Parallelization in time of numerical simulations of fully-developed plasma turbulence using the parareal algorithm

16 pages, 12 figures.It is shown that numerical simulations of fully-developed plasma turbulence can be successfully parallelized in time using the parareal algorithm. The result is far from trivial, and even unexpected, since the exponential divergence of Lagrangian trajectories as well as the extreme sensitivity to initial conditions characteristic of turbulence set these type of simulations apart from the much simpler systems to which the parareal algorithm has been applied to this day. It is also shown that the parallel gain obtainable with this method is very promising (close to an order of magnitude for the cases and implementations described), even when it scales with the number of processors quite differently to what is typical for spatial parallelization.Part of the research was carried out at the University of Alaska Fairbanks, funded by the DOE Office of Science Grant No. DE-FG02-04ER54741. Research was also carried out in part at Oak Ridge National Laboratory, managed by UT-Battelle LLC, for US DOE under Contract No. DE-AC05–00OR22725, and funded via the Seed Money Initiative Program.Publicad

PARAOPT: A parareal algorithm for optimality systems

The time parallel solution of optimality systems arising in PDE constraint optimization could be achieved by simply applying any time parallel algorithm, such as Parareal, to solve the forward and backward evolution problems arising in the optimization loop. We propose here a different strategy by devising directly a new time parallel algorithm, which we call ParaOpt, for the coupled forward and backward non-linear partial differential equations. ParaOpt is inspired by the Parareal algorithm for evolution equations, and thus is automatically a two-level method. We provide a detailed convergence analysis for the case of linear parabolic PDE constraints. We illustrate the performance of ParaOpt with numerical experiments both for linear and nonlinear optimality systems

A stencil-based implementation of Parareal in the C++ domain specific embedded language STELLA

In view of the rapid rise of the number of cores in modern supercomputers, time-parallel methods that introduce concurrency along the temporal axis are becoming increasingly popular. For the solution of time-dependent partial differential equations, these methods can add another direction for concurrency on top of spatial parallelization. The paper presents an implementation of the time-parallel Parareal method in a C++ domain specific language for stencil computations (STELLA). STELLA provides both an OpenMP and a CUDA backend for a shared memory parallelization, using the CPU or GPU inside a node for the spatial stencils. Here, we intertwine this node-wise spatial parallelism with the time-parallel Parareal. This is done by adding an MPI-based implementation of Parareal, which allows us to parallelize in time across nodes. The performance of Parareal with both backends is analyzed in terms of speedup, parallel efficiency and energy-to-solution for an advection-diffusion problem with a time-dependent diffusion coefficient

A hybrid parareal spectral deferred corrections method

The parareal algorithm introduced in 2001 by Lions, Maday, and Turinici is an iterative method for the parallelization of the numerical solution of ordinary differential equations or partial differential equations discretized in the temporal direction. The temporal interval of interest is partitioned into successive domains which are assigned to separate processor units. Each iteration of the parareal algorithm consists of a high accuracy solution procedure performed in parallel on each domain using approximate initial conditions and a serial step which propagates a correction to the initial conditions through the entire time interval. The original method is designed to use classical single-step numerical methods for both of these steps. This paper investigates a variant of the parareal algorithm first outlined by Minion and Williams in 2008 that utilizes a deferred correction strategy within the parareal iterations. Here, the connections between parareal, parallel deferred corrections, and a hybrid parareal-spectral deferred correction method are further explored. The parallel speedup and efficiency of the hybrid methods are analyzed, and numerical results for ODEs and discretized PDEs are presented to demonstrate the performance of the hybrid approach

In-Time Parallelization Of Atmospheric Chemical Kinetics

This work investigates the potential of an in-time parallelization of atmospheric chemical ki- netics. Its numerical calculation is one time-consuming step within the numerical prediction of the air quality. The widely used parallelization strategies only allow a limited potential level of parallelism. A higher level of parallelism within the codes will be necessary to enable benefits from future exa-scale computing architectures. In air quality prediction codes, chem- ical kinetics is typically considered to react in isolated boxes over short splitting intervals. This allows their trivial parallelization in space, which however is limited by the number of grid entities. This work pursues a parallelization beyond this trivial potential and investigates a parallelization across time using the so called “parareal algorithm”. The latter is an iterative prediction-correction scheme, whose efficiency strongly depends on the choice of the predictor. For that purpose, different options are being investigate and compared: Time-stepping schemes with fixed step size, adaptive time-stepping schemes and repro-models, functional representations, that map a given state to a later state in time. Only the choice of repromodels leads to a speed-up through parallelism, compared to the sequential reference for the scenarios considered here

Adaptive Multiple Shooting for Boundary Value Problems and Constrained Parabolic Optimization Problems

Subject of this thesis is the development of adaptive techniques for multiple shooting methods. The focus is on the application to optimal control problems governed by parabolic partial differential equations. In order to retain as much freedom as possible in the later choice of discretization schemes, the details of both direct and indirect multiple shooting variants are worked out on an abstract function space level. Therefore, shooting techniques do not constitute a way of discretizing a problem. A thorough examination of the connections between the approaches provides an overview of different shooting formulations and enables their comparison for both linear and nonlinear problems. We extend current research by considering additional constraints on the control variable in the multiple shooting context. An optimization problem is developed which includes so-called box constraints in the multiple shooting context. Several modern algorithms treating control constraints are adapted to the requirements of shooting methods. The modified algorithms permit an extended comparison of the different shooting approaches. The efficiency of numerical methods can often be increased by developing grid adaptation techniques. While adaptive discretization schemes can be readily transferred to the multiple shooting context, questions of conditioning and stability make it difficult to develop adaptive features for shooting point distribution in multiple shooting processes. We concentrate on the design and comparison of two different approaches to shooting grid adaptation in the framework of ordinary differential equations. A residual-based adaptive algorithm is transferred to parabolic optimization problems with control constraints. The presented concepts and methods are verified by means of several examples, whereby theoretical results are numerically confirmed. We choose the test problems so that the simple shooting method becomes unstable and therefore a genuine multiple shooting technique is required