Computing Multivariate Fekete and Leja Points by Numerical Linear Algebra

Bos, Len; Marchi, Stefano De; Sommariva, Alvise; Vianello, Marco

doi:10.1137/090779024

Cited by 87 publications

(118 citation statements)

References 26 publications

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“…The AFP are good approximation of the true Fekete points as proved in [4] and they are determined by a "simple" numerical procedure which turns out to be equivalent to the QR factorization with column pivoting of the transposed of the rectangular Vandermonde matrix associated to the approximation process.…”

Section: Approximate Fekete Pointsmentioning

confidence: 99%

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A New Quasi-Monte Carlo Technique Based on Nonnegative Least Squares and Approximate Fekete Points

Bittante

Marchi

Elefante

2016

Numer. Math. Theory Methods Appl.

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Abstract. The computation of integrals in higher dimensions and on general domains, when no explicit cubature rules are known, can be "easily" addressed by means of the quasi-Monte Carlo method. The method, simple in its formulation, becomes computationally inefficient when the space dimension is growing and the integration domain is particularly complex. In this paper we present two new approaches to the quasi-Monte Carlo method for cubature based on nonnegative least squares and approximate Fekete points. The main idea is to use less points and especially good points for solving the system of the moments. Good points are here intended as points with good interpolation properties, due to the strict connection between interpolation and cubature. Numerical experiments show that, in average, just a tenth of the points should be used mantaining the same approximation order of the quasi-Monte Carlo method. The method has been satisfactory applied to 2 and 3-dimensional problems on quite complex domains.

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Section: Approximate Fekete Pointsmentioning

confidence: 99%

“…For details about the AFP algorithm and its Matlab implementation we suggest the readers to refer to the papers [4,5,28]. Here we simply recall that at the web page http://www.math.unipd.it/∼marcov/CAAsoft.html once can find all the necessary scripts for polynomial fitting and interpolation on WAMs.…”

Section: Approximate Fekete Pointsmentioning

confidence: 99%

A New Quasi-Monte Carlo Technique Based on Nonnegative Least Squares and Approximate Fekete Points

Bittante

Marchi

Elefante

2016

Numer. Math. Theory Methods Appl.

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“…As a consequence of the considerations above (see [5] for a more detailed discussion), the computation of Discrete Extremal Sets can be done by few basic linear algebra operations, corresponding to the the LU factorization with row pivoting of the Vandermonde matrix (cf. [14]), and to the QR factorization with column pivoting of the transposed Vandermonde matrix (cf.…”

Section: Fekete Points and Discrete Extremal Setsmentioning

confidence: 99%

“…When the conditioning of the Vandermonde matrices is too high, the algorithms can still be used provided that a preliminary iterated orthogonalization, that is a change to a discrete orthogonal basis, is performed as in Section 2.1; cf. [4,5,15].…”

Section: Fekete Points and Discrete Extremal Setsmentioning

confidence: 99%

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Weakly Admissible Meshes and Discrete Extremal Sets

Bos¹

2011

NMTMA

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Efficient sampling for polynomial chaos‐based uncertainty quantification and sensitivity analysis using weighted approximate Fekete points

Burk

Narayan

Orr

2020

Numer Methods Biomed Eng

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Performing uncertainty quantification (UQ) and sensitivity analysis (SA) is vital when developing a patient‐specific physiological model because it can quantify model output uncertainty and estimate the effect of each of the model's input parameters on the mathematical model. By providing this information, UQ and SA act as diagnostic tools to evaluate model fidelity and compare model characteristics with expert knowledge and real world observation. Computational efficiency is an important part of UQ and SA methods and thus optimization is an active area of research. In this work, we investigate a new efficient sampling method for least‐squares polynomial approximation, weighted approximate Fekete points (WAFP). We analyze the performance of this method by demonstrating its utility in stochastic analysis of a cardiovascular model that estimates changes in oxyhemoglobin saturation response. Polynomial chaos (PC) expansion using WAFP produced results similar to the more standard Monte Carlo in quantifying uncertainty and identifying the most influential model inputs (including input interactions) when modeling oxyhemoglobin saturation, PC expansion using WAFP was far more efficient. These findings show the usefulness of using WAFP based PC expansion to quantify uncertainty and analyze sensitivity of a oxyhemoglobin dissociation response model. Applying these techniques could help analyze the fidelity of other relevant models in preparation for clinical application.

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Computing Multivariate Fekete and Leja Points by Numerical Linear Algebra

Cited by 87 publications

References 26 publications

A New Quasi-Monte Carlo Technique Based on Nonnegative Least Squares and Approximate Fekete Points

A New Quasi-Monte Carlo Technique Based on Nonnegative Least Squares and Approximate Fekete Points

Weakly Admissible Meshes and Discrete Extremal Sets

Efficient sampling for polynomial chaos‐based uncertainty quantification and sensitivity analysis using weighted approximate Fekete points

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