In this work we discuss the dynamically orthogonal (DO) approximation of time dependent partial differential equations with random data. The approximate solution is expanded at each time instant on a time dependent orthonormal basis in the physical domain with a fixed and small number of terms. Dynamic equations are written for the evolution of the basis as well as the evolution of the stochastic coefficients of the expansion. We analyze the case of a linear parabolic equation with random data and derive a theoretical bound for the approximation error of the S-terms DO solution by the corresponding S-terms best approximation, i.e., the truncated Sterms Karhunen-Loève expansion at each time instant. The bound is applicable on the largest time interval in which the best S-terms approximation is continuously time differentiable. Properties of the DO approximations are analyzed on simple cases of deterministic equations with random initial data. Numerical tests are presented that confirm the theoretical bound and show potentials and limitations of the proposed approach.
Introduction.Many physical and engineering problems can be properly described by mathematical models, typically of differential type. However, in many situations, the input parameters may be affected by uncertainty due, e.g., to measurement errors, limited data availability, or intrinsic variability of the phenomenon itself. A convenient way to characterize uncertainty consists in describing the uncertain parameters as random variables or space and/or time varying random fields. Starting from a suitable partial differential equation (PDE) model, the aim of the uncertainty quantification is to assess the effects of the uncertainty by computing the statistics of the solutions or of some quantities of interest. Several approaches have been proposed and analyzed in the last decades. We name the Monte Carlo method [6], quasi-Monte Carlo [35] and the corresponding multilevel versions [34], or the approaches based on deterministic approximations of the parameters-to-solution map (response function) such as the generalized polynomial chaos [42,7,25] in its Galerkin [40,8] and collocation versions [2,1,3,23].In this work we focus on a reduced basis method to approximate the solution. We consider a general (linear or nonlinear) type of time dependent PDE with random data in which the randomness can appear in the operator as a random parameter or forcing term as well as in the initial datum. A possible approach to approximate the solution
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