We investigate the possibility for using boundary measurements to recover a sparse source term $f(x)$ in the potential equation. This work is motivated by the observation that standard methods typically suggest that internal sinks and sources are located close to the boundary and hence fail to produce adequate results. That is, the large null space of the associated forward operator is not “correctly handled” by the classical regularization techniques.

Provided that weighted sparsity regularization is used, we derive criteria which assure that several sinks ($f(x)<0$) and sources ($f(x)>0$) can be identified. Furthermore, we present two cases for which these criteria always are fulfilled: a) well-separated sources and sinks, and b) many sources or sinks located at the boundary plus one interior source/sink. Our approach is such that the linearity of the associated forward operator is preserved in the discrete formulation. The theory is therefore conveniently developed in terms of Euclidean spaces, and it can be applied to a wide range of problems. In particular, it can be applied to both isotropic and anisotropic cases. We present a series of numerical experiments.

This work extends the results presented at the "Symposium on Inverse Problems" in Potsdam in September 2022: The theory for the single source case is generalized to the several sources and sinks situation, we do not employ any box constraints and the analysis is carried out for the potential equation instead of focusing on the screened Poisson equation or the Helmholtz equation.

We consider an inverse problem of the wave equation on a Lorentzian manifold, a type of semi-Riemannian manifold. This kind of equation is obtained by linearizing the Einstein equation and is known as the equation satisfied by gravitational waves. In this talk, we prove a global Lipschitz stability for the inverse source problem of determining a source term in the equation. Sobolev spaces on manifolds, semigeodesic coordinates, and Carleman estimates, which are important tools in geometric analysis, will also be discussed.

We study the inverse source problem to determine the strength of a random acoustic source from correlation data. More precisely, the data of the inverse problems are correlations of random time-harmonic acoustic waves measured on a surface surrounding a region of random, uncorrelated sources. Such a model is used in experimental aeroacoustics to determine the strength of sound sources [1]. Uniqueness has been previously established [1,2]. In this talk we report on logarithmic stability results and logarithmic convergence rates for the Tikhonov regularisation applied to the inverse source problem by establishing a variational source condition under Sobolev type smoothness assumption. We also present logarithmic instability estimates using an entropy argument. Furthermore, we will show numerical experiments supporting our theoretical results.

[1] T. Hohage, H.-G. Raumer, C. Spehr. Uniqueness of an inverse source problem in experimental aeroacoustics. Inverse Problems, 36(7):075012, 2020.

[2] A. J. Devaney. The inverse problem for random sources. Journal of Mathematical Physics, 20(8):1687–1691, 1979.

MEG and EEG source analysis is frequently used in presurgical evaluation of pharmacoresistant epilepsy patients. The localization quality depends, among other aspects, on the selected inverse and forward approaches and their respective parameter choices. In my talk, I will present new forward and inverse approaches and their application for the identification of the epileptogenic zone in focal epilepsy. The forward approaches are based on the finite element method (FEM). The inverse approaches contain beamforming, hierarchical Bayesian modeling (HBM) and standard dipole scanning techniques. I will discuss advantages and disadvantages of those approaches and compare their performance in a retrospective evaluation study with patients of focal epilepsy.

[1] Neugebauer, F., Antonakakis, M., Unnwongse, K., Parpaley, Y., Wellmer, J., Rampp, S., Wolters, C.H., Validating EEG, MEG and Combined MEG and EEG Beamforming for an Estimation of the Epileptogenic Zone in Focal Cortical Dysplasia. Brain Sci:114, 2022. https://doi.org/10.3390/brainsci12010114.

[2] Aydin, Ü., Rampp, S., Wollbrink, A., Kugel, H., Cho, J.-H., Knösche, T.R.,Grova, C., Wellmer, J., Wolters, C.H., Zoomed MRI guided by combined EEG/MEG source analysis: A multimodal approach for optimizing presurgical epilepsy work-up and its application in a multi-focal epilepsy patient case study, Brain Topography, 30(4):417-433, 2017. https://doi.org/10.1007/s10548-017-0568-9.