Adaptive sparse grids

Hegland, Markus

doi:10.21914/anziamj.v44i0.685

Cited by 98 publications

(78 citation statements)

References 11 publications

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“…Instead, we will allow more general index sets [18,28,39] in the summation of (1) and try to choose them properly. To this end, we will consider the selection of the whole index set as an optimization problem, i.e.…”

Section: Dimension-adaptive Quadraturementioning

confidence: 99%

Dimension?Adaptive Tensor?Product Quadrature

2003

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We consider the numerical integration of multivariate functions defined over the unit hypercube. Here, we especially address the high-dimensional case, where in general the curse of dimension is encountered. Due to the concentration of measure phenomenon, such functions can often be well approximated by sums of lower-dimensional terms. The problem, however, is to find a good expansion given little knowledge of the integrand itself.The dimension-adaptive quadrature method which is developed and presented in this paper aims to find such an expansion automatically. It is based on the sparse grid method which has been shown to give good results for low-and moderatedimensional problems. The dimension-adaptive quadrature method tries to find important dimensions and adaptively refines in this respect guided by suitable error estimators. This leads to an approach which is based on generalized sparse grid index sets. We propose efficient data structures for the storage and traversal of the index sets and discuss an efficient implementation of the algorithm.The performance of the method is illustrated by several numerical examples from computational physics and finance where dimension reduction is obtained from the Brownian bridge discretization of the underlying stochastic process.

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Section: Dimension-adaptive Quadraturementioning

confidence: 99%

Dimension?Adaptive Tensor?Product Quadrature

2003

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“…A standard refinement approach is simply to grow an isotropic simplex of side length n. [13] and [16] instead suggest a series of greedy refinements that customize the Smolyak algorithm to a particular problem.…”

Section: Dimension Adaptivitymentioning

confidence: 99%

Adaptive Smolyak Pseudospectral Approximations

Conrad¹,

Marzouk²

2013

SIAM J. Sci. Comput.

140

166

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Abstract. Polynomial approximations of computationally intensive models are central to uncertainty quantification. This paper describes an adaptive method for non-intrusive pseudospectral approximation, based on Smolyak's algorithm with generalized sparse grids. We rigorously analyze and extend the non-adaptive method proposed in [6], and compare it to a common alternative approach for using sparse grids to construct polynomial approximations, direct quadrature. Analysis of direct quadrature shows that O(1) errors are an intrinsic property of some configurations of the method, as a consequence of internal aliasing. We provide precise conditions, based on the chosen polynomial basis and quadrature rules, under which this aliasing error occurs. We then establish theoretical results on the accuracy of Smolyak pseudospectral approximation, and show that the Smolyak approximation avoids internal aliasing and makes far more effective use of sparse function evaluations. These results are applicable to broad choices of quadrature rule and generalized sparse grids. Exploiting this flexibility, we introduce a greedy heuristic for adaptive refinement of the pseudospectral approximation. We numerically demonstrate convergence of the algorithm on the Genz test functions, and illustrate the accuracy and efficiency of the adaptive approach on a realistic chemical kinetics problem.Key words. Smolyak algorithms, sparse grids, orthogonal polynomials, pseudospectral approximation, approximation theory, uncertainty quantification AMS subject classifications. 41A10, 41A63, 47A80, 65D15, 65D321. Introduction. A central issue in the field of uncertainty quantification is understanding the response of a model to random inputs. When model evaluations are computationally intensive, techniques for approximating the model response in an efficient manner are essential. Approximations may be used to evaluate moments or the probability distribution of a model's outputs, or to evaluate sensitivities of model outputs with respect to the inputs [21,41,33]. Approximations may also be viewed as surrogate models to be used in optimization [22] or inference [23], replacing the full model entirely.Often one is faced with black box models that can only be evaluated at designated input points. We will focus on constructing multivariate polynomial approximations of the inputoutput relationship generated by such a model; these approximations offer fast convergence for smooth functions and are widely used. One common strategy for constructing a polynomial approximation is interpolation, where interpolants are conveniently represented in Lagrange form [1,42]. Another strategy is projection, particularly orthogonal projection with respect to some inner product. The results of such a projection are conveniently represented with the corresponding family of orthogonal polynomials [4,21,43]. When the inner product is chosen according to the input probability measure, this construction is known as the (finite dimensional) polynomial chaos expansion (PCE) [14,32,...

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“…The combination technique has been studied extensively [2,5,1,8]. Let Ω = [0, 1] , for k ∈ N we define h k := 2 −k and…”

Section: The Combination Technique and Error Splittingsmentioning

confidence: 99%

Combination technique coefficients via error splittings

Harding

2016

ANZIAMJ

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We investigate a new way of choosing combination coefficients for the sparse grid combination technique. Previous work considered choosing coefficients such that the interpolation error of sufficiently smooth functions is minimised. We instead obtain an error bound using an error splitting model of approximation error and seek coefficients which minimise this. With minor modification this approach can also yield extrapolations. There are also potential applications to fault tolerance where new coefficients are required when a solution becomes unavailable due to a fault. We test the approach numerically on a scalar advection problem and compare with classical combinations from the literature.

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Adaptive sparse grids

Cited by 98 publications

References 11 publications

Dimension?Adaptive Tensor?Product Quadrature

Dimension?Adaptive Tensor?Product Quadrature

Adaptive Smolyak Pseudospectral Approximations

Combination technique coefficients via error splittings

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