Numerische Simulation und Optimierung stationärer Gasflüsse
Tuesday, 19.10.21, 14:15-15:15, Raum 226, Hermann-Herder-Str. 10
European transmission gas networks are expected to play a major role for the transition to green energy supply as transition technology and for transporting and storing green gas as generated from green energy sources. They are endangered by a wide range of potential disruptions, for example ageing and corrosion, physical damage through construction works (third party interference) and attacks, cyber-attacks and coordinated attack vectors. Hence the determination of the effects of local damage events on the\nsupply of consumers and network elements is required for a variety of individual events and combinations of events. Against this background, (0-dimensional) simulations are\nused to predict and model gas pressures at nodes and gas volume flows for each pipeline. This can be realized by solving algebraic equations using numerical optimization\nwith the aim of minimizing a given objective function subject to equality and inequality constraints. Representative network examples within the context of the EU project SecureGas are generated from published sample grids, inspection of existing maps and literature on gas transport on the level of European transmission grids. Thus, gas grids with realistic lengths, diameters and pressure boundary conditions, the external and internal inflows, outflows and possibly also storage capabilities are generated. Using such representative networks, network models are created for which the effects of potential\ndisruptions are calculated predictively and systematically. For this purpose, the number of nodes not supplied and the pressure loss in nodes that are no longer supplied sufficiently are determined given defined full, single or multiple disruptions.
Finite element methods for 1D few-particle quantum dynamics
Tuesday, 26.10.21, 14:15-15:15, Raum 226, Hermann-Herder-Str. 10
The dynamics of one-dimensional few-particle quantum systems are key to understand the phenomenological differences between single- and many-body systems, and ultimately the transition to the thermodynamic limit. While experimentally such systems become increasingly controllable, exact numerical approaches are feasible but challenging.\n\nIn this talk, we start with a short introduction to the theoretical description of closed quantum systems. We then demonstrate a numerical treatment of two or three indistinguishable, interacting bosons in a one-dimensional trapping potential, by diagonalization of the many-body Hamiltonian after discretization in an appropriate finite element basis. Along the way, we analyse the convergence properties of our approach and briefly comment on (questions concerning) the mathematical structure of the problem.
Asymptotic Stability in a free boundary model of cell motion
Tuesday, 9.11.21, 14:15-15:15, Raum 226, Hermann-Herder-Str. 10
We introduce a free boundary model of the onset of motion of a living cell (e.g. a keratocyte) driven by myosin contraction, with focus on a transition from unstable radial stationary states to stable asymmetric moving states. This model generalizes a previous 1D model (Truskinovsky et al.) by combining a Keller-Segel model, a Hele-Shaw boundary condition, and the Young-Laplace law with a nonlocal regularizing term, which precludes blow-up or collapse by ensuring that membrane-cortex interaction is sufficiently strong. We show that this model has a family of asymmetric traveling wave solutions bifurcating from a family of stationary states. Our goal is to establish observable steady cell motion with constant velocity. Mathematically, this amounts to proving stability of the traveling wave solutions, which requires generalization of the standard notion of stability. Our main result is establishing nonlinear asymptotic stability of traveling solutions. To this end, we derive an explicit asymptotic formula for the stability-determining eigenvalue from asymptotic expansions in small speed. This formula greatly simplifies the numerical computation of the sign of this eigenvalue and reveals the physics underlying onset of the cell motion and stability of moving states. If time permits, we will discuss work in progress on fingering instability in multicellular tissue spreading.\n\nThis is joint work with V. Rybalko (Verkin Institute, Ukraine) and C. A. Safsten (Penn State, PhD student).
An obstacle problem for the p−elastic energy
Tuesday, 16.11.21, 14:15-15:15, Raum 226, Hermann-Herder-Str. 10
In this talk we seek to minimize the p-elastic curvature energy\nE(u) := \bintgraph(u) |κ|^p ds\namong all graphs u ∈ W^2,p (0, 1)∩ W0^1,p(0, 1) that satisfy the obstacle constraint u(x) ≥ ψ(x) for all x ∈ [0, 1]. Here ψ ∈ C^0([0, 1]) is an obstacle function. The energy functional imposes three major challenges that we need to overcome:\n\n1. Lack of coercivity.\n\n2. Loss of regularity on the coincidence set {u = ψ}.\n\n3. (For p > 2:) Degeneracy of the Euler-Lagrange equation.\n\nWe will develop methods to examine all three phenomena. A key ingredient for\nthis analysis goes back to L. Euler: One can find a substitution that makes the\nEuler-Lagrange equation elliptic.\n\nFinally, we are able to show sharp existence (and non-existence) results and\ndiscuss the optimal regularity of minimizers.
A convergent algorithm for the interaction of mean curvature flow and diffusion
Monday, 22.11.21, 12:15-13:15, Raum 226, Hermann-Herder-Str. 10
\nIn this talk we will present an evolving surface finite elment algorithm for the interaction of forced mean curvature flow and a diffusion process on the surface.\nThe evolving surface finite element discretisation is analysed for the evolution of a closed two-dimensional surface governed by the above coupled geometric PDE system. The coupled system is inspired by the gradient flow of a coupled energy, we will use this model for introductury purposes.\nWe will present two algorithms, based on a system coupling the diffusion equation to evolution equations for geometric quantities in the velocity law for the surface. In some sense, these algorithms return home, since they were heavily inspired by the works of Professor Dziuk.\nFor one of the numerical methods we will give some insights into the stability estimates which are used to prove optimal-order \(H1\)-norm error estimates for finite elements of degree at least two.\nWe will present numerical experiments illustrating the convergence behaviour and demonstrating the qualitative properties of the flow: preservation of mean convexity, loss of convexity, weak maximum principles, and the occurrence of self-intersections.\nThe talk is based on joint work with C.~M.~Elliott (Warwick) and H.~Garcke (Regensburg).
