International Journal for Uncertainty Quantification, 4 (3): 225–242 (2014) A MULTI-FIDELITY STOCHASTIC COLLOCATION METHOD FOR PARABOLIC PARTIAL DIFFERENTIAL EQUATIONS WITH RANDOM INPUT DATA Maziar Raissi & Padmanabhan Seshaiyer * Department of Mathematical Sciences, George Mason University, 4400 University Drive, MS: 3F2, Planetary Hall, Fairfax, Virginia 22030, USA Original Manuscript Submitted: 05/07/2013; Final Draft Received: 09/03/2013 Over the last few years there have been dramatic advances in the area of uncertainty quantification. In particular, we have seen a surge of interest in developing efficient, scalable, stable, and convergent computational methods for solving differential equations with random inputs. Stochastic collocation (SC) methods, which inherit both the ease of implementation of sampling methods like Monte Carlo and the robustness of nonsampling ones like stochastic Galerkin to a great deal, have proved extremely useful in dealing with differential equations driven by random inputs. In this work we propose a novel enhancement to stochastic collocation methods using deterministic model reduction techniques. Linear parabolic partial differential equations with random forcing terms are analysed. The input data are assumed to be represented by a finite number of random variables. A rigorous convergence analysis, supported by numerical results, shows that the proposed technique is not only reliable and robust but also efficient. KEY WORDS: collocation, stochastic partial differential equations, sparse grid, smolyak algorithm, finite element, proper orthogonal decomposition, multifidelity 1. INTRODUCTION The effectiveness of stochastic partial differential equations (SPDEs) in modeling complex systems is a well-known fact. One can name wave propagation [1], diffusion through heterogeneous random media [2], randomly forced Burg- ers and Navier-Stokes equations (see e.g., [3–6] and the references therein) as a couple of examples. Currently, Monte Carlo is one of the most widely used tools in simulating models driven by SPDEs. However, Monte Carlo simulations are generally very expensive. To meet this concern, methods based on the Fourier analysis with respect to the Gaussian (rather than the Lebesgue) measure, have been investigated in recent decades. More specifically, the Cameron–Martin version of the Wiener Chaos expansion (see, e.g., [7, 8] and the references therein) is among the earlier efforts. Some- times, the Wiener Chaos expansion (WCE for short) is also referred to as the Hermite polynomial chaos expansion. The term polynomial chaos was coined by Nobert Wiener [9]. In Wieners’ work, Hermite polynomials served as an orthogonal basis. The validity of the approach was then proved in [7]. There is a long history of using WCE as well as other polynomial chaos expansions in problems in physics and engineering. See, e.g., [10–13], etc. Applications of the polynomial chaos to stochastic PDEs considered in the literature typically deal with stochastic input generated by a finite number of random variables (see, e.g. [14–17]). This assumption is usually introduced either directly or via a representation of the stochastic input by a truncated Karhunen–Lo` eve (KL) expansion. Stochastic finite element meth- ods based on the Karhunen–Lo` eve expansion and Hermite polynomial chaos expansion [14, 15] have been developed by Ghanem and other authors. Karniadakis et al. generalized this idea to other types of randomness and polynomials [16, 18, 19]. The stochastic finite element procedure often results in a set of coupled deterministic equations which requires additional effort to be solved. To resolve this issue, the stochastic collocation (SC) method was introduced. * Correspond to Padmanabhan Seshaiyer, E-mail: pseshaiy@gmu.edu, URL: http://math.gmu.edu/˜ pseshaiy/ 2152–5080/14/$35.00 c  2014 by Begell House, Inc. 225