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Articles written by Tartakovsky, D.M.

  1. Xiu, D. and Tartakovsky, D.M.. "Numerical Methods for Differential Equations in Random Domains" SIAM Journal on Scientific Computing. 28 (3). 2006. pp. 1167--1185.

    Physical phenomena in domains with rough boundaries play an important role in a variety of applications. Often the topology of such boundaries cannot be accurately described in all of its relevant detail due to either insufficient data or measurement errors or both. This topological uncertainty can be efficiently handled by treating rough boundaries as random fields, so that an underlying physical phenomenon is described by deterministic or stochastic differential equations in random domains. To deal with this class of problems, we propose a novel computational framework, which is based on using stochastic mappings to transform the original deterministic/stochastic problem in a random domain into a stochastic problem in a deterministic domain. The latter problem has been studied more extensively, and existing analytical/numerical techniques can be readily applied. In this paper, we employ both a stochastic Galerkin method and Monte Carlo simulations to solve the transformed stochastic problem. We demonstrate our approach by applying it to an elliptic problem in single- and double-connected random domains, and comment on the accuracy and convergence of the numerical methods.


  2. Tartakovsky, D.M. and Xiu, D.. "Stochastic analysis of transport in tubes with rough walls" Journal of Computational Physics. 217 (1). 2006. pp. 248--259.

    Flow and transport in tubes with rough surfaces play an important role in a variety of applications. Often the topology of such surfaces cannot be accurately described in all of its relevant details due to either insufficient data or measurement errors or both. In such cases, this topological uncertainty can be efficiently handled by treating rough boundaries as random fields, so that an underlying physical phenomenon is described by deterministic or stochastic differential equations in random domains. To deal with this class of problems, we use a computational framework, which is based on stochastic mappings to transform the original deterministic/stochastic problem in a random domain into a stochastic problem in a deterministic domain. The latter problem has been studied more extensively and existing analytical/numerical techniques can be readily applied. In this paper, we employ both a generalized polynomial chaos and Monte Carlo simulations to solve the transformed stochastic problem. We use our approach to describe transport of a passive scalar in Stokes' flow and to quantify the corresponding predictive uncertainty.