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Some remarks on the jump condition appearing in Theorems 1.1 and 1.2 in the article [1] and their implications are given., Comment: 7 pages. arXiv admin note: text overlap with arXiv:1902.05191

In this paper, we consider the monodomain model of cardiac electrophysiology. After an analysis of the well-posedness of the forward problem, we show that perfectly insulating regions (modeling ischemic regions in the cardiac tissue) can be uniquely determined by partial boundary measurements of the potential.

The electrical impedance tomography (EIT) problem of estimating the unknown conductivity distribution inside a domain from boundary current or voltage measurements requires the solution of a nonlinear inverse problem. Sparsity promoting hierarchical Bayesian models have been shown to be very effective in the recovery of almost piecewise constant solutions in linear inverse problems. We demonstrate that by exploiting linear algebraic considerations it is possible to organize the calculation for the Bayesian solution of the nonlinear EIT inverse problem via finite element methods with sparsity promoting priors in a computationally efficient manner. The proposed approach uses the Iterative Alternating Sequential (IAS) algorithm for the solution of the linearized problems. Within the IAS algorithm, a substantial reduction in computational complexity is attained by exploiting the low dimensionality of the data space and an adjoint formulation of the Tikhonov regularized solution that constitutes part of the iterative updating scheme. Numerical tests illustrate the computational efficiency of the proposed algorithm. The paper sheds light also on the convexity properties of the objective function of the maximum a posteriori (MAP) estimation problem.

The paper studies an imaging problem in the diffusive ultrasound-modulated bioluminescence tomography with partial boundary measurement in an anisotropic medium. Assuming plane-wave modulation, we transform the imaging problem to an inverse problem with internal data, and derive a reconstruction procedure to recover the bioluminescent source. Subsequently, an uncertainty quantification estimate is established to assess the robustness of the reconstruction. To facilitate practical implementation, we discretize the diffusive model using the staggered grid scheme, resulting in a discrete formulation of the UMBLT inverse problem. A discrete reconstruction procedure is then presented along with a discrete uncertainty quantification estimate. Finally, the reconstruction procedure is quantitatively validated through numerical examples to demonstrate the efficacy and reliability of the proposed approach and estimates.

In this paper we develop in detail the geometric constructions that lead to many uniqueness results for the determination of polyhedral sets, typically scatterers, by a finite minimal number of measurements. We highlight how unique continuation and a suitable reflection principle are enough to proceed with the constructions, without any other assumption on the underlying partial differential equation or the boundary condition. We also aim to keep the geometric constructions and their proofs as simple as possible. To illustrate the applicability of this theory, we show how several uniqueness results present in the literature immediately follow from our arguments. Indeed we believe that this theory may serve as a roadmap for establishing similar uniqueness results for other partial differential equations or boundary conditions., Comment: 21 pages

By using some deep tools from microlocal analysis, the authors of the papers (Ann. of Math., 165 (2007), 567--591, J. Amer. Math. Soc., 23 (2010), 655--691; Invent. Math., 178 (2009), 119--171; Duke Math. J., 158(2011), 83--120) have successfully established various Carleman estimates for elliptic operators that possess limiting Carleman weight. In this study, we revisit these problems and present a unified and fundamental approach for deriving these estimates. The main tool we employ is an elementary pointwise estimate for second-order elliptic operators.

Study of a simple single-trace transmission example shows how an extended source formulation of full-waveform inversion can produce an optimization problem without spurious local minima ("cycle skipping"), hence efficiently solvable via Newton-like local optimization methods. The data consist of a single trace extracted from a causal pressure field, propagating in a homogeneous fluid according to linear acoustics, and recorded at a given distance from a transient point energy source. The source intensity ("wavelet") is presumed quasi-impulsive, with zero energy for time lags greater than a specified maximum lag. The inverse problem is: from the recorded trace, recover both the sound velocity or slowness and source wavelet with specified support, so that the data is fit with prescribed RMS relative error. The least-squares objective function has multiple large residual minimizers. The extended inverse problem permits source energy to spread in time, and replaces the maximum lag constraint by a weighted quadratic penalty. A companion paper shows that for proper choice of weight operator, any stationary point of the extended objective produces a good approximation of the global minimizer of the least squares objective, with slowness error bounded by a multiple of the maximum lag and the assumed noise level. This paper summarizes the theory developed in the companion paper and presents numerical experiments demonstrating the accuracy of the predictions in concrete instances. We also show how to dynamically adjust the penalty scale during iterative optimization to improve the accuracy of the slowness estimate., Comment: 40 pages, 32 figures

