# American Institute of Mathematical Sciences

ISSN:
1930-8337

eISSN:
1930-8345

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## Inverse Problems and Imaging

April 2021 , Volume 15 , Issue 2

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2021, 15(2): 185-199 doi: 10.3934/ipi.2020060 +[Abstract](1044) +[HTML](229) +[PDF](375.28KB)
Abstract:

We consider the dynamical inverse problem for the wave equation on a metric tree graph and describe the dynamical Leaf Peeling (LP) method. The main step of the method is recalculating the response operator from the original tree to a peeled tree. The LP method allows us to recover the connectivity, potential function on a tree graph and the lengths of its edges from the response operator given on a finite time interval.

2021, 15(2): 201-228 doi: 10.3934/ipi.2020061 +[Abstract](1147) +[HTML](210) +[PDF](528.41KB)
Abstract:

We investigate an empirical Bayesian nonparametric approach to a family of linear inverse problems with Gaussian prior and Gaussian noise. We consider a class of Gaussian prior probability measures with covariance operator indexed by a hyperparameter that quantifies regularity. By introducing two auxiliary problems, we construct an empirical Bayes method and prove that this method can automatically select the hyperparameter. In addition, we show that this adaptive Bayes procedure provides optimal contraction rates up to a slowly varying term and an arbitrarily small constant, without knowledge about the regularity index. Our method needs not the prior covariance, noise covariance and forward operator have a common basis in their singular value decomposition, enlarging the application range compared with the existing results. A simple simulation example is given that illustrates the effectiveness of the proposed method.

2021, 15(2): 229-256 doi: 10.3934/ipi.2020062 +[Abstract](1237) +[HTML](225) +[PDF](816.01KB)
Abstract:

Many inverse problems are concerned with the estimation of non-negative parameter functions. In this paper, in order to obtain non-negative stable approximate solutions to ill-posed linear operator equations in a Hilbert space setting, we develop two novel non-negativity preserving iterative regularization methods. They are based on fixed point iterations in combination with preconditioning ideas. In contrast to the projected Landweber iteration, for which only weak convergence can be shown for the regularized solution when the noise level tends to zero, the introduced regularization methods exhibit strong convergence. There are presented convergence results, even for a combination of noisy right-hand side and imperfect forward operators, and for one of the approaches there are also convergence rates results. Specifically adapted discrepancy principles are used as a posteriori stopping rules of the established iterative regularization algorithms. For an application of the suggested new approaches, we consider a biosensor problem, which is modelled as a two dimensional linear Fredholm integral equation of the first kind. Several numerical examples, as well as a comparison with the projected Landweber method, are presented to show the accuracy and the acceleration effect of the novel methods. Case studies of a real data problem indicate that the developed methods can produce meaningful featured regularized solutions.

2021, 15(2): 257-270 doi: 10.3934/ipi.2020063 +[Abstract](1125) +[HTML](226) +[PDF](372.89KB)
Abstract:

A spectral problem occurring in description of small transverse vibrations of a star graph of Stieltjes strings is considered. At all but one pendant vertices Dirichlet conditions are imposed which mean that these vertices are clamped. One vertex (the root) can move with damping in the direction orthogonal to the equilibrium position of the strings. We describe the spectrum of such spectral problem. The corresponding inverse problem lies in recovering the values of point masses and the lengths of the intervals between the masses using the spectrum and some other parameters. We propose conditions on a sequence of complex numbers and a collection of real numbers to be the spectrum of a problem we consider and the lengths of the edges, correspondingly.

2021, 15(2): 271-283 doi: 10.3934/ipi.2020064 +[Abstract](1010) +[HTML](203) +[PDF](383.82KB)
Abstract:

We investigate the inverse scattering problem of the perturbed biharmonic operator by studying the recovery process of the magnetic field \begin{document}${\mathbf{A}}$\end{document} and the potential field \begin{document}$V$\end{document}. We show that the high-frequency asymptotic of the scattering amplitude of the biharmonic operator uniquely determines \begin{document}${\rm{curl}}\ {\mathbf{A}}$\end{document} and \begin{document}$V-\frac{1}{2}\nabla\cdot{\mathbf{A}}$\end{document}. We study the near-field scattering problem and show that the high-frequency asymptotic expansion up to an error \begin{document}$\mathcal{O}(\lambda^{-4})$\end{document} recovers above two quantities with no additional information about \begin{document}${\mathbf{A}}$\end{document} and \begin{document}$V$\end{document}. We also establish stability estimates for \begin{document}${\rm{curl}}\ {\mathbf{A}}$\end{document} and \begin{document}$V-\frac{1}{2}\nabla\cdot{\mathbf{A}}$\end{document}.

2021, 15(2): 285-314 doi: 10.3934/ipi.2020065 +[Abstract](1099) +[HTML](194) +[PDF](6625.31KB)
Abstract:

In this paper we address the identification of defects by the Linear Sampling Method in half-waveguides which are related to each other by junctions. Firstly a waveguide which is characterized by an abrupt change of properties is considered, secondly the more difficult case of several half-waveguides related to each other by a junction of complex geometry. Our approach is illustrated by some two-dimensional numerical experiments.

