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Selected Recent Applications of Sparse Grids

Published online by Cambridge University Press:  03 March 2015

Benjamin Peherstorfer ,
Christoph Kowitz ,
Dirk Pflüger  and
Hans-Joachim Bungartz
Benjamin Peherstorfer*
Affiliation:
Scientific Computing Group, Dept. of Computer Science, Boltzmannstr. 3, 85748 Garching, Germany
Christoph Kowitz
Affiliation:
Institute for Advanced Study, Technische Universität München, Lichtenbergstr. 2a, 85748 Garching, Germany
Dirk Pflüger
Affiliation:
Institute for Parallel and Distributed Systems, University of Stuttgart, Universitätsstr. 38, 70569 Stuttgart, Germany
Hans-Joachim Bungartz
Affiliation:
Scientific Computing Group, Dept. of Computer Science, Boltzmannstr. 3, 85748 Garching, Germany
*
*Email address: pehersto@in.tum.de (B. Peherstorfer)
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Abstract

Sparse grids have become a versatile tool for a vast range of applications reaching from interpolation and numerical quadrature to data-driven problems and uncertainty quantification. We review four selected real-world applications of sparse grids: financial product pricing with the Black-Scholes model, interactive exploration of simulation data with sparse-grid-based surrogate models, analysis of simulation data through sparse grid data mining methods, and stability investigations of plasma turbulence simulations.

Keywords

65M6065M2262P0565D0562H3065F15Sparse gridshierarchical decompositionadaptivitycombination technique
Type
Research Article
Information
Numerical Mathematics: Theory, Methods and Applications , Volume 8 , Issue 1 , February 2015 , pp. 47 - 77
Copyright
Copyright © Global-Science Press 2015 

