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Perturbed Block Toeplitz matrices and the non-Hermitian skin effect in dimer systems of subwavelength resonators
(2023)SAM Research ReportThe aim of this paper is fourfold: (i) to obtain explicit formulas for the eigenpairs of perturbed tridiagonal block Toeplitz matrices; (ii) to make use of such formulas in order to provide a mathematical justification of the non-Hermitian skin effect in dimer systems by proving the condensation of the system's bulk eigenmodes at one of the edges of the system; (iii) to show the topological origin of the non-Hermitian skin effect for dimer ...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 -
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 -
A mathematical theory of resolution limits for super-resolution of positive sources
(2023)SAM Research ReportThe superresolving capacity for number and location recoveries in the super-resolution of positive sources is analyzed in this work. Specifically, we introduce the computational resolution limit for respectively the number detection and location recovery in the one-dimensional super-resolution problem and quantitatively characterize their dependency on the cutoff frequency, signal-to-noise ratio, and the sparsity of the sources. As a ...Report -
The non-Hermitian skin effect with three-dimensional long-range coupling
(2023)SAM Research ReportWe study the non-Hermitian skin effect in a three-dimensional system of finitely many subwavelength resonators with an imaginary gauge potential. We introduce a discrete approximation of the eigenmodes and eigenfrequencies of the system in terms of the eigenvectors and eigenvalues of the so-called gauge capacitance matrix, which is a dense matrix due to long-range interactions in the system. Based on translational invariance of this matrix ...Report -
Mathematical foundation of sparsity-based multi-illumination super-resolution
(2022)SAM Research ReportIt is well-known that the resolution of traditional optical imaging system is limited by the so-called Rayleigh resolution or diffraction limit, which is of several hundreds of nanometers. By employing fluorescence techniques, modern microscopic methods can resolve point scatterers separated by a distance much lower than the Rayleigh resolution limit. Localization-based fluorescence subwavelength imaging techniques such as PALM and STORM ...Report -
Dynamic super-resolution in particle tracking problems
(2022)SAM Research ReportParticle tracking in a live cell environment is concerned with reconstructing the trajectories, locations, or velocities of the targeting particles, which holds the promise of revealing important new biological insights. The standard approach of particle tracking consists of two steps: first reconstructing statically the source locations in each time step, and second applying tracking techniques to obtain the trajectories and velocities. ...Report -
Nearly optimal resolution estimate for the two-dimensional super-resolution and a new algorithm for direction of arrival estimation with uniform rectangular array
(2022)SAM Research ReportIn this paper, we develop a new technique to obtain nearly optimal estimates of the computational resolution limits for two-dimensional super-resolution problems. Our main contributions are fivefold: (i) Our work improves the resolution estimates for number detection and location recovery in two-dimensional super-resolution problems to nearly optimal; (ii) As a consequence, we derive a stability result for a sparsity-promoting algorithm ...Report -
Super-resolution of positive near-colliding point sources
(2023)SAM Research ReportIn this paper, we analyze the capacity of super-resolution of one-dimensional positive sources. We consider resolving of positive point sources where some nodes are closely spaced and forming a cluster, while the rest of the nodes are well separated. For both the cluster and the non-cluster nodes, we estimate the minimax error rates for reconstructing the nodes and recovering the corresponding amplitudes. Our numerical experiments show ...Report -
Spectra and pseudo-spectra of tridiagonal k-Toeplitz matrices and the topological origin of the non-Hermitian skin effect
(2024)SAM Research ReportWe establish new results on the spectra and pseudo-spectra of tridiagonal k-Toeplitz operators and matrices. In particular, we prove the connection between the winding number of the eigenvalues of the symbol function and the exponential decay of the associated eigenvectors (or pseudo-eigenvectors). Our results elucidate the topological origin of the non-Hermitian skin effect in general one-dimensional polymer systems of subwavelength ...Report