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Inverse Problems & Imaging

February 2019 , Volume 13 , Issue 1

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Hyperpriors for Matérn fields with applications in Bayesian inversion
Lassi Roininen, Mark Girolami, Sari Lasanen and Markku Markkanen
2019, 13(1): 1-29 doi: 10.3934/ipi.2019001 +[Abstract](1707) +[HTML](164) +[PDF](2476.56KB)
Abstract:

We introduce non-stationary Matérn field priors with stochastic partial differential equations, and construct correlation length-scaling with hyperpriors. We model both the hyperprior and the Matérn prior as continuous-parameter random fields. As hypermodels, we use Cauchy and Gaussian random fields, which we map suitably to a desired correlation length-scaling range. For computations, we discretise the models with finite difference methods. We consider the convergence of the discretised prior and posterior to the discretisation limit. We apply the developed methodology to certain interpolation, numerical differentiation and deconvolution problems, and show numerically that we can make Bayesian inversion which promotes competing constraints of smoothness and edge-preservation. For computing the conditional mean estimator of the posterior distribution, we use a combination of Gibbs and Metropolis-within-Gibbs sampling algorithms.

Inverse problems for the heat equation with memory
Sergei A. Avdonin, Sergei A. Ivanov and Jun-Min Wang
2019, 13(1): 31-38 doi: 10.3934/ipi.2019002 +[Abstract](1826) +[HTML](185) +[PDF](321.32KB)
Abstract:

We study inverse boundary problems for one dimensional linear integro-differential equation of the Gurtin-Pipkin type with the Dirichlet-to-Neumann map as the inverse data. Under natural conditions on the kernel of the integral operator, we give the explicit formula for the solution of the problem with the observation on the semiaxis t>0. For the observation on finite time interval, we prove the uniqueness result, which is similar to the local Borg-Marchenko theorem for the Schrödinger equation.

Magnetic moment estimation and bounded extremal problems
Laurent Baratchart, Sylvain Chevillard, Douglas Hardin, Juliette Leblond, Eduardo Andrade Lima and Jean-Paul Marmorat
2019, 13(1): 39-67 doi: 10.3934/ipi.2019003 +[Abstract](1340) +[HTML](142) +[PDF](3153.37KB)
Abstract:

We consider the inverse problem in magnetostatics for recovering the moment of a planar magnetization from measurements of the normal component of the magnetic field at a distance from the support. Such issues arise in studies of magnetic material in general and in paleomagnetism in particular. Assuming the magnetization is a measure with L2-density, we construct linear forms to be applied on the data in order to estimate the moment. These forms are obtained as solutions to certain extremal problems in Sobolev classes of functions, and their computation reduces to solving an elliptic differential-integral equation, for which synthetic numerical experiments are presented.

A partial inverse problem for the Sturm-Liouville operator on the lasso-graph
Chuan-Fu Yang and Natalia Pavlovna Bondarenko
2019, 13(1): 69-79 doi: 10.3934/ipi.2019004 +[Abstract](1296) +[HTML](210) +[PDF](350.7KB)
Abstract:

The Sturm-Liouville operator with singular potentials on the lasso graph is considered. We suppose that the potential is known a priori on the boundary edge, and recover the potential on the loop from a part of the spectrum and some additional data. We prove the uniqueness theorem and provide a constructive algorithm for the solution of this partial inverse problem.

Recovering two coefficients in an elliptic equation via phaseless information
Vladimir G. Romanov and Masahiro Yamamoto
2019, 13(1): 81-91 doi: 10.3934/ipi.2019005 +[Abstract](1134) +[HTML](150) +[PDF](355.95KB)
Abstract:

