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title:
Conjugate gradient and GMRES methods for parameterized linear random algebraic equations |
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name:
Hakansson |
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first name:
Par
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location/conference:
SPDE09
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PRESENTATION-link:
http://www.dfg-spp1324.de/download/spde09/material/hakansson.pdf |
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abstract:
Conjugate gradient and GMRES methods for parameterized linear random algebraic equations
We present preconditioned conjugate gradient methods for solving linear
random algebraic equations that arise from discretization of stochastic partial
differential equations (SPDE) [1]. Hence, by finding solution to SPDE’s we
can look into uncertainty propagation in mechanical problems. The random
dimension is parameterized in a polynomial chaos basis. Preconditioned it-
erative schemes are presented for equations with symmetric positive definite
and general nonsymmetric coefficient matrices. We show that for the special
case of symmetric positive definite coefficient matrices, the proposed formu-
lation of the conjugate gradient algorithm has a mathematical equivalence
with the Ghanem-Spanos polynomial chaos projection scheme. We illustrate
and solve a problem in computing the residual at accurate PC-order needed
to safely determine convergence of solution. Numerical results are presented
for a steady-state stochastic diffusion equation to illustrate the performance
of the proposed numerical methods [2]. |
