A block MINRES algorithm based on the band Lanczos method
- 172 Downloads
We develop a block minimum residual (MINRES) algorithm for symmetric indefinite matrices. This version is built upon the band Lanczos method that generates one basis vector of the block Krylov subspace per iteration rather than a whole block as in the block Lanczos process. However, we modify the method such that the most expensive operations are still performed in a block fashion. The benefit of using the band Lanczos method is that one can detect breakdowns from scalar values arising in the computation, allowing for a handling of breakdown which is straightforward to implement. We derive a progressive formulation of the MINRES method based on the band Lanczos process and give some implementation details. Specifically, a simple reordering of the steps allows us to perform many of the operations at the block level in order to take advantage of communication efficiencies offered by the block Lanczos process. This is an important concern in the context of next-generation super computing applications. We also present a technique allowing us to maintain the block size by replacing dependent Lanczos vectors with pregenerated random vectors whose orthogonality against all Lanczos vectors is maintained. Numerical results illustrate the performance on some sample problems. We present experiments that show how the relationship between right-hand sides can affect the performance of the method.
KeywordsSymmetric matrices Block methods Krylov subspace methods Minimum residual methods Band Lanczos High-performance computing
Unable to display preview. Download preview PDF.
- 4.Birk, S., Frommer, A.: A deflated conjugategate gradient method for multiple right-hand sides and multiple shifts. (In preparation)Google Scholar
- 7.Freund, R. W.: Computation of matrix Padé approximations of transfer functions via a Lanczos-type process, in Approximation Theory VIII, Vol. 1: Approximation and Interpolation. In: Charles K. Chui, Larry L. Schumaker (eds.) , pp 215–222. World Scientific Publishing Co., River Edge, NJ (1995)Google Scholar
- 8.Freund, R.W., Malhotra, M.: A block QMR algorithm for non-Hermitian linear systems with multiple right-hand sides. In: Proceedings of the Fifth Conference of the International Linear Algebra Society (Atlanta, GA, 1995), vol. 254, 1997, pp. 119–157Google Scholar
- 13.Hoemmen, M.: Communication-avoiding krylov subspace methods. PhD thesis, University of California Berkeley (2010)Google Scholar
- 14.Langou, J.: Iterative methods for solving linear systems with multiple right-hand sides. PhD thesis, CERFACS, France (2003)Google Scholar
- 15.Loher, D.: Reliable nonsymmetric block Lanczos algorithms, PhD thesis, Diss. no. 16337. ETH Zurich, Zurich, Switzerland (2006)Google Scholar
- 18.Parks, M.L., Soodhalter, K.M., Szyld, D.B.: Block krylov subspace recycling. (In Preparation)Google Scholar
- 21.Saad, Y.: Iterative methods for sparse linear systems, 2nd ed. SIAM, Philadelphia (2003)Google Scholar
- 23.Schmelzer, T.: Block Krylov methods for Hermitian linear systems. Master’s thesis (2004)Google Scholar
- 26.Soodhalter, K.M.: A block MINRES algorithm based on the band lanczos method, Tech. Report. arXiv:1301.2102v2 (2013)
- 27.Vital, B.: Etude de quelques méthodes de résolution de problèmes linéaires de grande taille sur multiprocesseur. PhD thesis, Université de Rennes (1990)Google Scholar