Convergence of Sparse Collocation for Functions of Countably Many Gaussian Random Variables (with Application to Elliptic PDEs)

Ernst, Oliver G.; Sprungk, Björn; Tamellini, Lorenzo

doi:10.1137/17m1123079

Cited by 41 publications

(61 citation statements)

References 42 publications

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“…for the orthonormal Hermite polynomials H n , n ≥ 0, defined in (2.6). The proof is based on the Cramér inequality, e.g., in [1], that is made aware from [19,Lemma 14], and the Markoff's theorem, e.g., in [48].…”

Section: Convergence Analysismentioning

confidence: 99%

“…Proof. The bound is a result of [19,Proposition 18], which states that there exists a constant C such that N p ≤ CN 2 .…”

Section: Convergence Analysismentioning

confidence: 99%

“…More recently, a dimension-independent convergence rate of the polynomial chaos (based on Hermite polynormials) approximation for an elliptic problem with lognormal coefficients is obtained in [28], whose convergence rate is improved in [3]. A convergence result based on [3] is obtained in [19] for a sparse collocation method.…”

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confidence: 99%

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Sparse quadrature for high-dimensional integration with Gaussian measure

Chen

2018

ESAIM: M2AN

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In this work we analyze the dimension-independent convergence property of an abstract sparse quadrature scheme for numerical integration of functions of high-dimensional parameters with Gaussian measure. Under certain assumptions of the exactness and the boundedness of univariate quadrature rules as well as the regularity of the parametric functions with respect to the parameters, we obtain the convergence rate O(N −s ), where N is the number of indices, and s is independent of the number of the parameter dimensions. Moreover, we propose both an a-priori and an a-posteriori schemes for the construction of a practical sparse quadrature rule and perform numerical experiments to demonstrate their dimension-independent convergence rates.

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Section: Convergence Analysismentioning

confidence: 99%

“…Proof. The bound is a result of [19,Proposition 18], which states that there exists a constant C such that N p ≤ CN 2 .…”

Section: Convergence Analysismentioning

confidence: 99%

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Sparse quadrature for high-dimensional integration with Gaussian measure

Chen

2018

ESAIM: M2AN

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“…j=1 . The quadrature points should be chosen according to the underlying probability measure ρ; see, e.g., [5,46]. Moreover, for refinement purposes it is advantageous if the quadrature points are chosen to be "nested", i.e., {t β,j }…”

Section: Multi-index Stochastic Collocation (Misc)mentioning

confidence: 99%

IGA-based multi-index stochastic collocation for random PDEs on arbitrary domains

Beck

Tamellini

Tempone

2019

Computer Methods in Applied Mechanics and Engineering

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This paper proposes an extension of the Multi-Index Stochastic Collocation (MISC) method for forward uncertainty quantification (UQ) problems in computational domains of shape other than a square or cube, by exploiting isogeometric analysis (IGA) techniques. Introducing IGA solvers to the MISC algorithm is very natural since they are tensor-based PDE solvers, which are precisely what is required by the MISC machinery. Moreover, the combination-technique formulation of MISC allows the straight-forward reuse of existing implementations of IGA solvers. We present numerical results to showcase the effectiveness of the proposed approach. Highlights• Isogeometric solvers used in a MISC framework for forward UQ problems.• The combination-technique formulation of the method allows straight-forward reuse of legacy IGA solvers.

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“…To overcome this bottleneck, state-of-the-art approaches rely on algorithms for the adaptive construction of sparse approximations [8][9][10][16][17][18].Adaptive sparse approximation algorithms for Lagrange interpolation methods typically employ nested sequences of univariate interpolation nodes. While not strictly necessary [19], nested node sequences are helpful for the efficient construction of sparse grid interpolation algorithms. Nested node sequences are readily available for the well studied cases of uniformly or normally distributed random variables (RVs) [20,21], but not for more general probability measures.…”

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confidence: 99%

Approximation and Uncertainty Quantification of Systems with Arbitrary Parameter Distributions Using Weighted Leja Interpolation

Loukrezis

Gersem

2020

Algorithms

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Approximation and uncertainty quantification methods based on Lagrange interpolation are typically abandoned in cases where the probability distributions of one or more system parameters are not normal, uniform, or closely related distributions, due to the computational issues that arise when one wishes to define interpolation nodes for general distributions. This paper examines the use of the recently introduced weighted Leja nodes for that purpose. Weighted Leja interpolation rules are presented, along with a dimension-adaptive sparse interpolation algorithm, to be employed in the case of high-dimensional input uncertainty. The performance and reliability of the suggested approach is verified by four numerical experiments, where the respective models feature extreme value and truncated normal parameter distributions. Furthermore, the suggested approach is compared with a well-established polynomial chaos method and found to be either comparable or superior in terms of approximation and statistics estimation accuracy.Journal Not Specified 2020, xx, 1 2 of 21 interpolation schemes [5,6] or generalized polynomial chaos (gPC) [7], where the latter approach is typically based on least squares (LS) regression [8][9][10] or pseudo-spectral projection methods [11][12][13], or, less commonly, on interpolation [14]. Assuming a smooth input-output dependence, spectral UQ methods provide fast convergence rates, in some cases even of exponential order. However, their performance decreases rapidly for an increasing number of input parameters due to the curse of dimensionality [15]. To overcome this bottleneck, state-of-the-art approaches rely on algorithms for the adaptive construction of sparse approximations [8][9][10][16][17][18].Adaptive sparse approximation algorithms for Lagrange interpolation methods typically employ nested sequences of univariate interpolation nodes. While not strictly necessary [19], nested node sequences are helpful for the efficient construction of sparse grid interpolation algorithms. Nested node sequences are readily available for the well studied cases of uniformly or normally distributed random variables (RVs) [20,21], but not for more general probability measures. In principle, one could derive nested interpolation (or quadrature) rules tailored to an arbitrary probability density function (PDF) [22,23], however, this is a rather cumbersome task for which dedicated analyses are necessary each and every time a new PDF is considered.Diversely, the main requirement in gPC approximations, either full or sparse, is to find a complete set of polynomials that are orthogonal to each other with respect to the input parameter PDFs. In the case of arbitrary PDFs, such polynomials can be numerically constructed [24][25][26]. As a result, despite the fact that according to a number of studies Lagrange interpolation methods enjoy an advantage over gPC-based methods with respect to the error-cost ratio [27][28][29], they are abandoned whenever the input parameter PDFs are not uniform, or normal, or closely ...

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Convergence of Sparse Collocation for Functions of Countably Many Gaussian Random Variables (with Application to Elliptic PDEs)

Cited by 41 publications

References 42 publications

Sparse quadrature for high-dimensional integration with Gaussian measure

Sparse quadrature for high-dimensional integration with Gaussian measure

IGA-based multi-index stochastic collocation for random PDEs on arbitrary domains

Approximation and Uncertainty Quantification of Systems with Arbitrary Parameter Distributions Using Weighted Leja Interpolation

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