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We propose an adaptive sparse grid stochastic collocation approach based upon Leja interpolation sequences for approximation of parameterized functions with high-dimensional parameters. Leja sequences are arbitrarily granular (any number of nodes may be added to a current sequence, producing a new sequence) and thus are a good choice for the univariate composite rule used to construct adaptive sparse grids in high dimensions. When undertaking stochastic collocation one is often interested in constructing weighted approximation where the weights are determined by the probability densities of the random variables. This paper establishes that a certain weighted formulation of one-dimensional Leja sequences produces a sequence of nodes whose empirical distribution converges to the corresponding limiting distribution of the Gauss quadrature nodes associated with the weight function. This property is true even for unbounded domains. We apply the Leja sparse grid approach to several high-dimensional problems and demonstrate that Leja sequences are often superior to more standard sparse grid constructions (e.g., Clenshaw-Curtis), at least for interpolatory metrics. Introduction.Stochastic collocation has become a standard tool for nonintrusively quantifying uncertainty in simulation models that are subject to a degree of uncertainty or randomness. Sources of uncertainty can include physical stochastic processes, parametric uncertainty, or model form uncertainty [54,40,39]. If the sources of uncertainty can be parameterized by a set of random variables, the approaches used for uncertainty quantification (UQ) frequently reduce to computational methods that describe the behavior of a model with respect to those random variables.In this paper we will focus on the use of stochastic collocation methods that utilize sparse grid interpolants [46,56,22,9] to approximate the dependence of model output on unknown model parameters. Sparse grid stochastic collocation involves constructing an ensemble of random variables realizations, solving a deterministic physical system for each corresponding realization on the sparse grid, and using the resulting model output to build an approximation of the model response to the uncertain parameters. Once constructed, the sparse grid can be evaluated inexpensively to predict the variability of the physical model with respect to the random parameters.Obtaining the ensemble of model solutions is usually the most expensive part of the collocation procedure since the model under consideration is often very complicated

In this paper we discuss a numerical approach for the stochastic collocation method with multifidelity simulation models. The method we consider was recently proposed in [A. Narayan, C. Gittelson, and D. Xiu, SIAM J. Sci. Comput., 36 (2014), pp. A495-A521] to combine the computational efficiency of low-fidelity models with the high accuracy of high-fidelity models. This method is able to produce more accurate results at a much reduced simulation cost. The purpose of this paper includes (1) a presentation of the detailed implementation of the method developed by [A. Narayan, C. Gittelson, and D. Xiu, SIAM J. Sci. Comput., 36 (2014), pp. A495-A521], which is largely theoretical; (2) an adaptation of that method to handle multifidelity scenarios that are of more practical interest; (3) a closer examination of the method via a set of more comprehensive benchmark examples including several two-dimensional stochastic PDEs with high-dimensional random parameters; and (4) a more detailed investigation of the strengths of the multifidelity approach in [A. Narayan, C. Gittelson, and D. Xiu, SIAM J. Sci. Comput., 36 (2014), pp. A495-A521]. Specifically, we present a numerical algorithm to construct accurate simulation results when multiple low-fidelity models are available. We suggest that trifidelity simulations with a low-fidelity, a medium-fidelity, and a high-fidelity model would be sufficient for most practical problems. The availability of more simulation models (i.e., more than three) would not present much improvement under the current framework.

We propose, theoretically investigate, and numerically validate an algorithm for the Monte Carlo solution of least-squares polynomial approximation problems in a collocation framework. Our investigation is motivated by applications in the collocation approximation of parametric functions, which frequently entails construction of surrogates via orthogonal polynomials. A standard Monte Carlo approach would draw samples according to the density defining the orthogonal polynomial family. Our proposed algorithm instead samples with respect to the (weighted) pluripotential equilibrium measure of the domain, and subsequently solves a weighted least-squares problem, with weights given by evaluations of the Christoffel function. We present theoretical analysis to motivate the algorithm, and numerical results that show our method is superior to standard Monte Carlo methods in many situations of interest.

In this paper we propose an algorithm for recovering sparse orthogonal polynomials using stochastic collocation. Our approach is motivated by the desire to use generalized polynomial chaos expansions (PCE) to quantify uncertainty in models subject to uncertain input parameters. The standard sampling approach for recovering sparse polynomials is to use Monte Carlo (MC) sampling of the density of orthogonality. However MC methods result in poor function recovery when the polynomial degree is high. Here we propose a general algorithm that can be applied to any admissible weight function on a bounded domain and a wide class of exponential weight functions defined on unbounded domains. Our proposed algorithm samples with respect to the weighted equilibrium measure of the parametric domain, and subsequently solves a preconditioned 1 -minimization problem, where the weights of the diagonal preconditioning matrix are given by evaluations of the Christoffel function. We present theoretical analysis to motivate the algorithm, and numerical results that show our method is superior to standard Monte Carlo methods in many situations of interest. Numerical examples are also provided that demonstrate that our proposed Christoffel Sparse Approximation algorithm leads to comparable or improved accuracy even when compared with Legendre and Hermite specific algorithms.

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