Wellposedness and Finite Element Convergence for the Joule Heating Problem with Mixed Boundary Conditions.
The stationary Joule heating problem is a two way coupled system of non-linear partial differential equations modelling the heat and electrical potential in a body. The electrical current acts as a heat source in a resistive material while the temperature feeds back to the electrical potential through the electrical conductivity. Joule heating is important in many micro-electromechanical systems, where the effect is used to achieve very exact positioning at the micro scale. In applications boundary conditions of mixed type are typically used. In this talk we present the existence proof for finite energy solutions of the Joule heating problem in three dimensions with mixed boundary conditions, using only very mild assumptions on the computational domain and the data. In particular, we show how previously established results can be extended to mixed boundary conditions. Furthermore, we prove strong convergence (of subsequences in case of non-unique exact solutions) of conforming finite element approximations. Under the additional assumption of a so-called creased domain together with a sufficiently weak temperature dependency in the electrical conductivity we also prove optimal global regularity estimates together with local estimates guaranteeing smooth solutions away from the boundary given smooth data. We further discuss a priori and a posteriori error bounds for conforming finite element approximations on shape regular meshes.
The presented material is joint work with Axel Målqvist (Uppsala, Sweden).
We consider a parabolic model for the evolution of an interface in random medium. The local velocity of the interface is governed by line tension and a competition between a constant external driving force F>0 and a heterogeneous random field f(x,y,ω), which describes the interaction of the interface with its environment. To be precise, let (Ω,F,P) be a probability space, ω∈Ω. We consider the evolution equation ∂tu(x,t,ω) = Δu(x,t,ω) - f(x,u(x,t,ω),ω) + F with zero initial condition. The random field f>0 has the form of localized smooth obstacles of random strength.
In particular, we are interested in the macroscopic, homogenized behavior of solutions to the evolution equation and their dependence on F. We prove that, under some assumptions on f, we have existence of a non-negative stationary solution for F small enough. This means that all solutions to the evolution equation become stuck if the driving force is not sufficiently large. The proof relies on a percolation argument. Given stronger assumptions on f, but still without a uniform bound on the obstacle strength, we also show that for large enough F the interface will propagate with a finite velocity.
The two results combined show the emergence of a rate-independent hysteresis in systems subject to a viscous microscopic evolution law through the interaction with a random environment.
Generalized Sampling and Infinite Dimensional Compressed Sensing
I will discuss a generalization of the Shannon Sampling Theorem that allows for reconstruction of signals in arbitrary bases. Not only can one reconstruct in arbitrary bases, but this can also be done in a completely stable way. When extra information is
available, such as sparsity or compressibility of the signal in a particular bases, one may reduce the number of samples dramatically. This is done via Compressed Sensing techniques, however, the usual finite-dimensional framework is not sufficient. To overcome this obstacle I'll introduce the concept of Infinite Dimensional Compressed Sensing.
Sensitivity and Out-of-sample Error in Data Assimilation
"Data Assimilation'' is one of many names for the following problem: Given a history of observations as well as a dynamical model, find trajectories which are, on the one hand, consistent with the model, and on the other hand, consistent with the observations. Attaining both objectives at the same time is essentially never possible (nor in fact desired) in reality, since our models are invariably simplifications. Any data assimilation algorithm should therefore allow for deviations from the proposed model equations as well as from the observations. How we might trade off between these two is the subject of this talk. It is shown how we can still find ``good'' trajectories, where the measure of goodness obviously cannot be just the deviation from the observations. Rather, a measure similar to the out-of-sample error from statistical learning is considered. The connection between the out-of-sample error and the sensitivity is elucidated, including some numerical examples.
On the Inverse Problems for the Coupled Continuum Pipe Flow model for flows in karst aquifers
We investigate two inverse problems for the coupled continuum pipe flow (CCPF) model which describes the fluid flows in karst
aquifers. After generalizing the well-posedness of the forward problem to the anisotropic exchange rate case which is a space-dependent variable, we present the uniqueness of this parameter by measuring the Cauchy data. Besides, the uniqueness of the geometry of the conduit by the Cauchy data is verified as well. These results enhance the practicality of the CCPF model.