Communication-aware adaptive parareal with application to a nonlinear hyperbolic system of partial dierential equations

In the strong scaling limit, the performance of conventional spatial domain decomposition techniques for the parallel solution of PDEs saturates. When sub-domains become small, halo-communication and other overheard come to dominate. A potential path beyond this scaling limit is to introduce domain-decomposition in time, with one such popular approach being the Parareal algorithm which has received a lot of attention due to its generality and potential scalability. Low efficiency, particularly on convection dominated problems, has however limited the adoption of the method. In this paper we introduce a new strategy, Communication Aware Adaptive Parareal (CAAP) to overcome some of the challenges. With CAAP, we choose time-subdomains short enough that convergence of the Parareal algorithm is quick, yet long enough that the overheard of communicating time-subdomain interfaces does not induce a new limit to parallel speed-up. Furthermore, we propose an adaptive work scheduling algorithm that overlaps consecutive Parareal cycles and decouples the number of time-subdomains and number of active node-groups in an efficient manner to allow for comparatively high parallel eciency. We demonstrate the viability of CAAP trough the parallel-in-time integration of a hyperbolic system of PDEs in the form of the two-dimensional nonlinear shallow-water wave equation solved using a 3rd order accurate WENO-RK discretization. For the computational cheap approximate operator needed as a preconditioner in the Parareal corrections we use a lower order Roe type discretization. Time-parallel integration of purely hyperbolic type evolution problems is traditionally considered impractical. Trough large-scale numerical experiments we demonstrate that with CAAP, it is possible not only to obtain time-parallel speedup on this class of evolution problems, but also that we may obtain parallel acceleration beyond what is possible using conventional spatial domain-decomposition techniques alone. The approach is widely applicable for parallel-in-time integration over long time domains, regardless of the class of evolution problem

Scaling and Resilience in Numerical Algorithms for Exascale Computing

The first Petascale supercomputer, the IBM Roadrunner, went online in 2008. Ten years later, the community is now looking ahead to a new generation of Exascale machines. During the decade that has passed, several hundred Petascale capable machines have been installed worldwide, yet despite the abundance of machines, applications that scale to their full size remain rare. Large clusters now routinely have 50.000+ cores, some have several million. This extreme level of parallelism, that has allowed a theoretical compute capacity in excess of a million billion operations per second, turns out to be difficult to use in many applications of practical interest. Processors often end up spending more time waiting for synchronization, communication, and other coordinating operations to complete, rather than actually computing. Component reliability is another challenge facing HPC developers. If even a single processor fail, among many thousands, the user is forced to restart traditional applications, wasting valuable compute time. These issues collectively manifest themselves as low parallel efficiency, resulting in waste of energy and computational resources. Future performance improvements are expected to continue to come in large part due to increased parallelism. One may therefore speculate that the difficulties currently faced, when scaling applications to Petascale machines, will progressively worsen, making it difficult for scientists to harness the full potential of Exascale computing. The thesis comprises two parts. Each part consists of several chapters discussing modifications of numerical algorithms to make them better suited for future Exascale machines. In the first part, the use of Parareal for Parallel-in-Time integration techniques for scalable numerical solution of partial differential equations is considered. We propose a new adaptive scheduler that optimize the parallel efficiency by minimizing the time-subdomain length without making communication of time-subdomains too costly. In conjunction with an appropriate preconditioner, we demonstrate that it is possible to obtain time-parallel speedup on the nonlinear shallow water equation, beyond what is possible using conventional spatial domain-decomposition techniques alone. The part is concluded with the proposal of a new method for constructing Parallel-in-Time integration schemes better suited for convection dominated problems. In the second part, new ways of mitigating the impact of hardware failures are developed and presented. The topic is introduced with the creation of a new fault-tolerant variant of Parareal. In the chapter that follows, a C++ Library for multi-level checkpointing is presented. The library uses lightweight in-memory checkpoints, protected trough the use of erasure codes, to mitigate the impact of failures by decreasing the overhead of checkpointing and minimizing the compute work lost. Erasure codes have the unfortunate property that if more data blocks are lost than parity codes created, the data is effectively considered unrecoverable. The final chapter contains a preliminary study on partial information recovery for incomplete checksums. Under the assumption that some meta knowledge exists on the structure of the data encoded, we show that the data lost may be recovered, at least partially. This result is of interest not only in HPC but also in data centers where erasure codes are widely used to protect data efficiently

