arXiv subjects

Roland Griesmaier

Publications and source records attributed to Roland Griesmaier.

13 recordsLinked to original sources

arXiv2026

Domain derivative and shape reconstruction for an inverse backscattering problem for the wave equation

We consider an inverse backscattering problem for time-dependent acoustic waves with a compactly supported penetrable scattering obstacle in unbounded three-dimensional free space. Assuming that dynamic backscattering far field data of scattered waves corresponding to a few time-dependent incident plane waves are available, the goal is to reconstruct the shape of the scattering obstacle. We establish Fréchet differentiability of the time-dependent far field pattern with respect to the shape of the scattering obstacle and give a characterization of the associated temporal domain derivative in terms of its Laplace transform. This characterization is then utilized in an efficient implementation of a regularized Gauß-Newton method for the inverse backscattering problem using convolution quadrature and a boundary element method. Numerical examples demonstrate potentials and limitations of the algorithm.

math.NA↗

arXiv2026

Optimal Design of Tubular Perfectly Conducting Objects in Electromagnetic Chirality

This work is about the shape optimization of long tubular objects in electromagnetic chirality (em-chirality). Em-chirality is a property of individual scattering objects or metamaterials describing their qualitatively different response to electromagnetic waves of opposite polarization handedness. The optimization is performed by a Newton-type iterative maximization of a regularized em-chirality measure with respect to the scatterer's shape. In this context, the differentiability of the object-to-far field operator map is analyzed rigorously, thereby extending previously known results on the domain derivative to the far field operator setting. Our optimal design algorithm is based on the electric field integral equation, which is employed both for the evaluation of scattered fields and for the computation of the domain derivative. Our implementation is done via the boundary element method. The numerical examples presented in this work yield strongly em-chiral scattering objects capable of exciting higher-order modes beyond the dipole regime with nonintuitive shapes that expand the known set of highly em-chiral scattering objects.

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arXiv2026

Localizing acoustic and electromagnetic waves in space and time

We study time-dependent acoustic and electromagnetic waves governed by the scalar wave equation or Maxwell's equations in a bounded three-dimensional domain. We establish the existence of time-dependent boundary excitations that can be prescribed on any open subset of the boundary of the domain such that the associated waves are strongly localized in space in the sense that they possess arbitrarily large norms in a given subdomain and on a given time-interval, while remaining arbitrarily small in any other given subdomain for all times. Similarly, we also show the existence of boundary data such that the associated waves are strongly localized in time in the sense that they possess arbitrarily large norms in a given subdomain and on a given time-interval, while remaining arbitrarily small on the same subdomain but on any other prescribed time-interval. In case that we have access to the possibly inhomogeneous coefficients in the wave equation or in the Maxwell system, we also give explicit constructions to obtain boundary data that generate these localized waves, and we comment on possible applications.

math.AP↗

arXiv2026

Monotonicity-based regularization of inverse medium scattering for shape reconstruction

We consider the scattering of time-harmonic plane waves by a compactly supported inhomogeneous scattering obstacle governed by the Helmholtz equation. Given far field observations of the scattered fields corresponding to plane wave incident fields for all possible incident and observation directions we study the inverse problem to recover the support of the scatterer. We propose a qualitative monotonicity-based regularization scheme which combines monotonicity-based shape reconstruction with one-step linearization to reconstruct a discrete approximation of the shape of the scatterer from noisy far field data. The purpose of the one-step linearization is to stabilize the monotonicity approach to shape reconstruction. We show that the monotonicity-based regularization scheme recovers the correct shape of the scatterer for noise-free data. Furthermore, we establish that the solution of the monotonicity-based regularization converges to the exact solution as the noise level tends to zero. We present numerical examples to illustrate our theoretical findings.

math.NA↗

arXiv2022

Maximizing the electromagnetic chirality of thin metallic nanowires at optical frequencies

Electromagnetic waves impinging on three-dimensional helical metallic metamaterials have been shown to exhibit chiral effects of large magnitude both theoretically and in experimental realizations. Chirality here describes different responses of scatterers, materials, or metamaterials to left and right circularly polarized electromagnetic waves. These differences can be quantified in terms of electromagnetic chirality measures. In this work we consider the optimal design of thin metallic free-form nanowires that possess measures of electromagnetic chirality as large as fundamentally possible. We focus on optical frequencies and use a gradient based optimization scheme to determine the optimal shape of highly chiral thin silver and gold nanowires. The electromagnetic chirality measures of our optimized nanowires exceed that of traditional metallic helices. Therefore, these should be well suited as building blocks of novel metamaterials with an increased chiral response. We discuss a series of numerical examples, and we evaluate the performance of different optimized designs.

math.OC↗

arXiv2022

The factorization method and Capon's method for random source identification in experimental aeroacoustics

Experimental aeroacoustics is concerned with the estimation of acoustic source power distributions, which are for instance caused by fluid structure interactions on scaled aircraft models inside a wind tunnel, from microphone array measurements of associated sound pressure fluctuations. In the frequency domain aeroacoustic sound propagation can be modelled as a random source problem for a convected Helmholtz equation. This article is concerned with the inverse random source problem to recover the support of an uncorrelated aeroacoustic source from correlations of observed pressure signals. We show that a variant of the factorization method from inverse scattering theory can be used for this purpose. We also discuss a surprising relation between the factorization method and a commonly used beamforming algorithm from experimental aeroacoustics, which is known as Capon's method or as the minimum variance method. Numerical examples illustrate our theoretical findings.

