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We obtain new analytic results for the problem of the recovery of a doped region D in semiconductor devices from the total flux of electrons/holes through a part of the boundary for various applied potentials on some com- plementa...
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We obtain new analytic results for the problem of the recovery of a doped region D in semiconductor devices from the total flux of electrons/holes through a part of the boundary for various applied potentials on some com- plementary part of the boundary. We consider the stationary two-dimensional case and we use the index of the gradient of solutions of the linear elliptic equation modeling a unipolar device. Under mild assumptions we prove local uniqueness of smooth D and global uniqueness of polygonal D satisfying some geometrical (star-shapednedness or convexity in some direction) assumptions. We design a nonlinear minimization algorithm for numerical solution and we demonstrate its effectiveness on some basic examples. An essential ingredient of this algorithm is a numerical solution of the direct problem by using single layer potentials.
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We are concerned with the reconstruction of both the heat coefficient and the source appearing in a parabolic equation from observations of the solution on the boundary. We propose two methods. The first uses a single point on the...
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We are concerned with the reconstruction of both the heat coefficient and the source appearing in a parabolic equation from observations of the solution on the boundary. We propose two methods. The first uses a single point on the boundary and a final time overdetermination, while the second uses only the observations over a small region on the boundary only. These methods are based on properties of the principal Dirichlet eigenfunction and Holmgren's theorem. (C) 2015 Elsevier Inc. All rights reserved.
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For the Regge problem, which can be characterized by a Sturm-Liouville problem with a Dirichlet boundary condition and an eigenparameter dependent boundary condition, it is known that the potential can be uniquely determined by al...
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For the Regge problem, which can be characterized by a Sturm-Liouville problem with a Dirichlet boundary condition and an eigenparameter dependent boundary condition, it is known that the potential can be uniquely determined by all eigenvalues. In this paper, we prove that if the potential is known a prior on a subinterval of the whole interval, then infinitely many eigenvalues can be missing for the recovery of the potential. That is, the fraction of all eigenvalues can uniquely determine the potential on the whole interval. Moreover, the relationship between the proportion of the missing eigenvalues and the length of the subinterval on the given potential is revealed.
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A hybrid method is proposed to locate the boundaries and determine the dielectric constants of multiple homogeneous scatterers by postprocessing the results of the linear sampling method (LSM). The hybrid method is composed of thr...
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A hybrid method is proposed to locate the boundaries and determine the dielectric constants of multiple homogeneous scatterers by postprocessing the results of the linear sampling method (LSM). The hybrid method is composed of three steps combining the capabilities of the LSM, the multiple signal classification (MUSIC) algorithm, and the generalized multipole technique (GMT). Initially, a new measure of the noise level of the multistatic response (MSR) matrix is proposed. Subsequently, the LSM is applied using the Tikhonov regularization. Then, the scattered field is back-propagated into the investigation domain using the GMT, where in order to ensure the stability, a MUSIC-based approach is exploited to determine the optimum reconstructing multipole sources. The case of limited scattered field data is also addressed by introducing a modification of the Papoulis–Gerchberg algorithm (PGA). Finally, the boundary and the dielectric constant of each scatterer are obtained simultaneously by solving an inverse boundary value problem through an optimization. The proposed hybrid approach is capable of efficiently approximating the dielectric constant of multiple homogeneous dielectric scatterers in a parallel manner. The effectiveness of the proposed method and the accuracy of the solutions are demonstrated by applying the method to both numerical and measured data.
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We introduce in this study an algorithm for the imaging of faults and of slip fields on those faults. The physics of this problem are modeled using the equations of linear elasticity. We define a regularized functional to be minim...
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We introduce in this study an algorithm for the imaging of faults and of slip fields on those faults. The physics of this problem are modeled using the equations of linear elasticity. We define a regularized functional to be minimized for building the image. We first prove that the minimum of that functional converges to the unique solution of the related fault inverse problem. Due to inherent uncertainties in measurements, rather than seeking a deterministic solution to the fault inverse problem, we then consider a Bayesian approach. The randomness involved in the unknown slip is teased out by assuming independence of the priors, and we show how the regularized error functional introduced earlier can be used to recover the probability density of the geometry parameter. The advantage of this Bayesian approach is that we obtain a way of quantifying uncertainties as part of our final answer. On the downside, this approach leads to a very large computation which we implemented on a parallel platform. After showing how this algorithm performs on simulated data, we apply it to measured data. The data was recorded during a slow slip event in Guerrero, Mexico.
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This work is concerned with the inverse problem for ocean acoustics modeled by a multilayered waveguide with a finite depth. We provide explicit formulae to locate the layers, including the seabed, and reconstruct the speed of sou...
