2021-06-16: Marcus Grote, University of Basel
Stabilized leapfrog based local time-stepping method for wave propagation
For the time integration of second-order wave equations, the leapfrog (LF) method probably remains to this day the most popular numerical method. Based on a centered finite difference approximation of the second-order time derivative,
it is second-order accurate, explicit, time-reversible, and, for linear problems, conserves (a discrete version of) the energy for all time. For the spatial discretization of partial differential equations, finite element methods (FEM) provide a flexible approach, which easily accommodates a varying mesh size or polynomial degree. Not only do FEM permit the use of high-order polynomials, necessary to capture the oscillatory nature of wave phenomena and keep numerical dispersion (''pollution error'') minimal, but they are also apt at locally resolving small geometric features or material interfaces. Hence the combined FEM and LF based numerical discretization of second-order wave equations has proved a versatile and highly efficient approach, be it for the simulation of acoustic or elastic waves.
Local mesh refinement, however, can cause a severe bottleneck for the LF method, or any other standard explicit scheme, due to the stringent CFL stability condition on the time-step imposed by the smallest element in the mesh. Even when the locally refined region consists of a few small elements only, those elements will dictate a tiny time-step throughout the entire computational domain for stability. To overcome the crippling effect of a few small elements, without sacrificing its
high efficiency or explicitness elsewhere, a leapfrog based explicit local time-stepping (LF-LTS) method was proposed for the time integration of second-order wave equations. Recently, optimal convergence rates were proved for a conforming FEM discretization, albeit under a CFL stability condition where the global time-step depends on the smallest elements in the mesh. In general one cannot improve upon that stability constraint, as the LF-LTS method may become unstable at certain
discrete values of the time-step. To remove those critical values for the time-step, we apply a slight modification to the original LF-LTS method which nonetheless preserves its desirable properties: it is fully explicit, second-order accurate,
satisfies a three-term (leapfrog like) recurrence relation, and conserves the energy. The new stabilized LF-LTS method
also yields optimal convergence rates for a standard conforming FE discretization, yet under a CFL condition
where the time-step no longer depends on the mesh size inside the locally refined region.
Joint work with: Simon Michel (Univ. of Basel) and Stefan Sauter (Univ. of Zurich)
2021-06-09: Ulrik Skre Fjordholm, University of Oslo
Numerical methods for conservation laws on graphs
We consider a set of scalar conservation laws on a graph. Based on a choice of stationary states of the problem – analogous to the constants in Kruzkhov's entropy condition – we establish the uniqueness and stability of entropy solutions. For rather general flux functions we establish the convergence of an easy-to-implement Engquist–Osher-type finite volume method.
This is joint work with Markus Musch and Nils Henrik Risebro (University of Oslo).
2021-06-02: André Massing, NTNU
Stabilized Cut Discontinuous Galerkin Methods
To ease the burden of mesh generation in finite-element based simulation pipelines, novel so called unfitted finite element methods have gained much attention in recent years. The main idea is to embed the domain of interest into a structured but unfitted background mesh where the domain boundary can cut through the mesh in an arbitrary fashion. Unfitted finite-element based methods typically suffer from stability and conditioning problems caused by the presence of small cut elements. In this talk we develop both theoretically and practically a novel cut Discontinuous Galerkin framework (CutDG) by combining stabilization techniques from the cut finite element method with the classical interior penalty discontinuous Galerkin methods for elliptic and hyperbolic problems. To cope with robustness problems caused by small cut elements, we introduce ghost penalties in the vicinity of the embedded boundary to stabilize certain (semi-)norms associated with the relevant partial differential operators. A few abstract assumptions on the ghost penalties are identified enabling us to derive geometrically robust and optimal a priori error and condition number estimates for the both elliptic and stationary advection-reaction problems which hold irrespective of the particular cut configuration. Possible realizations of suitable ghost penalties are discussed. The theoretical results are corroborated by a number of computational studies for various approximation orders and for two-and three-dimensional test problems.
2021-05-26: Sven-Erik Ekström, Uppsala University
Matrix-less methods: approximating eigenvalues and eigenvectors without the matrix
Discretizing partial differential equations (PDE) typically gives rise to structured matrices. Exploiting these, often hidden, structures can be very beneficial when analysing these matrices or when constructing fast solvers.
In this talk we first present an overview of the theory of generalised locally Toeplitz (GLT) sequences; we introduce the notions of matrix sequences and symbols, and how one can approximate eigenvalues and singular values of matrices belonging to these sequences.
Then, we discuss the so-called matrix-less methods (not to be confused with matrix-free methods). These methods are extremely fast and efficient compared with conventional methods for computing eigenvalues.
Illustrative numerical examples will be presented, from model problems and PDE discretizations, both for Hermitian and non-Hermitian matrix sequences.
Finally, we show preliminary results for approximating the eigenvalues of discretization matrices of variable coefficient problems, and the computation of eigenvectors of Hermitian Toeplitz matrices.
2021-05-19: Sara Zahedi, KTH
A cut finite element method for incompressible two-phase flows
I will present a computational technique, a space-time Cut Finite Element Method (CutFEM), that can be used for simulations of two-phase flow problems. With CutFEM we develop a strategy for accurately approximating solutions to Partial Differential Equations (PDEs) in complex geometries that are arbitrarily located with respect to a fixed background mesh. We consider the time-dependent Navier-Stokes equations involving two immiscible incompressible fluids with different viscosities, densities, and with surface tension. Due to surface tension effects at the interface separating the two fluids and different fluid viscosities the pressure may be discontinuous and the velocity field may have a kink across the interface. I will address several challenges that computational methods for such simulations must handle, such as how to accurately capture discontinuities across an evolving interface and how to compute the mean curvature vector needed for the surface tension force with a convenient implementation allowing the interface to be arbitrary located with respect to a fixed mesh.
2021-05-12: Des Higham, The University of Edinburgh
Should We Be Perturbed About Deep Learning?
Many commentators are asking whether current AI solutions are sufficiently robust, resilient, and trustworthy; and how such issues should be quantified and addressed. I believe that numerical analysts can contribute to the debate. In part 1 of this talk I will look at the common practice of using low precision floating point formats to speed up computation time. I will focus on evaluating the softmax and log-sum-exp functions, which play an important role in many classification tools. Here, across widely used packages we see mathematically equivalent but computationally different formulations; these variations have been designed in an effort to avoid overflow and underflow. I will show that classical rounding error analysis gives insights into their floating point accuracy, and suggests a method of choice. In part 2 I will look at a bigger picture question concerning sensitivity to adversarial attacks in deep learning. Adversarial attacks are deliberate, targeted perturbations to input data that have a dramatic effect on the output; for example, a traffic "Stop" sign on the roadside can be misinterpreted as a speed limit sign when minimal graffiti is added. The vulnerability of systems to such interventions raises questions around security, privacy and ethics, and there has been a rapid escalation of attack and defence strategies. I will consider a related higher-level question: under realistic assumptions, do adversarial examples always exist with high probability? I will also introduce and discuss the idea of a stealth attack: an undetectable, targeted perturbation to the trained network itself.
Part 1 is joint work with Pierre Blanchard (ARM) and Nick Higham (Manchester).
Part 2 is joint work with Alexander Gorban and Ivan Tyukin (Leicester).
2021-05-05: Erik Jansson, Chalmers & GU