verfürth

2023

Jahresber. Dtsch. Math. Ver.

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Multiscale high-contrast media can cause astonishing wave propagation phenomena through resonance effects. For instance, waves could be exponentially damped independent of the incident angle or waves could be re-focused as through a lense. In this review article, we discuss the numerical treatment of wave propagation through multiscale high-contrast media at the example of the Helmholtz equation. First, we briefly summarize the findings of analytical homogenization theory, which inspire the design of numerical methods and indicate interesting regimes for simulation. In the main part, we discuss two different classes of numerical multiscale methods and focus on how to treat especially high-contrast media. Some elements of a priori error analysis are discussed as well. Various numerical simulations showcase the applicability of the numerical methods to explore unusual wave phenomena, for instance exponential damping and lensing with flat interfaces.

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Mathematical analysis of transmission properties of electromagnetic meta-materials

Cited by 4 publications

References 31 publications

A relaxed localized trust-region reduced basis approach for optimization of multiscale problems

A relaxed localized trust-region reduced basis approach for optimization of multiscale problems

A relaxed localized trust-region reduced basis approach for optimization of multiscale problems

Numerical Multiscale Methods for Waves in High-Contrast Media

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