2021, 15(2): 315-338 doi: 10.3934/ipi.2020070 +[Abstract](1539) +[HTML](264) +[PDF](2837.91KB)
Abstract:

Seeking the convex hull of an object (or point set) is a very fundamental problem arising from various tasks. In this work, we propose a variational approach based on the level-set representation for convex hulls of 2-dimensional objects. This method can adapt to exact and inexact convex hull problems. In addition, this method can compute multiple convex hulls simultaneously. In this model, the convex hull is characterized by the zero sublevel-set of a level-set function. For the exact case, we require the zero sublevel-set to be convex and contain the whole given object, where the convexity is characterized by the non-negativity of Laplacian of the level-set function. Then, the convex hull can be obtained by minimizing the area of the zero sublevel-set. For the inexact case, instead of requiring all the given points are included, we penalize the distance from all given points to the zero sublevel-set. Especially, the inexact model can handle the convex hull problem of the given set with outliers very well, while most of the existing methods fail. An efficient numerical scheme using the alternating direction method of multipliers is developed. Numerical examples are given to demonstrate the advantages of the proposed methods.

2021, 15(2): 339-366 doi: 10.3934/ipi.2020071 +[Abstract](1581) +[HTML](200) +[PDF](2380.89KB)
Abstract:

In this paper, we propose new operator-splitting algorithms for the total variation regularized infimal convolution (TV-IC) model [6] in order to remove mixed Poisson-Gaussian (MPG) noise. In the existing splitting algorithm for TV-IC, an inner loop by Newton method had to be adopted for one nonlinear optimization subproblem, which increased the computation cost per outer loop. By introducing a new bilinear constraint and applying the alternating direction method of multipliers (ADMM), all subproblems of the proposed algorithms named as BCA (short for Bilinear Constraint based ADMM algorithm) and BCA\begin{document}$_{f}$\end{document} (short for a variant of BCA with \begin{document}${\bf f}$\end{document}ully splitting form) can be very efficiently solved. Especially for the proposed BCA\begin{document}$_{f}$\end{document}, they can be calculated without any inner iterations. The convergence of the proposed algorithms are investigated, where particularly, a Huber type TV regularizer is adopted to guarantee the convergence of BCA\begin{document}$_f$\end{document}. Numerically, compared to existing primal-dual algorithms for the TV-IC model, the proposed algorithms, with fewer tunable parameters, converge much faster and produce comparable results meanwhile.

2021, 15(2): 367-386 doi: 10.3934/ipi.2020072 +[Abstract](1571) +[HTML](151) +[PDF](447.08KB)
Abstract:

In this paper we will show the duality between the range test (RT) and no-response test (NRT) for the inverse boundary value problem for the Laplace equation in \begin{document}$\Omega\setminus\overline D$\end{document} with an unknown obstacle \begin{document}$D\Subset\Omega$\end{document} whose boundary \begin{document}$\partial D$\end{document} is visible from the boundary \begin{document}$\partial\Omega$\end{document} of \begin{document}$\Omega$\end{document} and a measurement is given as a set of Cauchy data on \begin{document}$\partial\Omega$\end{document}. Here the Cauchy data is given by a unique solution \begin{document}$u$\end{document} of the boundary value problem for the Laplace equation in \begin{document}$\Omega\setminus\overline D$\end{document} with homogeneous and inhomogeneous Dirichlet boundary condition on \begin{document}$\partial D$\end{document} and \begin{document}$\partial\Omega$\end{document}, respectively. These testing methods are domain sampling methods to estimate the location of the obstacle using test domains and the associated indicator functions. Also both of these testing methods can test the analytic extension of \begin{document}$u$\end{document} to the exterior of a test domain. Since these methods are defined via some operators which are dual to each other, we could expect that there is a duality between the two methods. We will give this duality in terms of the equivalence of the pre-indicator functions associated to their indicator functions. As an application of the duality, the reconstruction of \begin{document}$D$\end{document} using the RT gives the reconstruction of \begin{document}$D$\end{document} using the NRT and vice versa. We will also give each of these reconstructions without using the duality if the Dirichlet data of the Cauchy data on \begin{document}$\partial\Omega$\end{document} is not identically zero and the solution to the associated forward problem does not have any analytic extension across \begin{document}$\partial D$\end{document}. Moreover, we will show that these methods can still give the reconstruction of \begin{document}$D$\end{document} if we a priori knows that \begin{document}$D$\end{document} is a convex polygon and it satisfies one of the following two properties: all of its corner angles are irrational and its diameter is less than its distance to \begin{document}$\partial\Omega$\end{document}.

2021 Impact Factor: 1.483
5 Year Impact Factor: 1.462
2021 CiteScore: 2.6