Volume 22, Issue 6

In this paper we present a simplified model of tumor growth and an associated inverse problem. Our model consists in describing the evolution of a population of cancer cells and the density of oxygen. The growth rate depends on the concentration of oxygen. The oxygen distribution is computed by means of a diffusion equation, the source term being localized on the blood vessels. We consider the inverse problem that consists in recovering the position of the blood vessels assuming the distribution of tumor cells. We use an adjoint method. Results relative to idealized clinical cases are discussed.

We present a shape optimization algorithm to estimate the ice thickness distribution within a two-dimensional, non-sliding mountain glacier, given a transient surface geometry and a mass-balance distribution. The approach is based on the minimization of the surface topography misfit at the end of the glacier's evolution in the shallow ice approximation of ice flow. Neither filtering of the surface topography where its gradient vanishes nor interpolation of the basal shear stress is involved. Novelty of the presented shape optimization algorithm is the use of surface topography and mass-balance only within a time-dependent Lagrangian approach for moving-boundary glaciers. On real-world inspired geometries, it is shown to produce estimations of even better quality in smaller time than the recently proposed steady and transient inverse methods. A sensitivity analysis completes the study and evinces the method's higher susceptibility to perturbations in the surface topography than in surface mass-balance or rate factor.

This paper concerns the inverse problem of retrieving the principal coefficient in a Korteweg–de Vries (KdV) equation from boundary measurements of a single solution. The Lipschitz stability of this inverse problem is obtained using a new global Carleman estimate for the linearized KdV equation. The proof is based on the Bukhgeĭm–Klibanov method.

The aim of electrical impedance tomography is to form an image of the conductivity distribution inside an unknown body using electric boundary measurements. The computation of the image from measurement data is a non-linear ill-posed inverse problem and calls for a special regularized algorithm. One such algorithm, the so-called D-bar method, is improved in this work by introducing new computational steps that remove the so far necessary requirement that the conductivity should be constant near the boundary. The numerical experiments presented suggest two conclusions. First, for most conductivities arising in medical imaging, it seems the previous approach of using a best possible constant near the boundary is sufficient. Second, for conductivities that have high contrast features at the boundary, the new approach produces reconstructions with smaller quantitative error and with better visual quality.

The regularization properties of the total generalized variation (TGV) functional for the solution of linear inverse problems by means of Tikhonov regularization are studied. Considering the associated minimization problem for general symmetric tensor fields, the well-posedness is established in the space of symmetric tensor fields of bounded deformation, a generalization of the space of functions of bounded variation. Convergence for vanishing noise level is shown in a multiple regularization parameter framework in terms of the naturally arising notion of TGV-strict convergence. Finally, some basic properties, in particular non-equivalence for different parameters, are discussed for this notion.