TBA
Tuesday, 23.11.21, 14:15-15:15, https://uni-freiburg.zoom.us/j/61095147552?pwd=VDNSdnRnMVFCbVgxTVJ0QWNmeU0yQT09#success
TBA
https://conferencekuwert.github.io/directions/
Monday, 29.11.21, 00:00-01:00, KG1 : Platz der Universität 3
Geometric PDEs in Freiburg: A conference in honor of the 60th Birthday of Ernst Kuwert. \n\n
Adaptive finite elements in convex optimisation
Tuesday, 11.1.22, 02:15-03:15, https://uni-freiburg.zoom.us/j/64858555488?pwd=eGgvUHErS1VZUm43bXl1SXJCMlloUT09
Abstract: The advantages of adaptive discretizations are well-documented, however, many convex optimization algorithms are not able to utilize these advantages. Taking a step to address this, in this talk we will analyze how the FISTA algorithm behaves with inexact discretizations. In doing so, we also prove new convergence results beyond the capabilities of the original algorithm. We will finish with numerical experiments which demonstrate the potential for improved efficiency by using adaptive finite elements in convex optimization problems.\n
HOMOGENIZATION OF DISCRETE THIN STRUCTURES
Tuesday, 25.1.22, 14:15-15:15, https://uni-freiburg.zoom.us/j/62669086921?pwd=bEdlZDNQU2plREZ3aEJ6RFpTOWNuQT09
We investigate discrete thin objects which are described by a subset \(X\) of \(\bmathbb{Z}^d\btimes \b{0,\bdots, T-1 \b}^k\), for some \(T\bin\bmathbb{N}\) and \(d,k\bgeq 1\). We only require that \(X\) is a connected graph and periodic in the first \(d\)-directions.\nWe consider quadratic energies on \(X\) and we perform a discrete-to-continuum and dimension-reduction process for such energies.\nWe show that, upon scaling of the domain and of the energies by a small parameter \(\bvarepsilon\), the scaled energies \(\bGamma\)-converges to a \(d\)-dimensional functional. The main technical points are a dimension-lowering coarse-graining process and a discrete version of the p-connectedness approach by Zhikov.\nThis is a joint work with A. Braides.
Goal-oriented adaptive FEMs with optimal computational complexity
Tuesday, 1.2.22, 14:15-15:15, https://uni-freiburg.zoom.us/j/62025301433?pwd=TURsRWs3NHN6T0lCNGZsekFKMGNtQT09
We consider a linear symmetric and elliptic PDE and a linear goal functional. We design \na goal-oriented adaptive finite element method (GOAFEM), which steers the adaptive \nmesh-refinement as well as the approximate solution of the arising linear systems by \nmeans of a contractive iterative solver like the optimally preconditioned conjugate gradient \nmethod (PCG). We prove linear convergence of the proposed adaptive algorithm with optimal \nalgebraic rates with respect to the number of degrees of freedom as well as the computational cost.
Neural network approximations for high-dimensional PDEs
Tuesday, 8.2.22, 14:15-15:15, https://uni-freiburg.zoom.us/j/64146027041?pwd=dCtxbTBqbjI4MjlOcFV2WDJ5ODI2dz09
Most of the numerical approximation methods for PDEs in the scientific literature suffer from the so-called curse of dimensionality (CoD) in the sense that the number of computational operations employed in the corresponding approximation scheme to obtain an approximation precision \(\bvarepsilon > 0\) grows exponentially in the PDE dimension and/or the reciprocal of \(\bvarepsilon\). Recently, certain deep learning based approximation methods for PDEs have been proposed and various numerical simulations for such methods suggest that deep neural network (DNN) approximations might have the capacity to indeed overcome the CoD in the sense that the number of real parameters used to describe the approximating DNNs grows at most polynomially in both the PDE dimension \(d \bin \bN\) and the reciprocal of the prescribed approximation accuracy \(\bvarepsilon > 0\). In this talk we show that solutions of suitable Kolmogorov PDEs can be approximated by DNNs without the CoD.
Three-dimensional, homogenized PDE/ODE model for bone fracture healing
Tuesday, 15.2.22, 11:10-12:10, Raum 226, Hermann-Herder-Str. 10 & und Online auf Zoom: https://un1-freiburg.zoom.us/j/64612030148?pwd=eUF6WTk4bVVVb1g4YnhucFhOUS9jZz09
We present a three-dimensional, homogenized PDE/ODE model for bone\nfracture healing in the presence of a porous, bio-resorbable scaffold\nand an associated PDE constrained optimization problem concerning the\noptimal scaffold density distribution for an ideal healing environment.\nThe model is analyzed mathematically and a well-posedness result is\nprovided. Concerning the optimization problem, we show the existence of\nan optimal scaffold design. We touch delicate regularity results for\nelliptic and parabolic equations with mixed boundary conditions which\nare crucial for the analysis of the optimal control problem, extending\nresults from the literature. Numerical simulations for the PDE/ODE\nsystem and the PDE constrained optimization problem are presented,\nillustrating the effect of stress-shielding on optimal scaffold design.