Tag - Applied mathematics

After recalling basic facts from quantum information theory (density matrices, partial trace, purification), I'll present two examples of random states: the induced Gaussian state and the uniform pure one on the Euclidean sphere. Afterwards, I'll introduce the dynamical version of the latter which leads to the problem of computing the joint law of the moduli of a given set of its coordinates. The first approach to solve this problem is based on solving a heat equation in the simplex but lacks the knowledge of boundary conditions and a natural choice of basis. The second approach is rather based on direct computations by means of unitary spherical harmonics and gives rise to the so-called Jacobi polynomials in the simplex.

What do these two themes have in common? Both are treated variationally, both deal with energies of different dimensionalities, and concepts of geometric measure theory prevail in both.

Phase Separation in Heterogeneous Media

Modern technologies and biological systems, such as temperature-responsive polymers and lipid rafts, take advantage of engineered inclusions, or natural heterogeneities of the medium, to obtain novel composite materials with specific physical properties. To model such situations using a variational approach based on the gradient theory of phase transitions, the potential and the wells may have to depend on the spatial position, even in a discontinuous way, and different regimes should be considered. In the critical case case where the scale of the small heterogeneities is of the same order of the scale governing the phase transition and the wells are fixed, the interaction between homogenization and the phase transitions process leads to an anisotropic interfacial energy. The supercritical case for fixed wells is also addressed, now leading to an isotropic interfacial energy. In the subcritical case with moving wells, where the heterogeneities of the material are of a larger scale than that of the diffuse interface between different phases, it is observed that there is no macroscopic phase separation and that thermal fluctuations play a role in the formation of nanodomains. This is joint work with Riccardo Cristoferi (Radboud University, The Netherlands) and Likhit Ganedi (Aachen University, Germany), based on previous results also obtained with Adrian Hagerty (USA) and Cristina Popovici (USA).

Learning Training Schemes for Image Denoising

Due to their ability to handle discontinuous images while having a well-understood behavior, regularizations with total variation (TV) and total generalized variation (TGV) are some of the best known methods in image denoising. However, like other variational models including a fidelity term, they crucially depend on the choice of their tuning parameters. A remedy is to choose these automatically through multilevel approaches, for example by optimizing performance on noisy/clean image training pairs. Such methods with spacedependent parameters which are piecewise constant on dyadic grids are considered, with the grid itself being part of the minimization. Existence of minimizers for discontinuous parameters is established, and it is shown that box constraints for the values of the parameters lead to existence of finite optimal partitions. Improved performance on some representative test images when compared with constant optimized parameters is demonstrated. This is joint work with Elisa Davoli (TU Wien, Austria), Jose Iglesias (U. Twente, The Netherlands) and Rita Ferreira (KAUST, Saudi Arabia)

Phase change in materials is important in many sciences and engineering applications such as metal processing, crystal growth, industrial furnaces, among others. In all the applications that involve high temperature, radiation greatly influences the thermal features and it cannot be neglected. Experimental predictions of the impact of radiation in the heat conduction can be very demanding. Therefore, numerical modeling can play a crucial role and provide accurate and effective thermal predictions. This presentation covers two main topics. First, we present accurate numerical methods for solving radiation–conduction heat transfer in participating media. The mathematical model is based on the SP_N approximations. The numerical methods are based on a mixed finite element formulation and a time-dependent mesh adaptation algorithm to enhance the accuracy of the solutions. Second, we present mathematical models and numerical methods developed for phase change problems. Two- and three-dimensional numerical simulations are considered to show the effectiveness of the overall methodology.