For fixed \begin{document} $y \in \mathbb{R}^3$ \end{document}, we consider the equation \begin{document} $L u+k^2u = - δ(x-y), \>x \in \mathbb{R}^3$ \end{document}, where \begin{document} $L=\text{div}(n(x)^{-2}\nabla)+q(x)$ \end{document}, \begin{document} $k >0$ \end{document} is a frequency, \begin{document} $n(x)$ \end{document} is a refraction index and \begin{document} $q(x)$ \end{document} is a potential. Assuming that the refraction index \begin{document} $n(x)$ \end{document} is different from \begin{document} $1$ \end{document} only inside a bounded compact domain \begin{document} $Ω$ \end{document} with a smooth boundary \begin{document} $S$ \end{document} and the potential \begin{document} $q(x)$ \end{document} vanishes outside of the same domain, we study an inverse problem of finding both coefficients inside \begin{document} $Ω$ \end{document} from some given information on solutions of the elliptic equation. Namely, it is supposed that the point source located at point \begin{document} $y \in S$ \end{document} is a variable parameter of the problem. Then for the solution \begin{document} $u(x,y,k)$ \end{document} of the above equation satisfying the radiation condition, we assume to be given the following phaseless information \begin{document} $f(x,y,k)=|u(x,y,k)|^2$ \end{document} for all \begin{document} $x,y \in S$ \end{document} and for all \begin{document} $k≥ k_0>0$ \end{document}, where \begin{document} $k_0$ \end{document} is some constant. We prove that this phaseless information uniquely determines both coefficients \begin{document} $n(x)$ \end{document} and \begin{document} $q(x)$ \end{document} inside \begin{document} $Ω$ \end{document}.

The regularized monotonicity method: Detecting irregular indefinite inclusions
Henrik Garde and Stratos Staboulis
2019, 13(1): 93-116 doi: 10.3934/ipi.2019006 +[Abstract](1271) +[HTML](158) +[PDF](1007.28KB)
Abstract:

In inclusion detection in electrical impedance tomography, the support of perturbations (inclusion) from a known background conductivity is typically reconstructed from idealized continuum data modelled by a Neumann-to-Dirichlet map. Only few reconstruction methods apply when detecting indefinite inclusions, where the conductivity distribution has both more and less conductive parts relative to the background conductivity; one such method is the monotonicity method of Harrach, Seo, and Ullrich [17,15]. We formulate the method for irregular indefinite inclusions, meaning that we make no regularity assumptions on the conductivity perturbations nor on the inclusion boundaries. We show, provided that the perturbations are bounded away from zero, that the outer support of the positive and negative parts of the inclusions can be reconstructed independently. Moreover, we formulate a regularization scheme that applies to a class of approximative measurement models, including the Complete Electrode Model, hence making the method robust against modelling error and noise. In particular, we demonstrate that for a convergent family of approximative models there exists a sequence of regularization parameters such that the outer shape of the inclusions is asymptotically exactly characterized. Finally, a peeling-type reconstruction algorithm is presented and, for the first time in literature, numerical examples of monotonicity reconstructions for indefinite inclusions are presented.

Nonconvex TGV regularization model for multiplicative noise removal with spatially varying parameters
Hanwool Na, Myeongmin Kang, Miyoun Jung and Myungjoo Kang
2019, 13(1): 117-147 doi: 10.3934/ipi.2019007 +[Abstract](1838) +[HTML](236) +[PDF](3461.13KB)
Abstract:

In this article, we introduce a novel variational model for the restoration of images corrupted by multiplicative Gamma noise. The model incorporates a convex data-fidelity term with a nonconvex version of the total generalized variation (TGV). In addition, we adopt a spatially adaptive regularization parameter (SARP) approach. The nonconvex TGV regularization enables the efficient denoising of smooth regions, without staircasing artifacts that appear on total variation regularization-based models, and edges and details to be conserved. Moreover, the SARP approach further helps preserve fine structures and textures. To deal with the nonconvex regularization, we utilize an iteratively reweighted \begin{document}$\ell_1$\end{document} algorithm, and the alternating direction method of multipliers is employed to solve a convex subproblem. This leads to a fast and efficient iterative algorithm for solving the proposed model. Numerical experiments show that the proposed model produces better denoising results than the state-of-the-art models.

Note on Calderón's inverse problem for measurable conductivities
Matteo Santacesaria
2019, 13(1): 149-157 doi: 10.3934/ipi.2019008 +[Abstract](1108) +[HTML](154) +[PDF](341.8KB)
Abstract:

The unique determination of a measurable conductivity from the Dirichlet-to-Neumann map of the equation \begin{document} ${\rm{div}} (σ \nabla u) = 0$ \end{document} is the subject of this note. A new strategy, based on Clifford algebras and a higher dimensional analogue of the Beltrami equation, is here proposed. This represents a possible first step for a proof of uniqueness for the Calderón problem in three and higher dimensions in the \begin{document} $L^\infty$ \end{document} case.