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References

[1] Achatz, S.. Adaptive finite Dünngitter-Elemente höherer Ordnung für elliptische partielle Differentialgleichungen mit variablen Koeffizienten. PhD thesis, Technische Universität München, 2003.Google Scholar
[2] Angioni, C., Peeters, A., Jenko, F., and Dannert, T.. Collisionality dependence of density peaking in quasilinear gyrokinetic calculations. Physics of Plasmas, 12(11), 2005.Google Scholar
[3] Balder, R.. Adaptive Verfahren für elliptische und paraboliche Differentialgleichungen auf dunnen Gittern. PhD thesis, Technische Universitat München, 1994.Google Scholar
[4] Benk, J. and Pflüger, D.. Hybrid parallel solutions of the Black-Scholes PDE with the truncated combination technique. In High Performance Computing and Simulation (HPCS), 2012 International Conference on, pages 678–683, 2012.CrossRefGoogle Scholar
[5] Bishop, C.M.. Pattern Recognition and Machine Learning. Springer, 2007.Google Scholar
[6] Bohn, B., Garcke, J., Iza-Teran, R., Paprotny, A., Peherstorfer, B., Schepsmeier, U., and Thole, C.-A.. Analysis of car crash simulation data with nonlinear machine learning methods. In Proceedings of the International Conference on Computational Science, ICCS 2013, 2013.Google Scholar
[7] Brizard, A. and Hahm, T.. Foundations of nonlinear gyrokinetic theory. Reviews of Modern Physics, 79(2):421–468, 2007.CrossRefGoogle Scholar
[8] Bungartz, H.-J.. Finite Elements of Higher Order on Sparse Grids. Habilitationsschrift, Technische Universität München, 1998.Google Scholar
[9] Bungartz, H.-J. and Griebel, M.. Sparse grids. Acta Numerica, 13:1–123, 2004.CrossRefGoogle Scholar
[10] Bungartz, H.-J., Griebel, M., Röschke, D., and Zenger, C.. Pointwise convergence of the combination technique for Laplace’s equation. East-West J. Numer. Math, 2:21–45, 1994.Google Scholar
[11] Bungartz, H.-J., Griebel, M., Röschke, D., and Zenger, C.. Two proofs of convergence for the combination technique for the efficient solution of sparse grid problems. In Keyes, D.E. and Xu, J., editors, Domain Decomposition Methods in Scientific and Engineering Computing, DDM7, Contemp. Math. 180, pages 15–20. American Mathematical Society,1994.Google Scholar
[12] Bungartz, H.-J., Griebel, M., and Rüde, U.. Extrapolation, combination, and sparse grid techniques for elliptic boundary value problems. Comput. Methods Appl. Mech. Engrg, 116:243–252, 1994.CrossRefGoogle Scholar
[13] Bungartz, H.-J., Heinecke, A., Pflüger, D., and Schraufstetter, S.. Option pricing with a direct adaptive sparse grid approach. Journal of Computational and Applied Mathematics, 236(15):3741–3750, 2011.CrossRefGoogle Scholar
[14] Bungartz, H.-J., Heinecke, A., Pflüger, D., and Schraufstetter, S.. Parallelizing a Black-Scholes solver based on finite elements and sparse grids. Concurrency and Computation: Practice and Experience, 2012.Google Scholar
[15] Butnaru, D.. Computational Steering with Reduced Complexity. PhD thesis, Technische Universität München, 2013.Google Scholar
[16] Butnaru, D., Buse, G., and Pflüger, D.. A parallel and distributed surrogate model implementation for computational steering. In Proceeding ofthe 11th International Symposium on Parallel and Distributed Computing … ISPDC 2012. IEEE, 2012.CrossRefGoogle Scholar
[17] Butnaru, D., Peherstorfer, B., Pflüger, D., and Bungartz, H.-J.. Fast insight into high-dimensional parametrized simulation data. In 11th International Conference on Machine Learning and Applications (ICMLA). IEEE, 2012.Google Scholar
[18] Butnaru, D., Pflüger, D., and Bungartz, H.-J.. Towards high-dimensional computational steering of precomputed simulation data using sparse grids. In Proceedings ofthe International Conference on Computational Science (ICCS) 2011, volume 4 of Procedia CS, pages 56–65. Tsukaba, Japan, Springer, 2011.Google Scholar
[19] Dijkema, T., Schwab, C., and Stevenson, R.. An adaptive wavelet method for solving high-dimensional elliptic PDEs. Constructive Approximation, 30(3):423–455, 2009.CrossRefGoogle Scholar
[20] Ester, M., Kriegel, H.-P., Sander, J., and Xu, X.. A density-based algorithm for discovering clusters in large spatial databases with noise. In 2nd International Conference on Knowledge Discoveryand Data Mining, 1996.Google Scholar
[21] Faber, G.. Über stetige Funktionen. MathematischeAnnalen, 66(1):81–94, 1908.Google Scholar
[22] Fang, Y.. The combination approximation method. PhD thesis, Australian National University, 2012.Google Scholar
[23] Feuersänger, C.. Duünngitterverfahren für hochdimensionale elliptische partielle Differ-entialgleichungen. Diplomarbeit, Institut für Numerische Simulation, Universität Bonn, 2005.Google Scholar
[24] Feuersänger, C.. Sparse Grid Methods for Higher Dimensional Approximation. PhD thesis, Institut für Numerische Simulation, Universität Bonn, 2010.Google Scholar
