This is a report on a joint work with Isabelle Catto, Norbert Mauser and Saber Trabelsi. The Multiconfiguration time dependent Hartree Fock Method (MCTDHF) is a nonlinear approximation of a linear system of /N/ quantum particles with binary interaction. It combines the principle of the Hartree Fock and the Galerkin approximation. The main difficulty is the introduction of a global (in space) density matrix $\Gamma(t) $ which may degenerate. By construction this approximation formally preserves the mass and the energy of the system. The conservation of energy can be used to balance the singularities Coulomb potential and to provide sufficient conditions for the global in time invertibility of $\Gamma(t)$.

In numerical computations this matrix is very often regularized (changed into $\Gamma(t) +\epsilon(t)$). In this situation the energy is no more conserved
and the mathematical analysis done in $L^2$ relies on Strichartz type estimates.

Since the seminal paper by Perona and Malik nonlinear diffusions have successfully been used for various image processing tasks. They also have attracted steadfast interest in the mathematical community. In this talk we will give an historical overview of the developments on the Perona-Malik equation and describe two new nonlinear diffusions which resolve the main mathematical shortcomings of Perona-Malik without sacrificing but rather enhancing the cherished practical qualities of the Perona-Malik model. The new equations, while
well-posed and purely diffusive, exhibit a non trivial dynamical behavior, which makes them mathematically interesting and practically
effective.

In inverse problems, when the forward map is a smoothing
(regularizing) operator,
the inverse map is usually unbounded. Thus only the low frequency
component of the object of interest is accessible from noisy measurements.
In many inverse problems however, the neglected high frequency component may
significantly affect the measured data. Using simple scaling arguments,
we characterize the influence of the high frequency component.
We will then show how to eliminate the effect of the high frequency
component in a one-dimensional inverse spectral problem to
obtain better reconstructions of the low frequency component of
the unknown. Numerical results with synthetic data will be presented.