Cardinal Interpolation with Radial Basis Functions: An Integral Transform Approach

Buhmann, Martin D.

doi:10.1007/978-3-0348-7298-0_6

Cited by 4 publications

(8 citation statements)

References 3 publications

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“…The regression model

is used to predict the state

by computing

This method is called POD with interpolation due to this regression function acting on the latent variables. Common choices for the interpolatory map are radial basis functions, 40 Gaussian process, 41 , 42 cubic splines, 43 , 44 or artificial neural networks. 45 , 46 We are going to show how to compute an efficient approximation of such a map by exploiting only the directions of maximal variations in a multi‐fidelity setting, without the need to perform additional simulations.…”

Section: Reduced Order Modelsmentioning

confidence: 99%

A multifidelity approach coupling parameter space reduction and nonintrusive POD with application to structural optimization of passenger ship hulls

Tezzele

Fabris

Sidari

et al. 2022

Numerical Meth Engineering

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Nowadays, the shipbuilding industry is facing a radical change toward solutions with a smaller environmental impact. This can be achieved with low emissions engines, optimized shape designs with lower wave resistance and noise generation, and by reducing the metal raw materials used during the manufacturing. This work focuses on the last aspect by presenting a complete structural optimization pipeline for modern passenger ship hulls which exploits advanced model order reduction techniques to reduce the dimensionality of both input parameters and outputs of interest. We introduce a novel approach which incorporates parameter space reduction through active subspaces into the proper orthogonal decomposition with interpolation method. This is done in a multi‐fidelity setting. We test the whole framework on a simplified model of a midship section and on the full model of a passenger ship, controlled by 20 and 16 parameters, respectively. We present a comprehensive error analysis and show the capabilities and usefulness of the methods especially during the preliminary design phase, finding new unconsidered designs while handling high dimensional parameterizations.

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“…The regression model

is used to predict the state

by computing

Section: Reduced Order Modelsmentioning

confidence: 99%

A multifidelity approach coupling parameter space reduction and nonintrusive POD with application to structural optimization of passenger ship hulls

Tezzele

Fabris

Sidari

et al. 2022

Numerical Meth Engineering

View full text Add to dashboard Cite

show abstract

“…Using the form of uj's Fourier transform (3.9), we get that this is the same as f (3.12) We require an auxiliary result in order to bound this expression. The following lemma which is a consequence of Lemma 7 in [3] shows that we get <j>(t) ~ (1 4-\\t||)~n~2. An application of the theorem we just proved shows that, if for instance h = 0(<5 3n/2+3 ), then we get convergence order O(/i 2/3 ) for the approximant in any dimension.…”

Section: -2£«$(x)*(x -Jh) + J2 Eandm^mw* -Jh)mentioning

confidence: 86%

“…We want to apply the results in were already considered in [11,3], but they were not of compact support there, i.e., the weight g was not compactly supported. We require 0 > 0 everywhere.…”

Section: The Radial Basis Functionsmentioning

confidence: 99%

“…Thus the result is clearly of compact support. Similar radial basis functions that are related to multiply monotone functions and their Williamson representation [18] were already considered in [11,3], but they were not of compact support there, i.e., the weight g was not compactly supported. We require 0 > 0 everywhere.…”

Section: The Radial Basis Functionsmentioning

confidence: 99%

“…Viewed as functions 4>, they are piecewise polynomial functions which have nonnegative, nontrivial Fourier transform, when transformed as functions <f> m K"< This means, by a famous theorem of Bochner, that the interpolation matrices for interpolation at the centres are positive definite if the centres are distinct and lie in R\ Therefore they are called positive definite functions. For his purpose, Wu makes essential use of nonnegativity of the Fourier transform of <j>, if <j> is taken from a set of certain multiply monotone functions, see also [11,3]. Specifically, he creates positive definite radial functions <f> in one dimension first, and then uses a certain differentiation operator, applied to 4>, to lift these univariate functions that give rise to positive definite interpolation matrices to higher dimensions (and lower smoothness).…”

Section: Introductionmentioning

confidence: 99%

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Radial functions on compact support

Buhmann

1998

Proceedings of the Edinburgh Mathematical Society

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In this paper, radial basis functions that are compactly supported and give rise to positive definite interpolation matrices for scattered data are discussed. They are related to the well-known thin plate spline radial functions which are highly useful in applications for gridfree approximation methods. Also, encouraging approximation results for the compactly supported radial functions are shown.1991 Mathematics subject classification: 41A05, 41A25, 41A63, 42A82, 65D05, 65D07.

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Novel methodologies for solving the inverse unsteady heat transfer problem of estimating the boundary heat flux in continuous casting molds

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Barral

Quintela

et al. 2022

Numerical Meth Engineering

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In this article, we investigate the estimation of the transient mold‐slab heat flux in continuous casting molds given some thermocouples measurements in the mold plates. Mathematically, we can see this problem as the estimation of a Neumann boundary condition given pointwise state observations in the interior of the domain. We formulate it in a deterministic inverse problem setting. After introducing the industrial problem, we present the mold thermal model and related assumptions. Then, we formulate the boundary heat flux estimation problem in a deterministic inverse problem setting using a sequential approach according to the sequentiality of the temperature measurements. We consider different formulations of the inverse problem. For each one, we develop novel direct methodologies exploiting a space parameterization of the heat flux and the linearity of the mold model. We construct these methods to be divided into a computationally expensive offline phase that can be computed before the process starts, and a cheaper online phase to be performed during the casting process. To conclude, we test the performance of the proposed methods in two benchmark cases.

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Cardinal Interpolation with Radial Basis Functions: An Integral Transform Approach

Cited by 4 publications

References 3 publications

A multifidelity approach coupling parameter space reduction and nonintrusive POD with application to structural optimization of passenger ship hulls

A multifidelity approach coupling parameter space reduction and nonintrusive POD with application to structural optimization of passenger ship hulls

Radial functions on compact support

Novel methodologies for solving the inverse unsteady heat transfer problem of estimating the boundary heat flux in continuous casting molds

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