Abstract
In this article, we examine an efficient solution of the stochastic Galerkin (SG) matrix equations coming from the Darcy flow problem with uncertain material parameters on given interfaces. The solution of the SG system of equations, here represented as matrix equations, is usually a very challenging task. A relatively new approach to the solution of the SG matrix equations is the reduced basis (RB) solver, which looks for the low-rank representation of the solution. The construction of the RB is usually done iteratively and consists of multiple solutions of systems of equations. We aim to speed up the process using the deflated conjugate gradients (DCG). Other contributions of this work are a modified specific construction of the RB without the need of Cholesky factor and an adaptive choice of the candidate vectors for the expansion of the RB. The proposed approach allows an efficient parallel implementation.
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References
Additive Schwarz Preconditioners. In: Marsden, J.E., Sirovich, L., Antman, S.S. (eds.) The Mathematical Theory of Finite Element Methods, vol. 15, pp. 175–214. Springer, New York (2008). https://doi.org/10.1007/978-0-387-75934-08
Babuška, I., Nobile, F., Tempone, R.: A stochastic collocation method for elliptic partial differential equations with random input data. SIAM J. Numer. Anal. 45(3), 1005–1034 (2007). https://doi.org/10.1137/050645142
Béreš, M., Domesová, S.: The stochastic galerkin method for darcy flow problem with log-normal random field coefficients. Adv. Electr. Electron. Eng. 15(2) (2017). https://doi.org/10.15598/aeee.v15i2.2280
Hoeksema, R.J., Kitanidis, P.K.: Analysis of the spatial structure of properties of selected aquifers. Water Resour. Res. 21(4), 563–572 (1985). https://doi.org/10.1029/WR021i004p00563
Khoromskij, B.N., Schwab, C.: Tensor-structured galerkin approximation of parametric and stochastic elliptic PDEs. SIAM J. Sci. Comput. 33(1), 364–385 (2011). https://doi.org/10.1137/100785715
Lord, G.J., Powell, C.E., Shardlow, T.: An Introduction to Computational Stochastic PDEs, 1 edn., No. 50 in Cambridge Texts in Applied Mathematics. Cambridge University Press, New York (2014)
Matthies, H.G., Zander, E.: Solving stochastic systems with low-rank tensor compression. Linear Algebra Appl. 436(10), 3819–3838 (2012). https://doi.org/10.1016/j.laa.2011.04.017
Nouy, A.: A generalized spectral decomposition technique to solve a class of linear stochastic partial differential equations. Comput. Methods Appl. Mech. Eng. 196(45–48), 4521–4537 (2007). https://doi.org/10.1016/j.cma.2007.05.016
Powell, C.E., Silvester, D., Simoncini, V.: An efficient reduced basis solver for stochastic galerkin matrix equations. SIAM J. Sci. Comput. 39(1), A141–A163 (2017). https://doi.org/10.1137/15M1032399
Saad, Y., Yeung, M., Erhel, J., Guyomarc’h, F.: A deflated version of the conjugate gradient algorithm. SIAM J. Sci. Comput. 21(5), 1909–1926 (2000). https://doi.org/10.1137/S1064829598339761
Ullmann, E.: A kronecker product preconditioner for stochastic galerkin finite element discretizations. SIAM J. Sci. Comput. 32(2), 923–946 (2010). https://doi.org/10.1137/080742853
Acknowledgments
This work was supported by The Ministry of Education, Youth and Sports from the National Programme of Sustainability (NPS II) project “IT4Innovations excellence in science - LQ1602”. The work was also partially supported by Grant of SGS No. SP2018/68 and by Grant of SGS No. SP2018/161, VšB - Technical University of Ostrava, Czech Republic.
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Béreš, M. (2020). An Efficient Reduced Basis Construction for Stochastic Galerkin Matrix Equations Using Deflated Conjugate Gradients. In: Zelinka, I., Brandstetter, P., Trong Dao, T., Hoang Duy, V., Kim, S. (eds) AETA 2018 - Recent Advances in Electrical Engineering and Related Sciences: Theory and Application. AETA 2018. Lecture Notes in Electrical Engineering, vol 554. Springer, Cham. https://doi.org/10.1007/978-3-030-14907-9_18
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