In this article, a general model for 1D thermal wave interference is derived for multi-layered coating systems on a thermally thick substrate using the same principles as for the well established one-layered and two-layered coating cases. Using the lock-in thermography principle, an illumination source modulates the surface of those systems periodically by a planar, sinusoidal wave form with a fixed frequency. The coating systems absorb the optical energy on its surface and convert it into thermal energy, resulting in the propagation of a spatially and temporally periodic thermal wave with the same frequency. These thermal waves, originating at the surface, are reflected and transmitted at each interface leading to infinitely many wave trains that need to be tracked in order to formulate the final surface temperature as a superposition of all these waves. The heat transfer inside the object depends on the layer thickness of each coating, but also on the thermal properties of each layer material. The goal is to have a mathematical and physical model which describes the phase angle data measured by an infrared camera. Having these data, the main objective of this paper is to determine the thickness of each coating layer. In practice, the thermal properties of the layers usually are unknown, which makes the process even more difficult. For that reason, this article presents a concept to determine the thermal properties in advance., Comment: 16 pages, 8 figures

The time dependent experimental data are always collected at discrete grids with respect to the time t. The step size h of such a grid is always separated from zero by a certain positive number. The same is true for all computations, which are always done on discrete grids with their grid step sizes being not too small. These applied considerations prompt us to introduce a new type of Ill-Posed Problems and Coefficient Inverse Problems (CIP)for parabolic equations. In these problems the t-derivatives of corresponding parabolic operators are written in finite differences with the grid step size being separated from zero. We call this the "t-finite difference framework" (TFD). We address three long standing open questions within the TFD framework. Finally, a numerical method is developed for the CIP of monitoring epidemics. The global convergence of this method is proven.

We study the inverse medium scattering problem to reconstruct the unknown inhomogeneous medium from the far-field patterns of scattered waves. The inverse scattering problem is generally ill-posed and nonlinear, and the iterative optimization method is often adapted. A natural iterative approach to this problem is to place all available measurements and mappings into one long vector and mapping, respectively, and to iteratively solve the linearized large system equation using the Tikhonov regularization method, which is called the Levenberg-Marquardt scheme. However, this is computationally expensive because we must construct the larger system equations when the number of available measurements increases. In this paper, we propose two reconstruction algorithms based on the Kalman filter. One is the algorithm equivalent to the Levenberg-Marquardt scheme, and the other is inspired by the Extended Kalman Filter. For the algorithm derivation, we iteratively apply the Kalman filter to the linearized equation for our nonlinear equation. Our proposed algorithms sequentially update the state and the weight of the norm for the state space, which avoids the construction of a large system equation and retains the information of past updates. Finally, we provide numerical examples to demonstrate our proposed algorithms., Comment: 24 pages, 9 figures. arXiv admin note: text overlap with arXiv:2110.09518, arXiv:2110.08773

The limited angle Radon transform is notoriously difficult to invert due to the ill-posedness. In this work, we give a mathematical explanation that the data-driven approach based on deep neural networks can reconstruct more information in a stable way compared to traditional methods.

Adaptive Waveform Inversion applied to transient transmitted wave data yields estimates of index of refraction (or wave velocity) similar to those obtained by travel time inversion, provided that the data contain a single smooth wavefront., Comment: 11 pages

We determine the conductivity of the interior of a body using electrical measurements on its surface. We assume only that the conductivity is bounded below by a positive constant and that the conductivity and surface are Lipschitz continuous. To determine the conductivity we first solve an associated integral equation locally, finding solutions in $H^1(B)$, where $B$ is a ball that properly contains the body. A key ingredient is to equip this Sobolev space with an equivalent norm which depends on two auxiliary parameters that can be chosen to yield a contraction., Comment: 23 pages

We establish a double logarithmic stability inequality for the problem of determining the initial data in an IBVP for the wave equation outside a non-trapping obstacle from two localized measurements.

