In this paper we consider the continuous–time nonlinear filtering problem, which has an infinite–dimensional solution in general, as proved by Chaleyat–Maurel and Michel. There are few examples of nonlinear systems for which the optimal filter is finite dimensional, in particular Kalman’s, Beneš’, and Daum’s filters. In the present paper, we construct new classes of scalar nonlinear filtering p...

We are developing a regularization technique for Perona–Malik diffusion equations that relies on multiresolution techniques. The main result of this paper is to show that the chosen discretization overcomes the ill-posedness of the nonlinear Perona–Malik model. The resulting algorithm is tested and the results are compared with pixel–based methods. keywords, phrases: Nonlinear diffusion, regula...

Theoretical and experimental studies related to manipulation of pattern formation in self-organizing reaction-diffusion processes by appropriate control stimuli become increasingly important both in chemical engineering and cellular biochemistry. In a model study, we demonstrate here exemplarily the application of an efficient nonlinear model predictive control (NMPC) algorithm to real-time opt...

In this paper we consider the continuous–time nonlinear filtering problem, which has an infinite–dimensional solution in general, as proved by Chaleyat–Maurel and Michel. There are few examples of nonlinear systems for which the optimal filter is finite dimensional, in particular Kalman’s, Beneš’, and Daum’s filters. In the present paper, we construct new classes of scalar nonlinear filtering p...

In nonlinear disordered Hamiltonian lattices, where there are no propagating phonons, the spreading of energy is of subdiffusive nature. Recently, the universality class of the subdiffusive spreading according to the nonlinear diffusion equation (NDE) has been suggested and checked for one-dimensional lattices. Here, we apply this approach to two-dimensional strongly nonlinear lattices and find...

A lattice Boltzmann model for convection-diffusion equation with nonlinear convection and isotropic-diffusion terms is proposed through selecting equilibrium distribution function properly. The model can be applied to the common real and complex-valued nonlinear evolutionary equations, such as the nonlinear Schrödinger equation, complex Ginzburg-Landau equation, Burgers-Fisher equation, nonline...

In this paper we present a novel nonlinear video diffusion approach based on the fusion of information in audio and video channels. Both modalities are efficiently combined into a diffusion coefficient that integrates the basic assumption in this domain, i.e. related events in audio and video channels occur approximately at the same time. The proposed diffusion coefficient depends thus on an es...

In high-density gas discharges in which the electrons, but not the ions, are confined by a magnetic field, the radial density profile is often observed to have a triangular shape, as if the ionization were concentrated on axis. However, such profiles can also result from nonlinear diffusion, in which the diffusion coefficient varies inversely with density.

These notes are the lecture notes of the course Symmetry and nonlinear diffusion flows http://verso.mat.uam.es/∼difusion.nolineal/pdes-geometry-lectures/ Lectures on PDEs and Geometry Universidad Autónoma de Madrid (Spain) The material has been collected from various papers and publications. The goal is to provide additional details on the proofs and references, without covering the most genera...