Veranstaltungen

The work on parameter orthogonalisation by Cox and Reid (1987) is presented as an inducement of abstract population-level sparsity. The latter is taken as a unifying theme for the talk, in which sparsity-inducing parameterisations or data transformations are sought. Examples from some of my work are framed in this light, with emphasis on recent developments addressing the following question: for a given statistical problem, not obviously sparse in its natural formulation, can a sparsity-inducing reparametrisation be deduced? In general, the solution strategy for the problem of exact or approximate sparsity inducement appears to be context specific and may entail, for instance, solving one or more partial differential equation, or specifying a parametrised path through data-transformation or parametrisation space.

Quantum systems typically reach thermal equilibrium when in weak contact with a large external bath. Understanding the speed of this thermalisation is a challenging problem, especially in the context of quantum many-body systems where direct calculations are intractable. The usual way of bounding the speed of this process is by estimating the spectral gap of the dissipative generator, but this does not always yield a reasonable estimate for the thermalisation time. When the system satisfies instead a modified logarithmic Sobolev inequality (MLSI), the thermalisation time is at most logarithmic in the system size, yielding wide-ranging applications to the study of many-body in and out-of-equilibrium quantum systems, such as stability against local perturbations (in the generator), efficient preparation of Gibbs states (the equilibria of these processes), etc. In this talk, I will present an overview on a strategy to prove that a system satisfies a MLSI provided that correlations decay sufficiently fast between spatially separated regions on the Gibbs state of a local, commuting Hamiltonian in any dimension. I will subsequently review the current state of the art for Davies dissipative generators thermalising to such Gibbs states.

Many engineering problems require the calculation of certain quantities of interest, which are usually defined by linear functionals depending on the solution of a partial differential equation. Examples include the local or global mean value of a temperature, or the magnetic flux density at a given point of an electromagnetic device. In this talk, we focus on estimating the error of such functionals using a wide variety of numerical methods (finite elements, discontinuous galerkin and finite volumes), within a unified framework for elliptic and parabolic problems. The key point lies in solving a dual problem and using guaranteed equilibrated estimators for the primal and the dual problems, computed using flux and potential reconstructions. In all cases, we prove that the goal-oriented error can be split into a fully computable estimator and a remainder term that can be bounded above by computable energy- based estimators. We present some numerical tests to underline the capability of this goal-oriented estimator in different contexts : reaction-diffusion problems, heat equation, and harmonic formulation of eddy-current problems.

Nonlinear dynamical systems often exhibit rich and complicated recurrent dynamics. Understanding these dynamics is challenging, particularly in higher dimensions where visualization is limited. Additionally, in many applications, time series data is all that is available. In this talk, I present an algebraic topological tool to identify elementary oscillations and the transitions between them. More precisely, we introduce the cycling signature which is constructed by taking persistent homology of time series segments in a suitable ambient space. Oscillations in a time series can then be identified by analyzing the cycling signatures of its segments.