Abstract
This paper describes how reinforcement learning can be used to select from a wide variety of preconditioned solvers for sparse linear systems. This approach provides a simple way to consider complex metrics of goodness, and makes it easy to evaluate a wide range of preconditioned solvers. A basic implementation recommends solvers that, when they converge, generally do so with no more than a 17% overhead in time over the best solver possible within the test framework. Potential refinements of, and extensions to, the system are discussed.
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Benzi, M.: Preconditioning techniques for large linear systems: A survey. J. of Comp. Physics 182(2), 418–477 (2002)
Saad, Y., van der Vorst, H.A.: Iterative solution of linear systems in the 20th century. J. Comput. Appl. Math. 123(1-2), 1–33 (2000)
Bhowmick, S., Eijkhout, V., Freund, Y., Fuentes, E., Keyes, D.: Application of machine learning to the selection of sparse linear solvers. International Journal of High Performance Computing Applications (submitted, 2006)
Holloway, A.L., Chen, T.-Y.: Neural networks for predicting the behavior of preconditioned iterative solvers. In: Shi, Y., van Albada, G.D., Dongarra, J., Sloot, P.M.A. (eds.) ICCS 2007. LNCS, vol. 4487, pp. 302–309. Springer, Heidelberg (2007)
Xu, S., Zhang, J.: Solvability prediction of sparse matrices with matrix structure-based preconditioners. In: Proc. Preconditioning 2005, Atlanta, Georgia (2005)
Xu, S., Zhang, J.: SVM classification for predicting sparse matrix solvability with parameterized matrix preconditioners. Technical Report 450-06, University of Kentucky (2006)
George, T., Sarin, V.: An approach recommender for preconditioned iterative solvers. In: Proc. Preconditioning 2007, Toulouse, France (2007)
Ramakrishnan, N., Ribbens, C.J.: Mining and visualizing recommendation spaces for elliptic PDEs with continuous attributes. ACM Trans. on Math. Softw. 26(2), 254–273 (2000)
Sutton, R.S., Barto, A.G.: Reinforcement Learning: An Introduction. MIT Press, Cambridge (1998)
Barrett, R., Berry, M., Chan, T.F., Demmel, J., Donato, J., Dongarra, J., Eijkhout, V., Pozo, R., Romine, C., van der Vorst, H.: Templates for the solution of linear systems: Building blocks for iterative methods. SIAM, Philadelphia (1994)
Duff, I.S., Koster, J.: The design and use of algorithms for permuting large entries to the diagonal of sparse matrices. SIAM J. Matrix Anal. Appl. 20(4), 889–901 (1999)
Duff, I.S., Koster, J.: On algorithms for permuting large entries to the diagonal of a sparse matrix. SIAM J. Matrix Anal. Appl. 22(4), 973–996 (2001)
Cuthill, E., McKee, J.: Reducing the bandwidth of sparse symmetric matrices. In: Proc. of the 24th Natl. Conf. of the ACM, pp. 157–172 (1969)
Davis, T., Gilbert, J., Larimore, S., Ng, E.: Algorithm 836: COLAMD, a column approximate minimum degree ordering algorithm. ACM Trans. on Math. Softw. 30(3), 377–380 (2004)
Davis, T., Gilbert, J., Larimore, S., Ng, E.: A column approximate minimum degree ordering algorithm. ACM Trans. on Math. Softw. 30(3), 353–376 (2004)
Chen, T.-Y.: ILUTP_Mem: A space-efficient incomplete LU preconditioner. In: Laganá, A., Gavrilova, M.L., Kumar, V., Mun, Y., Tan, C.J.K., Gervasi, O. (eds.) ICCSA 2004. LNCS, vol. 3046, pp. 31–39. Springer, Heidelberg (2004)
Saad, Y., Schultz, M.H.: GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM J. Sci. Stat. Comput. 7(3), 856–869 (1986)
Xu, S., Zhang, J.: A data mining approach to matrix preconditioning problem. Technical Report 433-05, University of Kentucky (2005)
Lazzareschi, M., Chen, T.-Y.: Using performance profiles to evaluate preconditioners for iterative methods. In: Gavrilova, M.L., Gervasi, O., Kumar, V., Tan, C.J.K., Taniar, D., Laganá, A., Mun, Y., Choo, H. (eds.) ICCSA 2006. LNCS, vol. 3982, pp. 1081–1089. Springer, Heidelberg (2006)
Davis, T.: University of Florida sparse matrix collection. NA Digest 92(42), October 16, 1994 and NA Digest 96(28) July 23, 1996, and NA Digest 97(23) June 7 (1997) http://www.cise.ufl.edu/research/sparse/matrices/
Manteuffel, T.A.: An incomplete factorization technique for positive definite linear systems. Mathematics of Computation 34, 473–497 (1980)
Kanerva, P.: Sparse Distributed Memory. MIT Press, Cambridge (1988)
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Kuefler, E., Chen, TY. (2008). On Using Reinforcement Learning to Solve Sparse Linear Systems. In: Bubak, M., van Albada, G.D., Dongarra, J., Sloot, P.M.A. (eds) Computational Science – ICCS 2008. ICCS 2008. Lecture Notes in Computer Science, vol 5101. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-69384-0_100
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DOI: https://doi.org/10.1007/978-3-540-69384-0_100
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