Schnelle Löser für Partielle Differentialgleichungen

This workshop was well attended by 52 participants with broad geographic representation from 11 countries and 3 continents. It was a nice blend of researchers with various backgrounds

Toward fault-tolerant parallel-in-time integration with PFASST

We introduce and analyze different strategies for the parallel-in-time integration method PFASST to recover from hard faults and subsequent data loss. Since PFASST stores solutions at multiple time steps on different processors, information from adjacent steps can be used to recover after a processor has failed. PFASST's multi-level hierarchy allows to use the coarse level for correcting the reconstructed solution, which can help to minimize overhead. A theoretical model is devised linking overhead to the number of additional PFASST iterations required for convergence after a fault. The potential efficiency of different strategies is assessed in terms of required additional iterations for examples of diffusive and advective type

Parallel-in-time integration of astro- and geo- physical flows; application of Parareal to kinematic dynamos and Rayleigh-Bénard convection

The precise mechanisms responsible for the natural dynamos in the Earth and Sun are still not fully understood. Numerical simulations of natural dynamos are extremely computationally intensive, and are carried out in parameter regimes many orders of magnitude away from real conditions. Parallelization in space is a common strategy to speed up simulations on high performance computers, but eventually hits a scaling limit. Additional directions of parallelization are desirable to utilise the high number of processor cores now available. Parallel-in-time methods can deliver speed up in addition to that offered by spatial partitioning but have not yet been applied to dynamo simulations. This thesis investigates the feasibility of using Parallel-in-time integration to speed up numerical simulations of dynamos. We concentrate on applying the non-intrusive Parareal algorithm to two sub-problems of natural dynamos: kinematic dynamos and Rayleigh-Bénard convection (RBC). We perform real-world scaling tests on high performance computing (HPC) facilities using the open source Dedalus spectral solver. The kinematic dynamo problem prescribes a fluid flow and observes how the magnetic field changes over time. We investigate the time independent Roberts and time dependent Galloway-Proctor 2.5D dynamos over a range of magnetic Reynolds numbers. Speed ups beyond those possible from spatial parallelisation are found in both cases. Results for the Galloway-Proctor flow are promising, with Parareal efficiency found to be close to 0.3, while Roberts flow results are less efficient, with efficiencies < 0.1. Parallel in space and time speed ups of 300 were found for 1600 cores for the Galloway-Proctor flow, with total parallel efficiency of 0.16. Convective motions are thought to be the source of dynamo action in the Earth and Sun. RBC is the archetypal problem for convection studies, and is also a fundamental problem of fluid dynamics, with many applications to geophysical, astrophysical, and industrial flows. We investigate Parareal for Rayleigh numbers Ra = 10⁵, 10⁶ and 10⁷, finding limited speed up in all cases for up to ~20 processors, whilst performance and convergence of Parareal degrades as Ra increases. We summarise our results for the kinematic dynamos + RBC, and discuss their relevance and implications on Parallel-in-time simulations for the full dynamo problem

Novel Integration in Time Methods via Deferred Correction Formulations and Space-Time Parallelization

A major avenue of research in numerical analysis is creating algorithms in order to decrease the amount of computational time in numerical simulations while maintaining high accuracy. Notably when modeling PDE systems, much effort has been focused in creating methods that undergo the spatial calculations very quickly and accurately. Even with these results, simulations may still take too long, limiting the robustness of a numerical model. Hence, a new research direction is to create methods that decrease runtime by focusing on the temporal direction. The subject of this dissertation is the development of algorithms that decrease runtime by taking acount of temporal properties, and when possible coupling both temporal spatial properties, of time-dependent differential equations.Doctor of Philosoph

Space-time balancing domain decomposition

No separate or additional fees are collected for access to or distribution of the work.In this work, we propose two-level space-time domain decomposition preconditioners for parabolic problems discretized using finite elements. They are motivated as an extension to space-time of balancing domain decomposition by constraints preconditioners. The key ingredients to be defined are the subassembled space and operator, the coarse degrees of freedom (DOFs) in which we want to enforce continuity among subdomains at the preconditioner level, and the transfer operator from the subassembled to the original finite element space. With regard to the subassembled operator, a perturbation of the time derivative is needed to end up with a well-posed preconditioner. The set of coarse DOFs includes the time average (at the space-time subdomain) of classical space constraints plus new constraints between consecutive subdomains in time. Numerical experiments show that the proposed schemes are weakly scalable in time, i.e., we can efficiently exploit increasing computational resources to solve more time steps in the same total elapsed time. Further, the scheme is also weakly space-time scalable, since it leads to asymptotically constant iterations when solving larger problems both in space and time. Excellent wall clock time weak scalability is achieved for space-time parallel solvers on some thousands of coresPeer ReviewedPostprint (published version

