The Combination Technique for the Initial Value Problem in Linear Gyrokinetics

We propose the use of sparse grids to accelerate particlein-cell (PIC) schemes. By using the so-called 'combination technique' from the sparse grids literature, we are able to dramatically increase the size of the spatial cells in multi-dimensional PIC schemes while paying only a slight penalty in grid-based error. The resulting increase in cell size allows us to reduce the statistical noise in the simulation without increasing total particle number. We present initial proof-of-principle results from test cases in two and three dimensions that demonstrate the new scheme's efficiency, both in terms of computation time and memory usage. arXiv:1607.06516v1 [physics.comp-ph]

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“…If we again assume that particle steps dominate the complexity of the PIC scheme, then we find that the complexity scales as (29) κ…”

Section: Merging Pic With Sparse Gridsmentioning

confidence: 97%

Sparse grid techniques for particle-in-cell schemes

Ricketson

Cerfon

2016

Plasma Phys. Control. Fusion

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“…Sparse grids have previously been introduced to plasma physics in the context of solving an eigenvalue problem in linear gyrokinetics with the sparse grid combination technique [13]. For this application the sparse grid method is particularly suited since field-aligned coordinates are used and nonlinear effects are not considered.…”

Section: Introductionmentioning

confidence: 99%

Sparse Grids for the Vlasov–Poisson Equation

Kormann

Sonnendrücker

2016

Lecture Notes in Computational Science and Engineering

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The Vlasov-Poisson equation models the evolution of a plasma in an external or self-consistent electric field. The model consists of an advection equation in six dimensional phase space coupled to Poisson's equation. Due to the high dimensionality and the development of small structures the numerical solution is quite challenging. For two or four dimensional Vlasov problems, semi-Lagrangian solvers have been successfully applied. Introducing a sparse grid, the number of grid points can be reduced in higher dimensions. In this paper, we present a semi-Lagrangian Vlasov-Poisson solver on a tensor product of two sparse grids. In order to defeat the problem of poor representation of Gaussians on the sparse grid, we introduce a multiplicative delta-f method and separate a Gaussian part that is then handled analytically. In the semi-Lagrangian setting, we have to evaluate the hierarchical surplus on each mesh point. This interpolation step is quite expensive on a sparse grid due to the global nature of the basis functions. In our method, we use an operator splitting so that the advection steps boil down to a number of one dimensional interpolation problems. With this structure in mind we devise an evaluation algorithm with constant instead of logarithmic complexity per grid point. Results are shown for standard test cases and in four dimensional phase space the results are compared to a full-grid solution and a solution on the four dimensional sparse grid.

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“…High scalability to 10K cores has been reported [12]. Due to the relative smoothness of the solution, the SGCT has yielded good results in producing a relatively accurate solution on important problem sets [13].…”

Section: The Sparse Grid Combination Techniquementioning

confidence: 95%

“…The first application of the SGCT to GENE was reported in [13]. Under this scheme, component grid instances of GENE were run, with their respective outputs written to data files that were subsequently combined to compute the SGCT solution.…”

Section: Related Workmentioning

confidence: 99%

A fault-tolerant gyrokinetic plasma application using the sparse grid combination technique

Ali

Strazdins

Harding

et al. 2015

2015 International Conference on High Performance Computing &Amp; Simulation (HPCS)

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Applications performing ultra-large scale simulations via solving PDEs require very large computational systems for their timely solution. Studies have shown the rate of failure grows with the system size and these trends are likely to worsen in future machines as less reliable components are used to reduce the energy cost. Thus, as systems, and the problems solved on them, continue to grow, the ability to survive failures is becoming a critical aspect of algorithm development. The sparse grid combination technique (SGCT) is a cost-effective method for solving time-evolving PDEs, especially for higherdimensional problems. It can also be easily modified to provide algorithm-based fault tolerance for these problems. In this paper, we show how the SGCT can produce a fault-tolerant version of the GENE gyrokinetic plasma application, which evolves a 5D complex density field over time. We use an alternate component grid combination formula to recover data from lost processes. User Level Failure Mitigation (ULFM) MPI is used to recover the processes, and our implementation is robust over multiple failures and recovery for both process and node failures. An acceptable degree of modification of the application is required. Results using the SGCT on two of the fields' dimensions show competitive execution times with acceptable error (within 0.1%), compared to the same simulation with a single full resolution grid. The benefits improve when the SGCT is used over three dimensions. Our experiments show that the GENE application can successfully recover from multiple process failures, and applying the SGCT the corresponding number of times minimizes the error for the lost sub-grids. Application recovery overhead via ULFM MPI increases from ∼1.5s at 64 cores to ∼5s at 2048 cores for a oneoff failure. This compares favourably to using GENE's in-built checkpointing with job restart in conjunction with the classical SGCT on failure, which have overheads four times as large for a single failure, excluding the backtrack overhead. An analysis for a long-running application taking into account checkpoint backtrack times indicates a reduction in overhead of over an order of magnitude.

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The Combination Technique for the Initial Value Problem in Linear Gyrokinetics

Cited by 17 publications

References 15 publications

Sparse grid techniques for particle-in-cell schemes

Sparse grid techniques for particle-in-cell schemes

Sparse Grids for the Vlasov–Poisson Equation

A fault-tolerant gyrokinetic plasma application using the sparse grid combination technique

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