Inverse scattering problem for quasi-linear perturbation of the biharmonic operator on the line
Teemu Tyni and Valery Serov
2019, 13(1): 159-175 doi: 10.3934/ipi.2019009 +[Abstract](1186) +[HTML](161) +[PDF](412.59KB)
Abstract:

We consider an inverse scattering problem of recovering the unknown coefficients of quasi-linearly perturbed biharmonic operator on the line. These unknown complex-valued coefficients are assumed to satisfy some regularity conditions on their nonlinearity, but they can be discontinuous or singular in their space variable. We prove that the inverse Born approximation can be used to recover some essential information about the unknown coefficients from the knowledge of the reflection coefficient. This information is the jump discontinuities and the local singularities of the coefficients.

A reference ball based iterative algorithm for imaging acoustic obstacle from phaseless far-field data
Heping Dong, Deyue Zhang and Yukun Guo
2019, 13(1): 177-195 doi: 10.3934/ipi.2019010 +[Abstract](1338) +[HTML](148) +[PDF](542.74KB)
Abstract:

In this paper, we consider the inverse problem of determining the location and the shape of a sound-soft obstacle from the modulus of the far-field data for a single incident plane wave. By adding a reference ball artificially to the inverse scattering system, we propose a system of nonlinear integral equations based iterative scheme to reconstruct both the location and the shape of the obstacle. The reference ball technique causes few extra computational costs, but breaks the translation invariance and brings information about the location of the obstacle. Several validating numerical examples are provided to illustrate the effectiveness and robustness of the proposed inversion algorithm.

Simultaneously recovering potentials and embedded obstacles for anisotropic fractional Schrödinger operators
Xinlin Cao, Yi-Hsuan Lin and Hongyu Liu
2019, 13(1): 197-210 doi: 10.3934/ipi.2019011 +[Abstract](1375) +[HTML](129) +[PDF](421.86KB)
Abstract:

Let \begin{document}$A∈{\rm{Sym}}(n× n)$\end{document} be an elliptic 2-tensor. Consider the anisotropic fractional Schrödinger operator \begin{document}$\mathscr{L}_A^s+q$\end{document}, where \begin{document}$\mathscr{L}_A^s: = (-\nabla·(A(x)\nabla))^s$\end{document}, \begin{document}$s∈ (0, 1)$\end{document} and \begin{document}$q∈ L^∞$\end{document}. We are concerned with the simultaneous recovery of \begin{document}$q$\end{document} and possibly embedded soft or hard obstacles inside \begin{document}$q$\end{document} by the exterior Dirichlet-to-Neumann (DtN) map outside a bounded domain \begin{document}$Ω$\end{document} associated with \begin{document}$\mathscr{L}_A^s+q$\end{document}. It is shown that a single measurement can uniquely determine the embedded obstacle, independent of the surrounding potential \begin{document}$q$\end{document}. If multiple measurements are allowed, then the surrounding potential \begin{document}$q$\end{document} can also be uniquely recovered. These are surprising findings since in the local case, namely \begin{document}$s = 1$\end{document}, both the obstacle recovery by a single measurement and the simultaneous recovery of the surrounding potential by multiple measurements are long-standing problems and still remain open in the literature. Our argument for the nonlocal inverse problem is mainly based on the strong uniqueness property and Runge approximation property for anisotropic fractional Schrödinger operators.

A connection between uniqueness of minimizers in Tikhonov-type regularization and Morozov-like discrepancy principles
Vinicius Albani and Adriano De Cezaro
2019, 13(1): 211-229 doi: 10.3934/ipi.2019012 +[Abstract](1280) +[HTML](158) +[PDF](442.81KB)
Abstract:

We state sufficient conditions for the uniqueness of minimizers of Tikhonov-type functionals. We further explore a connection between such results and the well-posedness of Morozov-like discrepancy principle. Moreover, we find appropriate conditions to apply such results to the local volatility surface calibration problem.

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