Near optimal polynomial regression on norming meshes

Bos, Len; Pianzola, Federico; Vianello, Marco

doi:10.1109/sampta45681.2019.9030910

Cited by 2 publications

(2 citation statements)

References 23 publications

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“…Following References [18,19], we can however effectively compute a design which has the same G-efficiency of u(k) but a support with a cardinality not exceeding N 2m = dim(P d 2m (X)), where in many applications N 2m card(X), obtaining a remarkable compression of the near optimal design. The theoretical foundation is a generalized version [15] of Tchakaloff Theorem [20] on positive quadratures, which asserts that for every measure on a compact set Ω ⊂ R d there exists an algebraic quadrature formula exact on P d n (Ω)), with positive weights, nodes in Ω and cardinality not exceeding N n = dim(P d n (Ω).…”

Section: Computing Near G-optimal Compressed Designsmentioning

confidence: 99%

dCATCH—A Numerical Package for d-Variate near G-Optimal Tchakaloff Regression via Fast NNLS

2020

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We provide a numerical package for the computation of a d-variate near G-optimal polynomial regression design of degree m on a finite design space X ⊂ R d , by few iterations of a basic multiplicative algorithm followed by Tchakaloff-like compression of the discrete measure keeping the reached G-efficiency, via an accelerated version of the Lawson-Hanson algorithm for Non-Negative Least Squares (NNLS) problems. This package can solve on a personal computer large-scale problems where c a r d ( X ) × dim ( P 2 m d ) is up to 10 8 – 10 9 , being dim ( P 2 m d ) = 2 m + d d = 2 m + d 2 m . Several numerical tests are presented on complex shapes in d = 3 and on hypercubes in d > 3 .

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Section: Computing Near G-optimal Compressed Designsmentioning

confidence: 99%

dCATCH—A Numerical Package for d-Variate near G-Optimal Tchakaloff Regression via Fast NNLS

2020

Self Cite

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“…Such metrics can be moment matching conditions [37,33] or matrix determinants [53], but a wide variety of such metrics can be devised [41,25]. More recently, compression techniques based on algorithms inspired by Carathéodory's Theorem and Tchakaloff's theorem have been used to reduce a very large discrete set of samples into a smaller set of samples over which least squares procedures are stable [45,34,10]. These procedures aim to generate parsimonious least squares designs that matches moments of a large discrete sample set instead of a continous density.…”

Section: Generation Of Designs Via Optimizationmentioning

confidence: 99%

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

Burk,

Narayan,

Orr

2020

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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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Near optimal polynomial regression on norming meshes

Abstract: We connect the approximation theoretic notions of polynomial norming mesh and Tchakaloff-like quadrature to the statistical theory of optimal designs, obtaining near optimal polynomial regression at a near optimal number of sampling locations on domains with different shapes.

Cited by 2 publications

References 23 publications

dCATCH—A Numerical Package for d-Variate near G-Optimal Tchakaloff Regression via Fast NNLS

dCATCH—A Numerical Package for d-Variate near G-Optimal Tchakaloff Regression via Fast NNLS

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

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