math.NA↗

arXiv2022

Inverse medium scattering for a nonlinear Helmholtz equation

We discuss a time-harmonic inverse scattering problem for a nonlinear Helmholtz equation with compactly supported inhomogeneous scattering objects that are described by a nonlinear refractive index in unbounded free space. Assuming the knowledge of a nonlinear far field operator, which maps Herglotz incident waves to the far field patterns of corresponding solutions of the nonlinear scattering problem, we show that the nonlinear index of refraction is uniquely determined. We also generalize two reconstruction methods, a factorization method and a monotonicity method, to recover the support of such nonlinear scattering objects. Numerical results illustrate our theoretical findings.

math.AP↗

arXiv2021

Monotonicity in inverse medium scattering on unbounded domains

We discuss a time-harmonic inverse scattering problem for the Helmholtz equation with compactly supported penetrable and possibly inhomogeneous scattering objects in an unbounded homogeneous background medium, and we develop a monotonicity relation for the far field operator that maps superpositions of incident plane waves to the far field patterns of the corresponding scattered waves. We utilize this monotonicity relation to establish novel characterizations of the support of the scattering objects in terms of the far field operator. These are related to and extend corresponding results known from factorization and linear sampling methods to determine the support of unknown scattering objects from far field observations of scattered fields. An attraction of the new characterizations is that they only require the refractive index of the scattering objects to be above or below the refractive index of the background medium locally and near the boundary of the scatterers. An important tool to prove these results are so-called localized wave functions that have arbitrarily large norm in some prescribed region while at the same time having arbitrarily small norm in some other prescribed region. We present numerical examples to illustrate our theoretical findings.

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arXiv2020

An asymptotic representation formula for scattering by thin tubular structures and an application in inverse scattering

We consider the scattering of time-harmonic electromagnetic waves by a penetrable thin tubular scattering object in three-dimensional free space. We establish an asymptotic representation formula for the scattered wave away from the thin tubular scatterer as the radius of its cross-section tends to zero. The shape, the relative electric permeability and the relative magnetic permittivity of the scattering object enter this asymptotic representation formula by means of the center curve of the thin tubular scatterer and two electric and magnetic polarization tensors. We give an explicit characterization of these two three-dimensional polarization tensors in terms of the center curve and of the two two-dimensional polarization tensor for the cross-section of the scattering object. As an application we demonstrate how this formula may be used to evaluate the residual and the shape derivative in an efficient iterative reconstruction algorithm for an inverse scattering problem with thin tubular scattering objects. We present numerical results to illustrate our theoretical findings. Mathematics subject classifications (MSC2010): 35C20, (65N21, 78A46)

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arXiv2016

Uncertainty principles for inverse source problems, far field splitting and data completion

Starting with far field data of time-harmonic acoustic or electromagnetic waves radiated by a collection of compactly supported sources in two-dimensional free space, we develop criteria and algorithms for the recovery of the far field components radiated by each of the individual sources, and the simultaneous restoration of missing data segments. Although both parts of this inverse problem are severely ill-conditioned in general, we give precise conditions relating the wavelength, the diameters of the supports of the individual source components and the distances between them, and the size of the missing data segments, which guarantee that stable recovery in presence of noise is possible. The only additional requirement is that a priori information on the approximate location of the individual sources is available. We give analytic and numerical examples to confirm the sharpness of our results and to illustrate the performance of corresponding reconstruction algorithms, and we discuss consequences for stability and resolution in inverse source and inverse scattering problems.

math.AP↗

arXiv2016

A multifrequency MUSIC algorithm for locating small inhomogeneities in inverse scattering

We consider an inverse scattering problem for time-harmonic acoustic or electromagnetic waves. The goal is to localize several small penetrable objects embedded inside an otherwise homogeneous background medium from observations of far field patterns of scattered fields corresponding to plane wave incident fields with one fixed incident direction but several different frequencies. Taking advantage of the smallness of the scatterers with respect to wave length we utilize an asymptotic representation formula for the far field pattern to design and analyze a MUSIC-type reconstruction method for this setup. We establish lower bounds on the number of frequencies and receiver directions required to recover the number and the positions of the scatterers for a given configuration by the reconstruction algorithm. Furthermore we apply the method to the practically interesting case of multifrequency backscattering data. Numerical examples are presented to document the potentials and limitations of this approach.

math.NA↗

arXiv2015

Polarization tensors of planar domains as functions of the admittivity contrast

(Electric) polarization tensors describe part of the leading order term of asymptotic voltage perturbations caused by low volume fraction inhomogeneities of the electrical properties of a medium. They depend on the geometry of the support of the inhomogeneities and on their admittivity contrast. Corresponding asymptotic formulas are of particular interest in the design of reconstruction algorithms for determining the locations and the material properties of inhomogeneities inside a body from measurements of current flows and associated voltage potentials on the body's surface. In this work we consider the two-dimensional case only and provide an analytic representation of the polarization tensor in terms of spectral properties of the double layer integral operator associated with the support of simply connected conductivity inhomogeneities. Furthermore, we establish that an (infinitesimal) simply connected inhomogeneity has the shape of an ellipse, if and only if the polarization tensor is a rational function of the admittivity contrast with at most two poles whose residues satisfy a certain algebraic constraint. We also use the analytic representation to provide a proof of the so-called Hashin-Shtrikman bounds for polarization tensors; a similar approach has been taken previously by Golden and Papanicolaou and Kohn and Milton in the context of anisotropic composite materials.

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arXiv2012

A note on analyticity properties of far field patterns

In scattering theory the far field pattern describes the directional dependence of a time-harmonic wave scattered by an obstacle or inhomogeneous medium, when observed sufficiently far away from these objects. Considering plane wave excitations, the far field pattern can be written as a function of two variables, namely the direction of propagation of the incident plane wave and the observation direction, and it is well-known to be separately real analytic with respect to each of them. We show that the far field pattern is in fact a jointly real analytic function of these two variables.

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