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1. Inverse initial data reconstruction for Maxwell's equations via time-dimensional reduction method

   Le, T T; Van, C B; Dang, T D; Nguyen, L H (June 2025, preprint arXiv:2506.20777)

   We study an inverse problem for the time-dependent Maxwell system in an inhomogeneous and anisotropic medium. The objective is to recover the initial electric field $$\mathbf{E}_0$$ in a bounded domain $$\Omega \subset \mathbb{R}^3$$, using boundary measurements of the electric field and its normal derivative over a finite time interval. Informed by practical constraints, we adopt an under-determined formulation of Maxwell's equations that avoids the need for initial magnetic field data and charge density information. To address this inverse problem, we develop a time-dimension reduction approach by projecting the electric field onto a finite-dimensional Legendre polynomial-exponential basis in time. This reformulates the original space-time problem into a sequence of spatial systems for the projection coefficients. The reconstruction is carried out using the quasi-reversibility method within a minimum-norm framework, which accommodates the inherent non-uniqueness of the under-determined setting. We prove a convergence theorem that ensures the quasi-reversibility solution approximates the true solution as the noise and regularization parameters vanish. Numerical experiments in a fully three-dimensional setting validate the method's performance. The reconstructed initial electric field remains accurate even with $$10\%$$ noise in the data, demonstrating the robustness and applicability of the proposed approach to realistic inverse electromagnetic problems. 

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2. Nonlinear interaction of waves in elastodynamics and an inverse problem

   https://doi.org/10.1007/s00208-018-01796-y  

   de Hoop, Maarten; Uhlmann, Gunther; Wang, Yiran (January 2019, Mathematische Annalen)

   We consider nonlinear elastic wave equations generalizing Goldberg’s five constants model. We analyze the nonlinear interaction of two distorted plane waves and characterize the possible nonlinear responses. Using the boundary measurements of the nonlinear responses, we solve the inverse problem of determining elastic parameters from the displacement-to-traction map. 

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3. Hyper-differential sensitivity analysis for inverse problems constrained by partial differential equations

   https://doi.org/10.1088/1361-6420/abaf63  

   Sunseri, Isaac; Hart, Joseph; van Bloemen Waanders, Bart; Alexanderian, Alen (December 2020, Inverse Problems)

   Abstract High fidelity models used in many science and engineering applications couple multiple physical states and parameters. Inverse problems arise when a model parameter cannot be determined directly, but rather is estimated using (typically sparse and noisy) measurements of the states. The data is usually not sufficient to simultaneously inform all of the parameters. Consequently, the governing model typically contains parameters which are uncertain but must be specified for a complete model characterization necessary to invert for the parameters of interest. We refer to the combination of the additional model parameters (those which are not inverted for) and the measured data states as the ‘complementary parameters’. We seek to quantify the relative importance of these complementary parameters to the solution of the inverse problem. To address this, we present a framework based on hyper-differential sensitivity analysis (HDSA). HDSA computes the derivative of the solution of an inverse problem with respect to complementary parameters. We present a mathematical framework for HDSA in large-scale PDE-constrained inverse problems and show how HDSA can be interpreted to give insight about the inverse problem. We demonstrate the effectiveness of the method on an inverse problem by estimating a permeability field, using pressure and concentration measurements, in a porous medium flow application with uncertainty in the boundary conditions, source injection, and diffusion coefficient. 

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4. Reconstruction of the doping profile in Vlasov–Poisson system

   https://doi.org/10.1088/1361-6420/ad7c78  

   Lai, Ru-Yu; Li, Qin; Sun, Weiran (September 2024, Inverse Problems)

   Abstract We study the inverse problem of recovering the doping profile in the stationary Vlasov–Poisson equation, given the knowledge of the incoming and outgoing measurements at the boundary of the domain. This problem arises from identifying impurities in the semiconductor manufacturing. Our result states that, under suitable assumptions, the doping profile can be uniquely determined through an asymptotic formula of the electric field that it generates. 

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5. An inverse acoustic-elastic interaction problem with phased or phaseless far-field data

   https://doi.org/10.1088/1361-6420/ab693e  

   Dong, H.; Lai, J.; Li, P. (January 2020, Inverse problems)

   Consider the scattering of a time-harmonic acoustic plane wave by a bounded elastic obstacle which is immersed in a homogeneous acoustic medium. This paper is concerned with an inverse acoustic-elastic interaction problem, which is to determine the location and shape of the elastic obstacle by using either the phased or phaseless far-field data. By introducing the Helmholtz decomposition, the model problem is reduced to a coupled boundary value problem of the Helmholtz equations. The jump relations are studied for the second derivatives of the single-layer potential in order to deduce the corresponding boundary integral equations. The well-posedness is discussed for the solution of the coupled boundary integral equations. An efficient and high order Nyström-type discretization method is proposed for the integral system. A numerical method of nonlinear integral equations is developed for the inverse problem. For the case of phaseless data, we show that the modulus of the far-field pattern is invariant under a translation of the obstacle. To break the translation invariance, an elastic reference ball technique is introduced. We prove that the inverse problem with phaseless far-field pattern has a unique solution under certain conditions. In addition, a numerical method of the reference ball technique based nonlinear integral equations is proposed for the phaseless inverse problem. Numerical experiments are presented to demonstrate the effectiveness and robustness of the proposed methods. 

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