Abstract.An approximately globally convergent numerical method for a 1-D Coefficient Inverse Problem for a hyperbolic PDE is applied to image dielectric constants of targets from blind experimental data. The data were collected in the field by the Forward Looking Radar of the US Army Research Laboratory. A posteriori analysis has revealed that computed and tabulated values of dielectric constants are in a good agreement. Convergence analysis is presented.1. Introduction. In this paper we test the 1-D version [31] of the numerical method of recent publications [5,6,7,8,9,10,11,12,27,28,32,48] for the case when the time resolved backscattering electric signal is measured experimentally in the field. Measurements were conducted by the Forward Looking Radar built in US Army Research Laboratory (ARL). All kinds of clutter were present at the site of data collection. The data are severely limited. The goal of this radar is to detect and possibly identify shallow explosive-like targets. Prior to this effort, the focus of the ARL team was on the image processing rather than on the target detection and identification [37]. The current data processing procedure of ARL delivers only the energy information. The algorithm of this paper computes values of dielectric constants of targets using those data. These values represent a new, surprising and quite useful dimension of information for the ARL team. A hope is that these values might be helpful in the target detection and identification process.The UNCC/ChalmersGU team has worked only with the most challenging case of blind experimental data. "Blind" means that first computations were made by the UNCC/ChalmersGU team without any knowledge of correct answers. Next, computational results were sent to the ARL team. The ARL team has compared a posteriori those results with the reality and then revealed correct answers to the UNCC/ChalmersGU team. The performance of the algorithm of above cited publications for transmitted blind experimental data was presented in [27], see Tables 5 and 6 there. Images of [27] were refined in the follow up publication [9] using the adaptivity technique of [3,4].In the above cited works a new numerical method was developed for some Multidimensional Coefficient Inverse Problems (MCIPs) for a hyperbolic PDE with single measurement data. "Single measurement" means that either only a single position of the point source or only a single direction of the incident plane wave is considered. Because of many dangers on the battlefield, the single measurement arrangement is the most suitable one for military applications. There were two goals of those publications: Goal 1. To develop such a numerical method, which would have a rigorous guarantee obtaining a good approximation for the exact solution of a CIP without using an advanced knowledge of neither a small neighborhood of that solution nor of the background medium in the domain of interest.Goal 2. This method should demonstrate a good performance on both computationally simulated and experimental data....

A new numerical method is proposed for a 1-D inverse medium scattering problem with multi-frequency data. This method is based on the construction of a weighted cost functional. The weight is a Carleman Weight Function (CWF). In other words, this is the function, which is present in the Carleman estimate for the undelying differential operator. The presence of the CWF makes this functional strictly convex on any a priori chosen ball with the center at {0} in an appropriate Hilbert space. Convergence of the gradient minimization method to the exact solution starting from any point of that ball is proven. Computational results for both computationally simulated and experimental data show a good accuracy of this method. of [11], which is based on Carleman estimates and which was originally designed in [11] only for proofs of uniqueness theorems for CIPs, also see the recent survey in [27].Another version of the GCM with a CWF in it was recently developed in [6] for a CIP for the hyperbolic equation w tt = ∆w + a (x) w + f (x, t) , where a (x) is the unknown coefficient. This GCM was tested numerically in [7]. In [6,7] non-vanishing conditions are imposed: it is assumed that either f (x, 0) = 0 or w (x, 0) = 0 or w t (x, 0) = 0 in the entire domain of interest. Similar assumptions are imposed in [10,29] for the GCM of the second type. On the other hand, we consider in the current paper, so as in [24,25,26,33], the fundamental solution of the corresponding PDE. The differences between the fundamental solutions of those PDEs and solutions satisfying non-vanishing conditions cause quite significant differences between [24, 25, 26, 33] and [6, 7, 10, 29] of corresponding versions of the GCM of the second type.Recently, the idea of the GCM of the second type was extended to the case of ill-posed Cauchy problems for quasilinear PDEs, see the theory in [28] and some extensions and numerical examples in [4,30].CIPs of wave propagation are a part of a bigger subfield, Inverse Scattering Problems (ISPs). ISPs attract a significant attention of the scientific community. In this regard we refer to some direct methods which successfully reconstruct positions, sizes and shapes of scatterers without iterations [12,13,20,22,36,37,38,45]. We also refer to [3,37,41,42] for some other ISPs in the frequency domain. In addition, we cite some other numerical methods for ISPs considered in [2,5,46].As to the CIPs with multiple measurement, i.e. the Dirichlet-to-Neumann map data, we mention recent works [1,21,43] and references cited therein, where reconstruction

