Computation of induced orthogonal polynomial distributions

Narayan, Akil

doi:10.1553/etna_vol50s71

Cited by 16 publications

(15 citation statements)

References 18 publications

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“…On the other hand, using a biased measure defined by multiplying an existing underlying measure by κ (which is how µ is defined) has become popular in recent years, and was first investigated for least-squares problems in [16,5]. A measure constructed in this way is called an induced measure, and computational algorithms for sampling from several multivariate induced distributions exist [22], and sampling from a multivariate induced measure for gPC approximations is considered in [19] using a different strategy to construct the basis Φ j . Notice that in all above mentioned references, the input density is known in prior.…”

Section: 3mentioning

confidence: 99%

“…However, an explicit form for the equilibrium measure is not known in general; alternatively the induced distribution introduced in [5] is an attractive sampler for its optimal stability properties. Algorithms for generating samples from fairly general classes of induced distributions are also available [22]. However, such algorithms are less helpful when the underlying distribution is not known.…”

Section: Introductionmentioning

confidence: 99%

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Sparse Approximation of Data-Driven Polynomial Chaos Expansions: An Induced Sampling Approach

Guo¹,

Narayan²,

Liu³

et al. 2020

CMR

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One of the open problems in the field of forward uncertainty quantification (UQ) is the ability to form accurate assessments of uncertainty having only incomplete information about the distribution of random inputs. Another challenge is to efficiently make use of limited training data for UQ predictions of complex engineering problems, particularly with high dimensional random parameters. We address these challenges by combining data-driven polynomial chaos expansions with a recently developed preconditioned sparse approximation approach for UQ problems. The first task in this two-step process is to employ the procedure developed in (Ahlfeld et al. 2016, [1]) to construct an "arbitrary" polynomial chaos expansion basis using a finite number of statistical moments of the random inputs. The second step is a novel procedure to effect sparse approximation via 1 minimization in order to quantify the forward uncertainty. To enhance the performance of the preconditioned 1 minimization problem, we sample from the so-called induced distribution, instead of using Monte Carlo (MC) sampling from the original, unknown probability measure. We demonstrate on test problems that induced sampling is a competitive and often better choice compared with sampling from asymptotically optimal measures (such as the equilibrium measure) when we have incomplete information about the distribution. We demonstrate the capacity of the proposed induced sampling algorithm via sparse representation with limited data on test functions, and on a Kirchoff plating bending problem with random Young's modulus.

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Section: 3mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Sparse Approximation of Data-Driven Polynomial Chaos Expansions: An Induced Sampling Approach

Guo¹,

Narayan²,

Liu³

et al. 2020

CMR

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“…In this section we present methods for discretizing the domain based on the support of the parameters and their distributions. The survey builds on a recent review of sampling techniques for polynomial least squares [34] and from more recent texts including Narayan [48,49] and Jakeman and Narayan [37]. Intuitively, the simplest sampling strategy for polynomial least squares is to generate random Monte-Carlo type samples based on the joint density ρ(ζ ζ ζ ).…”

Section: Selecting a Sampling Strategymentioning

confidence: 99%

“…In [35], the authors devise an MCMC strategy which seeks to find µ, thereby explicitly minimizing their coherence parameter (a weighted form of K ∞ ). In [48], formulations for computing µ via induced distributions is detailed. In our numerical experiments investigating the condition numbers of matrices obtained via such induced samples-in a similar vein to the Figures 3-the condition numbers were found to be comparable to those from Christoffel sampling.…”

Section: Coherence and Induced Samplingmentioning

confidence: 99%

Quadrature Strategies for Constructing Polynomial Approximations

Seshadri

Iaccarino

Ghisu

2018

Uncertainty Modeling for Engineering Applications

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Finding suitable points for multivariate polynomial interpolation and approximation is a challenging task. Yet, despite this challenge, there has been tremendous research dedicated to this singular cause. In this paper, we begin by reviewing classical methods for finding suitable quadrature points for polynomial approximation in both the univariate and multivariate setting. Then, we categorize recent advances into those that propose a new sampling approach and those centered on an optimization strategy. The sampling approaches yield a favorable discretization of the domain, while the optimization methods pick a subset of the discretized samples that minimize certain objectives. While not all strategies follow this two-stage approach, most do. Sampling techniques covered include subsampling quadratures, Christoffel, induced and Monte Carlo methods. Optimization methods discussed range from linear programming ideas and Newton's method to greedy procedures from numerical linear algebra. Our exposition is aided by examples that implement some of the aforementioned strategies.

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“…The results in [11] suggest an exact sampling method and show optimal convergence estimates in the non-asymptotic case. The work in [27] provides efficient computational methods for exact sampling in the non-asymptotic case.…”

Section: Introductionmentioning

confidence: 99%

Weighted Approximate Fekete Points: Sampling for Least-Squares Polynomial Approximation

Guo¹,

Narayan²,

Yan³

et al. 2018

SIAM J. Sci. Comput.

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We propose and analyze a weighted greedy scheme for computing deterministic sample configurations in multidimensional space for performing least-squares polynomial approximations on L 2 spaces weighted by a probability density function. Our procedure is a particular weighted version of the approximate Fekete points method, with the weight function chosen as the (inverse) Christoffel function. Our procedure has theoretical advantages: when linear systems with optimal condition number exist, the procedure finds them. In the one-dimensional setting with any density function, our greedy procedure almost always generates optimally-conditioned linear systems. Our method also has practical advantages: our procedure is impartial to compactness of the domain of approximation, and uses only pivoted linear algebraic routines. We show through numerous examples that our sampling design outperforms competing randomized and deterministic designs when the domain is both low and high dimensional.Accurately estimating the coefficients f j of the expansion is important since these coefficients can be easily manipulated to infer revealing properties, such as statistical moments or parametric sensitivities. Many numerical techniques on how to obtain the polynomial coefficients in UQ problems have been developed in recent years. While early development often 1 The initial sample for the greedy method can be any point on R not coinciding with N − 1 isolated points; see Thereom 3.2 for details.

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Computation of induced orthogonal polynomial distributions

Cited by 16 publications

References 18 publications

Sparse Approximation of Data-Driven Polynomial Chaos Expansions: An Induced Sampling Approach

Sparse Approximation of Data-Driven Polynomial Chaos Expansions: An Induced Sampling Approach

Quadrature Strategies for Constructing Polynomial Approximations

Weighted Approximate Fekete Points: Sampling for Least-Squares Polynomial Approximation

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