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Effectively Subsampled Quadratures For Least Squares Polynomial Approximations [article]

Pranay Seshadri, Akil Narayan, Sankaran Mahadevan
2017 arXiv   pre-print
For polynomial interpolation using hyperbolic or total order sets, we then solve the following square least squares problem.  ...  For polynomial approximation, we use a column pruning heuristic that removes columns based on the highest total orders and then solves the tall least squares problem.  ...  On matrices Effectively subsampled quadratures In this section we describe polynomial least squares via effectively subsampled quadratures.  ... 
arXiv:1601.05470v4 fatcat:ggr52udoizfhll5tq4s4f23wja

Effective-Quadratures (EQ): Polynomials for Computational Engineering Studies

Pranay Seshadri, Geoffrey Parks
2017 Journal of Open Source Software  
The code also has routines for effectively subsampling an existing tensor grid for computing least squares approximations, based on the work of (Seshadri, Narayan, and Mahadevan 2016), which uses a QR  ...  Effective-Quadratures (EQ) is a suite of tools for generating polynomials for parametric computational studies.  ...  The code also has routines for effectively subsampling an existing tensor grid for computing least squares approximations, based on the work of (Seshadri, Narayan, and Mahadevan 2016), which uses a QR  ... 
doi:10.21105/joss.00166 fatcat:wla5xvzdlzaavjkktdwj2tr6wi

Quadrature Strategies for Constructing Polynomial Approximations [article]

Pranay Seshadri, Gianluca Iaccarino, Tiziano Ghisu
2018 arXiv   pre-print
In this paper, we begin by reviewing classical methods for finding suitable quadrature points for polynomial approximation in both the univariate and multivariate setting.  ...  Finding suitable points for multivariate polynomial interpolation and approximation is a challenging task.  ...  The authors are also grateful to Akil Narayan for numerous discussions on polynomial approximations and quadratures.  ... 
arXiv:1805.07296v1 fatcat:shrc53aqrbfszg3lruffa7o534

Subperiodic trigonometric subsampling: A numerical approach

Alvise Sommariva, Marco Vianello
2017 Applicable Analysis and Discrete Mathematics  
We show that Gauss-Legendre quadrature applied to trigonometric polynomials on subintervals of the period can be competitive with subperiodic trigonometric Gaussian quadrature.  ...  For example with intervals corresponding to few angular degrees, relevant for regional scale models on the earth surface, we see a subsampling ratio of one order of magnitude already at moderate trigonometric  ...  Observe that from the empirical formula (16) we expect in practice a subsampling effect (at least for large k) only for 0.54 ω < 1, that is ω < 1.85 or (13) (log scale) computed with g ε (u) (blue  ... 
doi:10.2298/aadm1702470s fatcat:5tse3bg6fnd65m5gszlqizolcm

Caratheodory-Tchakaloff Subsampling [article]

Federico Piazzon, Alvise Sommariva, Marco Vianello
2016 arXiv   pre-print
Least Squares.  ...  We also give an algorithm that computes the corresponding Caratheodory-Tchakaloff (CATCH) points and weights for polynomial spaces on compact sets and manifolds in 2D and 3D.  ...  In many situations CATCH subsampling can produce a high Compression Ratio, namely when N ≪ M like for example in polynomial least squares approximation [28] or in QMC (Quasi-Monte Carlo) integration  ... 
arXiv:1611.02065v2 fatcat:drpe2ygniveqnhqjbpgjmvk724

Caratheodory-Tchakaloff Least Squares

Federico Piazzon, Alvise Sommariva, Marco Vianello
2017 2017 International Conference on Sampling Theory and Applications (SampTA)  
Least Squares.  ...  We also give an algorithm that computes the corresponding Caratheodory-Tchakaloff (CATCH) points and weights for polynomial spaces on compact sets and manifolds in 2D and 3D. 2010 AMS subject classification  ...  This research has been accomplished within the RITA (Research ITalian network on Approximation).  ... 
doi:10.1109/sampta.2017.8024337 fatcat:kdepvkg4zzcw5n6tz7dbi35tdi

Weighted discrete least-squares polynomial approximation using randomized quadratures

Tao Zhou, Akil Narayan, Dongbin Xiu
2015 Journal of Computational Physics  
We discuss the problem of polynomial approximation of multivariate functions using discrete least squares collocation.  ...  We propose to construct the least squares approximation on points randomly and uniformly sampled from tensor product Gaussian quadrature points.  ...  We now present the method of random Gauss quadrature for least squares approximation.  ... 
doi:10.1016/j.jcp.2015.06.042 fatcat:3m4afbdm3bb23byzzc3hw7lxoi

Gaussian Quadrature for Kernel Features

Tri Dao, Christopher De Sa, Christopher Ré
2017 Advances in Neural Information Processing Systems  
We investigate some alternative schemes for constructing feature maps that are deterministic, rather than random, by approximating the kernel in the frequency domain using Gaussian quadrature.  ...  We show that deterministic feature maps can be constructed, for any γ > 0, to achieve error ε with O(eγ + ε-1/γ) samples as ε goes to 0.  ...  Government is authorized to reproduce and distribute reprints for Governmental purposes notwithstanding any copyright notation thereon.  ... 
pmid:29398882 pmcid:PMC5791159 fatcat:vlb7usdxdvb3hb6cnhise2bawa

Sensitivity Analysis of a Coupled Hydrodynamic-Vegetation Model Using the Effectively Subsampled Quadratures Method

