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