Emanuele Schiavi

ON A DOUBLY NONLINEAR PARABOLIC OBSTACLE PROBLEM MODELLING ICE SHEET DYNAMICS

This paper deals with the weak formulation of a free (moving) boundary problem arising in theoretical glaciology. Considering shallow ice sheet flow, we present the mathematical analysis and the numerical solution of the second order nonlinear degenerate parabolic equation modelling, in the isothermal case, the ice sheet non-Newtonian dynamics. An obstacle problem is then deduced and analyzed. The existence of a free boundary generated by the support of the solution is proved and its location and evolution are qualitatively described by using a comparison principle and an energy method. Then the solutions are numerically computed with a method of characteristics and a duality algorithm to deal with the resulting variational inequalities. The weak framework we introduce and its analysis (both qualitative and numerical) are not restricted to the simple physics of the ice sheet model we consider nor to the model dimension; they can be successfully applied to more realistic and sophisticated models related to other geophysical settings.

A theory of ice-sheet surges

A simplified model of a two-dimensional ice sheet is described. It includes basal ice sliding dependent on the basal water pressure, which itself is described by a simple theory of basal drainage. We show that this simple but sophisticated model predicts surges of the ice mass in realistic circumstances, and we describe these surges by solving the problem numerically. We also are able to describe some parts of the surge analytically. The numerical solution of the model is a delicate matter, and highlights pitfalls to be avoided if more complicated models are to be solved successfully.

Evaluation of the Influence of Multiple Environmental Factors on the Biodegradation of Dibenzofuran, Phenanthrene, and Pyrene by a Bacterial Consortium Using an Orthogonal Experimental Design

For a bioremediation process to be effective, we suggest to perform preliminary studies in laboratory to describe and characterize physicochemical and biological parameters (type and concentration of nutrients, type and number of microorganisms, temperature) of the environment concerned. We consider that these studies should be done by taking into account the simultaneous interaction between different factors. By knowing the response capacity to pollutants, it is possible to select and modify the right treatment conditions to enhance bioremediation.

Parallel transmit pulse design for patients with deep brain stimulation implants

Specific absorption rate (SAR) amplification around active implantable medical devices during diagnostic MRI procedures poses a potential risk for patient safety. In this study, we present a parallel transmit (pTx) strategy that can be used to safely scan patients with deep brain stimulation (DBS) implants.

SAR reduction in 7T C-spine imaging using a "dark modes" transmit array strategy.

Local specific absorption rate (SAR) limits many applications of parallel transmit (pTx) in ultra high‐field imaging. In this Note, we introduce the use of an array element, which is intentionally inefficient at generating spin excitation (a “dark mode”) to attempt a partial cancellation of the electric field from those elements that do generate excitation. We show that adding dipole elements oriented orthogonal to their conventional orientation to a linear array of conventional loop elements can lower the local SAR hotspot in a C‐spine array at 7 T.

Automatic Segmentation of the Liver in CT Using Level Sets Without Edges

Liver volumetry is a required step for the planning of liver surgery and resection. It is generally based on Computerized tomography images, and segmentation of the liver is the most important step of the process. We propose an automatic segmentation algorithm based on a geometric level set method which provides an accurate segmentation of the liver, and requires no a priori information. We show results on different datasets, with and without a contrast agent. The segmentation is compared to manual delineation by a radiologist with good results.

On a degenerate parabolic/hyperbolic system in glaciology giving rise to a free boundary

The authors present and study a problem which models the evolution of the ice sheet in the Laurentide. They consider a one-dimensional problem in (3-dimensional) space which involves three parameters: the ice thickness h , the amount of water flux Q and the accumulated ice velocity ξ . Considering the mass conservation law, the momentum or balance equations, and introducing the special glaciology relations already described in the specialized literature, they write the coupled system involving these three unknowns. After some computations, they are led to some coupled system of parabolic and hyperbolic nonlinear and possibly degenerate equations. Initial and boundary conditions are introduced which correspond to the special case of the Hudson region. Replacing the data by piecewise constant approximations with respect to the time variable, the authors then present some stationary discretized coupled system for which they define the notion of weak solution. The purpose of this work is to obtain some existence result for this discretized system. This is done using an iterative scheme which decouples the three equations. An existence result is proved for each of these decoupled equations. The first equation (for the discretization of h ) is studied using the notion of super- and subsolution and comparison principles. The second equation (for the discretization of Q ) involves a maximal monotone graph and is studied, first by replacing this multivalued graph by some single-valued maximal monotone graph and then by passing to the limit. The study of the last equation is very easy. Then uniform estimates are established on these approximate solutions. This leads to an asymptotic result which finally proves the existence result for the original discretized system. The last part is devoted to the qualitative study of the function Q , which must be nonnegative. The authors prove the existence of a boundary layer, corresponding to the boundary of the region {Q>0} . The present work justifies earlier observations made by the second author when solving the problem numerically

Automatic Total Generalized Variation-Based DTI Rician Denoising

A bayesian model for Diffusion Tensorial Magnetic Resonance Images denosing and reconstruction is considered. This is based on a Tikhonov like-functional for Total Generalized Variation and Rician likelihood which is described in a variational framework. A primal-dual algorithm is implemented and accurate numerical solutions of the associated saddle-point formulation are computed. An automatic parameter selection rule is proposed to facilitate practical clinical usage and diagnostic of neurodegenerative disorders.

Box Relaxation Schemes in Staggered Discretizations for the Dual Formulation of Total Variation Minimization

In this paper, we propose some new box relaxation numerical schemes on staggered grids to solve the stationary system of partial differential equations arising from the dual minimization problem associated with the total variation operator. We present in detail the numerical schemes for the scalar case and its generalization to multichannel (vectorial) images. Then, we discuss their implementation in digital image denoising. The results outperform the resolution of the dual equation based on the gradient descent approach and pave the way for more advanced numerical strategies.

Vectorial total generalized variation for accelerated multi-channel multi-contrast MRI

PURPOSE To develop and implement an efficient reconstruction technique to improve accelerated multi-channel multi-contrast MRI. THEORY AND METHODS The vectorial total generalized variation (TGV) operator is used as a regularizer for the sensitivity encoding (SENSE) technique to improve image quality of multi-channel multi-contrast MRI. The alternating direction method of multipliers (ADMM) is used to efficiently reconstruct the data. The performance of the proposed method (MC-TGV-SENSE) is assessed on two healthy volunteers at several acceleration factors. RESULTS As demonstrated on the in vivo results, MC-TGV-SENSE had the lowest root-mean-square error (RMSE), highest structural similarity index, and best visual quality at all acceleration factors, compared to other methods under consideration. MC-TGV-SENSE yielded up to 17.3% relative RMSE reduction compared to the widely used total variation regularized SENSE. Furthermore, we observed that the reconstruction time of MC-TGV-SENSE is reduced by approximately a factor of two with comparable RMSEs by using the proposed ADMM-based algorithm as opposed to the more commonly used Chambolle-Pock primal-dual algorithm for the TGV-based reconstruction. CONCLUSION MC-TGV-SENSE is a better alternative than the existing reconstruction methods for accelerated multi-channel multi-contrast MRI. The proposed method exploits shared information among the images (MC), mitigates staircasing artifacts (TGV), and uses the encoding power of multiple receiver coils (SENSE).