Termine

In this talk, we investigate efficient and robust solution strategies for the incompressible Navier-Stokes equations, which are specifically designed to predict transient flow behaviors on modern hardware architectures. The main idea of the framework is to block the individual linear systems of equations at each time step into a single all-at-once saddle-point problem. To address the challenges associated with the zero block, we employ a global Schur complement technique, implicitly eliminating all velocity unknowns and leading to a problem for all pressure unknowns only. This so called pressure Schur complement (PSC) equation is then solved using a space-time multigrid algorithm with tailor-made preconditioners for smoothing purposes. However, as the Reynolds number increases, the accuracy of the PSC preconditioner deteriorates, leading to convergence issues in the overall multigrid solver. To overcome this disadvantage, we enhance the solution strategy by using the Augmented Lagrangian methodology in a global-in-time fashion. This approach guarantees a rapid convergence by drastically increasing the accuracy of the adapted PSC preconditioner by means of a strongly consistent modification of the velocity system matrix. For sufficiently large stabilization parameters, even the global-in-time Schur complement iteration by itself converges fast and makes the coarse grid acceleration of the multigrid algorithm redundant. The Augmented Lagrangian technique comes at the cost that the involved auxiliary problem for the velocity field becomes ill conditioned for large stabilization parameters so that standard iterative solvers prove to be practically unusable. This calls for more sophisticated iterative solvers, like a specialized multigrid solver utilizing modified intergrid transfer operators and block diagonal preconditioners. In the limit of large stabilization parameters, the Augmented Lagrangian approach implies that all intermediate velocity solutions satisfy the discrete incompressibility condition. A straightforward consequence is therefore the use of discretely divergence-free finite elements, eliminating the pressure variable in an a priori manner. Therefore, no saddle-point structure exists and a wide range of efficient and accurate preconditioners becomes applicable. After illustrating the potential of this approach for two-dimensional global-in-time problems, we give some insight into the rarely investigated extension to three dimensions.

TBA

Abstract: "We study the count of real ramified covers of P^1 with given ramification data. In particular, we establish a beautiful correspondence between classical and tropical real double Hurwitz numbers, mediated by monodromy representations and monodromy graphs."

I will present my attempt to apply the Hrushovski construction to ordered fields. I will introduce my result on the green points construction, which axiomatizes the real field with dense logarithmic spirals. I will focus on my recent progress on the raising to powers construction, an expansion of RCF by "power functions" on the unit circle, whose axiomatization consists of the Schanuel property and the EC-property. If time permits, I will verify the EC-property for the corresponding expansion of the real field.

Consider first a memoryless population model described by the usual branching process with a given mean reproduction matrix on a finite space of types. Motivated by the consequences of atavism in Evolutionary Biology, we are interested in a modification of the dynamics where individuals keep full memory of their forebearers, and procreation involves the reactivation of a gene picked at random on the ancestral lineage.
By comparing the spectral radii of the two mean reproduction matrices (with and without memory), we observe that, on average, the model with memory always grows at least as fast as the model without memory. The proof relies on analyzing a biased Markov chain on the space of memories, and the existence of a unique ergodic law is demonstrated through asymptotic coupling.

Markus Stroppel will give a lecture series on advanced topics in locally compact groups. The lectures take place on Monday at 14:15 in room SR 1D They will start on Monday, November 17. The planned topics are: * the scale function and tidy subgroups * automorphism groups of trees

Abstract: An E_1 Calabi-Yau object A in a symmetric monoidal infinity-category C is a dualizable E_1 algebra together with an S^1-cyclic trace that exhibits a self duality of A. Examples include the cochain complex of any closed oriented manifold. By work of Barkan and Steinebrunner, every E_1 Calabi-Yau object in C defines a 2-d open field theory with values in C, which are symmetric monoidal functors from the open bordism category on disks to C. In joint work with Barkan and Steinebrunner, we show that any open field theory F extends canonically to an open-closed field theory whose value at the circle is the THH of the E_1 Calabi-Yau object A associated to F. As a corollary, we obtain an action of the moduli spaces of surfaces on the THH of E_1 Calabi-Yau algebras. This provides a space level refinement of previous work of Costello (over Q) and Wahl (over Z). Time permitting, I will discuss ongoing work with Andrea Bianchi on canonical local to global extensions of TFTs associated with E_\infty Calabi-Yau objects.