Sparse approximation of multilinear problems with applications to kernel-based methods in UQ

Nobile, Fabio; Tempone, Raúl; Wolfers, Soeren

doi:10.1007/s00211-017-0932-4

Cited by 4 publications

(9 citation statements)

References 38 publications

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“…. , L}, but instead relies on a simple union bound (see Equation (29)). Thus, we could alternatively first create Γ L and then define all Γ l with l < L as subsets of it.…”

Section: Discussionmentioning

confidence: 99%

“…A common goal in uncertainty quantification [21] is the approximation of response surfaces γ → f (γ) := Q(u γ ) ∈ R, which describe how a quantity of interest Q of the solution u γ to some partial differential equation (PDE) depends on parameters γ ∈ Γ ⊂ R d of the PDE. The non-intrusive approach to this problem is to evaluate the response surface for finitely many values of γ and then to use an interpolation method, such as (tensor-)spline interpolation [8], kernel-based approximation (kriging) [29,13], or (global) polynomial approximation [21].…”

Section: Introductionmentioning

confidence: 99%

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Multilevel weighted least squares polynomial approximation

et al. 2020

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Weighted least squares polynomial approximation uses random samples to determine projections of functions onto spaces of polynomials. It has been shown that, using an optimal distribution of sample locations, the number of samples required to achieve quasi-optimal approximation in a given polynomial subspace scales, up to a logarithmic factor, linearly in the dimension of this space. However, in many applications, the computation of samples includes a numerical discretization error. Thus, obtaining polynomial approximations with a single level method can become prohibitively expensive, as it requires a sufficiently large number of samples, each computed with a sufficiently small discretization error. As a solution to this problem, we propose a multilevel method that utilizes samples computed with different accuracies and is able to match the accuracy of single-level approximations with reduced computational cost. We derive complexity bounds under certain assumptions about polynomial approximability and sample work. Furthermore, we propose an adaptive algorithm for situations where such assumptions cannot be verified a priori. Finally, we provide an efficient algorithm for the sampling from optimal distributions and an analysis of computationally favorable alternative distributions. Numerical experiments underscore the practical applicability of our method.

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“…. , L}, but instead relies on a simple union bound (see Equation (29)). Thus, we could alternatively first create Γ L and then define all Γ l with l < L as subsets of it.…”

Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Multilevel weighted least squares polynomial approximation

et al. 2020

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“…Proposition 3.8. [24,31] Suppose U = du j=1 U j , with U j ⊂ R bounded. Let k(θ) = k sepMat (θ) be a separable Matèrn covariance kernel with θ ∈ S, for some compact set S ⊆ (0, ∞) 2du+1 .…”

Section: Predictive Meanmentioning

confidence: 99%

“…For our further analysis, we now want to make use of the convergence results from [24], related results are also found in [31] and the references in [24]. For the design points D N , we will use Smolyak sparse grids [4].…”

Section: Predictive Meanmentioning

confidence: 99%

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Convergence of Gaussian Process Regression with Estimated Hyper-parameters and Applications in Bayesian Inverse Problems

Teckentrup¹

2019

Preprint

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This work is concerned with the convergence of Gaussian process regression. A particular focus is on hierarchical Gaussian process regression, where hyper-parameters appearing in the mean and covariance structure of the Gaussian process emulator are a-priori unknown, and are learnt from the data, along with the posterior mean and covariance. We work in the framework of empirical Bayes, where a point estimate of the hyper-parameters is computed, using the data, and then used within the standard Gaussian process prior to posterior update. We provide a convergence analysis that (i) holds for any continuous function f to be emulated; and (ii) shows that convergence of Gaussian process regression is unaffected by the additional learning of hyper-parameters from data, and is guaranteed in a wide range of scenarios. As the primary motivation for the work is the use of Gaussian process regression to approximate the data likelihood in Bayesian inverse problems, we provide a bound on the error introduced in the Bayesian posterior distribution in this context.

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Smolyak’s Algorithm: A Powerful Black Box for the Acceleration of Scientific Computations

Tempone

Wolfers

2018

Lecture Notes in Computational Science and Engineering

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We provide a general discussion of Smolyak's algorithm for the acceleration of scientific computations. The algorithm first appeared in Smolyak's work on multidimensional integration and interpolation. Since then, it has been generalized in multiple directions and has been associated with the keywords: sparse grids, hyperbolic cross approximation, combination technique, and multilevel methods. Variants of Smolyak's algorithm have been employed in the computation of highdimensional integrals in finance, chemistry, and physics, in the numerical solution of partial and stochastic differential equations, and in uncertainty quantification. Motivated by this broad and ever-increasing range of applications, we describe a general framework that summarizes fundamental results and assumptions in a concise application-independent manner.

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Sparse approximation of multilinear problems with applications to kernel-based methods in UQ

Cited by 4 publications

References 38 publications

Multilevel weighted least squares polynomial approximation

Multilevel weighted least squares polynomial approximation

Convergence of Gaussian Process Regression with Estimated Hyper-parameters and Applications in Bayesian Inverse Problems

Smolyak’s Algorithm: A Powerful Black Box for the Acceleration of Scientific Computations

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