We have divided this book into five main chapters. Chapter 1 gives the motivation for this book and the use of templates.Chapter 2 describes stationary and nonstationary iterative methods. In this chapter we present both historical development and state-of-the-art methods for solving some of the most challenging computational problems facing researchers.Chapter 3 focuses on preconditioners. Many iterative methods depend in part on preconditioners to improve performance and ensure fast convergence.Chapter 4 provides a glimpse of issues related to the use of iterative methods. This chapter, like the preceding, is especially recommended for the experienced user who wishes to have further guidelines for tailoring a specific code to a particular machine. It includes information on complex systems, stopping criteria, data storage formats, and parallelism.Chapter 5 includes overviews of related topics such as the close connection between the Lanczos algorithm and the Conjugate Gradient algorithm, block iterative methods, red/black orderings, domain decomposition methods, multigrid-like methods, and row-projection schemes.The Appendices contain information on how the templates and BLAS software can be obtained. A glossary of important terms used in the book is also provided.The field of iterative methods for solving systems of linear equations is in constant flux, with new methods and approaches continually being created, modified, tuned, and some eventually discarded. We expect the material in this book to undergo changes from time to time as some of these new approaches mature and become the state-of-the-art. Therefore, we plan to update the material included in this book periodically for future editions. We welcome your comments and criticisms of this work to help us in that updating process. Please send your comments and questions by email to templates@cs.utk.edu. AcknowledgementsThe authors gratefully acknowledge the valuable assistance of many people who commented on preliminary drafts of this book. In particular, we thank Loyce Adams, Bill Coughran, Matthew Fishler, Peter Forsyth, Roland Freund, Gene Golub, Eric Grosse, Mark Jones, David Kincaid, Steve Lee, Tarek Mathew, Noël Nachtigal, Jim Ortega, and David Young for their insightful comments. We also thank Geoffrey Fox for initial discussions on the concept of templates, and Karin Remington for designing the front cover.

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Magnetically confined plasmas can contain significant concentrations of nonthermal plasma particles arising from fusion reactions, neutral beam injection, and wave-driven diffusion in velocity space. Initial studies in one-dimensional and experimental results show that nonthermal energetic ions can significantly affect wave propagation and heating in the ion cyclotron range of frequencies. In addition, these ions can absorb power at high harmonics of the cyclotron frequency where conventional two-dimensional global-wave models are not valid. In this work, the all-orders global-wave solver AORSA [E. F. Jaeger et al., Phys. Rev. Lett. 90, 195001 (2003)] is generalized to treat non-Maxwellian velocity distributions. Quasilinear diffusion coefficients are derived directly from the wave fields and used to calculate energetic ion velocity distributions with the CQL3D Fokker-Planck code [R. W. Harvey and M. G. McCoy, Proceedings of the IAEA Technical Committee Meeting on Simulation and Modeling of Thermonuclear Plasmas, Montreal, Canada, 1992 (USDOC NTIS Document No. DE93002962)]. For comparison, the quasilinear coefficients can be calculated numerically by integrating the Lorentz force equations along particle orbits. Self-consistency between the wave electric field and resonant ion distribution function is achieved by iterating between the global-wave and Fokker-Planck solutions.

The next step toward fusion as a practical energy source is the design and construction of ITER [R. Aymar et al., Nucl. Fusion 41, 1301 (2001)], a device capable of producing and controlling the high-performance plasma required for self-sustaining fusion reactions, i.e., “burning plasma.” ITER relies in part on ion-cyclotron radio frequency power to heat the deuterium and tritium fuel to fusion temperatures. In order to heat effectively, the radio frequency wave fields must couple efficiently to the dense core plasma. Calculations in this paper support the argument that this will be the case. Three-dimensional full-wave simulations show that fast magnetosonic waves in ITER propagate radially inward with strong central focusing and little toroidal spreading. Energy deposition, current drive, and plasma flow are all highly localized near the plasma center. Very high resolution, two-dimensional calculations reveal the presence of mode conversion layers, where fast waves can be converted to slow ion cyclotron waves. When minority ions such as deuterium or helium-3 are used to damp the launched waves, these ions can be accelerated to high energies, forming suprathermal tails that significantly affect the wave propagation and absorption. By neglecting the toroidal localization of the waves and the finite radial excursion of the energetic particle orbits, the quasilinear evolution of these suprathermal ion tails can be simulated self-consistently in one spatial dimension and two velocity dimensions.

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Global wave solutions with self-consistent velocity distributions are calculated for ion cyclotron heating in non-Maxwellian plasmas. The all-orders spectral algorithm (AORSA) global wave solver is generalized to treat non-thermal velocity distributions arising from fusion reactions, neutral beam injection and wave driven diffusion in velocity space. Quasi-linear diffusion coefficients are derived directly from the wave electric field and used to calculate ion velocity distribution functions with the CQL3D Fokker–Planck code. Alternatively, the quasi-linear coefficients can be calculated numerically by integrating the Lorentz force equations along particle orbits. Self-consistency between the wave electric field and resonant ion distribution function is achieved by iterating between the global-wave and Fokker–Planck solutions.

To achieve exascale computing, fundamental hardware architectures must change. The most significant consequence of this assertion is the impact on the scientific applications that run on current high performance computing (HPC) systems, many of which codify years of scientific domain knowledge and refinements for contemporary computer systems. In order to adapt to exascale architectures, developers must be able to reason about new hardware and determine what programming models and algorithms will provide the best blend of performance and energy efficiency into the future. While many details of the exascale architectures are undefined, an abstract machine model is designed to allow application developers to focus on the aspects of the machine that are important or relevant to performance and code structure. These models are intended as communication aids between application developers and hardware architects during the co-design process. We use the term proxy architecture to describe a parameterized version of an abstract machine model, with the parameters added to elucidate potential speeds and capacities of key hardware components. These more detailed architectural models are formulated to enable discussion between the developers of analytic models and simulators and computer hardware architects. They allow for application performance analysis and hardware optimization opportunities. In this report our goal is to provide the application development community with a set of models that can help software developers prepare for exascale. In addition, use of proxy architectures, through the use of proxy architectures, we can enable a more concrete exploration of how well application codes map onto the future architectures.