Research seminar numerics
The numerics research seminar is aimed at students and researchers with an interest in current issues in numerics, scientific computing and optimization. Lectures on current topics are held at irregular intervals.
Summer term 2024
Wednesday, 03.07.2024 (11:00 a.m. in NC 02/99): Cu Cui (Heidelberg University)
Title: Efficient and High-Performance Finite Element Methods on GPUs
Abstract: Multigrid and domain decomposition methods are two classes of solvers or preprocessors that solve discretized elliptic partial differential equations with linear complexity in the number of unknowns. In this talk, we describe powerful smoothers based on domain decompositions with vertex patches as subdomains, which can be implemented very efficiently on the GPU by fast diagonalization. Applications to the Poisson and Stokes equations are demonstrated, highlighting the efficiency of the vertex-patch smoother.
Building on performance gains achieved from traditional CUDA Cores, we then focus on optimizing Nvidia Tensor Core kernels and analyzing the performance of tensor product operations in finite element methods. Tensor Cores, specifically for AI applications, are mixed-precision matrix multiplication and addition computational units that operate at speeds up to 312 TFlop/s on Nvidia A100 GPUs. We discuss the challenges of conducting numerical simulations using matrix-free method on this hardware and present optimization strategies through numerical examples. With half-precision computation, the solution to the Poisson problem is four times faster without sacrificing accuracy.
Thursday, 20.06.2024 (13:00 p.m. in IA 01/481): Pablo Herrera (PUC Chile, Faculty of Mathematics)
Title: DPG Methods for the quad-curl problem
Abstract: The quad-curl problem appears in several areas of physics and engineering when we study models governed by PDEs where a $\curl^4$ appears. Some examples are MHD with hyper-resistivity phenomena, inverse electromagnetic scattering theory, linear elasticity, or fluid dynamics [1]. The DPG method is a conforming method developed by L. Demkowicz and J. Gopalakrishnan provide inherent discrete inf-sup stability and a built-in local error estimator [2].
In this talk, we focus on the discontinuous Petrov-Galerkin (DPG) method with optimal test functions for the quad-curl problem in the ultraweak form (field variables being $L_2$ functions). We discuss regularity issues with the ultraweak formulation of the quad-curl problem and its discretization, and we prove the stability as a second-order PDE.
References:
[1] T. Führer, P. Herrera and N. Heuer. A DPG method for the quad-curl problem. Comput. Math. Appl., 148:221-238, 2023.
[2] L. F. Demkowicz and J. Gopalakrishnan. A class of discontinuous Petrov–Galerkin methods. Part ii. optimal test functions. Numer. Methods Partial Differential Equations., 27 (1):70–105, 2011.
Friday, 24.05.2024 (10:00 a.m. in IA 1/ 53): Timo Sprekeler (National University of Singapore)
Title: Homogenization of Hamilton-Jacobi-type equations: optimal rates and numerical effective Hamiltonians
Abstract: First, we discuss the optimal rate of convergence in periodic homogenization of viscous Hamilton-Jacobi equations, and the numerical approximation of the effective Hamiltonian. The numerical scheme is based on a finite element approximation of approximate corrector problems for Hamilton-Jacobi-Bellman (HJB) Hamiltonians. Thereafter, we study the approximation of the effective Hamiltonian corresponding to second-order HJB and Isaacs Hamiltonians in a framework surrounding a Cordes-type condition.
This is based on joint works with E. Kawecki (Mahindra Racing), J. Qian (Michigan State), H.V. Tran (Wisconsin), and Y. Yu (UC Irvine).
Thursday, 11.04.2024 (14:15 p.m. in IA 1/75): Jacques Xing (King's College London)
Title: Implementation of the Parareal Algorithm in the Nektar++ Spectral/hp Element Framework with Applications to Linear and Non-Linear Problems
Abstract: Time-parallel integration techniques are proposed as a potential solution to further increase concurrency and speed-up beyond the limits of strong scaling obtained from a pure spatial domain decomposition. Parareal is a non-intrusive, iterative-based approach, exploiting a fine and coarse solvers to achieve time-parallelism and can be applied to both linear and non-linear problems. The efficient implementation of the Parareal algorithm in the Nektar++ open-source framework is described in this work. Applications to multiple linear and non-linear problems are shown.
