A mass-conserving sparse grid combination technique with biorthogonal hierarchical basis functions for kinetic simulations

The exact numerical simulation of plasma turbulence is one of the assets and challenges in fusion research. For grid-based solvers, sufficiently fine resolutions are often unattainable due to the curse of dimensionality. The sparse grid combination technique provides the means to alleviate the curse of dimensionality for kinetic simulations. However, the hierarchical representation for the combination step with the state-of-the-art hat functions suffers from poor conservation properties and numerical instability. The present work introduces two new variants of hierarchical multiscale basis functions for use with the combination technique: the biorthogonal and full weighting bases. The new basis functions conserve the total mass and are shown to significantly increase accuracy for a finite-volume solution of constant advection. Further numerical experiments based on the combination technique applied to a semi-Lagrangian Vlasov--Poisson solver show a stabilizing effect of the new bases on the simulations

Fault-tolerant grid-based solvers: Combining concepts from sparse grids and MapReduce

A key issue confronting petascale and exascale computing is the growth in probability of soft and hard faults with increasing system size. A promising approach to this problem is the use of algorithms that are inherently fault tolerant. We introduce such an algorithm for the solution of partial differential equations, based on the sparse grid approach. Here, the solution of multiple component grids are efficiently combined to achieve a solution on a full grid. The technique also lends itself to a (modified) MapReduce framework on a cluster of processors, with the map stage corresponding to allocating each component grid for solution over a subset of the processors, and the reduce stage corresponding to their combination. We describe how the sparse grid combination method can be modified to robustly solve partial differential equations in the presence of faults. This is based on a modified combination formula that can accommodate the loss of one or two component grids. We also discuss accuracy issues associated with this formula. We give details of a prototype implementation within a MapReduce framework using the dynamic process features and asynchronous message passing facilities of MPI. Results on a two-dimensional advection problem show that the errors after the loss of one or two sub-grids are within a factor of 3 of the sparse grid solution in the presence of no faults. They also indicate that the sparse grid technique with four times the resolution has approximately the same error as a full grid, while requiring (for a sufficiently high resolution) much lower computation and memory requirements. We finally outline a MapReduce variant capable of responding to faults in ways other than re-scheduling of failed tasks. We discuss the likely software requirements for such a flexible MapReduce framework, the requirements it will impose on users’ legacy codes, and the system's runtime behavior.J. W. Larson, M. Hegland, B. Harding, S. Roberts, L. Stals, A. P. Rendell, P. Strazdins, M. M. Ali, C. Kowitz, R. Nobes, J. Southern, N. Wilson, M. Li, Y. Oish

Ion-scale turbulence in MAST: anomalous transport, subcritical transitions, and comparison to BES measurements

We investigate the effect of varying the ion temperature gradient (ITG) and toroidal equilibrium scale sheared flow on ion-scale turbulence in the outer core of MAST by means of local gyrokinetic simulations. We show that nonlinear simulations reproduce the experimental ion heat flux and that the experimentally measured values of the ITG and the flow shear lie close to the turbulence threshold. We demonstrate that the system is subcritical in the presence of flow shear, i.e., the system is formally stable to small perturbations, but transitions to a turbulent state given a large enough initial perturbation. We propose that the transition to subcritical turbulence occurs via an intermediate state dominated by low number of coherent long-lived structures, close to threshold, which increase in number as the system is taken away from the threshold into the more strongly turbulent regime, until they fill the domain and a more conventional turbulence emerges. We show that the properties of turbulence are effectively functions of the distance to threshold, as quantified by the ion heat flux. We make quantitative comparisons of correlation lengths, times, and amplitudes between our simulations and experimental measurements using the MAST BES diagnostic. We find reasonable agreement of the correlation properties, most notably of the correlation time, for which significant discrepancies were found in previous numerical studies of MAST turbulence.Comment: 67 pages, 37 figures. Submitted to PPC

A massively parallel combination technique for the solution of high-dimensional PDEs

The solution of high-dimensional problems, especially high-dimensional partial differential equations (PDEs) that require the joint discretization of more than the usual three spatial dimensions and time, is one of the grand challenges in high performance computing (HPC). Due to the exponential growth of the number of unknowns - the so-called curse of dimensionality, it is in many cases not feasible to resolve the simulation domain as fine as required by the physical problem. Although the upcoming generation of exascale HPC systems theoretically provides the computational power to handle simulations that are out of reach today, it is expected that this is only achievable with new numerical algorithms that are able to efficiently exploit the massive parallelism of these systems. The sparse grid combination technique is a numerical scheme where the problem (e.g., a high-dimensional PDE) is solved on different coarse and anisotropic computational grids (so-called component grids), which are then combined to approximate the solution with a much higher target resolution than any of the individual component grids. This way, the total number of unknowns being computed is drastically reduced compared to the case when the problem is directly solved on a regular grid with the target resolution. Thus, the curse of dimensionality is mitigated. The combination technique is a promising approach to solve high-dimensional problems on future exascale systems. It offers two levels of parallelism: the component grids can be computed in parallel, independently and asynchronously of each other; and the computation of each component grid can be parallelized as well. This reduces the demand for global communication and synchronization, which is expected to be one of the limiting factors for classical discretization techniques to achieve scalability on exascale systems. Furthermore, the combination technique enables novel approaches to deal with the increasing fault rates expected from these systems. With the fault-tolerant combination technique it is possible to recover from failures without time-consuming checkpoint-restart mechanisms. In this work, new algorithms and data structures are presented that enable a massively parallel and fault-tolerant combination technique for time-dependent PDEs on large-scale HPC systems. The scalability of these algorithms is demonstrated on up to 180225 processor cores on the supercomputer Hazel Hen. Furthermore, the parallel combination technique is applied to gyrokinetic simulations in GENE, a software for the simulation of plasma microturbulence in fusion devices

