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Inverse Problems & Imaging

August 2021 , Volume 15 , Issue 4

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A regularization operator for source identification for elliptic PDEs
Ole Løseth Elvetun and Bjørn Fredrik Nielsen
2021, 15(4): 599-618 doi: 10.3934/ipi.2021006 +[Abstract](519) +[HTML](212) +[PDF](1303.7KB)
Abstract:

We study a source identification problem for a prototypical elliptic PDE from Dirichlet boundary data. This problem is ill-posed, and the involved forward operator has a significant nullspace. Standard Tikhonov regularization yields solutions which approach the minimum \begin{document}$ L^2 $\end{document}-norm least-squares solution as the regularization parameter tends to zero. We show that this approach 'always' suggests that the unknown local source is very close to the boundary of the domain of the PDE, regardless of the position of the true local source.

We propose an alternative regularization procedure, realized in terms of a novel regularization operator, which is better suited for identifying local sources positioned anywhere in the domain of the PDE. Our approach is motivated by the classical theory for Tikhonov regularization and yields a standard quadratic optimization problem. Since the new methodology is derived for an abstract operator equation, it can be applied to many other source identification problems. This paper contains several numerical experiments and an analysis of the new methodology.

Cauchy problem of non-homogenous stochastic heat equation and application to inverse random source problem
Shuli Chen, Zewen Wang and Guolin Chen
2021, 15(4): 619-639 doi: 10.3934/ipi.2021008 +[Abstract](421) +[HTML](178) +[PDF](932.57KB)
Abstract:

In this paper, a Cauchy problem of non-homogenous stochastic heat equation is considered together with its inverse source problem, where the source term is assumed to be driven by an additive white noise. The Cauchy problem (direct problem) is to determine the displacement of random temperature field, while the considered inverse problem is to reconstruct the statistical properties of the random source, i.e. the mean and variance of the random source. It is proved constructively that the Cauchy problem has a unique mild solution, which is expressed an integral form. Then the inverse random source problem is formulated into two Fredholm integral equations of the first kind, which are typically ill-posed. To obtain stable inverse solutions, the regularized block Kaczmarz method is introduced to solve the two Fredholm integral equations. Finally, numerical experiments are given to show that the proposed method is efficient and robust for reconstructing the statistical properties of the random source.

Unique continuation property and Poincaré inequality for higher order fractional Laplacians with applications in inverse problems
Giovanni Covi, Keijo Mönkkönen and Jesse Railo
2021, 15(4): 641-681 doi: 10.3934/ipi.2021009 +[Abstract](411) +[HTML](175) +[PDF](579.63KB)
Abstract:

We prove a unique continuation property for the fractional Laplacian \begin{document}$ (-\Delta)^s $\end{document} when \begin{document}$ s \in (-n/2, \infty)\setminus \mathbb{Z} $\end{document} where \begin{document}$ n\geq 1 $\end{document}. In addition, we study Poincaré-type inequalities for the operator \begin{document}$ (-\Delta)^s $\end{document} when \begin{document}$ s\geq 0 $\end{document}. We apply the results to show that one can uniquely recover, up to a gauge, electric and magnetic potentials from the Dirichlet-to-Neumann map associated to the higher order fractional magnetic Schrödinger equation. We also study the higher order fractional Schrödinger equation with singular electric potential. In both cases, we obtain a Runge approximation property for the equation. Furthermore, we prove a uniqueness result for a partial data problem of the \begin{document}$ d $\end{document}-plane Radon transform in low regularity. Our work extends some recent results in inverse problems for more general operators.

Bragg scattering tomography
James W. Webber and Eric L. Miller
2021, 15(4): 683-721 doi: 10.3934/ipi.2021010 +[Abstract](386) +[HTML](192) +[PDF](1003.84KB)
Abstract:

Here we introduce a new forward model and imaging modality for Bragg Scattering Tomography (BST). The model we propose is based on an X-ray portal scanner with linear detector collimation, currently being developed for use in airport baggage screening. The geometry under consideration leads us to a novel two-dimensional inverse problem, where we aim to reconstruct the Bragg scattering differential cross section function from its integrals over a set of symmetric \begin{document}$ C^2 $\end{document} curves in the plane. The integral transform which describes the forward problem in BST is a new type of Radon transform, which we introduce and denote as the Bragg transform. We provide new injectivity results for the Bragg transform here, and describe how the conditions of our theorems can be applied to assist in the machine design of the portal scanner. Further we provide an extension of our results to \begin{document}$ n $\end{document}-dimensions, where a generalization of the Bragg transform is introduced. Here we aim to reconstruct a real valued function on \begin{document}$ \mathbb{R}^{n+1} $\end{document} from its integrals over \begin{document}$ n $\end{document}-dimensional surfaces of revolution of \begin{document}$ C^2 $\end{document} curves embedded in \begin{document}$ \mathbb{R}^{n+1} $\end{document}. Injectivity proofs are provided also for the generalized Bragg transform.

