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Functional Analysis and Applications

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Seminars

Previous academic years: 2010/11, 2009/10, 2008/09, 2007/08, 2006/07, 2005/06, 2004/05, 2003/04, 2002/03, 2001/02, 2000/01, 1999/2000, pre-1999

A reduced computational and geometrical framework for inverse problems in haemodynamics

The solution of inverse problems in cardiovascular mathematics is computationally expensive. Model reduction techniques might be needed to enable the solution of inverse problems also in other real-life applications. In this presentation we apply a domain parametrization technique to reduce both the geometrical and computational complexity of the forward problem, and replace the finite element solution of the incompressible Navier-Stokes equations by a computationally less expensive reduced basis approximation. This greatly reduces the cost of simulating the forward problem. We then consider the solution of inverse problems in both the deterministic sense, by solving a least-squares problem, and in the statistical sense, by using a Bayesian framework for quantifying uncertainty. Two inverse problems in haemodynamics are considered: (i) a simplified stenosed artery model for quantifying the risk of atherosclerosis by identifying the material parameters of the arterial wall based on pressure measurements; (ii) a simplified femoral bypass graft model for robust shape design under uncertain residual flow in the main arterial branch identified from pressure measurements. Joint work with Alfio Quarteroni, Gianluigi Rozza, Toni Lassila.

On some critical problems for the fractional Laplacian operator

Please click here for the abstract.

The fundamental theorem for surfaces in 3-dimensional Heisenberg group

Abstract: The fundamental theorem for hypersurfaces in a Euclidean space says that the equations of Gauss and Codazzi are exactly the integrability conditions for finding an isometric imbedding with prescribed metric and second fundamental form. In pseudohermitian geometry, we have the analogous fundamental theorem for surfaces in 3-dimensional Heisenberg group. This is a part of recent joint work with Jenn-Fang Hwang, Andrea Malchiodi, and Paul Yang.

Gradient theory for plasticity as the Γ-limit of a nonlinear dislocation energy

Since the motion of dislocations is regarded as the main cause of plastic deformation, a large literature is focused on the problem of deriving plasticity models from more fundamental dislocation models. The starting point of our derivation is a semi-discrete dislocation model. The main novelty of our approach is that we consider a nonlinear dislocation energy, whereas most mathematical and engineering papers treat only a quadratic dislocation energy. Our choice of a nonlinear stress-strain relation guarantees that the dislocation strain energy is well defined also in the vicinity of the dislocations, eliminating the need of the cut-off radius that is typical of the linear theories. The Γ-limit of our nonlinear dislocation energy as the length of the Burgers vector tends to zero is a strain-gradient model for plasticity and has the same form as the limit energy obtained by starting from a quadratic dislocation energy. Our result, however, is obtained by starting from a more physical model. (work in collaboration with Caterina Zeppieri)

Quasistatic evolution of cavities in nonlinear elasticity

Starting with a variational static model of cavitation in nonlinear elasticity, we propose a quasistatic model also based on global minimization. We prove the existence of a quasistatic evolution, which takes into account the non-interpenetration of matter and the irreversibility of the process of cavity formation.