Sparse Collocation Methods for Stochastic Interpolation and Quadrature

In this chapter, the authors survey the family of sparse stochastic collocation methods (SCMs) for partial differential equations with random input data. The SCMs under consideration can be viewed as a special case of the generalized stochastic finite element method (Gunzburger et al., Acta Numer 23:521–650, 2014), where the approximation of the solution dependences on the random variables is constructed using Lagrange polynomial interpolation. Relying on the “delta property” of the interpolation scheme, the physical and stochastic degrees of freedom can be decoupled, such that the SCMs have the same nonintrusive property as stochastic sampling methods but feature much faster convergence. To define the interpolation schemes or interpolatory quadrature rules, several approaches have been developed, including global sparse polynomial approximation, for which global polynomial subspaces (e.g., sparse Smolyak spaces (Nobile et al., SIAM J Numer Anal 46:2309–2345, 2008) or quasi-optimal subspaces (Tran et al., Analysis of quasi-optimal polynomial approximations for parameterized PDEs with deterministic and stochastic coefficients. Tech. Rep. ORNL/TM-2015/341, Oak Ridge National Laboratory, 2015)) are used to exploit the inherent regularity of the PDE solution, and local sparse approximation, for which hierarchical polynomial bases (Ma and Zabaras, J Comput Phys 228:3084–3113, 2009; Bungartz and Griebel, Acta Numer 13:1–123, 2004) or wavelet bases (Gunzburger et al., Lect Notes Comput Sci Eng 97:137–170, Springer, 2014) are used to accurately capture irregular behaviors of the PDE solution. All these method classes are surveyed in this chapter, including some novel recent developments. Details about the construction of the various algorithms and about theoretical error estimates of the algorithms are provided.