Project C6 • Uncertainty and monotonicity principles for inverse source and inverse scattering problems
Principal investigators
PD Dr. Tilo Arens
(7/2019 - )
Prof. Dr. Roland Griesmaier
(7/2019 - )
Project summary
Inverse source and inverse scattering problems are illposed inverse problems, where the goal is to recover information about sources or scatterers from remote observations of associated radiated or scattered waves. Structural information contained in the observed data can be used to establish uniqueness results, to devise new types of reconstruction algorithms, and to improve their stability. In this project we study two particular types of such structural information: uncertainty principles and monotonicity relations. The general aim is to develop novel reconstruction methods for inverse source and inverse scattering problems that are based on uncertainty and monotonicity principles, and to provide a rigorous analysis for these methods.
Uncertainty principles for inverse source problems
Classical uncertainty principles in signal processing have been fundamental for the development of algorithms and stability criteria for the sparse representation of functions, and for the development of compressed sensing. One goal of this project is to transfer some of the underlying ideas to inverse source and inverse scattering problems. First steps in that direction are the uncertainty principles for far fields of time-harmonic acoustic, electromagnetic, and elastic waves radiated by compactly supported sources. These are not only of theoretical interest but they have immediate practical applications. They have been utilized to provide stability criteria and reconstruction algorithms for the restoration of missing data segments, and for the recovery of far field components radiated by well-separated source components individually, i.e., far field splitting.
Figure 1. A wave radiated by three sources can be decomposed into the three waves radiated by each source individually.
Data completion is relevant if the far field cannot be observed all around the source, but measurement data are only available on some limited aperture. Far field splitting has been utilized to derive new reconstruction methods for inverse source problems, it has been applied to stabilize numerical algorithms for data completion, and it has been used as a theoretical tool to analyze resolution bounds for reconstruction algorithms for inverse source and inverse scattering problems beyond point spread functions.
Monotonicity in inverse scattering problems on unbounded domains
Efficient non-iterative reconstruction methods for recovering the location and the shape of unknown scattering objects from far field observations of scattered acoustic or electromagnetic waves are of high practical interest. Monotonicity-based shape reconstruction, which belongs to this class of methods, has originally been proposed for an inverse problem in electrical impedance tomography. The starting point for this method has been the observation that if $\sigma_1$ and $\sigma_2$ are positive functions representing electric conductivities in some bounded domain $\Omega\subset\mathbb{R}^d$ such that $\sigma_1\leq\sigma_2$, then the associated Neumann-to-Dirichlet operators $\Lambda_{\sigma_1}$ and $\Lambda_{\sigma_2}$ on $\partial\Omega$ satisfy $\Lambda_{\sigma_1}-\Lambda_{\sigma_2}\geq0$ in the sense that the self-adjoint compact linear operator $\Lambda_{\sigma_1}-\Lambda_{\sigma_2}$ is positive semidefinite. This observation has then been translated into a rigorously justified shape reconstruction algorithm. We explore this approach for shape reconstruction for inverse scattering problems on unbounded domains.
Figure 2. A torus and its reconstruction by a monotonicity method from scattered electromagnetic waves for different noise levels (same as Figure 2 from [AG23])
An advantage of the monotonicity based approach is that the theoretical justification of the associated reconstruction schemes applies to a large class of indefinite scattering objects, i.e., when the relative refractive index takes values larger and smaller than 1 inside the scattering objects. Continue reading.Collapse content.
Research and results
Monotonicity in inverse obstacle scattering. In [AG20] we considered an inverse obstacle scattering problem for the Helmholtz equation with obstacles that carry mixed Dirichlet and Neumann boundary conditions. We discussed far field operators that map superpositions of plane wave incident fields to far field patterns of scattered waves, and we derived monotonicity relations for the eigenvalues of these operators. These monotonicity relations were then used to establish a novel characterization of the support of mixed obstacles in terms of the corresponding far field operators. We applied this characterization in reconstruction schemes for shape reconstruction and object classification. In particular we have shown theoretically and in numerical experiments that mixed scattering configurations with sound-soft and sound-hard obstacles can be reconstructed and that the scatterers can be classified according to their boundary condition from the scattering data.
Monotonicity in inverse electromagnetic scattering. In [AG23] we developed monotonicity relations for an electromagnetic inverse scattering problem governed by Maxwell equations, and we applied them to establish novel rigorous characterizations of the shape of scattering objects in terms of the corresponding far field operators. We established the existence of electromagnetic fields that have arbitrarily large energy in some prescribed region, while at the same time having arbitrarily small energy in some other prescribed region. We have implemented the novel shape characterizations in reconstruction algorithms, and we have shown that scatterers with positive and negative permittivity contrast can be classified from scattering data based on the monotonicity principles.
Monotonicity for scattering in waveguides. In [AGZ22] we discussed extensions of the monotonicity relations for an inverse scattering problem in a wave guide. The main difference to the previous studies is that in the waveguide setting we worked with modal decompositions of a near field operator instead of a far field operator, and that we had to carefully distinguish propagating and evanescent modes. Monotonicity principles, the existence of localized wave functions, and monotonicity-based shape characterizations have been provided. We have also considered first estimates for the dimensions of the finite dimensional subspaces that have to be excluded in the monotonicity arguments due to the lack of coercivity of the time-harmonic Helmholtz equation. A numerical implementation of monotonicity-based reconstruction schemes has been discussed as well.
Inverse scattering problems for nonlinear Helmholtz equations. The linear Helmholtz equation is used to model the propagation of sound waves or electromagnetic waves of small amplitude in inhomogeneous isotropic media in the time-harmonic regime. However, if the magnitudes are large, then intensity-dependent material laws are required, and nonlinear Helmholtz equations are used. In [GKM22] we discussed an inverse scattering problem for a class of non-linear Helmholtz equations. Assuming the knowledge of a nonlinear far field operator, which maps superpositions of incident plane waves to the far field patterns of corresponding solutions of the nonlinear scattering problem, we established that the nonlinear index of refraction is uniquely determined. We have also generalized two algorithms for shape reconstruction, the inf-criterion and the monotonicity method, to the nonlinear scattering problem.
Random source identification in experimental aeroacoustics. Experimental aeroacoustics is concerned with the estimation of acoustic source power distributions, which are for instance caused by turbulent 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 modeled as a random source problem for a convected Helmholtz equation. In [GR22] we have considered the inverse random source problem to recover the support of an uncorrelated aeroacoustic source $Q$ from correlations of observed pressure signals. We have shown that the mathematical structure of the covariance operator of the aeroacoustic pressure signal is closely related to the structure of the Born approximation of the far field operator for the inverse medium scattering problem. This has been used to establish the theoretical foundation of the factorization method for support reconstruction in the aeroacoustic inverse source problem.