[25] Garcke, J.. Berechnung von Eigenwerten der stationären Schrödingergleichung mit der Kombinationstechnik. Diplomarbeit, Institut für Angewandte Mathematik, Universität Bonn, 1998.Google Scholar
[26] Garcke, J.. Regression with the optimised combination technique. In Cohen, W. and Moore, A., editors, Proceedings of the 23rd ICML ’06, pages 321–328. ACM Press, 2006.CrossRefGoogle Scholar
[27] Garcke, J.. An optimised sparse grid combination technique for eigenproblems. PAMM, 7(1):1022301–1022302, 2007.CrossRefGoogle Scholar
[28] Garcke, J.. A dimension adaptive sparse grid combination technique for machine learning. In Read, W., Larson, J.W., and Roberts, A.J., editors, Proceedings of the 13th Biennial Computational Techniques and Applications Conference, CTAC–2006, volume 48 of ANZIAM J., pages C725–C740, 2007.Google Scholar
[29] Garcke, J. and Griebel, M.. On the parallelization of the sparse grid approach for data mining. In Margenov, S., Wasniewski, J., and Yalamov, P., editors, Large-Scale Scientific Computations, Third International Conference, LSSC 2001, volume 2179 of Lecture Notes in Computer Science, pages 22–32. Springer, 2001.Google Scholar
[30] Garcke, J., Griebel, M., and Thess, M.. Data mining with sparse grids. Computing, 67(3):225–253, 2001.CrossRefGoogle Scholar
[31] Giles, M.. Multilevel Monte Carlo path simulation. Operations Research, 56:607–617, 2008.CrossRefGoogle Scholar
[32] Glasserman, P.. Monte Carlo methods in financial engineering. Springer, 2004.Google Scholar
[33] Griebel, M. and Holtz, M.. Dimension-wise integration of high-dimensional functions with applications to finance. J. Complexity, 26:455–489, 2010.CrossRefGoogle Scholar
[34] Griebel, M. and Oswald, P.. On additive Schwarz preconditioners for sparse grid discretiz tions. Numerische Mathematik, 66(1):449–463, 1993.CrossRefGoogle Scholar
[35] Griebel, M., Schneider, M., and Zenger, C.. A combination technique for the solution of sparse grid problems. In de Groen, P. and Beauwens, R., editors, Iterative Methods in Linear Algebra, pages 263–281. IMACS, Elsevier, North Holland, 1992.Google Scholar
[36] IIHarrar, D. and Osborne, M.. Computing eigenvalues of ordinary differential equations. ANZIAM Journal, 44(April):C313–C334, 2003.Google Scholar
[37] Hegland, M.. Adaptive sparse grids. In Burrage, K. and Sidje, R.B., editors, Proc. of 10th Computational Techniques and Applications Conference CTAC–2001, volume 44, pages C335–C353, 2003.Google Scholar
[38] Hegland, M., Garcke, J., and Challis, V.. The combination technique and some generalisations. Linear Algebra and its Applications, 420(2–3):249–275, 2007.CrossRefGoogle Scholar
[39] Hegland, M., Hooker, G., and Roberts, S.. Finite element thin plate splines in density estimation. ANZIAM Journal, 42:C712–C734, 2000.Google Scholar
[40] Heinecke, A. and Pflüger, D.. Emerging architectures enable to boost massively parallel data mining using adaptive sparse grids. International Journal ofParallel Programming, 41(3):357–399, 2013.CrossRefGoogle Scholar
[41] Heinecke, A., Schraufstetter, S., and Bungartz, H.-J.. A highly-parallel Black-Scholes solver based on adaptive sparse grids. International Journal ofComputer Mathematics, 89(9):1212–1238, 2012.Google Scholar
[42] Hernandez, V., Roman, J.E., and Vidal, V.. SLEPc: A scalable and flexible toolkit for the solution of eigenvalue problems. ACM Trans. Math. Software, 31(3):351–362, 2005.CrossRefGoogle Scholar
[43] Hinneburg, A. and Gabriel, H.-H.. Denclue 2.0: Fast clustering based on kernel density estimation. In Berthold, M., Shawe-Taylor, J., and Lavrac, N., editors, Advances in Intelligent Data Analysis VII, volume 4723 of Lecture Notes in Computer Science, pages 70–80. Springer, 2007.Google Scholar
[44] Jenko, F., Told, D., Görler, T., Citrin, J., Navarro, A., Bourdelle, C., Brunner, S., Con-way, G., Dannert, T., Doerk, H., Hatch, D., Haverkort, J., Hobirk, J., Hogeweij, G., Mantica, P., Pueschel, M., Sauter, O., Villard, L., Wolfrum, E., and the ASDEX Upgrade Team. Global and local gyrokinetic simulations of high-performance discharges in view of ITER. Nuclear Fusion, 53(7):073003, 2013.CrossRefGoogle Scholar
[45] Klimke, A.. Sparse Grid Interpolation Toolbox – user’s guide. Technical Report IANS report 2007/017, University of Stuttgart, 2007.Google Scholar
[46] Klimke, A. and Wohlmuth, B.. Algorithm 847: spinterp: Piecewise multilinear hierarchical sparse grid interpolation in MATLAB. ACM Transactions on Mathematical Software, 31(4), 2005.CrossRefGoogle Scholar
[47] Kowitz, C. and Hegland, M.. The sparse grid combination technique for computing eigenvalues in linear gyrokinetics. In Alexandrov, V.N., Lees, M., Krzhizhanovskaya, V.V., Don-garra, J., and Sloot, P.M.A., editors, ICCS, volume 18 of Procedia Computer Science, pages 449–458. Elsevier, 2013.CrossRefGoogle Scholar
