An Efficient Reduced Basis Construction for Stochastic Galerkin Matrix Equations Using Deflated Conjugate Gradients

  • Michal BérešEmail author
Conference paper
Part of the Lecture Notes in Electrical Engineering book series (LNEE, volume 554)


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.


Stochastic Galerkin method Reduced basis method Deflated conjugate gradients method Darcy flow problem 



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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Authors and Affiliations

  1. 1.FEECS, Department of Applied MathematicsVŠB - Technical university of OstravaOstrava - PorubaCzech Republic
  2. 2.IT4Innovations National Supercomputing CenterVŠB - Technical University of OstravaOstrava - PorubaCzech Republic
  3. 3.Institute of Geonics of the CASOstrava - PorubaCzech Republic

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