Tarandeep S. Kalra, Alfredo Aretxabaleta, Pranay Seshadri, Neil K. Ganju, Alexis Beudin
2017 Geoscientific Model Development Discussions  
In this study, we evaluate the sensitivity of the flow and wave dynamics to vegetation parameters using Sobol' indices and a least squares polynomial approach referred to as Effective Quadratures method  ...  The effect of vegetation has been incorporated into the Coupled-Ocean-Atmosphere-Wave-Sediment Transport (COAWST) Modeling System.  ...  Method for Sensitivity Analysis: Polynomial Least Squares Polynomial techniques are ubiquitous in the field of uncertainty quantification and model approximation.  ... 
doi:10.5194/gmd-2017-107 fatcat:l3637jq33vfafkvxsrfkamnvmi

Sensitivity analysis of a coupled hydrodynamic-vegetation model using the effectively subsampled quadratures method (ESQM v5.2)

Tarandeep S. Kalra, Alfredo Aretxabaleta, Pranay Seshadri, Neil K. Ganju, Alexis Beudin
2017 Geoscientific Model Development  
In this study, we evaluate the sensitivity of the flow and wave dynamics to vegetation parameters using Sobol' indices and a least squares polynomial approach referred to as the Effective Quadratures method  ...  The effect of vegetation has been incorporated into the Coupled Ocean–Atmosphere–Wave–Sediment Transport (COAWST) modeling system.  ...  We thank Jeremy Testa at the University of Maryland Center for Environmental Science for providing us guidance on the ranges of vegetation for sensitivity studies in the early stages of work.  ... 
doi:10.5194/gmd-10-4511-2017 fatcat:k6epejquubg77jxs46opnia5n4

Stochastic collocation methods via L_1 minimization using randomized quadratures [article]

Ling Guo, Akil Narayan, Tao Zhou, Yuhang Chen
2016 arXiv   pre-print
In this work, we discuss the problem of approximating a multivariate function via ℓ_1 minimization method, using a random chosen sub-grid of the corresponding tensor grid of Gaussian points.  ...  The independent variables of the function are assumed to be random variables, and thus, the framework provides a non-intrusive way to construct the generalized polynomial chaos expansions, stemming from  ...  Popular methods for achieving this include sparse grids approximation [2, 1, 3, 14, 16, 17, 26, 33] , pseudo orthogonal projection [37] , interpolation [30] , and least squares approach [40, 51, 9  ... 
arXiv:1602.00995v2 fatcat:75pfrhgz7jcxnlu6uae7pnvnhm

Optimal polynomial meshes and Caratheodory-Tchakaloff submeshes on the sphere [article]

Paul Leopardi, Alvise Sommariva, Marco Vianello
2016 arXiv   pre-print
From these we extract Caratheodory-Tchakaloff (CATCH) submeshes for compressed Least Squares fitting.  ...  Using the notion of Dubiner distance, we give an elementary proof of the fact that good covering point configurations on the 2-sphere are optimal polynomial meshes.  ...  Then, T 2n is a weakly admissible mesh for the sphere with cardinality N 2n = dim(P 3 2n (S 2 )) = (2n + 1) 2 , and (20) holds for the corresponding weighted Least Squares polynomial approximation L w  ... 
arXiv:1612.04952v1 fatcat:7c7atjgnszhfnnnohecuwk5qui

Gaussian Quadrature for Kernel Features [article]

Tri Dao, Christopher De Sa, Christopher Ré
2018 arXiv   pre-print
We investigate some alternative schemes for constructing feature maps that are deterministic, rather than random, by approximating the kernel in the frequency domain using Gaussian quadrature.  ...  We show that deterministic feature maps can be constructed, for any γ > 0, to achieve error ϵ with O(e^e^γ + ϵ^-1/γ) samples as ϵ goes to 0.  ...  One possible construction involves selecting random sample points ω i , and then solving (3) for the values of a i using a non-negative least squares (NNLS) algorithm.  ... 
arXiv:1709.02605v3 fatcat:zme7ke4irreknipx2kzjypbw2m

No-Trick (Treat) Kernel Adaptive Filtering using Deterministic Features [article]

Kan Li, Jose C. Principe
2019 arXiv   pre-print
Recently, deterministic, rather than random construction has been shown to outperform RFFs, by approximating the kernel in the frequency domain using Gaussian quadrature.  ...  Random features are prone to large variances in performance, especially for smaller dimensions.  ...  The range of ±1 has been shown to be effective for our simulation. A square-root formulation to reduce numerical error by improving precision and stability can also be used. C. Polynomial vs.  ... 
arXiv:1912.04530v1 fatcat:p6qo5tv7sjfzfjkmqqmizu4g3q

An overview of uncertainty quantification techniques with application to oceanic and oil-spill simulations

Mohamed Iskandarani, Shitao Wang, Ashwanth Srinivasan, W. Carlisle Thacker, Justin Winokur, Omar M. Knio
2016 Journal of Geophysical Research - Oceans  
An integral plume model for simulating the Deepwater Horizon oil-gas blowout provides examples for illustrating the different techniques.  ...  Three of these techniques use polynomial chaos (PC) expansions to construct the model proxy, but they differ in their approach to determining the expansions' coefficients; the fourth technique uses Gaussian  ...  Seshadri et al., Optimal quadrature subsampling for least squares polynomial approximations, arXiv:1601.05470, 2016] have proposed optimal and stable sampling strategies for least squares polynomial approximations  ... 
doi:10.1002/2015jc011366 fatcat:lsypgtfyungc3nfre4veowvjd4
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