A numerical study of a semi-Lagrangian Parareal method applied to the viscous Burgers equation

This is the author accepted manuscript. The final version is available from Springer via the DOI in this record.This work focuses on the Parareal parallelin-time method and its application to the viscous Burgers equation. A crucial component of Parareal is the coarse time stepping scheme, which strongly impacts the convergence of the parallel-in-time method. Three choices of coarse time stepping schemes are investigated in this work: explicit Runge-Kutta, implicit-explicit Runge-Kutta, and implicit Runge-Kutta with semiLagrangian advection. Manufactured solutions are used to conduct studies, which provide insight into the viability of each considered time stepping method for the coarse time step of Parareal. One of our main findings is the advantageous convergence behavior of the semi-Lagrangian scheme for advective flows.Schmitt: The work of this author is supported by the ’Excellence Initiative’ of the German Federal and State Governments and the Graduate School of Computational Engineering at Technische Universit¨at Darmstadt Peixoto: Acknowledges the Sao Paulo Research Foundation (FAPESP) under the grant number 2016/18445-7 and the National Science and Technology Development Council (CNPq) under grant number 441328/2014-

Resolução da equação da onda utilizando métodos multigrid espaço-tempo

Orientador: Prof. Dr. Marcio Augusto Villela PintoCoorientador: Prof. Dr. Sebastião Romero FrancoTese (doutorado) - Universidade Federal do Paraná, Setor de Ciências Exatas, Programa de Pós-Graduação em Métodos Numéricos em Engenharia. Defesa : Curitiba, 10/04/2023Inclui referências: p. 109-117Resumo: Neste trabalho apresenta-se a avaliação de diferentes formas de solução para problemas modelados pela equação da onda, para os casos 1D e 2D. Utiliza-se para discretização espacial, o Método das Diferenças Finitas ponderado por um parâmetro n em diferentes estágios de tempo, para obter-se um esquema de solução implícito. Com isso, propõe-se a utilização de diferentes varreduras no tempo, a fim de gerar métodos robustos e eficientes, desde a clássica Time-Stepping, até outra varredura que envolve simultaneamente o espaço e o tempo, como Waveform Relaxation. Neste trabalho, combina-se o método dos Subdomínios com a estratégia Waveform Relaxation para reduzir as fortes oscilações que ocorrem o início do processo iterativo. Obtém-se excelentes resultados ao aplicar o método Multigrid para esta classe de problemas, já que, melhora-se muito os fatores de convergência calculados a partir das soluções aproximadas do sistema de equações resultante das discretizações. Na verificação das metodologias propostas e suas características, apresentamse simulações de propagação de ondas envolvendo problemas uni e bidimensionais, onde analisa-se os erros de discretização, ordens efetiva e aparente, fator de convergência, ordens de complexidade e tempo computacional.Abstract: In this thesis presents the evaluation of different forms of solution for probelms modeled by the wave equation, for the 1D and 2D cases. The Finite Difference Method is used for the spatial discretization, weighted by a parameter n at different time steps, in order to obtain an implicit solution. With this, it is proposed the use of different sweeps in time, in order to generate robust and efficient methods, from the classical Time-Stepping, to other less usual sweep as Waveform Relaxation. In this work, the Subdomains method is combined with the Waveform Relaxation strategy to reduce the strong oscillations that occur early in the iterative process. Excellent results are obtained when applying the Multigrid method for this class of problems, since the convergence factors calculated from the approximate solutions of the system of equations resulting from the discretizations are greatly improved. In the verification of the proposed methodologies and their respective advantages, simulations of wave propagation involving one- and two-dimensional problems are presented, where the discretization errors, effective and apparent orders, convergence factor, complexity orders and computational time are analyzed

Scalable domain decomposition methods for finite element approximations of transient and electromagnetic problems

