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1. Solving the inverse scattering problem via Carleman-based contraction mapping

   Nguyen, P M; Nguyen, L H; Vu, H T (June 2024, preprint arXiv:2404.04145)

   This paper addresses the inverse scattering problem in a domain $$\Omega$$. The input data, measured outside $$\Omega$$, involve the waves generated by the interaction of plane waves with various directions and unknown scatterers that are fully occluded inside $$\Omega$$. The output of this problem is the spatial dielectric constant of these scatterers. Our approach to solving this problem consists of two primary stages. Initially, we eliminate the unknown dielectric constant from the governing equation, resulting in a system of partial differential equations. Subsequently, we develop the Carleman contraction mapping method to effectively tackle this system. It is noteworthy to highlight the robustness of this method. It does not require a precise initial guess of the true solution, and its computational cost is relatively inexpensive. Some numerical examples are presented. 

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2. Complex analytic dependence on the dielectric permittivity in ENZ materials: The photonic doping example

   https://doi.org/10.1002/cpa.22138  

   Kohn, Robert V; Venkatraman, Raghavendra (February 2024, Communications on Pure and Applied Mathematics)

   Motivated by the physics literature on “photonic doping” of scatterers made from “epsilon‐near‐zero” (ENZ) materials, we consider how the scattering of time‐harmonic TM electromagnetic waves by a cylindrical ENZ region \Omega x R is affected by the presence of a “dopant” D \subset \Omega in which the dielectric permittivity is not near zero. Mathematically, this reduces to analysis of a 2D Helmholtz equation div(a(x) grad u) + k^2 u = f with a piecewise‐constant, complex valued coefficient a(x) that is nearly infinite (say, a =1/\delta with \delta \approx 0) in \Omega \setminus D. We show (under suitable hypotheses) that the solution u depends analytically on \delta near 0, and we give a simple PDE characterization of the terms in its Taylor expansion. For the application to photonic doping, it is the leading‐order corrections in \delta that are most interesting: they explain why photonic doping is only mildly affected by the presence of losses, and why it is seen even at frequencies where the dielectric permittivity is merely small. Equally important: our results include a PDE characterization of the leading‐order electric field in the ENZ region as \delta approaches 0 , whereas the existing literature on photonic doping provides only the leading‐order magnetic field. 

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3. Inverse scattering without phase: Carleman convexification and phase retrieval via the Wentzel--Kramers--Brillouin approximation

   Le, T T; Nguyen, P M; Nguyen, L H (June 2025, Preprint arXiv:2506.21699)

   This paper addresses the challenging and interesting inverse problem of reconstructing the spatially varying dielectric constant of a medium from phaseless backscattering measurements generated by single-point illumination. The underlying mathematical model is governed by the three-dimensional Helmholtz equation, and the available data consist solely of the magnitude of the scattered wave field. To address the nonlinearity and servere ill-posedness of this phaseless inverse scattering problem, we introduce a robust, globally convergent numerical framework combining several key regularization strategies. Our method first employs a phase retrieval step based on the Wentzel--Kramers--Brillouin (WKB) ansatz, where the lost phase information is reconstructed by solving a nonlinear optimization problem. Subsequently, we implement a Fourier-based dimension reduction technique, transforming the original problem into a more stable system of elliptic equations with Cauchy boundary conditions. To solve this resulting system reliably, we apply the Carleman convexification approach, constructing a strictly convex weighted cost functional whose global minimizer provides an accurate approximation of the true solution. Numerical simulations using synthetic data with high noise levels demonstrate the effectiveness and robustness of the proposed method, confirming its capability to accurately recover both the geometric location and contrast of hidden scatterers. 

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4. Robustly Learning a Gaussian: Getting Optimal Error, Efficiently

   Diakonikolas, Ilias; Kamath, Ilias; Kane, Daniel M.; Li, Jerry; Moitra, Ankur; Stewart, Alistair (January 2018, Proceedings of the annual ACM-SIAM Symposium on Discrete Algorithms)

   We study the fundamental problem of learning the parameters of a high-dimensional Gaussian in the presence of noise — where an ε-fraction of our samples were chosen by an adversary. We give robust estimators that achieve estimation error O(ε) in the total variation distance, which is optimal up to a universal constant that is independent of the dimension. In the case where just the mean is unknown, our robustness guarantee is optimal up to a factor of and the running time is polynomial in d and 1/ε. When both the mean and covariance are unknown, the running time is polynomial in d and quasipolynomial in 1/ε. Moreover all of our algorithms require only a polynomial number of samples. Our work shows that the same sorts of error guarantees that were established over fifty years ago in the one-dimensional setting can also be achieved by efficient algorithms in high-dimensional settings. 

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5. Sharp Noisy Binary Search with Monotonic Probabilities

   https://doi.org/10.4230/LIPIcs.ICALP.2024.75  

   Gretta, Lucas; Price, Eric (January 2024, Schloss Dagstuhl – Leibniz-Zentrum für Informatik)

   Bringmann, Karl; Grohe, Martin; Puppis, Gabriele; Svensson, Ola (Ed.)

   We revisit the noisy binary search model of [Karp and Kleinberg, 2007], in which we have n coins with unknown probabilities p_i that we can flip. The coins are sorted by increasing p_i, and we would like to find where the probability crosses (to within ε) of a target value τ. This generalized the fixed-noise model of [Burnashev and Zigangirov, 1974], in which p_i = 1/2 ± ε, to a setting where coins near the target may be indistinguishable from it. It was shown in [Karp and Kleinberg, 2007] that Θ(1/ε² log n) samples are necessary and sufficient for this task. We produce a practical algorithm by solving two theoretical challenges: high-probability behavior and sharp constants. We give an algorithm that succeeds with probability 1-δ from 1/C_{τ, ε} ⋅ (log₂ n + O(log^{2/3} n log^{1/3} 1/(δ) + log 1/(δ))) samples, where C_{τ, ε} is the optimal such constant achievable. For δ > n^{-o(1)} this is within 1 + o(1) of optimal, and for δ ≪ 1 it is the first bound within constant factors of optimal. 

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