An Adaptive Parareal Algorithm

In this paper, we consider the problem of accelerating the numerical simulation of time dependent problems by time domain decomposition. The available algorithms enabling such decompositions present severe efficiency limitations and are an obstacle for the solution of large scale and high dimensional problems. Our main contribution is the improvement of the parallel efficiency of the parareal in time method. The parareal method is based on combining predictions made by a numerically inexpensive solver (with coarse physics and/or coarse resolution) with corrections coming from an expensive solver (with high-fidelity physics and high resolution). At convergence, the parareal algorithm provides a solution that has the fine solver's high-fidelity physics and high resolution In the classical version of parareal, the fine solver has a fixed high accuracy which is the major obstacle to achieve a competitive parallel efficiency. In this paper, we develop an adaptive variant of the algorithm that overcomes this obstacle. Thanks to this, the only remaining factor impacting performance becomes the cost of the coarse solver. We show both theoretically and in a numerical example that the parallel efficiency becomes very competitive when the cost of the coarse solver is small

Parallel-in-Time Simulation of an Electrical Machine using MGRIT

We apply the multigrid-reduction-in-time (MGRIT) algorithm to an eddy current simulation of a two-dimensional induction machine supplied by a pulse-width-modulation signal. To resolve the fast-switching excitations, small time steps are needed, such that parallelization in time becomes highly relevant for reducing the simulation time. The MGRIT algorithm is well suited for introducing time parallelism in the simulation of electrical machines using existing application codes, as MGRIT is a non-intrusive approach that essentially uses the same time integrator as a traditional time-stepping algorithm. We investigate effects of spatial coarsening on MGRIT convergence when applied to two numerical models of an induction machine, one with linear material laws and a full nonlinear model. Parallel results demonstrate significant speedup in the simulation time compared to sequential time stepping, even for moderate numbers of processors.Comment: 14 page

Multigrid reduction-in-time convergence for advection problems: A Fourier analysis perspective

A long-standing issue in the parallel-in-time community is the poor convergence of standard iterative parallel-in-time methods for hyperbolic partial differential equations (PDEs), and for advection-dominated PDEs more broadly. Here, a local Fourier analysis (LFA) convergence theory is derived for the two-level variant of the iterative parallel-in-time method of multigrid reduction-in-time (MGRIT). This closed-form theory allows for new insights into the poor convergence of MGRIT for advection-dominated PDEs when using the standard approach of rediscretizing the fine-grid problem on the coarse grid. Specifically, we show that this poor convergence arises, at least in part, from inadequate coarse-grid correction of certain smooth Fourier modes known as characteristic components, which was previously identified as causing poor convergence of classical spatial multigrid on steady-state advection-dominated PDEs. We apply this convergence theory to show that, for certain semi-Lagrangian discretizations of advection problems, MGRIT convergence using rediscretized coarse-grid operators cannot be robust with respect to CFL number or coarsening factor. A consequence of this analysis is that techniques developed for improving convergence in the spatial multigrid context can be re-purposed in the MGRIT context to develop more robust parallel-in-time solvers. This strategy has been used in recent work to great effect; here, we provide further theoretical evidence supporting the effectiveness of this approach

An Adaptive Parareal Algorithm

International audienceIn this paper, we consider the problem of accelerating the numerical simulation of time dependent problems by time domain decomposition. The available algorithms enabling such decompositions present severe efficiency limitations and are an obstacle for the solution of large scale and high dimensional problems. Our main contribution is the improvement of the parallel efficiency of the parareal in time method. The parareal method is based on combining predictions made by a numerically inexpensive solver (with coarse physics and/or coarse resolution) with corrections coming from an expensive solver (with high-fidelity physics and high resolution). At convergence, the parareal algorithm provides a solution that has the fine solver's high-fidelity physics and high resolution In the classical version of parareal, the fine solver has a fixed high accuracy which is the major obstacle to achieve a competitive parallel efficiency. In this paper, we develop an adaptive variant of the algorithm that overcomes this obstacle. Thanks to this, the only remaining factor impacting performance becomes the cost of the coarse solver. We show both theoretically and in a numerical example that the parallel efficiency becomes very competitive when the cost of the coarse solver is small

Space-Time Isogeometric Analysis of Parabolic Evolution Equations

We present and analyze a new stable space-time Isogeometric Analysis (IgA) method for the numerical solution of parabolic evolution equations in fixed and moving spatial computational domains. The discrete bilinear form is elliptic on the IgA space with respect to a discrete energy norm. This property together with a corresponding boundedness property, consistency and approximation results for the IgA spaces yields an a priori discretization error estimate with respect to the discrete norm. The theoretical results are confirmed by several numerical experiments with low- and high-order IgA spaces