Upcoming Events
Many events are currently organized online. Information on how to access these events can be found by clicking “more” below the respective entry.
- Friday, 24.03.2023, 13:00 (WIAS-Library)
- E-Coffee-Lecture
Heike Sill, WIAS:
KLR am WIAS
more ... Location
Weierstraß-Institut, Hausvogteiplatz 5-7, 10117 Berlin, R411
Host
WIAS Berlin
- Thursday, 23.03.2023, 16:00 (WIAS-ESH)
- Forschungsseminar Mathematische Modelle der Photonik
Lutz Mertenskötter, WIAS Berlin:
Laser linewidth estimation beyond the detector noise limit - Statistical inference from delayed self-heterodyne measurements
more ... Location
Weierstraß-Institut, Mohrenstr. 39, 10117 Berlin, Erdgeschoss, Erhard-Schmidt-Hörsaal
Further Informations
Forschungsseminar Mathematische Modelle der Photonik
Host
WIAS Berlin
- Thursday, 23.03.2023, 14:00 (WIAS-ESH)
- Seminar Numerische Mathematik
Dr. Gabriel R. Barrenechea, University of Strathclyde, UK:
Positivity-preserving discretisations in general meshes
more ... Location
Weierstraß-Institut, Mohrenstr. 39, 10117 Berlin, Erdgeschoss, Erhard-Schmidt-Hörsaal
Abstract
The quest for physical consistency in the discretisation of PDEs started as soon as the numerical methods started being proposed. By physical consistency we mean a discretisation that by design satisfies a property also satisfied by the continuous PDE. This property might be positivity of the discrete solution, or preservation of some bounds (e.g., concentrations should belong to the interval [0, 1]), or can also be energy preservation, or exactly divergence-free velocities for incompressible fluids. Regarding positivity preservation, this topic has been around since the pioneering work by Ph. Ciarlet in the late 1960s and early 1970s. In the context of finite element methods, it was shown in those early works (and not significantly improved since), that in order for a finite element method to preserve positivity the mesh needs to satisfy certain geometrical restrictions, e.g., in two space dimensions with simplicial elements the triangulation needs to be of Delaunay type (in higher dimensions or quadrilateral meshes the restrictions are more involved). Throughout the years several conclusions have been reached in this topic, but in the context of finite element methods the discretisations tend to be of first order in space. So, many important problems still remain open. In particular, one open problem is how to build a discretisation that will lead to a positive solution regardless of the geometry of the mesh and the order of the finite element method. In this talk I will review recent results addressing the last question posed in the last paragraph. More precisely, I will present a method that enforces bound-preservation (at the degrees of freedom) of the discrete solution. The method is built by first defining an algebraic projection onto the convex closed set of finite element functions that satisfy the bounds given by the solution of the PDE. Then, this projection is hardwired into the definition of the method by writing a discrete problem posed for this projected part of the solution. Since this process is done independently of the shape of the basis functions, and no result on the resulting finite element matrix is used, then the outcome is a finite element function that satisfies the bounds at the degrees of freedom. Another important observation to make is that this approach is related to variational inequalities, and this fact will be exploited in the error analysis. The core of the talk will be devoted to explaining the main idea in the context of linear (and nonlinear) reaction-diffusion equations. Then, I will explain the main difficulties encountered when extending this method to convection-diffusion equations, and, more importantly, to a finite element method defined in polytopal meshes. The results in this talk have been carried out in collaboration with Abdolreza Amiri (Strathclyde, UK), Emmanuil Geourgoulis (Heriot-Watt, UK and Athens, Greece), Tristan Pryer (Bath, UK), and Andreas Veeser (Milan, Italy).
Further Informations
Seminar Numerische Mathematik
Host
Humboldt-Universität zu Berlin
WIAS Berlin
- Wednesday, 22.03.2023, 10:00 (WIAS-406)
- Joint Research Seminar on Nonsmooth Variational Problems and Operator Equations / Mathematical Optimization
Dr. Constantin Christof, Technische Universität München:
On the identification and optimization of nonsmooth superposition operators in semilinear elliptic PDEs
more ... Location
Weierstraß-Institut, Mohrenstr. 39, 10117 Berlin, 4. Etage, Weierstraß-Hörsaal (Raum: 406)
Abstract
We study an infinite-dimensional optimization problem that aims to identify the Nemytskii operator in the nonlinear part of a prototypical semilinear elliptic partial differential equation which minimizes the distance between the PDE-solution and a given desired state. In contrast to previous works, we consider this identification problem in a low-regularity regime in which the function inducing the Nemytskii operator is a-priori only known to be an element of H1loc. This makes the studied problem class a suitable point of departure for the rigorous analysis of training problems for learning-informed PDEs in which an unknown superposition operator is approximated by means of a neural network with nonsmooth activation functions (ReLU, leaky-ReLU, etc.). We establish that, despite the low regularity of the controls, it is possible to derive a classical stationarity system for local minimizers and to solve the considered problem by means of a gradient projection method. It is also shown that the established first-order necessary optimality conditions imply that locally optimal superposition operators share various characteristic properties with commonly used activation functions: They are always sigmoidal, continuously differentiable away from the origin, and typically possess a distinct kink at zero.
Further Informations
Joint Research Seminar on Nonsmooth Variational Problems and Operator Equations
Host
WIAS Berlin