A new version of the convexification method is developed analytically and tested numerically for a 1-D coefficient inverse problem in the frequency domain. Unlike the previous version, this one does not use the so-called "tail function", which is a complement of a certain truncated integral with respect to the wave number. Globally strictly convex cost functional is constructed with the Carleman Weight Function. Global convergence of the gradient projection method to the correct solution is proved. Numerical tests are conducted for both computationally simulated and experimental data. 1 arXiv:1805.06025v1 [math.NA] 15 May 2018procedures, which would combine the currently used energy information with the estimates of dielectric constants. This combination, in turn might result in lower false alarm rates. Our targets are three dimensional ones of course. On the other hand, that radar can measure only one time dependent curve for each target. Thus, what can be done at most by any data inversion technique is to estimate a sort of an average of the dielectric constant for a target. We believe, however, that even these estimates might be useful for the goal of decreasing the false alarm rate. Thus, we model the wave propagation process by the 1-D Helmholtz equation.Convexification is the method, which constructs globally strictly convex weighted Tikhonov-like functionals either for Coefficient Inverse Problems (CIPs) or for ill-posed Cauchy problems for quasilinear PDEs. See our works [1,4,5,6,7,8,9] for CIPs and [10,11,12] for quasilinear PDEs. The key element of such a functional is the presence of the Carleman Weight Function (CWF). This is the weight function in the Carleman estimate for the principal part of the corresponding differential operator.Thus, convexification addresses the well known problem of multiple local minima and ravines of conventional Tikhonov-like functionals for CIPs, see, e.g. the paper [13] for a numerical example of multiple local minima. Convexification allows one to construct globally convergent numerical methods for CIPs. We call a numerical method for a CIP globally convergent if a theorem is proved, which claims that this method delivers at least one point in a sufficiently small neighborhood of the exact coefficient without any advanced knowledge of this neighborhood. Since conventional least squares Tikhonov functionals are non convex, then they usually have many local minima and ravines. This means that in order to obtain the correct solution, using such a functional, one needs to start the optimization process in a sufficiently small neighborhood of this solution, i.e. one should work with a locally convergent numerical method. However, such a small neighborhood is rarely known in applications.The convexification is a globally convergent numerical method, see Remark 4.1 in section 4. The idea of the convexification has roots in the method of Carleman estimates for CIPs. This method was originated in the work of Bukhgeim and Klibanov (1981) [14] as the tool of proofs of global uniqu...

Inverse scattering problems of the reconstructions of physical properties of a medium from boundary measurements are substantially challenging ones. This work aims to verify the performance on experimentally collected data of a newly developed convexification method for a 3D coefficient inverse problem for the case of unknown objects buried in a sandbox. The measured backscatter data are generated by a point source moving along an interval of a straight line and the frequency is fixed. Using a special Fourier basis, the method of this work strongly relies on a new derivation of a boundary value problem for a system of coupled quasilinear elliptic equations. This problem, in turn, is solved via the minimization of a Tikhonov-like functional weighted by a Carleman weight function. Different from the continuous case, our weighted cost functional in the partial finite difference does not need the penalty term to gain the global convergence analysis. The numerical verification is performed using experimental data, which are raw backscatter data of the electric field. These data were collected using a microwave scattering facility at The University of North Carolina at Charlotte.

A 3-D inverse medium problem in the frequency domain is considered. Another name for this problem is Coefficient Inverse Problem. The goal is to reconstruct spatially distributed dielectric constants from scattering data. Potential applications are in detection and identification of explosive-like targets. A single incident plane wave and multiple frequencies are used. A new numerical method is proposed. A theorem is proved, which claims that a small neigborhood of the exact solution of that problem is reached by this method without any advanced knowledge of that neighborhood. We call this property of that numerical method "global convergence". Results of numerical experiments for the case of the backscattering data are presented.

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Summary We explore diffuse formulations of Nitsche's method for consistently imposing Dirichlet boundary conditions on phase‐field approximations of sharp domains. Leveraging the properties of the phase‐field gradient, we derive the variational formulation of the diffuse Nitsche method by transferring all integrals associated with the Dirichlet boundary from a geometrically sharp surface format in the standard Nitsche method to a geometrically diffuse volumetric format. We also derive conditions for the stability of the discrete system and formulate a diffuse local eigenvalue problem, from which the stabilization parameter can be estimated automatically in each element. We advertise metastable phase‐field solutions of the Allen‐Cahn problem for transferring complex imaging data into diffuse geometric models. In particular, we discuss the use of mixed meshes, that is, an adaptively refined mesh for the phase‐field in the diffuse boundary region and a uniform mesh for the representation of the physics‐based solution fields. We illustrate accuracy and convergence properties of the diffuse Nitsche method and demonstrate its advantages over diffuse penalty‐type methods. In the context of imaging‐based analysis, we show that the diffuse Nitsche method achieves the same accuracy as the standard Nitsche method with sharp surfaces, if the inherent length scales, ie, the interface width of the phase‐field, the voxel spacing, and the mesh size, are properly related. We demonstrate the flexibility of the new method by analyzing stresses in a human vertebral body.

This study presents a development in quantitative structure–property relationships (QSPRs) for research in organic semiconductor materials by introducing a new structural descriptor called “degree of π-orbital overlap” based on two-dimensional structure information of aromatic molecules. Application of this method to predict the electronic properties of polycyclic aromatic hydrocarbon (PAH) molecules, which are known to be the core component of many organic semiconductor materials, is presented. Results demonstrated that QSPRs based on the new descriptor can predict rather accurate band gaps, ionization potentials and electron affinities for a large number of PAHs compared to those explicitly calculated by density functional theory method. This research opens new possibilities for developing QSPRs for other organic semiconductor classes in future.