Random Matrix Approach for Stochastic Flow Problems
Adhikari, S. and D. M. Tartakovsky
6th International Congress on Industrial and Applied Mathematics (ICIAM 2007),
Zurich, Switzerland, July 2007.
Consider diffusion (heat conduction) in a highly heterogeneous
environment whose diffusion coefficient (thermal conductivity) D
is sampled at a few locations throughout a computational domain.
The lack of sufficient information about D renders this problem
under-defined and, hence, ill-posed. The problem is routinely
regularized by treating D as a random field and the corresponding
diffusion equation as stochastic. A key feature of this and other
similar stochastic problems is that the randomness of system
parameters, such as D, is multiplicative. A solution of such
stochastic problems requires a closure approximation in all but a
few special cases. This is true for either analytical methods
(e.g., a closure by perturbation) or direct statistical methods
(e.g., the use of a finite number of realizations in Monte Carlo
simulations) or numerical methods (e.g., truncation of infinite
series in polynomial chaos expansions). The approximate nature of
such solutions formally limits their applicability to either
mildly heterogeneous or long-correlated environments.
We propose an alternative approach that, in many applications,
obviates the need for a closure approximation. It relies on a
representation of a stochastic partial differential equation as a
system of coupled linear random algebraic equations. To solve such
a problem is to find the inverse the corresponding random matrix.
We present an exact analytical method for the inverse of a real
symmetric (in general non-Gaussian) random matrix of arbitrary
dimension. The proposed method is based on random matrix theory
and utilizes the Jacobian of the underlying nonlinear matrix
transformation. For steady-state diffusion, exact expressions for
the mean and covariance of the system state is obtained exactly in
closed form.
BiBTeX Entry
@INPROCEEDINGS{cp36,
AUTHOR={S. Adhikari and D. M. Tartakovsky},
TITLE={Random matrix approach for stochastic flow problems},
BOOKTITLE={6th International Congress on Industrial and Applied Mathematics (ICIAM 2007)},
YEAR={2007},
Address={Zurich, Switzerland},
Month={July},
Note={}
}
by Sondipon Adhikari