Hyperkähler manifolds and their Betti numbers

Tags: TAG Events DMI, TAG Events Forschung Mathematik]]>

Propagation of regularity and mixing estimates for solutions of the incompressible continuity equation with Sobolev vector field

Tags: TAG Events Forschung Mathematik, TAG Events DMI]]>

Since the work by DiPerna and Lions (1989) the continuity and transport equation under mild regularity assumptions on the vector field have been extensively studied, becoming a florid research field. The applicability of this theory is very wide, especially in the study of partial differential equations and very recently also in the field of non-smooth geometry.

The aim of this talk is to give an overview of the quantitative side of the theory initiated by Crippa and De Lellis. We address the problem of mixing and propagation of regularity for solutions to the continuity equation drifted by Sobolev fields. The problem is well understood when the vector field enjoys a Sobolev regularity with integrability exponent p>1 and basically nothing is known (at the quantitative level) in the case p=1.

We present sharp regularity estimates for the case p>1 and new attempts to attack the challenging question in the case p=1. This is a join work with Quoc-Hung Nguyen.

]]>22.11.2018 14:15 - 22.11.2018 15:15

On bounded automorphisms of fields with operators

Tags: TAG Events DMI, TAG Events Forschung Mathematik]]>

Probabilistic numerical methods, Bayesian inference and multiscale inverse problems

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In the second part of the talk, we will discuss a Bayesian approach for inverse problems involving elliptic partial differential equations with multiple scales. Computing repeated forward problems in a multiscale context is computationnally too expensive and we propose a new strategy based on the use of "effective" forward models originating from homogenization theory. Convergence of the true posterior distribution for the parameters of interest towards the homogenized posterior is established via G-convergence for the Hellinger metric. A computational approach based on numerical homogenization and reduced basis methods is proposed for an efficient evaluation of the forward model in a Markov Chain Monte-Carlo procedure.

References:

A. Abdulle, G. Garegnani, Random time step probabilistic methods for uncertainty quantification in chaotic and geometric numerical integration, Preprint (2018), submitted for publication.

A. Abdulle, A. Di Blasio, Numerical homogenization and model order reduction for multiscale inverse problems, to appear in SIAM MMS.

A. Abdulle, A. Di Blasio, A Bayesian numerical homogenization method for elliptic multiscale inverse problems, Preprint (2018), submitted for publication.

For further information about the seminar, please visit this webpage. ]]>

Tags: TAG Events DMI, TAG Events Forschung Mathematik]]>

16:15 | Vortrag von Thomas Steiner, Basel: |

17:15 | Apéro mit Präsentation eines neuen Bandes der Euler-Edition durch das Bernoulli-Euler-Zentrum, das Departement Mathematik und Informatik der Universität Basel und Birkhäuser Basel Leonhardi Euleri Opera Omnia, Bd. IVA/8: Herausgegeben von Andreas Kleinert und Thomas Steiner |

18:15 | Vortrag von Andreas Kleinert, Halle/Saale:Eulers hallesche Korrespondenten |

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Finite Energy Weak Solutions of the Navier-Stokes-Korteweg equations

Tags: TAG Events Forschung Mathematik, TAG Events DMI]]>

In this talk I will present some results concerning the analysis of finite energy weak solutions of the Navier-Stokes-Korteweg equations, which model the dynamic of a viscous compressible fluid with diffuse interface. A general theory of global existence is still missing, however for some particular cases of physical interest I will present results regarding the global existence and the compactness of finite energy weak solutions. The talk is based on a series of joint works with Paolo Antonelli (GSSI - Gran Sasso Science Institute, L’Aquila).

]]>29.11.2018 14:15 - 29.11.2018 15:15

Tags: TAG Events DMI, TAG Events Forschung Mathematik]]>