Rational RBF-based partition of unity method for efficiently and accurately approximating 3D objects

Perracchione, Emma

doi:10.1007/s40314-018-0592-8

Cited by 6 publications

(3 citation statements)

References 24 publications

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“…The topic can be addressed in different approximation spaces, either by using spline spaces or radial basis functions which are particularly attractive in high dimension [19]. In particular, thanks to their low computational cost and also to their better control of conditioning, two stage schemes relying on local approximations have received a lot of attention, being formulated either as partition of unity interpolation or as two-stage approximation methods, see for example [3,5,6,7,15] and references therein.…”

Section: Introductionmentioning

confidence: 99%

THB-spline approximations for turbine blade design with local B-spline approximations

Bracco¹,

Giannelli²,

Großmann³

et al. 2020

Preprint

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We consider adaptive scattered data fitting schemes with truncated hierarchical B-splines (THB-splines) based on two-stage methods for the adaptive reconstruction of industrial models. The first stage of the scheme is devoted to the computation of local least squares B-spline approximations. A simple fairness functional is suitably exploited to handle data distributions with a varying density of points locally available. Hierarchical spline quasi-interpolation based on THB-splines is considered in the second stage of the method to construct the adaptive spline surface approximating the scattered data set. A selection of examples on geometric models representing components of aircraft turbine blades highlights the performance of the scheme. The tests include a scattered data set with voids and the adaptive reconstruction of a cylinder-like surface.

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Section: Introductionmentioning

confidence: 99%

THB-spline approximations for turbine blade design with local B-spline approximations

Bracco¹,

Giannelli²,

Großmann³

et al. 2020

Preprint

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show abstract

“…More precisely, we give a very general formulation of a rational RBF expansion and we investigate under which conditions it leads to problems that admit solutions. Then, since the general form of the rational approximant is not uniquely defined, following the idea first presented in [4] and then developed also in [5,6], we introduce additional constraints. The scheme, that reduces to a largest eigenvalue problem, is extended to work with the Partition of Unity (PU) method [7], allowing to overcome the usually high complexity costs of global methods, mainly due to the solution of large linear systems.…”

Section: Introductionmentioning

confidence: 99%

Fast and stable rational RBF-based partition of unity interpolation

Marchi

Martínez

Perracchione

2019

Journal of Computational and Applied Mathematics

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“…In many real-world problems, generally speaking, interpolation is considered a problem in high dimensions (Fassino and Moller 2016;Streletza et al 2016). This type of interpolation is also known as multivariate interpolation or spatial interpolation, which is the interpolation of the functions of more than one variable (Montero et al 2010;Cavoretto 2015;Perracchione 2018;Thacker et al 2010). A basic tool for overcoming the challenges in high-dimensional interpolation is through the usage of radial basis functions (RBFs).…”

Section: Introductionmentioning

confidence: 99%

An efficient method based on RBFs for multilayer data interpolation with application in air pollution data analysis

Esmaeilbeigi

Chatrabgoun

2019

Comp. Appl. Math.

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Multivariate interpolation is a fundamental and long-studied problem, which has numerous applications in mathematics, engineering, computer science, and the natural sciences. A basic tool for solving the problem of high-dimensional interpolation is through the usage of radial basis functions (RBFs). In fact, the combination of interpolation and RBFs can lead to very good properties in high dimensions. Unfortunately, the linear system of equations derived from the approximations of RBFs with a high order of convergence is ill-conditioned and unstable, and usually includes a full interpolation matrix. To solve such a system of equations, we face a very high computational cost if the dimension of the problems or the number of data points is increased. This can also lead to intense instability in the considered problem, and the condition number of the system of equations, as a measure of the ill-conditioning criterion, will be very large. To overcome these problems, this paper presents a layer-by-layer interpolation approach for solving scattered data approximation of an unknown multivariate function, where the information of this approximation problem has been given in certain layers. In this approach, by creating a layered structure, the computational cost is reduced and decreasing the condition number is also possible. This structure provides a possibility for encountering a much smaller linear system of equations. In other words, it increases the accuracy and the stability of the numerical structure of the considered interpolation. The new method can ensure the existence and the uniqueness of the solution for the multilayer interpolation problem. We also find that the layer-by-layer approach provides more numerically stable calculations than the traditional interpolation method. The new approach is applied for certain numerical examples in high dimensions and the obtained results confirm the high accuracy and the low computational cost of the proposed method. Finally, our approach is applied to explore one of the air pollution indexes, i.e., ozone concentration, which has been based on different stations in Tehran, Iran.

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Rational RBF-based partition of unity method for efficiently and accurately approximating 3D objects

Cited by 6 publications

References 24 publications

THB-spline approximations for turbine blade design with local B-spline approximations

THB-spline approximations for turbine blade design with local B-spline approximations

Fast and stable rational RBF-based partition of unity interpolation

An efficient method based on RBFs for multilayer data interpolation with application in air pollution data analysis

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