Akil Narayan

STOCHASTIC COLLOCATION METHODS ON UNSTRUCTURED GRIDS IN HIGH DIMENSIONS VIA INTERPOLATION

In this paper we propose a method for conducting stochastic collocation on arbitrary sets of nodes. To accomplish this, we present the framework of least orthogonal interpolation, which allows one to construct interpolation polynomials based on arbitrarily located grids in arbitrary dimensions. These interpolation polynomials are constructed as a subspace of the family of orthogonal polynomials corresponding to the probability distribution function on stochastic space. This feature enables one to conduct stochastic collocation simulations in practical problems where one cannot adopt some popular node selections such as sparse grids or cubature nodes. We present in detail both the mathematical properties of the least orthogonal interpolation and its practical implementation algorithm. Numerical benchmark problems are also presented to demonstrate the efficacy of the method.

Multivariate Discrete Least-Squares Approximations with a New Type of Collocation Grid

In this work, we discuss the problem of approximating a multivariate function by discrete least-squares projection onto a polynomial space using a specially designed deterministic point set. The independent variables of the function are assumed to be random variables, stemming from the motivating application of uncertainty quantification. Our deterministic points are inspired by a theorem due to Andre Weil. We first work with the Chebyshev measure and consider the approximation in Chebyshev polynomial spaces. We prove the stability and an optimal convergence estimate, provided the number of points scales quadratically with the dimension of the polynomial space. A possible application for quantifying epistemic uncertainties is then discussed. We show that the point set asymptotically equidistributes to the product-Chebyshev measure, allowing us to propose a weighted least-squares framework and extending our method to more general polynomial approximations. Numerical examples are given to confirm the theore...

A STOCHASTIC COLLOCATION ALGORITHM WITH MULTIFIDELITY MODELS

We present a numerical method for utilizing stochastic models with differing fideli- ties to approximate parameterized functions. A representative case is where a high-fidelity and a low-fidelity model are available. The low-fidelity model represents a coarse and rather crude ap- proximation to the underlying physical system. However, it is easy to compute and consumes little simulation time. On the other hand, the high-fidelity model is a much more accurate representation of the physics but can be highly time consuming to simulate. Our approach is nonintrusive and is therefore applicable to stochastic collocation settings where the parameters are random variables. We provide sufficient conditions for convergence of the method, and present several examples that are of practical interest, including multifidelity approximations and dimensionality reduction.

A Generalized Sampling and Preconditioning Scheme for Sparse Approximation of Polynomial Chaos Expansions

We propose an algorithm for recovering sparse orthogonal polynomial expansions via collocation. A standard sampling approach for recovering sparse polynomials uses Monte Carlo sampling, from the density of orthogonality, which results in poor function recovery when the polynomial degree is high. Our proposed approach aims to mitigate this limitation by sampling with respect to the weighted equilibrium measure of the parametric domain and subsequently solves a preconditioned

A Christoffel function weighted least squares algorithm for collocation approximations

\ell^1

Computational Aspects of Stochastic Collocation with Multifidelity Models

-minimization problem, where the weights of the diagonal preconditioning matrix are given by evaluations of the Christoffel function. Our algorithm can be applied to a wide class of orthogonal polynomial families on bounded and unbounded domains, including all classical families. 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 to demonstrate that our proposed algor...

Minimal multi-element stochastic collocation for uncertainty quantification of discontinuous functions

We propose, theoretically investigate, and numerically validate an algorithm for the Monte Carlo solution of least-squares polynomial approximation problems in a collocation frame- work. Our method is motivated by generalized Polynomial Chaos approximation in uncertainty quantification where a polynomial approximation is formed from a combination of orthogonal polynomials. A standard Monte Carlo approach would draw samples according to the density of orthogonality. Our proposed algorithm samples with respect to the equilibrium measure of the parametric 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.

Constructing nested nodal sets for multivariate polynomial interpolation

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 st...

Weighted discrete least-squares polynomial approximation using randomized quadratures

We propose a multi-element stochastic collocation method that can be applied in high-dimensional parameter space for functions with discontinuities lying along manifolds of general geometries. The key feature of the method is that the parameter space is decomposed into multiple elements defined by the discontinuities and thus only the minimal number of elements are utilized. On each of the resulting elements the function is smooth and can be approximated using high-order methods with fast convergence properties. The decomposition strategy is in direct contrast to the traditional multi-element approaches which define the sub-domains by repeated splitting of the axes in the parameter space. Such methods are more prone to the curse-of-dimensionality because of the fast growth of the number of elements caused by the axis based splitting. The present method is a two-step approach. Firstly a discontinuity detector is used to partition parameter space into disjoint elements in each of which the function is smooth. The detector uses an efficient combination of the high-order polynomial annihilation technique along with adaptive sparse grids, and this allows resolution of general discontinuities with a smaller number of points when the discontinuity manifold is low-dimensional. After partitioning, an adaptive technique based on the least orthogonal interpolant is used to construct a generalized Polynomial Chaos surrogate on each element. The adaptive technique reuses all information from the partitioning and is variance-suppressing. We present numerous numerical examples that illustrate the accuracy, efficiency, and generality of the method. When compared against standard locally-adaptive sparse grid methods, the present method uses many fewer number of collocation samples and is more accurate.

A generalization of the Wiener rational basis functions on infinite intervals: Part I–derivation and properties

We present a robust method for choosing multivariate polynomial interpolation nodes. Our algorithm is an optimization method to greedily minimize a measure of interpolant sensitivity, a variant of a weighted Lebesgue function. Nodes are therefore chosen that tend to control oscillations in the resulting interpolant. This method can produce an arbitrary number of nodes and is not constrained by the dimension of a complete polynomial space. Our method is therefore flexible: nested nodal sets are produced in spaces of arbitrary dimensions, and the number of nodes added at each stage can be arbitrary. The algorithm produces a nodal set given a probability measure on the input space, thus parameterizing interpolants with respect to finite measures. We present examples to show that the method yields nodal sets that behave well with respect to standard interpolation diagnostics: the Lebesgue constant, the Vandermonde determinant, and the Vandermonde condition number. We also show that a nongreedy version of the ...