Numerical Methods for Partial Differential Equations Seminar

Abstract: In this talk we will discuss the paper "Neumann Networks for Linear Inverse Problems in Imaging" by D. Gilton, G. Ongie y R. Wilett.

Abstract:

Respiratory diseases are the third leading cause of death in the world. In chronic respiratory diseases, most of the symptoms can be explained by drastic changes in the mechanical properties and deformation mechanisms occurring in the lung tissue and airways at a regional, rather than whole-organ, level. The need for a regional mechanical analysis has motivated the development of computational tools for the assessment of regional lung mechanics from medical images, which hold promise in the creation of more effective diagnostic tools in respiratory medicine.

In this talk, I will introduce the problem of image registration and biomechanical analysis in the context of regional lung mechanics. Then, I will present a mixed formulation of the image registration problem that allows for a seamless analysis of the stresses and strain in the lung tissue. Finite-element schemes will be presented to solve both the primal and mixed formulations of the image registration problem, along with their corresponding numerical analysis. I will end the presentation with some applications of the image-based biomechanical tools we have developed in the study of ventilator-induced lung injury.

This work is a joint collaboration with Nicolás Barnafi (former UC student, now at Politecnico di Milano) and Gabriel Gatica (Universidad de Concepción).

Abstract: We consider an adaptive algorithm from Congreve & Wihler (JCAM 311, 2017) for nonlinear PDEs with strongly monotone operator. The analysis also includes the iterative and inexact solution of the arising nonlinear systems by means of the Picard iteration. Hence, the algorithm steers the local refinement of the FEM triangulation as well as the accuracy of the employed nonlinear solver. We prove that the adaptive scheme guarantees convergence with optimal algebraic rates with respect to the number of elements. Moreover, we prove that the overall computational cost is optimal.

The talk is based on joint work with Gregor Gantner, Dirk Praetorius, and Bernhard Stiftner.

Abstract: Rough linear functionals (such as Dirac Delta distributions) often appear on the right-hand side of variational formulations of PDEs. As they live in negative Sobolev spaces, they dramatically affect adaptive finite element procedures to approximate the solution of the involved PDE.

In this talk we propose an alternative that, in a first step, computes a projection of the rough functional over piecewise polynomial spaces, up to a desired precision in a negative norm sense. The projection (being $L^p$-regular) is then used as the right-hand side of a regularized problem for which adaptive Galerkin methods performs better. A complete error analysis of the proposed methodology will be shown.

Abstract: For a given PDE problem, the choice of a variational formulation is critical for the design of stable Galerkin discretizations. Contrary to other schemes, the discontinuous Petrov-Galerkin method with optimal test functions (DPG method) inherits its stability from the well-posedness of the variational formulation. Therefore, the use of DPG approximations gives full flexibility in the choice of a variational formulation. This allows for creating specific formulations depending on the variables of interest and their norms.

After giving a brief introduction to the DPG method we will illustrate this paradigm in the case of the Kirchhoff-Love plate bending model. Some of its variables, for instance the bending moments, possess interesting regularity properties that are difficult to deal with both at the continuous and discrete levels.

This work is a joint collaboration with Thomas Führer (UC, Santiago) and Antti Niemi (U of Oulo, Finland).