Table of contents

Recently, we developed an approach for solving the computed tomography (CT) interior problem based on the high-order TV (HOT) minimization, assuming that a region-of-interest (ROI) is piecewise polynomial. In this paper, we generalize this finding from the CT field to the single-photon emission computed tomography (SPECT) field, and prove that if an ROI is piecewise polynomial, then the ROI can be uniquely reconstructed from the SPECT projection data associated with the ROI through the HOT minimization. Also, we propose a new formulation of HOT, which has an explicit formula for any n-order piecewise polynomial function, while the original formulation has no explicit formula for n ⩾ 2. Finally, we verify our theoretical results in numerical simulation, and discuss relevant issues.

The inverse problem which arises in the study of the integrable PDE proposed by V Novikov is solved for a class of discrete densities. The method of solution relies on the use of Cauchy biorthogonal polynomials. The explicit formulas are obtained directly from the analysis on the real axis without any additional transformations to the 'string'-type boundary value problems known from prior works.

A sphere excited by an interior point source or a point dipole gives a simplified yet realistic model for studying a variety of applications in medical imaging. We suppose that there is an exterior field (transmission problem) and that the total field on the sphere is known. We give analytical inversion algorithms for determining the interior physical characteristics of the sphere as well as the location, strength and orientation of the source/dipole. We start with static problems (Laplace's equation) and then proceed to acoustic problems (Helmholtz equation).

The total variation (TV) model with a fidelity term of the generalized Kullback–Leibler (KL) divergence is a classical method for Poissonian image deblurring. In this paper, we propose a new TV-KL model with a spatially dependent regularization parameter. This model is able to preserve small details of images while homogeneous regions still remain sufficiently smooth. The automated selection of the regularization parameter is based on the local discrepancy function. The corresponding minimization problem with a spatially adapted regularization parameter can be solved efficiently by the split Bregman method. Numerical experiments demonstrate that the proposed algorithm has the potential to enhance regions of images containing detail and remove Poisson noise simultaneously, which leads to an improvement in the signal-to-noise ratio and the mean absolute error for deblurring results.

This paper is devoted to a geometrical inverse problem associated with a fluid–structure system. More precisely, we consider the interaction between a moving rigid body and a viscous and incompressible fluid. Assuming a low Reynolds regime, the inertial forces can be neglected and, therefore, the fluid motion is modelled by the Stokes system. We first prove the well posedness of the corresponding system. Then we show an identifiability result: with one measure of the Cauchy forces of the fluid on one given part of the boundary and at some positive time, the shape of a convex body and its initial position are identified.

We study the inverse source problem for the eddy current approximation of Maxwell equations. As for the full system of Maxwell equations, we show that a volume current source cannot be uniquely identified by knowledge of the tangential components of the electromagnetic fields on the boundary, and we characterize the space of non-radiating sources. On the other hand, we prove that the inverse source problem has a unique solution if the source is supported on the boundary of a subdomain or if it is the sum of a finite number of dipoles. We address the applicability of this result for the localization of brain activity from electroencephalography and magnetoencephalography measurements.

We consider abstract operator equations Fu = y, where F is a compact linear operator between Hilbert spaces U and V, which are function spaces on closed, finite-dimensional Riemannian manifolds, respectively. This setting is of interest in numerous applications such as computer vision and non-destructive evaluation. In this work, we study the approximation of the solution of the ill-posed operator equation with Tikhonov-type regularization methods. We state well-posedness, stability, convergence and convergence rates of the regularization methods. Moreover, we study in detail the numerical analysis and the numerical implementation. Finally, we provide for three different inverse problems numerical experiments.

We propose a new framework to solve inverse obstacle problems with a Dirichlet condition, consisting in the construction of a decreasing sequence of open domains that contain the searched obstacle. We provide a theoretical justification of this new methodology, infer from it a new algorithm based on the coupling of the quasi-reversibility technique and a level-set method, and illustrate the functionality of the algorithm with the help of numerical experiments in 2D.

We describe a novel approach to the inversion of elasto-static tiltmeter measurements to monitor planar hydraulic fractures propagating within three-dimensional elastic media. The technique combines the extended Kalman filter (EKF), which predicts and updates state estimates using tiltmeter measurement time-series, with a novel implicit level set algorithm (ILSA), which solves the coupled elasto-hydrodynamic equations. The EKF and ILSA are integrated to produce an algorithm to locate the unknown fracture-free boundary. A scaling argument is used to derive a strategy to tune the algorithm parameters to enable measurement information to compensate for unmodeled dynamics. Synthetic tiltmeter data for three numerical experiments are generated by introducing significant changes to the fracture geometry by altering the confining geological stress field. Even though there is no confining stress field in the dynamic model used by the new EKF-ILSA scheme, it is able to use synthetic data to arrive at remarkably accurate predictions of the fracture widths and footprints. These experiments also explore the robustness of the algorithm to noise and to placement of tiltmeter arrays operating in the near-field and far-field regimes. In these experiments, the appropriate parameter choices and strategies to improve the robustness of the algorithm to significant measurement noise are explored.

Weyl theory for Dirac systems with rectangular matrix potentials is non-classical. The corresponding Weyl functions are rectangular matrix functions. Furthermore, they are non-expansive in the upper semi-plane. Inverse problems are studied for such Weyl functions, and some results are new even for the square Weyl functions. High-energy asymptotics of Weyl functions and Borg–Marchenko-type uniqueness results are derived too.

In this paper, we establish global Carleman estimates for the heat and Schrödinger equations on a network. The heat equation is considered on a general tree and the Schrödinger equation on a star-shaped tree. The Carleman inequalities are used to prove the Lipschitz stability for an inverse problem consisting in retrieving a stationary potential in the heat (resp. Schrödinger) equation from boundary measurements.

This paper is concerned with the numerical solution of nonlinear ill-posed operator equations involving convex constraints. We study a Newton-type method which consists in applying linear Tikhonov regularization with convex constraints to the Newton equations in each iteration step. Convergence of this iterative regularization method is analyzed if both the operator and the right-hand side are given with errors and all error levels tend to zero. Our study has been motivated by the joint estimation of object and phase in 4Pi microscopy, which leads to a semi-blind deconvolution problem with nonnegativity constraints. The performance of the proposed algorithm is illustrated both for simulated and for three-dimensional experimental data.