This paper introduces a highly adaptive and automated approach for generating Finite Element (FE) discretization for a given realistic multi-compartment human head model obtained through magnetic resonance imaging (MRI) dataset. We aim at obtaining accurate tetrahedral FE meshes for electroencephalographic source localization. We present recursive solid angle labeling for the surface segmentation of the model and then adapt it with a set of smoothing, inflation, and optimization routines to further enhance the quality of the FE mesh. The results show that our methodology can produce FE mesh with an accuracy greater than 1 millimeter, significant with respect to both their 3D structure discretization outcome and electroencephalographic source localization estimates. FE meshes can be achieved for the human head including complex deep brain structures. Our algorithm has been implemented using the open Matlab-based Zeffiro Interface toolbox with it effective time-effective parallel computing system.

There are two main approaches to solve inverse coefficient determination problems for wave equations: the Boundary Control method and an approach based on geometric optics. These notes focus on the Boundary Control method, but we will have a brief look at the geometric optics as well., Comment: These lecture notes were written for CIRM SMF School Spectral Theory, Control and Inverse Problems, November 2022

We establish in dimension $3$ a stability inequality for the problem of determining the potential in the Schr\"odinger equation from boundary measurements in the case where the potential belongs to $L^s$ with $s\in (2,3)$.

In this paper, we analyze the nonlinear single pixel X-ray transform $K$ and study the reconstruction of $f$ from the measurement $Kf$. Different from the well-known X-ray transform, the transform $K$ is a nonlinear operator and uses a single detector that integrates all rays in the space. We derive stability estimates and an inversion formula of $K$. We also consider the case where we integrate along geodesics of a Riemannian metric. Moreover, we conduct several numerical experiments to corroborate the theoretical results.

This paper is concerned with reconstruction issue of inverse obstacle problems governed by partial differential equations and consists of two parts. (i) The first part considers the foundation of the probe and enclosure methods for an impenetrable obstacle embedded in a medium governed by the stationary Schr\"odinger equation. Under a general framework, some natural estimates for a quantity computed from a pair of the Dirichlet and Neumann data on the outer surface of the body occupied by the medium are given. The estimates enables us to derive almost immediately the necessary asymptotic behaviour of indicator functions for both methods. (ii) The second one considers the realization of the enclosure method for a penetrable obstacle embedded in an absorbing medium governed by the stationary Schr\"odinger equation. The unknown obstacle considered here is modeled by a perturbation term added to the background complex-valued $L^{\infty}$-potential. Under a jump condition on the term across the boundary of the obstacle and some kind of regularity for the obstacle surface including Lipschitz one as a special case, the enclosure method using infinitely many pairs of the Dirichlet and Neumann data is established., Comment: 32pages. This is the final version

In this paper, we study the inverse acoustic medium scattering problem to reconstruct the unknown inhomogeneous medium from far field patterns of scattered waves. We propose the reconstruction scheme based on the Kalman filter, which becomes possible to sequentially estimate the inhomogeneous medium. We also show that in the linear inverse problem, the estimation for the Kalman filter is equivalent to that for the Tikhonov regularization. Finally, we give numerical examples to demonstrate our proposed method., Comment: 21 pages, 5 figures

In this paper, we study the inverse medium scattering problem to reconstruct unknown inhomogeneous medium from far field patterns of scattered waves. In the first part of our work, the linear inverse scattering problem was discussed, while in the second part, we deal with the nonlinear problem. The main idea is to apply the linear Kalman filter to the linearized problem. There are several ways to linearize, which introduce two reconstruction algorithms. Finally, we give numerical examples to demonstrate our proposed method., Comment: 22 pages, 6 figures

We establish stability inequalities for the problem of determining the potential, appearing in a Sch\"odinger equation, from partial boundary data in the high frequency limit. These stability inequalities hold under the assumption that the potential is known near the boundary of the domain under consideration.

We consider the inverse problem of the simultaneous identification of the coefficients $\sigma$ and $q$ of the equation div$(\sigma\nabla u) + qu=0$ from the knowledge of the complete Cauchy data pairs. We assume that $\sigma=\gamma A$ where $A$ is a given matrix function and $\gamma, q$ are unknown piecewise affine scalar functions. No sign, nor spectrum condition on $q$ is assumed. We derive a result of global Lipschitz stability in dimension $n\geq 3$. The proof relies on the method of singular solutions and on the quantitative estimates of unique continuation., Comment: pp. 38