17 October 2024, Thursday, 14:15, in seminar room E2.304

Title:
Dynamical low-rank approximation: From radiation therapy to quantum mechanics

Abstract:
Many problems in physics, engineering, and computer science require substantial computational and memory resources. Three prominent examples are radiation therapy, quantum mechanics, and neural networks. This is because evolving the wave function in quantum mechanics in time, calculating the radiation therapy dose, or training neural network weights involves solving large,
often prohibitively complex, matrix or tensor differential equations.

However, it has been observed that solutions in all three fields often exhibit low-rank structures. To efficiently evolve solutions as low-rank representations, we employ dynamical low-rank approximation (DLRA), as introduced in [Koch and Lubich (2007)]. DLRA is closely related to the Dirac–Frenkel time-dependent variational principle (see, e.g., [Dirac (1930), Beck (2000)]), representing solutions through low-rank matrix or tensor factorizations and deriving time evolution equations for the low-rank factors. Since these equations are highly stiff, novel numerical methods tailored to the geometry of low-rank tensor manifolds are necessary. I will discuss the development and analysis of new time integration methods that can efficiently simulate problems in radiation therapy, quantum mechanics, and neural network training while preserving essential structures of the original full-rank dynamics.

25 April 2024, Thursday, 14:15, in seminar room D1.320

Title:
Moving mesh split form discontinuous Galerkin methods to solve conservation laws

Abstract:
The construction of high order nodal discontinuous Galerkin (DG) methods to solve conservation laws and related equations includes the approximation of non-linear flux terms in the volume integrals. These terms can lead to aliasing and stability issues in the DG approximation. Aliasing issues arises when the flux terms are composed of products of polynomials, and are then interpolated by a polynomial basis of a lower order than the product of the polynomials. The split form DG framework provides a tool to construct the numerical approximation in a way that aliasing issues caused by the discretization of the DG volume integrals are avoided. In particular the split form DG framework can be used to construct entropy stable DG methods.
On the other hand the r-adaptive method or moving mesh method involves the re-distribution of the mesh nodes in regions of rapid variation of the solution. In comparison with h-adaptive discretizations, where the mesh is refined and coarsened by changing the number of elements in the tessellation, the r-adaptive method has some advantages, e.g. no hanging nodes appear and the number of elements does not change. In this talk, the focus is on the construction of moving mesh entropy stable split form DG methods. Thereby, a proper methodology to compute the grid point distribution to move the mesh will be not discussed. Numerical experiments for the three dimensional compressible Euler equations will be presented to show the capability of the moving mesh split form DG methods.

16 May 2024, Thursday, 14:00 c.t., in seminar room D1.320

Title:
The linearly implicit two-step BDF method for harmonic maps into spheres

Abstract:
After recalling the notion of harmonic maps into spheres, we discuss two variational formulations of the corresponding Euler–Lagrange equations. The second variational formulation leads easily to a linearization of the nonlinear equation. Subsequently, we focus on the gradient flow approach and recall known results for the linearly implicit Euler method, namely, energy decay (stability) and constraint violation properties.
Our contribution concerns the application of the linearly implicit two-step BDF method to the gradient flow problem. We devise a projection-free iterative scheme for the approximation of harmonic maps that is unconditionally energy stable and provides a second-order accuracy of the constraint violation under a mild, sharp discrete regularity condition. The considered problem serves as a model for partial differential equations with holonomic constraint.
For the performance of the method, illustrated via the computation of stationary harmonic maps and bending isometries.
The talk is based on joint work with Sören Bartels and Christian Palus (Albert-Ludwigs-Universität Freiburg).

27 June 2024, Thursday, 14:15, in seminar room D1.320

Title:
 

Abstract:

1. February 2024, Thursday, 14:15, in seminar room E2.304

Title:
A full discretization of the Cahn--Hilliard equation with optimal order error estimates

Abstract:
The Cahn--Hilliard equation with Cahn--Hilliard type dynamic boundary conditions models the phase separation process of binary mixtures in a bounded domain with non-permeable walls. This was proposed by Goldstein et al. (2011) and has since attracted a lot of attention of scientists. Our goal is to analyze the error of the discretization of this model by finite elements and the BDF method. We will look at techniques on how to adapt energy estimates to this fully discrete setting which were developed by Lubich et al. (2013) and on how to adapt a system of energy estimates first used by Kovács et al. (2020), to get a convergence result with optimal order error estimates.

