title:
Conjugate gradient and GMRES methods for parameterized linear random algebraic equations 
name:
Hakansson 
first name:
Par

location/conference:
SPDE09

PRESENTATIONlink:
http://www.dfgspp1324.de/download/spde09/material/hakansson.pdf 
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 GhanemSpanos polynomial chaos projection scheme. We illustrate
and solve a problem in computing the residual at accurate PCorder needed
to safely determine convergence of solution. Numerical results are presented
for a steadystate stochastic diffusion equation to illustrate the performance
of the proposed numerical methods [2]. 