The main object of study of this thesis is the development of scalable and robust solvers based on domain decomposition (DD) methods for the linear systems arising from the finite element (FE) discretization of transient and electromagnetic problems. The thesis commences with a theoretical review of the curl-conforming edge (or Nédélec) FEs of the first kind and a comprehensive description of a general implementation strategy for h- and p- adaptive elements of arbitrary order on tetrahedral and hexahedral non-conforming meshes. Then, a novel balancing domain decomposition by constraints (BDDC) preconditioner that is robust for multi-material and/or heterogeneous problems posed in curl-conforming spaces is presented. The new method, in contrast to existent approaches, is based on the definition of the ingredients of the preconditioner according to the physical coefficients of the problem and does not require spectral information. The result is a robust and highly scalable preconditioner that preserves the simplicity of the original BDDC method. When dealing with transient problems, the time direction offers itself an opportunity for further parallelization. Aiming to design scalable space-time solvers, first, parallel-in-time parallel methods for linear and non-linear ordinary differential equations (ODEs) are proposed, based on (non-linear) Schur complement efficient solvers of a multilevel partition of the time interval. Then, these ideas are combined with DD concepts in order to design a two-level preconditioner as an extension to space-time of the BDDC method. The key ingredients for these new methods are defined such that they preserve the time causality, i.e., information only travels from the past to the future. The proposed schemes are weakly scalable in time and space-time, i.e., one can efficiently exploit increasing computational resources to solve more time steps in (approximately) the same time-to-solution. All the developments presented herein are motivated by the driving application of the thesis, the 3D simulation of the low-frequency electromagnetic response of High Temperature Superconductors (HTS). Throughout the document, an exhaustive set of numerical experiments, which includes the simulation of a realistic 3D HTS problem, is performed in order to validate the suitability and assess the parallel performance of the High Performance Computing (HPC) implementation of the proposed algorithms.L’objecte principal d’estudi d’aquesta tesi és el desenvolupament de solucionadors escalables i robustos basats en mètodes de descomposició de dominis (DD) per a sistemes lineals que sorgeixen en la discretització mitjançant elements finits (FE) de problemes transitoris i electromagnètics. La tesi comença amb una revisió teòrica dels FE d’eix (o de Nédélec) de la primera família i una descripció exhaustiva d’una estratègia d’implementació general per a elements h- i p-adaptatius d’ordre arbitrari en malles de tetraedres i hexaedres noconformes. Llavors, es presenta un nou precondicionador de descomposició de dominis balancejats per restricció (BDDC) que és robust per a problemes amb múltiples materials i/o heterogenis definits en espais curl-conformes. El nou mètode, en contrast amb els enfocaments existents, està basat en la definició dels ingredients del precondicionador segons els coeficients físics del problema i no requereix informació espectral. El resultat és un precondicionador robust i escalable que preserva la simplicitat del mètode original BDDC. Quan tractem amb problemes transitoris, la direcció temporal ofereix ella mateixa l’oportunitat de seguir explotant paral·lelisme. Amb l’objectiu de dissenyar precondicionadors en espai-temps, primer, proposem solucionadors paral·lels en temps per equacions diferencials lineals i no-lineals, basats en un solucionador eficient del complement de Schur d’una partició multinivell de l’interval de temps. Seguidament, aquestes idees es combinen amb conceptes de DD amb l’objectiu de dissenyar precondicionadors com a extensió a espai-temps dels mètodes de BDDC. Els ingredients clau d’aquests nous mètodes es defineixen de tal manera que preserven la causalitat del temps, on la informació només viatja de temps passats a temps futurs. Els esquemes proposats són dèbilment escalables en temps i en espai-temps, és a dir, es poden explotar eficientment recursos computacionals creixents per resoldre més passos de temps en (aproximadament) el mateix temps transcorregut de càlcul. Tots els desenvolupaments presentats aquí són motivats pel problema d’aplicació de la tesi, la simulació de la resposta electromagnètica de baixa freqüència dels superconductors d’alta temperatura (HTS) en 3D. Al llarg del document, es realitza un conjunt exhaustiu d’experiments numèrics, els quals inclouen la simulació d’un problema de HTS realista en 3D, per validar la idoneïtat i el rendiment paral·lel de la implementació per a computació d’alt rendiment dels algorismes proposatsPostprint (published version

On improving the efficiency of ADER methods

The (modern) arbitrary derivative (ADER) approach is a popular technique for the numerical solution of differential problems based on iteratively solving an implicit discretization of their weak formulation. In this work, focusing on an ODE context, we investigate several strategies to improve this approach. Our initial emphasis is on the order of accuracy of the method in connection with the polynomial discretization of the weak formulation. We demonstrate that precise choices lead to higher-order convergences in comparison to the existing literature. Then, we put ADER methods into a Deferred Correction (DeC) formalism. This allows to determine the optimal number of iterations, which is equal to the formal order of accuracy of the method, and to introduce efficient $p$-adaptive modifications. These are defined by matching the order of accuracy achieved and the degree of the polynomial reconstruction at each iteration. We provide analytical and numerical results, including the stability analysis of the new modified methods, the investigation of the computational efficiency, an application to adaptivity and an application to hyperbolic PDEs with a Spectral Difference (SD) space discretization