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Journal of Scientific Computing

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10-09-2018

On the Algebraic Construction of Sparse Multilevel Approximations of Elliptic Tensor Product Problems

Authors: Helmut Harbrecht, Peter Zaspel

Published in: Journal of Scientific Computing | Issue 2/2019

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Abstract

We consider the solution of elliptic problems on the tensor product of two physical domains as for example present in the approximation of the solution covariance of elliptic partial differential equations with random input. Previous sparse approximation approaches used a geometrically constructed multilevel hierarchy. Instead, we construct this hierarchy for a given discretized problem by means of the algebraic multigrid method. Thereby, we are able to apply the sparse grid combination technique to problems given on complex geometries and for discretizations arising from unstructured grids, which was not feasible before. Numerical results show that our algebraic construction exhibits the same convergence behaviour as the geometric construction, while being applicable even in black-box type PDE solvers.

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Balder, R., Zenger, C.: The solution of multidimensional real Helmholtz equations on sparse grids. SIAM J. Sci. Comput. 17(3), 631–646 (1996)MathSciNetCrossRefMATH

Bramble, J., Pasciak, J., Xu, J.: Parallel multilevel preconditioners. Math. Comput. 55, 1–22 (1990)MathSciNetCrossRefMATH

Bungartz, H.J.: A multigrid algorithm for higher order finite elements on sparse grids. ETNA. Electron. Trans. Numer. Anal. 6, 63–77 (1997)MathSciNetMATH

Bungartz, H.J., Griebel, M.: Sparse grids. Acta Numer. 13, 1–123 (2004)MathSciNetCrossRefMATH

Dahmen, W.: Wavelet and multiscale methods for operator equations. Acta Numer. 6, 55–228 (1997)MathSciNetCrossRefMATH

Falgout, R.D., Yang, U.M.: hypre: A library of high performance preconditioners. In: Sloot, P.M.A., Hoekstra, A.G., Tan, C.J.K., Dongarra, J.J. (eds.) Computational Science – ICCS 2002, pp. 632–641. Springer, Berlin, Heidelberg (2002)CrossRef

Griebel, M.: Multilevelmethoden als Iterationsverfahren über Erzeugendensystemen. Teubner Skripten zur Numerik. B.G. Teubner, Stuttgart (1993)

Griebel, M.: Multilevel algorithms considered as iterative methods on semidefinite systems. SIAM Int. J. Sci. Stat. Comput. 15(3), 547–565 (1994)MathSciNetCrossRefMATH

Griebel, M., Harbrecht, H.: On the convergence of the combination technique. In: Garcke, J., Pflüger, D. (eds.) Sparse Grids and Applications - Stuttgart 2014. Lecture Notes in Computational Science and Engineering, vol. 97, pp. 55–74. Springer, Berlin (2014)

10.

Griebel, M., Oswald, P.: Greedy and randomized versions of the multiplicative Schwarz method. Linear Algebr. Appl. 7, 1596–1610 (2012)MathSciNetCrossRefMATH

11.

Griebel, M., Schneider, M., Zenger, C.: A combination technique for the solution of sparse grid problems. In: de Groen, P., Beauwens, R. (eds.) Iterative Methods in Linear Algebra, pp. 263–281. IMACS, Elsevier, North Holland (1992)

12.

Harbrecht, H.: A finite element method for elliptic problems with stochastic input data. Appl. Numer. Math. 60(3), 227–244 (2010)MathSciNetCrossRefMATH

13.

Harbrecht, H., Peters, M., Schneider, R.: On the low-rank approximation by the pivoted Cholesky decomposition. Appl. Numer. Math. 62(4), 428–440 (2012)

14.

Harbrecht, H., Peters, M., Siebenmorgen, M.: Combination technique based \(k\)-th moment analysis of elliptic problems with random diffusion. J. Comput. Phys. 252(C), 128–141 (2013)MathSciNetCrossRefMATH

15.

Harbrecht, H., Schneider, R., Schwab, C.: Multilevel frames for sparse tensor product spaces. Numer. Math. 110(2), 199–220 (2008)MathSciNetCrossRefMATH

16.

Hegland, M., Garcke, J., Challis, V.: The combination technique and some generalisations. Linear Algebr. Appl. 420(2), 249–275 (2007)

17.

Oswald, P.: Multilevel finite element approximation. Theory and applications. Teubner Skripten zur Numerik. B.G. Teubner, Stuttgart (1994)

18.

Ruge, J., Stüben, K.: Algebraic multigrid (AMG). In: McCormick, S. (ed.) Multigrid Methods, Frontiers in Applied Mathematics, vol. 5. SIAM, Philadelphia (1986)

19.

Schwab, C., Todor, R.A.: Sparse finite elements for elliptic problems with stochastic loading. Numer. Math. 95(4), 707–734 (2003)MathSciNetCrossRefMATH

20.

Schwab, C., Todor, R.A.: Sparse finite elements for stochastic elliptic problems: higher order moments. Computing 71(1), 43–63 (2003)MathSciNetCrossRefMATH

21.

Stüben, K.: A review of algebraic multigrid. J. Comput. Appl. Math. 128(1–2), 281–309 (2001). Numerical Analysis 2000. Vol. VII: Partial Differential EquationsMathSciNetCrossRefMATH

22.

Trottenberg, U., Schuller, A.: Multigrid. Academic Press Inc, Orlando (2001)MATH

23.

Yang, U.M.: On long-range interpolation operators for aggressive coarsening. Numer. Linear Algebr. Appl. 17(2–3), 453–472 (2010)MathSciNetMATH

24.

Zaspel, P.: Subspace correction methods in algebraic multi-level frames. Linear Algebr. Appl. 488, 505–521 (2016)

25.

Zeiser, A.: Fast matrix-vector multiplication in the sparse-grid Galerkin method. SIAM J. Sci. Comput. 47(3), 328–346 (2011)MathSciNetCrossRefMATH

Title: On the Algebraic Construction of Sparse Multilevel Approximations of Elliptic Tensor Product Problems
Authors: Helmut Harbrecht
Peter Zaspel
Publication date: 10-09-2018
Publisher: Springer US
Published in: Journal of Scientific Computing / Issue 2/2019
Print ISSN: 0885-7474
Electronic ISSN: 1573-7691
DOI: https://doi.org/10.1007/s10915-018-0807-6

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