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Guest lectures and seminars - Page 7

Time and place: , Niels Henrik Abels hus, 9th floor

A peculiarity of nonlinear hyperbolic problems is that they must be interpreted as limits of second-order equations with vanishing viscosity. Despite not explicitly being present in the hyperbolic case, diffusion is needed, e. g., at discontinuities or to avoid the occurrence of nonphysical states. In the case of gas dynamics, for instance, dissipation corresponds to the production of thermodynamic entropy. To solve hyperbolic problems numerically, one needs to adapt these ideas to the discrete setting. Standard high-order methods, however, do not incorporate the appropriate amounts of artificial viscosity because these need to be chosen adaptively based on the solution. Among the high-resolution schemes capable of doing so are the recently proposed monolithic convex limiting (MCL) techniques [1] to be discussed in this talk. They offer a way to enforce physical admissibility, entropy stability, and discrete maximum principles for conservation laws. These methods can also be generalized to systems of balance laws in a well-balanced manner [2]. In addition to second-order finite element methods, extensions to high-order discontinuous Galerkin (DG) schemes shall also be presented [3]. Numerical examples for the so-called KPP problem, the nonconservative shallow water system, and the compressible Euler equations will be shown. An overview of MCL and other property-preserving methods can be found in our recently published book [4].

Time and place: , NHA 723 and Online
Time and place: , NHA B1120

We prove that (logarithmic, Nygaard completed) prismatic and (logarithmic) syntomic cohomology are representable in the category of logarithmic motives. As an application, we immediately obtain Gysin maps for prismatic and syntomic cohomology, and we precisely identify their cofibers. In the second part of the talk we develop a descent technique that we call saturated descent, inspired by the work of Niziol on log K-theory. Using this, we prove crystalline comparison theorems for log prismatic cohomology, log Segal conjectures and log analogues of the Breuil-Kisin prismatic cohomology, from which we get Gysin maps for the Ainf cohomology.

Time and place: , Erling Sverdrups plass, Niels Henrik Abels hus, 8th floor
This paper considers hypothesis testing in semiparametric models which may be non – regular for certain values of a (potentially infinite dimensional) nuisance parameter. In such models no (locally) regular estimator of the parameter of interest exists. The situation for testing is somewhat different: I establish that C(α) – style test statistics achieve their limiting distributions in a (locally) regular manner under mild conditions, leading to tests with correct size in situations where standard tests fail to control size. Additionally, I characterise the appropriate limit experiment in which to study local (asymptotic) optimality of tests in the case where the efficient information matrix is singular. This permits the generalisation of classical power bounds to the non – regular case. I provide appropriate statements of these bounds and give conditions under which they are attained by the proposed C(α) – style tests. Three examples are worked out in detail.
Time and place: , Niels Henrik Abels hus, 9th floor

We combine a pressure correction scheme with interior penalty discontinuous Galerkin (dG) discretisation to solve the time-dependent Navier–Stokes equations. We prove unconditional energy stability and a priori error estimates for the velocity. With duality arguments, optimal L2 error rates are obtained. Convergence of the discrete pressure is also established.  Further, we propose a splitting scheme,  integrating the pressure correction approach, for the Cahn–Hilliard–Navier–Stokes system  The numerical analysis of dG combined with this scheme is discussed. Namely, we show well--posedness, stability, and error estimates. Numerical results with manufactured solutions display our theoretical findings, and a spinodal decomposition example portrays the robustness of our approach.