Gyrokinetic simulation of multimode plasma turbulence

Durch Mikroturbulenz verursachter Wärme- und Teilchentransport in magnetisch eingeschlossenen Hochtemperaturplasmen ist eines der drängendsten Probleme der Fusionsforschung.In dieser Arbeit werden die gyrokinetischen Gleichungen, die magnetisierte Plasmen bei fusionsrelevanten Parametern beschreiben, für allgemeine magnetische Geometrien unter Berücksichtigung von Stößen präsentiert, weiterhin werden Aspekte der numerischen Implementierung in den massiv parallelen Plasmaturbulenz-Code GENE diskutiert. Nichthermitescher Entartungen in lineare Modenübergängen werden mit Hilfe eines Eigenwertlösers untersucht. Statistische Untersuchungen der ExB-Nichtlinearität im Fall reiner Trapped Electron Mode (TEM) Turbulenz zeigen, dass diese durch einen Diffusionsterm approximiert werden kann, was ein schon bekanntes quasilineares Transportmodell stützt. Es werden Übergänge zwischen TEM- und Ion Temperature Gradient (ITG) Turbulenz untersucht, eine Koexistenz führt zu interessanten Effekten beim Teilchentransport. Abschließend werden verschiedene Aspekte der ITG-Turbulenz im Stellarator W7-X mit adiabatischen und kinetischen Elektronen diskutiert

Confinement physics for a steady state net electric burning spherical tokamak

Spherical tokamaks have many desirable properties that make them a suitable candidate for a compact fusion reactor. Such a device could accelerate the timeline of fusion and reduce capital costs, allowing fusion to have a more significant impact on the world. The feasibility of a compact spherical tokamak able to generate net electricity needs to be examined as well as the modelling tools currently available. Extrapolating to reactor relevant conditions requires a great deal of trust in these models. This work begins by identifying steady state plasma equilibria and applying empirical limits to characterise the available parameter space for a given machine design and scale. This is done with a consistent calculation of the neoclassical currents, allowing for the auxiliary current drive requirements to be determined. A baseline scenario was identified with a major radius of 2.5m and fusion power of 1.1GW. An important result found is that a minimum current drive efficiency is required given the empirical limits used. Neutral beam injection was found to have a sufficient current drive efficiency, with 94MW of power needed to drive all the required current. The validity of reduced physics neutral beam models was also examined and it was found that reasonable predictions were made provided the beams were aligned with the magnetic field. The performance of a tokamak is generally limited by the turbulent transport so the linear gyrokinetic stability of a baseline ST reactor plasma scenario was investigated. The baseline equilibrium showed some desirable properties as the electron scale turbulence was found to be stable. In the ion scale, kinetic ballooning modes and micro-tearing modes were found to co-exist on multiple flux surfaces. Through exploring the drives of these modes it was possible to optimise the equilibrium to minimise their growth rates. Moreover, the credibility of quasi-linear transport models was explored with a new tool developed that is better able to capture the instabilities in this regime, though further development is still needed

Software for Exascale Computing - SPPEXA 2016-2019

This open access book summarizes the research done and results obtained in the second funding phase of the Priority Program 1648 "Software for Exascale Computing" (SPPEXA) of the German Research Foundation (DFG) presented at the SPPEXA Symposium in Dresden during October 21-23, 2019. In that respect, it both represents a continuation of Vol. 113 in Springer’s series Lecture Notes in Computational Science and Engineering, the corresponding report of SPPEXA’s first funding phase, and provides an overview of SPPEXA’s contributions towards exascale computing in today's sumpercomputer technology. The individual chapters address one or more of the research directions (1) computational algorithms, (2) system software, (3) application software, (4) data management and exploration, (5) programming, and (6) software tools. The book has an interdisciplinary appeal: scholars from computational sub-fields in computer science, mathematics, physics, or engineering will find it of particular interest

Kontextsensitive Modellhierarchien für Quantifizierung der höherdimensionalen Unsicherheit

We formulate four novel context-aware algorithms based on model hierarchies aimed to enable an efficient quantification of uncertainty in complex, computationally expensive problems, such as fluid-structure interaction and plasma microinstability simulations. Our results show that our algorithms are more efficient than standard approaches and that they are able to cope with the challenges of quantifying uncertainty in higher-dimensional, complex problems.Wir formulieren vier kontextsensitive Algorithmen auf der Grundlage von Modellhierarchien um eine effiziente Quantifizierung der Unsicherheit bei komplexen, rechenintensiven Problemen zu ermöglichen, wie Fluid-Struktur-Wechselwirkungs- und Plasma-Mikroinstabilitätssimulationen. Unsere Ergebnisse zeigen, dass unsere Algorithmen effizienter als Standardansätze sind und die Herausforderungen der Quantifizierung der Unsicherheit in höherdimensionalen, komplexen Problemen bewältigen können