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Fast Magnetic Resonance Electrical Impedance Tomography with highly undersampled data
(2016)Research ReportReport -
Bound states in the continuum and Fano resonances in subwavelength resonator arrays
(2021)SAM Research ReportWhen wave scattering systems are subject to certain symmetries, resonant states may decouple from the far-field continuum; they remain localized to the structure and cannot be excited by incident waves from the far field. In this work, we use layer-potential techniques to prove the existence of such states, known as bound states in the continuum, in systems of subwavelength resonators. When the symmetry is slightly broken, this resonant ...Report -
Mathematical framework for abdominal electrical impedance tomography to assess fatness
(2016)Research ReportReport -
Analysis of a Monte-Carlo Nystrom method
(2021)SAM Research ReportThis paper considers a Monte-Carlo Nystrom method for solving integral equations of the second kind, whereby the values (z(yi))1⩽i⩽N of the solution z at a set of N random and independent points (yi)1⩽i⩽N are approximated by the solution (zN,i)1⩽i⩽N of a discrete, N-dimensional linear system obtained by replacing the integral with the empirical average over the samples (yi)1⩽i⩽N. Under the unique assumption that the integral equation ...Report -
Functional analytic methods for discrete approximations of subwavelength resonator systems
(2021)SAM Research ReportWe survey functional analytic methods for studying subwavelength resonator systems. In particular, rigorous discrete approximations of Helmholtz scattering problems are derived in an asymptotic subwavelength regime. This is achieved by re-framing the Helmholtz equation as a non-linear eigenvalue problem in terms of integral operators. In the subwavelength limit, resonant states are described by the eigenstates of the generalized capacitance ...Report -
Elastic Scattering Coefficients and Enhancement of Nearly Elastic Cloaking
(2016)Research ReportReport -
Wave interaction with subwavelength resonators
(2020)SAM Research ReportThe aim of this review is to cover recent developments in the mathematical analysis of subwavelength resonators. The use of sophisticated mathematics in the field of metamaterials is reported, which provides a mathematical framework for focusing, trapping, and guiding of waves at subwavelength scales. Throughout this review, the power of layer potential techniques combined with asymptotic analysis for solving challenging wave propagation ...Report -
Stability of the non-Hermitian skin effect
(2023)SAM Research ReportThis paper shows that the skin effect in systems of non-Hermitian subwavelength resonators is robust with respect to random imperfections in the system. The subwavelength resonators are highly contrasting material inclusions that resonate in a low-frequency regime. The non-Hermiticity is due to the introduction of an imaginary gauge potential, which leads to a skin effect that is manifested by the system's eigenmodes accumulating at one ...Report -
Edge modes in subwavelength resonators in one dimension
(2023)SAM Research ReportWe present the mathematical theory of one-dimensional infinitely periodic chains of subwavelength resonators. We analyse both Hermitian and non-Hermitian systems. Subwavelength resonances and associated modes can be accurately predicted by a finite dimensional eigenvalue problem involving a capacitance matrix. We are able to compute the Hermitian and non-Hermitian Zak phases, showing that the former is quantised and the latter is not. ...Report -
A mathematical theory of super-resolution and diffraction limit
(2023)SAM Research ReportThis paper is devoted to elucidating the essence of super-resolution and deals mainly with the stability of super-resolution and the diffraction limit. The first discovery is two location-amplitude identities characterizing the relations between source locations and amplitudes in the super-resolution problem. These identities allow us to directly derive the super-resolution capability for number, location, and amplitude recovery in the ...Report