Polynomial chaos representation of a stochastic preconditioner. (English) Zbl 1122.74541
Summary: A method is developed in this paper to accelerate the convergence in computing the solution of stochastic algebraic systems of equations. The method is based on computing, via statistical sampling, a polynomial chaos decomposition of a stochastic preconditioner to the system of equations. This preconditioner can subsequently be used in conjunction with either chaos representations of the solution or with approaches based on Monte Carlo sampling. In addition to presenting the supporting theory, the paper also presents a convergence analysis and an example to demonstrate the significance of the proposed algorithm.
MSC:
| 74S30 | Other numerical methods in solid mechanics (MSC2010) |
| 74H50 | Random vibrations in dynamical problems in solid mechanics |
Software:
LIPSOLReferences:
| [1] | . Stochastic Finite Elements: A Spectral Approach. Springer: New York, 1991. · doi:10.1007/978-1-4612-3094-6 |
| [2] | Soize, Journal of the Acoustical Society of America 109 pp 1979– (2001) |
| [3] | Soize, Journal of Sound and Vibration 263 pp 893– (2003) |
| [4] | Babuska, Computer Methods in Applied Mechanics and Engineering 191 pp 4093– (2002) |
| [5] | Soize, SIAM Journal on Scientific Computing 26 pp 395– (2005) |
| [6] | Schueller, Computers and Structures 79 pp 2225– (2001) |
| [7] | Ghanem, Physica D 133 pp 137– (1999) |
| [8] | . Matrix Computation (2nd edn). The John Hopkins University Press: Baltimore, MD, 1989. |
| [9] | Iterative methods for sparse linear systems. Preconditioning Techniques, Chapter 10. PWS Publishing Company: Massachusetts, 1996. |
| [10] | , . Incomplete Cholesky factorization with sparsity pattern modification. Technical Report 394, Indiana University-Bloomington, Bloomington, IN 47405, 1994. |
| [11] | Solving large-scale linear programs by interior-point methods under the MATLAB environment. Technical Report TR96-01, Department of Mathematics and Statistics, University of Maryland, Baltimore County, 1996. |
| [12] | Zhang, SIAM Journal on Matrix Analysis and Applications 22 pp 925– (2000) |
| [13] | Benzi, SIAM Journal on Scientific Computing 17 pp 1135– (1996) |
| [14] | , et al. Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods. SIAM: Philadelphia, PA, 1994. · doi:10.1137/1.9781611971538 |
| [15] | Pellissetti, Journal of Advances in Engineering Software 31 pp 607– (2002) |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