Existence and stability of electromagnetic Stekloff eigenvalues with a trace class modification
Samuel Cogar
2021, 15(4): 723-744 doi: 10.3934/ipi.2021011 +[Abstract](319) +[HTML](171) +[PDF](455.49KB)
Abstract:

A recent area of interest is the development and study of eigenvalue problems arising in scattering theory that may provide potential target signatures for use in nondestructive testing of materials. We consider a generalization of the electromagnetic Stekloff eigenvalue problem that depends upon a smoothing parameter, for which we establish two main results that were previously unavailable for this type of eigenvalue problem. First, we use the theory of trace class operators to prove that infinitely many eigenvalues exist for a sufficiently high degree of smoothing, even for an absorbing medium. Second, we leverage regularity results for Maxwell's equations in order to establish stability results for the eigenvalues with respect to the material coefficients, and we show that this generalized class of Stekloff eigenvalues converges to the standard class as the smoothing parameter approaches zero.

A sampling type method in an electromagnetic waveguide
Shixu Meng
2021, 15(4): 745-762 doi: 10.3934/ipi.2021012 +[Abstract](361) +[HTML](185) +[PDF](3332.76KB)
Abstract:

We propose a sampling type method to image scatterer in an electromagnetic waveguide. The waveguide terminates at one end and the measurements are on the other end and in the far field. The imaging function is based on integrating the measurements and a known function over the measurement surface directly. The design and analysis of such imaging function are based on a factorization of a data operator given by the measurements. We show by analysis that the imaging function peaks inside the scatterer, where the coercivity of the factorized operator and the design of the known function play a central role. Finally, numerical examples are provided to demonstrate the performance of the imaging method.

Synthetic-Aperture Radar image based positioning in GPS-denied environments using Deep Cosine Similarity Neural Networks
Seonho Park, Maciej Rysz, Kaitlin L. Fair and Panos M. Pardalos
2021, 15(4): 763-785 doi: 10.3934/ipi.2021013 +[Abstract](387) +[HTML](242) +[PDF](10005.15KB)
Abstract:

Navigating unmanned aerial vehicles in precarious environments is of great importance. It is necessary to rely on alternative information processing techniques to attain spatial information that is required for navigation in such settings. This paper introduces a novel deep learning-based approach for navigating that exclusively relies on synthetic aperture radar (SAR) images. The proposed method utilizes deep neural networks (DNNs) for image matching, retrieval, and registration. To this end, we introduce Deep Cosine Similarity Neural Networks (DCSNNs) for mapping SAR images to a global descriptive feature vector. We also introduce a fine-tuning algorithm for DCSNNs, and DCSNNs are used to generate a database of feature vectors for SAR images that span a geographic area of interest, which, in turn, are compared against a feature vector of an inquiry image. Images similar to the inquiry are retrieved from the database by using a scalable distance measure between the feature vector outputs of DCSNN. Methods for reranking the retrieved SAR images that are used to update position coordinates of an inquiry SAR image by estimating from the best retrieved SAR image are also introduced. Numerical experiments comparing with baselines on the Polarimetric SAR (PolSAR) images are presented.

Preconditioned Douglas-Rachford type primal-dual method for solving composite monotone inclusion problems with applications
Yixuan Yang, Yuchao Tang, Meng Wen and Tieyong Zeng
2021, 15(4): 787-825 doi: 10.3934/ipi.2021014 +[Abstract](437) +[HTML](186) +[PDF](4059.99KB)
Abstract:

This paper is concerned with the monotone inclusion involving the sum of a finite number of maximally monotone operators and the parallel sum of two maximally monotone operators with bounded linear operators. To solve this monotone inclusion, we first transform it into the formulation of the sum of three maximally monotone operators in a proper product space. Then we derive two efficient iterative algorithms, which combine the partial inverse method with the preconditioned Douglas-Rachford splitting algorithm and the preconditioned proximal point algorithm. Furthermore, we develop an iterative algorithm, which relies on the preconditioned Douglas-Rachford splitting algorithm without using the partial inverse method. We carefully analyze the theoretical convergence of the proposed algorithms. Finally, in order to demonstrate the effectiveness and efficiency of these algorithms, we conduct numerical experiments on a novel image denoising model for salt-and-pepper noise removal. Numerical results show the good performance of the proposed algorithms.

2019  Impact Factor: 1.373

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