Winter term 2023/24
Wednesday, 13.12.2023 (13:00 p.m. in IA 02/105): Tuan Ahn Dao (Uppsala University)
Title: Viscous regularization and structure-preserving schemes for ideal MHD
Abstract: This talk discusses viscosity methods for the ideal MHD equations. For the choice of the viscous flux, I will talk about the monolithic parabolic flux and motivate why it is a good choice. To be more precise, at the PDE level, the monolithic parabolic regularization of the equations of ideal magnetohydrodynamics (MHD) is compatible with all the generalized entropies, fulfills the minimum entropy principle, and preserves the positivity of density and internal energy. The artificial viscosity is constructed using an entropy viscosity method. Several numerical examples will be demonstrated.
The second part of the talk is about our new structure-preserving method to approximate the compressible ideal Magnetohydrodynamics (MHD) equations. This technique addresses the MHD equations using a non-divergence formulation, where the contributions of the magnetic field to the momentum and total mechanical energy are treated as source terms. Our approach uses the Marchuk-Strang splitting technique and involves three distinct components: a compressible Euler solver, a source-system solver, and an update procedure for the total mechanical energy. The scheme allows for significant freedom on the choice of Euler’s equation solver, while the magnetic field is discretized using a curl-conforming finite element space, yielding exact preservation of the involution constraints. The method provably preserves invariant domain properties, including positivity of density, positivity of internal energy, and the minimum principle of the specific entropy. In our approach, the CFL condition does not depend on magnetosonic wavespeeds, but only on the usual maximum wavespeed from Euler’s system.
Summer term 2023
Wednesday, 20.09.2023 (11:00 a.m.): Tileuzhan Mukhamet (Master student at RUB)
Title: An arbitrary Lagrangian Eulerian Discontinuous Galerkin method for the Vlasov equation with a strong magnetic field
Friday, 11.08.2023 (11:00 a.m., IA 1/53): Mahima Yadav (PhD student at RUB)
Title: On discrete ground states of rotating Bose-Einstein condensates
Abstract: The talk focuses on the study of ground states of Bose-Einstein condensates in a rotating frame. The ground states are described as the constrained minimizers of the Gross-Pitaevskii energy functional with an angular momentum term. The problem is discretized using Lagrange finite element spaces of arbitrary polynomial order and the approximation properties of the corresponding numerical approximations are presented, taking into account the missing uniqueness of ground states which is mainly caused by the invariance of the energy functional under complex phase shifts. Error estimates of optimal order are shown for the L2- and H1-norm, as well as for the ground state energy and chemical potential.
Thursday, 11.05.2023 (11:15 a.m.): Paul Wilhelm (RWTH Aachen)
Title: NuFI: An implicit Lagrangian Flow reconstruction scheme for the Vlasov equation
Abstract: The Vlasov equation is a high-dimensional partial differential equation arising from kinetic theory and used to model the behaviour of plasma flows in collision-less and strongly non-equilibrium regimes. We present a novel approach, the numerical flow iteration (NuFI), which evaluates the numerical solution via storing the three-dimensional electric potentials, using these to iteratively reconstruct the Lagrangian flow and directly evaluating the initial data via the method of characteristics. This reduces the total memory-requirement by several orders of magnitude and allows to shift workload from frequent memory access to computations on the fly, i.e., yielding high flop/Byte-rates, which is favourable on modern compute-architectures. Furthermore using the Lagrangian formulation one conserves desired properties like $L^p$-norms and kinetic entropy exactly, as well as total energy up to time-integration error. We demonstrate the accuracy and scalability of the new approach on several test-cases in up to six dimensions showing computations done on a workstation as well as a GPU-cluster.
Thursday, 27.04.2023 (11:00 a.m.): Ivo Dravins (RUB)
Title: Preconditioning for block matrices with square blocks
Abstract: Linear systems of equations appear in one way or another in almost every scientific and engineering problem. They are so ubiquitous that, in addition to solving linear problems, also non-linear problems are typically reduced to a sequence of linear ones. The availability of modern large-scale computational resources motivates the development and the use of well parallelizable efficient solvers with a limited memory footprint. For many problems, these properties can be achieved by the employment of iterative solution methods combined with preconditioning techniques. We explore the design of preconditioners for block-matrices with square blocks. This form of matrices occurs in many applications, encountered for instance when numerically solving partial differential equations, ordinary differential equations and others.
We focus on two classes of problems, one being optimal control problems within the PDE-constrained optimization framework, and the other being fully implicit Runge-Kutta time-stepping schemes. Both necessitate the solution of large and sparse linear systems, for which we employ preconditioned Krylov subspace methods. The main topic of the talk is on the design of preconditioners, although the entire solution procedure is explored.