14 December 2023, Thursday, 14:15, in seminar room E2.304

Title:
Adaptivity for parabolic surface PDEs on stationary domains

Abstract:
The field of application for surface PDEs steadily increases. The numerical analysis of surface PDEs had it's kick off 1988 with the convergence proof of the surface finite element by Dziuk. In contrast to standard finite elements the main challenges arise from the geometric nature of the problem. One should notice that the domain can usually not be triangulated directly. This leads to the introduction of lift operators which aim to sustain the comparability of functions on the hyper surface and on its approximation as a polyhedron. In the presentation we will give an overview of the numerical analysis of surface PDEs and introduce an adaptive algorithm for a parabolic surface PDE on a stationary surface. First we introduce some technical results on residual based error indicators. Afterwards we take a look on the proposed algorithm, especially we focus on refinement and coarsening strategies.

Zoom:
Topic: Oberseminar "Numerics for PDEs"
Time: This is a recurring meeting Meet anytime

Join Zoom Meeting
https://uni-paderborn-de.zoom-x.de/j/61806528444?pwd=WEUySnNqSVlqSTFxUk9kRjBHeWVZQT09

Meeting ID: 618 0652 8444
Passcode: 308931

23 November 2023, Thursday, 14:15, in seminar room E2.304.

Title:
The origin of Fourier series

Abstract:
-

Link:
https://math.uni-paderborn.de/ag/arbeitsgruppe-numerik-und-steuerung/colloquium


Zoom:
Topic: Applied Mathematics Colloquium

Join Zoom Meeting
https://uni-paderborn-de.zoom-x.de/j/61650043657?pwd=TWZVTmM4cTlnR25vYkZRREhYMDlTQT09

Meeting ID: 616 5004 3657
Passcode: 639371

16 November 2023, Thursday, 14:15, in seminar room E2.304.

Title:
Time-dependent electromagnetic scattering from dispersive materials

Abstract:
This talk discusses time-dependent electromagnetic scattering problems in the context of dispersive material laws. We consider the numerical treatment of a scattering problem in which a retarded material law, for a causal and passive homogeneous material, determines the wave-material interaction in the scatterer. The resulting problem is nonlocal in time in the interior of the scatterer and is posed on an unbounded domain. Well-posedness of the scattering problem is shown using a formulation based on time-dependent boundary integral equations, which is fully formulated on the surface of the scatterer. Discretizing the boundary integral equation by convolution quadrature in time and boundary elements in space yields a provably stable and convergent method that is fully parallel in time and space. Under regularity assumptions on the exact solution we present error bounds with explicit convergence rates in time and space. Numerical experiments illustrate the theoretical results and show the effectiveness of the method.


Link:
https://math.uni-paderborn.de/ag/arbeitsgruppe-numerik-und-steuerung/colloquium

14 November 2023, Tuesday, 9:15, in seminar room D2.314 (Medienraum)

Title:
Numerical approximation for mean curvature flow of surfaces with boundary

Abstract:
In this talk we will briefly review strategies to obtain numerical approximations for mean curvature flow with boundary conditions. We will focus mainly on the Dirichlet boundary case.

We will review how some existing algorithms can be adapted to the case of surfaces with boundary and look at some numerical experiments in this direction. Finally, we will discuss some ideas and advances in the search for error estimates.

9 November 2023, Thursday, 14:15, in seminar room E2.304

Title:
Algorithms for mean curvature flow and "What happens when we let a numerical analyst to the operating table?"

Abstract:
In this talk we will briefly review some theoretical and numerical results for geometric flows, in particular focusing on mean curvature flow, and briefly discuss some theoretical results for flows evolving through singularities.

For the most part of the talk we would like to discuss a new surgical method, which can be applied in real-life scenarios where the patient is still maturing. The discussed approach is inspired by some classical ideas in the field by the cutting-edge procedures developed by Huisken and Sinestrari, and Brendle and Huisken. Naturally, our surgical method is based on previous works in the field where important patient data need to be tracked along the process.
We will describe the surgical process in detail, and share detailed results and recordings from test-operations.