[48] Kowitz, C. and Hegland, M.. An opticom method for computing eigenpairs. In Garcke, J. and Pflüger, D., editors, Sparse Grids and Applications – Munich 2012, volume 97 of Lecture Notes in Computational Science and Engineering, pages 239–253. Springer, 2014.CrossRefGoogle Scholar
[49] Lederer, H., Tisma, R., and Hatzky, R.. Application enabling in deisa: Petascaling of plasma turbulence codes. Parallel Computing: Architectures, Algorithms and Applications, 2008.Google Scholar
[50] Mei, L. and Thole, C.-A.. Data analysis for parallel car-crash simulation results and model optimization. Sim. Modelling Practice and Theory, 16(3):329–337, 2008.CrossRefGoogle Scholar
[51] Merz, F., Kowitz, C., Romero, E., Roman, J., and Jenko, F.. Multi-dimensional gyrokinetic parameter studies based on eigenvalue computations. Computer Physics Communications, 183(4):922–930, 2012.CrossRefGoogle Scholar
[52] Osborne, M.R.. A new method for the solution of eigenvalue problems. The Computer Journal, 7(3):228–232, 1964.CrossRefGoogle Scholar
[53] Peherstorfer, B.. Model Order Reduction of Parametrized Systems with Sparse Grid Learning Techniques. PhD thesis, Technische Universität München, 2013.Google Scholar
[54] Peherstorfer, B., Pflüge, D., and Bungartz, H.. Density estimation with adaptive sparse grids for large data sets. In Proceedings ofthe 2014 SIAM International Conference on Data Mining, pages 443–451. SIAM, 2014.CrossRefGoogle Scholar
[55] Peherstorfer, B., Pflüger, D., and Bungartz, H.-J.. A sparse-grid-based out-of-sample extension for dimensionality reduction and clustering with Laplacian eigenmaps. In Wang, D. and Reynolds, M., editors, AI 2011: Advances in Artificial Intelligence, volume 7106 of Lecture Notes in Computer Science, pages 112–121. Springer, 2011.Google Scholar
[56] Peherstorfer, B., Pflüger, D., and Bungartz, H.-J.. Clustering based on density estimation with sparse grids. In KI2012: Advances in Artificial Intelligence, volume 7526 of Lecture Notes in Computer Science. Springer, 2012.Google Scholar
[57] Peherstorfer, B., Zimmer, S., and Bungartz, H.-J.. Model reduction with the reduced basis method and sparse grids. In Garcke, J. and Griebel, M., editors, Sparse Grids and Applications, volume 88 of Lecture Notes in Computational Science and Engineering. Springer, 2013.Google Scholar
[58] Petersdorff, T. and Schwab, C.. Sparse finite element methods for operator equations with stochastic data. Applications of Mathematics, 51(2):145–180, 2006.CrossRefGoogle Scholar
[59] Pflaum, C.. Diskretisierung elliptischer Differentialgleichungen mit dünnen Gittern. PhD thesis, Technische Universität München, 1996.Google Scholar
[60] Pflaum, C. and Zhou, A.. Error analysis of the combination technique. Numerische Mathematik, 84(2):327–350, 1999.CrossRefGoogle Scholar
[61] Pflüger, D.. Spatially Adaptive Sparse Grids for High-Dimensional Problems. Hut, Verlag Dr., 2010.Google Scholar
[62] Pflüger, D.. Spatially adaptive refinement. In Garcke, J. and Griebel, M., editors, Sparse Grids and Applications, Lecture Notes in Computational Science and Engineering, pages 243–262. Springer, 2012.CrossRefGoogle Scholar
[63] Pflüger, D., Peherstorfer, B., and Bungartz, H.-J.. Spatially adaptive sparse grids for high-dimensional data-driven problems. Journal of Complexity, 26(5):508–522, 2010.CrossRefGoogle Scholar
[64] Rannacher, R.. Finite element solution of diffusion problems with irregular data. Nu-merische Mathematik, 43:309–327, 1984.CrossRefGoogle Scholar
[65] Reisinger, C. and Wittum, G.. Efficient hierarchical approximation of high-dimensional option pricing problems. SIAM Journal on Scientific Computing, 29(1):440–458, 2007.CrossRefGoogle Scholar
[66] Roman, J.E., Kammerer, M., Merz, F., and Jenko, F.. Fast eigenvalue calculations in a massively parallel plasma turbulence code. Parallel Computing, 36(5–6):339–358, 2010.CrossRefGoogle Scholar
[67] Stork, A., Thole, C.-A., Klimenko, S., Nikitin, I., Nikitina, L., and Astakhov, Y.. Towards interactive simulation in automotive design. The Visual Computer, 24(11):947–953, 2008.CrossRefGoogle Scholar
[68] Wesson, J.. Tokamaks. Oxford University Press, third edition, 2004.Google Scholar
[69] Yserentant, H.. On the multi-level splitting of finite element spaces. Numerische Mathematik, 49(4):379–412, 1986.CrossRefGoogle Scholar
[70] Zeiser, A.. Fast matrix-vector multiplication in the sparse-grid Galerkin method. Journal of Scientific Computing, 47(3):328–346, 2011.CrossRefGoogle Scholar
[71] Zenger, C.. Sparse grids. In Hackbusch, W., editor, Parallel Algorithms for Partial Differential Equations, volume 31 of Notes on Numerical Fluid Mechanics, pages 241–251. Vieweg, 1991.Google Scholar