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This work is concerned with the inverse problem for ocean acoustics modeled by a multilayered waveguide with a finite depth. We provide explicit formulae to locate the layers, including the seabed, and reconstruct the speed of sound and the densities in each layer from measurements collected on the surface of the waveguide. We proceed in two steps. First, we use Gaussian type excitations on the upper surface of the waveguide and then from the corresponding scattered fields, collected on the same surface, we recover the boundary spectral data of the related 1D spectral problem. Second, from these spectral data, we reconstruct the values of the normal derivatives of the singular solutions, of the original waveguide problem, on that upper surface. Finally, we derive formulae to reconstruct the layers from these values based on the asymptotic expansion of these singular solutions in terms of the source points.
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We consider boundary measurements for the wave equation on a bounded domain M ? □~2or on a compact Riemannian surface, and introduce a method to locate a discontinuity in the wave speed. Assuming that the wave speed consist of an...
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We consider boundary measurements for the wave equation on a bounded domain M ? □~2or on a compact Riemannian surface, and introduce a method to locate a discontinuity in the wave speed. Assuming that the wave speed consist of an inclusion in a known smooth background, the method can determine the distance from any boundary point to the inclusion. In the case of a known constant background wave speed, the method reconstructs a set contained in the convex hull of the inclusion and containing the inclusion. Even if the background wave speed is unknown, the method can reconstruct the distance from each boundary point to the inclusion assuming that the Riemannian metric tensor determined by the wave speed gives simple geometry in M. The method is based on reconstruction of volumes of domains of influence by solving a sequence of linear equations. For τ ∈C(?M) the domain of influence M(τ) is the set of those points on the manifold from which the distance to some boundary point x is less than τ(x).
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We consider the problem of detecting three-dimensional inclusions from quasi-backscattering far field data generated by an incident field of time-harmonic fixed frequency plane waves modeled with the Born approximation. We assume ...
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We consider the problem of detecting three-dimensional inclusions from quasi-backscattering far field data generated by an incident field of time-harmonic fixed frequency plane waves modeled with the Born approximation. We assume only partial far field data is known and use a sampling-type method to reconstruct small obstacles and extended spherical obstacles. In particular, at the location of a device transmitting an incident wave, we assume far field data is collected only along a line extending a short distance from the transmitting device. Several numerical examples are provided to demonstrate the effectiveness of the approach.
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We develop new efficient algorithms for a class of inverse problems of gravimetry to recover an anomalous volume mass distribution (measure) in the sense that we design fast local level-set methods to simultaneously reconstruct bo...
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We develop new efficient algorithms for a class of inverse problems of gravimetry to recover an anomalous volume mass distribution (measure) in the sense that we design fast local level-set methods to simultaneously reconstruct both unknown domain and varying density of the anomalous measure from modulus of gravity force rather than from gravity force itself. The equivalent-source principle of gravitational potential forces us to consider only measures of the form μ = f χ_D, where f is a density function and D is a domain inside a closed set in R~n. Accordingly, various constraints are imposed upon both the density function and the domain so that well-posedness theories can be developed for the corresponding inverse problems, such as the domain inverse problem, the density inverse problem, and the domain-density inverse problem. Starting from uniqueness theorems for the domain-density inverse problem, we derive a new gradient from the misfit functional to enforce the directional-independence constraint of the density function and we further introduce a new labeling function into the level-set method to enforce the geometrical constraint of the corresponding domain;consequently, we are able to recover simultaneously both unknown domain and varying density from given modulus of gravity force. Our fast level-set method is built upon localizing the level-set evolution around a narrow band near the zero level-set and upon accelerating numerical modeling by novel low-rank matrix multiplication. Numerical results demonstrate that uniqueness theorems are crucial for solving the inverse problem of gravimetry and will be impactful on gravity prospecting. To the best of our knowledge, our inversion algorithm is the first of such for the domain-density inverse problem since it is based upon the conditional well-posedness theory of the inverse problem.
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We consider the problem of reconstructing an even polynomial potential from one set of spectral data of a Sturm-Liouville problem. We show that we can recover an even polynomial of degree 2m from m + 1 given Taylor coefficients of...
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We consider the problem of reconstructing an even polynomial potential from one set of spectral data of a Sturm-Liouville problem. We show that we can recover an even polynomial of degree 2m from m + 1 given Taylor coefficients of the characteristic function whose zeros are the eigenvalues of one spectrum. The idea here is to represent the solution as a power series and identify the unknown coefficients from the characteristic function. We then compute these coefficients by solving a nonlinear algebraic system, and provide numerical examples at the end. Because of its algebraic nature, the method applies also to non self-adjoint problems.
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