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Sparse grid
About: Sparse grid is a research topic. Over the lifetime, 1013 publications have been published within this topic receiving 20664 citations.
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TL;DR: In this paper, a unified approach to error estimates of periodic interpolation on equidistant, full, and sparse grids for functions from a scale of function spaces which includes L 2-Sobolev spaces, the Wiener algebra and the Korobov spaces is presented.
Abstract: We present a unified approach to error estimates of periodic interpolation on equidistant, full, and sparse grids for functions from a scale of function spaces which includes L 2-Sobolev spaces, the Wiener algebra and the Korobov spaces.
16 citations
TL;DR: This work presents several approaches to fault tolerance using the combination technique, and describes a novel approach in which the solution is computed on additional grids which are used for alternative combinations if other grids experience failure.
Abstract: One of the challenges for efficiently and effectively using petascale and exascale computers is the handling of run-time errors. Without such robustness, applications developed for these machines will have little chance of completing successfully. The sparse grid combination technique approximates the solution to a given problem by taking the linear combination of its solution on multiple grids. It is successful in many high performance computing applications due to its ability to tackle the curse of dimensionality. We present several approaches to fault tolerance using the combination technique. The first of these is implemented within the MapReduce model in order to utilise the existing fault tolerance of this framework. In addition, we present a method which utilises the redundancy shared by solutions on different grids. Finally, we describe a novel approach in which the solution is computed on additional grids which are used for alternative combinations if other grids experience failure. We include some results based on the solution of the 2D scalar advection PDE.
References S. Balay, J. Brown, K. Buschelman, W. D. Gropp, D. Kaushik, M. G. Knepley, L. C. McInnes, B. F. Smith, and H. Zhang. PETSc Web page (2012). http://www.mcs.anl.gov/petsc . G. Bosilca, R. Delmas, J. Dongarra, and J. Langou. Algorithm-based fault tolerance applied to high performance computing. Journal of Parallel and Distributed Computing , 69(4):410--416 (2009). doi:10.1016/j.jpdc.2008.12.002 . H. J. Bungartz and M. Griebel. Sparse grids. Acta Numerica , 13:147--269 (2004). doi:10.1017/S0962492904000182 . F. Cappello. Fault Tolerance in Petascale/Exascale Systems: Current Knowledge, Challenges and Research Opportunities. International Journal of High Performance Computing Applications , 23(3):212--226, (2009). doi:10.1177/1094342009106189 . J. Dean and S. Ghemawat. MapReduce: Simplified data processing on large clusters. Communications of the ACM , 51(1):107--113 (2008). doi:10.1145/1327452.1327492 . J. Garcke. Sparse grids in a nutshell. In J. Garcke and M. Griebel, editors, Sparse grids and applications , volume 88 of Lecture Notes in Computational Science and Engineering , pages 57--80. Springer (2013). doi:10.1007/978-3-642-31703-3_3 . M. Griebel, M. Schneider, and C. Zenger. A combination technique for the solution of sparse grid problems. In P. de Groen and R. Beauwens, editors, Iterative Methods in Linear Algebra , pages 263--281. IMACS, Elsevier, North Holland (1992). Zbl 0785.65101. M. Hegland. Adaptive sparse grids. ANZIAM Journal , 44:C335--C353 (2003). http://journal.austms.org.au/ojs/index.php/ANZIAMJ/article/view/685 . K.-H. Huang and J. A. Abraham. Algorithm-based fault tolerance for matrix operations. Computers, IEEE Transactions on , C-33(6):518--528 (1984). doi:10.1109/TC.1984.1676475 . C. Zenger. Sparse Grids. In W.Hackbusch, editor, Parrallel Algorithms for Partial Differential Equations, Proceedings of the Sixth GAMM-Seminar, Kiel, 1990, volume 31 of Notes on Num. Fluid Mech. , Vieweg--Verlag, 31:241--251 (1991). Zbl 0763.65091.
16 citations
TL;DR: An uncertainty propagation analysis method is developed based on an extended sparse grid technique and maximum entropy principle, aiming at improving the solving accuracy of the high-order moments and hence the fitting accuracy ofThe probability density function (PDF) of the system response.
Abstract: In this paper, an uncertainty propagation analysis method is developed based on an extended sparse grid technique and maximum entropy principle, aiming at improving the solving accuracy of the high-order moments and hence the fitting accuracy of the probability density function (PDF) of the system response The proposed method incorporates the extended Gauss integration into the uncertainty propagation analysis Moreover, assisted by the Rosenblatt transformation, the various types of extended integration points are transformed into the extended Gauss-Hermite integration points, which makes the method suitable for any type of continuous distribution Subsequently, within the sparse grid numerical integration framework, the statistical moments of the system response are obtained based on the transformed points Furthermore, based on the maximum entropy principle, the obtained first four-order statistical moments are used to fit the PDF of the system response Finally, three numerical examples are investigated to demonstrate the effectiveness of the proposed method, which includes two mathematical problems with explicit expressions and an engineering application with a black-box model
15 citations
01 Jan 2013
TL;DR: Evaluating the performance of different combination schemes suggests, that the sparse grid combination technique is a feasible method to compute eigenvalues of the gyrokinetic eigenvalue problem.
Abstract: Using the five-dimensional gyrokinetic equations for simulations of hot fusion plasmas requires discretizations with a lot degrees of freedom due to the curse of dimensionality. The sparse grid combination technique could be one remedy to this problem, since it is dividing the original problem into smaller subproblems, which can be solved independently. For this study, the gyrokinetic code GENE has been used for solving the gyrokinetic eigenvalue problem using the combination technique. The performance of different combination schemes is used on a test problem and evaluated with respect to accuracy in retrieving an eigenvalue and the computational effort required. This evaluation suggests, that the sparse grid combination technique is a feasible method to compute eigenvalues of the gyrokinetic eigenvalue problem.
15 citations
01 Jan 1996
TL;DR: The parallel numerical solution of the Navier-Stokes equations with the sparse grid combination method with parallel implementation on an IBM SP2 computer and its application to turbulence simulation is reported.
Abstract: The parallel numerical solution of the Navier-Stokes equations with the sparse grid combination method was studied. This algorithmic concept is based on the independent solution of many problems with reduced size and their linear combination. The algorithm for three-dimensional problems is described and its application to turbulence simulation is reported. Statistical results on a pipe flow for Reynolds number Re cl = 6950 are presented and compared with results obtained from other numerical simulations and physical experiments. Its parallel implementation on an IBM SP2 computer is also discussed.
15 citations