Parallel tensor sampling in the hierarchical Tucker format
Tóm tắt
We consider the problem of uncertainty quantification for extreme scale parameter dependent problems where an underlying low rank property of the parameter dependency is assumed. For this type of dependency the hierarchical Tucker format offers a suitable framework to approximate a given output function of the solutions of the parameter dependent problem from a number of samples that is linear in the number of parameters. In particular we can a posteriori compute the mean, variance or other interesting statistical quantities of interest. In the extreme scale setting it is already assumed that the underlying fixed-parameter problem is distributed and solved for in parallel. We provide in addition a parallel evaluation scheme for the sampling phase that allows us on the one hand to combine several solves and on the other hand parallelise the sampling.
Tài liệu tham khảo
Ballani, J., Grasedyck, L., Kluge, M.: Black box approximation of tensors in hierarchical Tucker format. Linear Algeb. Appl. 438, 639–657 (2013)
Ballani, J., Grasedyck, L.: Hierarchical tensor approximation of output quantities of parameter-dependent PDEs, is accepted for SIAM J. UQ, preprint 385, RWTH Aachen. www.igpm.rwth-aachen.de/forschung/preprints/385 (2014)
Bebendorf, M.: Approximation of boundary element matrices. Numer. Math. 86, 565–589 (2000)
Caflisch, R.: Monte Carlo and quasi-Monte Carlo methods. Acta Numer. 7, 1–49 (1998)
Giles, M.: Multilevel Monte Carlo methods. Springer, Berlin (2013)
Goreinov SA, Tyrtyshnikov EE (2001) The maximal-volume concept in approximation of low-rank matrices, Contemp. Math. 280
Goreinov, S.A., Tyrtyshnikov, E.E., Zamarashkin, N.L.: A theory of pseudoskeleton approximations. Linear Algeb. Appl. 261, 1–22 (1997)
Goreinov, S., Zamarashkin, N., Tyrtyshnikov, E.: Pseudo-skeleton approximations by matrices of maximal volume. Math. Notes 62, 515–519 (1997)
Grasedyck, L.: Hierarchical singular value decomposition of tensors. SIAM J. Matrix Anal. Appl. 31, 2029–2054 (2010)
Hackbusch, W.: Tensor Spaces and Numerical Tensor Calculus. Springer, Berlin (2012)
Hackbusch, W., Kühn, S.: A new scheme for the tensor representation. J. Fourier Anal. Appl. 15, 706–722 (2009)
Heisig, M., Lieckfeldt, R., Wittum, G., Mazurkevich, G., Lee, G.: Non steady-state descriptions of drug permeation through stratum corneum. I. The biphasic brick-and-mortar model. Pharm. Res. 13, 421–426 (1996)
Nägel, A., Hansen, S., Neumann, D., Lehr, C.-M., Schaefer, U.F., Wittum, G., Heisig, M.: In-silico model of skin penetration based on experimentally determined input parameters. Part II: Mathematical modelling of in-vitro diffusion experiments. Identification of critical input parameters. Eur. J. Pharm. Biopharm. 68, 368–379 (2008)
Schwab, C., Gittelson, C.: Sparse tensor discretizations of high-dimensional parametric and stochastic PDEs. Acta Num. 20, 291–467 (2011)
Vogel, A., Reiter, S., Rupp, M., Nägel, A., Wittum, G.: UG 4—a novel flexible software system for simulating pde based models on high performance computers. Comp. Vis. Sci. 16, 165–179 (2013)