Uncertainty quantification in risk analysis has become a key application. In this context, computing the diagonal of inverse covariance matrices is of paramount importance. Standard techniques, that employ matrix factorizations, incur a cubic cost which quickly becomes intractable with the current explosion of data sizes. In this work we reduce this complexity to quadratic with the synergy of two algorithms that gracefully complement each other and lead to a radically different approach. First, we turned to stochastic estimation of the diagonal. This allowed us to cast the problem as a linear system with a relatively small number of multiple right hand sides. Second, for this linear system we developed a novel, mixed precision, iterative refinement scheme, which uses iterative solvers instead of matrix factorizations. We demonstrate that the new framework not only achieves the much needed quadratic cost but in addition offers excellent opportunities for scaling at massively parallel environments. We based our implementation on BLAS 3 kernels that ensure very high processor performance. We achieved a peak performance of 730 TFlops on 72 BG/P racks, with a sustained performance 73% of theoretical peak. We stress that the techniques presented in this work are quite general and applicable to several other important applications. Copyright © 2009 ACM.
|Original language||English (US)|
|Title of host publication||Proceedings of the 2nd Workshop on High Performance Computational Finance - WHPCF '09|
|Publisher||Association for Computing Machinery (ACM)|
|State||Published - 2009|
Bibliographical noteKAUST Repository Item: Exported on 2020-10-01
Acknowledgements: The authors gratefully acknowledge Prof. Thomas Lippertand the Julich Supercomputing Center for kindly grantingaccess to their 72 rack BG/P cluster. We also would likethe thank for the support of WatsonShaheen - an 8 rackBlue Gene/P Supercomputer at IBM's T.J. Watson ResearchCenter that is jointly owned and managed by the KingAbdullah University of Science and Technology (KAUST)and IBM.
This publication acknowledges KAUST support, but has no KAUST affiliated authors.