Quantics-TT Collocation Approximation of Parameter-Dependent and Stochastic Elliptic PDES

Computational Methods in Applied Mathematics

Volumn 10, Issue 4, 2010, Pages 376-394

  Khoromskij, B N ^a   Oseledets, I ^b  

^a MAX PLANCK INSTITUTE FOR MATHEMATICS IN THE SCIENCES   (Germany)

^b INSTITUTE OF NUMERICAL MATHEMATICS   (Russian Federation)

Author keywords

elliptic equations; preconditioners; quantics TT tensors; separable approximation; stochastic PDEs; tensor truncated iteration; the Karhunen Lo ve expansion

Indexed keywords

EID: 80053917258     PISSN: 16094840     EISSN: 16099389     Source Type: Journal    
DOI: 10.2478/cmam-2010-0023     Document Type: Article

References (29)

1. I. Babuska, F. Nobile, and R. Tempone, A Stochastic Collocation Method for Elliptic Partial Differential Equations with Random Input Data, SIAM Journal on Numerical Analysis, 45 (2007), pp. 1005-1034.
2. I. Babuska, R. Tempone, and G. E. Zouraris, Galerkin Finite Element Approximations of Stochastic Elliptic Partial Differential Equations, SIAM Journal on Numerical Analysis, 42 (2004), pp. 800-825.
3. B. W. Bader and T. G. Kolda, Efficient MATLAB Computations with Sparse and Factored Tensors, SIAM Journal on Scientific Computing, 30 (2008), p. 205.
4. G. Beylkin and M. J. Mohlenkamp, Algorithms for Numerical Analysis in High Dimensions, SIAM Journal on Scientific Computing, 26 (2005), p. 21-33.
5. A. Cohen, R. DeVore, and C. Schwab, Convergence rates of best N-term Galerkin approximations for a class of elliptic sPDEs, 2009.
6. V. de Silva and L.-H. Lim, Tensor Rank and the Ill-Posedness of the Best Low-Rank Approximation Problem, SIAM Journal on Matrix Analysis and Applications, 30 (2008), p. 1084.
7. I. Gavrilyuk, W. Hackbusch, and B. Khoromskij, Tensor-product approximation to elliptic and parabolic solution operators in higher dimensions, Computing, 74 (2005), pp. 131-157.
8. S. A. Goreinov, I. V. Oseledets, D. V. Savostyanov, E. E. Tyrtyshnikov, and N. L. Zamarashkin, How to find a good submatrix, in Matrix methods: theory, algorithms, applications, V. Olshevsky and E. Tyrtyshnikov, eds., NY, 2010, World Scientific Publishers, pp. 247-256.
9. L. Grasedyck, Polynomial Approximation in Hierarchical Tucker Format by Vector-Tensorization, 2010.
10. H. Harbrecht, R. Schneider, and C. Schwab, Sparse second moment analysis for elliptic problems in stochastic domains, Numerische Mathematik, 109 (2008), pp. 385-414.
11. W. Huang, M. R. Stan, K. Skadron, K. Sankaranarayanan, S. Ghosh, and S. Velusam, Compact thermal modeling for temperature-aware design, Annual ACM IEEE Design Automation Conference, (2004).
12. B. N. Khoromskij, O (d logN)-Quantics Approximation of Nd Tensors in High-Dimensional Numerical Modeling, 2009.
13. B. N. Khoromskij, Tensors-structured Numerical Methods in Scientific Computing: Survey on Recent Advances, December 2010.
14. B. N. Khoromskij, V. Khoromskaia, and H.-J. Flad, Numerical solution of the Hartree-Fock equation in multilevel tensor-structured format, SISC 2010, accepted.
15. B. N. Khoromskij and I. V. Oseledets, Quantics-TT approximation of elliptic solution operators in higher dimensions, J. Numer. Math., 2010, accepted.
16. B. N. Khoromskij and C. Schwab, Tensor-structured Galerkin approximation of parametric and stochastic elliptic PDEs, SISC 2010, accepted.
17. T. G. Kolda and B. W. Bader, Tensor Decompositions and Applications, SIAM Review, 51 (2009), p. 455.
18. D. Kressner and C. Tobler, Iterative solution of parameter-dependent linear Systems., 2010.
19. C. Lubich, From quantum to classical molecular dynamics: reduced models and numerical analysis, EMS, Zurich, 2008.
20. H. Matthies and A. Keese, Galerkin methods for linear and nonlinear elliptic stochastic partial differential equations, Computer Methods in Applied Mechanics and Engineering, 194 (2005), pp. 1295-1331.
21. H.-D. Meyer, F. Gatti, and G. A. Worth, Multidimensional Quantum Dynamics: MCTDH Theory and Applications, Wiley-VCH, Weinheim, 2009.
22. I. V. Oseledets, Compact matrix form of the d-dimensional tensor decomposition, 2009.
23. I. V. Oseledets, Tensors inside of matrices give logarithmic complexity, 2009.
24. I. V. Oseledets and E. E. Tyrtyshnikov, Breaking the Curse of Dimensionality, Or How to Use SVD in Many Dimensions, SIAM Journal on Scientific Computing, 31 (2009), p. 37-44.
25. E. B. Rudnyi and J. G. Korvink, Model Order Reduction of MEMS for Efficient Computer Aided Design and System Simulation, 2008.
26. C. Schwab and R. A. Todor, Sparse finite elements for elliptic problems with stochastic loading, Numerische Mathematik, 95 (2003), pp. 707-734.
27. R. A. Todor and C. Schwab, Convergence rates for sparse chaos approximations of elliptic problems with stochastic coefficients, IMA Journal of Numerical Analysis, 27 (2006), pp. 232-261.
28. G. Vidal, Efficient Classical Simulation of Slightly Entangled Quantum Computations, Physical Review Letters, 91 (2003).
29. H. Yserentant, Sparse grid spaces for the numerical solution of the electronic Schrödinger equation, Numerische Mathematik, 101 (2005), pp. 381-389.