Near-optimal data-independent point locations for radial basis function interpolation

Marchi, Stefano De; Schaback, Robert; Wendland, Holger

doi:10.1007/s10444-004-1829-1

Cited by 112 publications

(117 citation statements)

References 5 publications

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“…Note that the first set of equations simply forces s f,X to interpolate the data, while the second one is necessary to ensure a unique decomposition into the two terms in (14). This system needs to be solved only once and then yields an expression for s f,X in closed form, valid on the whole of T .…”

Section: Conditionally Positive Definite Kernelsmentioning

confidence: 99%

“…, u * n (x 0 ) determined by (13) and (12). Representation (14) was already noted by Matheron [50], and the corresponding equation system is known as dual kriging. The universal kriging interpolant can be also derived within a Bayesian framework.…”

Section: Ordinary and Universal Krigingmentioning

confidence: 99%

“…, u * n (x 0 ) determined by the equation systems (13) and (12). The difference to universal kriging lies in the interpretation of representation (14). While it is legitimate in universal kriging to interpret the second term in (14) as an approximate mean function and the first term as approximate deviation from it, such an interpretation would be wrong in intrinsic kriging where a mean function is not even defined.…”

Section: Intrinsic Krigingmentioning

confidence: 99%

“…The difference to universal kriging lies in the interpretation of representation (14). While it is legitimate in universal kriging to interpret the second term in (14) as an approximate mean function and the first term as approximate deviation from it, such an interpretation would be wrong in intrinsic kriging where a mean function is not even defined. covariance function symmetric and positive definite kernel ∼ of a weakly stationary random field translation-invariant kernel ∼ of a w. stat.…”

Section: Intrinsic Krigingmentioning

confidence: 99%

“…This is not surprising in the light of the preceding section were we noted that working with some smooth kernel will always guarantee optimal convergence rates whatever the particular form (and scaling) of this kernel, provided that the sampling locations are quasi-uniform. Consequently, more emphasis was put on the study of good configurations of the sampling locations [14,15,38] on the one hand, and edge correction strategies (see [25] for an overview) on the other hand to avoid the undesired oscillations near the boundaries that often come with smooth and flat kernels. Nevertheless several authors [6,27,26,62] have pointed out the big impact of the choice of e.g.…”

Section: Kernel Selection and Parameter Estimationmentioning

confidence: 99%

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Interpolation of spatial data – A stochastic or a deterministic problem?

2013

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Interpolation of spatial data is a very general mathematical problem with various applications. In geostatistics, it is assumed that the underlying structure of the data is a stochastic process which leads to an interpolation procedure known as kriging. This method is mathematically equivalent to kernel interpolation, a method used in numerical analysis for the same problem, but derived under completely different modelling assumptions. In this article we present the two approaches and discuss their modelling assumptions, notions of optimality and the different concepts to quantify the interpolation accuracy. Their relation is much closer than has been appreciated so far, and even results on convergence rates of kernel interpolants can be translated to the geostatistical framework. We sketch the different answers obtained in the two fields concerning the issue of kernel misspecification, present some methods for kernel selection, and discuss the scope of these methods with a data example from the computer experiments literature.

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Section: Conditionally Positive Definite Kernelsmentioning

confidence: 99%

Section: Ordinary and Universal Krigingmentioning

confidence: 99%

Section: Intrinsic Krigingmentioning

confidence: 99%

Section: Intrinsic Krigingmentioning

confidence: 99%

Section: Kernel Selection and Parameter Estimationmentioning

confidence: 99%

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Interpolation of spatial data – A stochastic or a deterministic problem?

2013

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A diagonal offset algorithm for the polynomial point interpolation method

Kanber

Bozkurt

2007

Commun. Numer. Meth. Engng.

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SUMMARYA diagonal offset algorithm is presented to overcome the singular moment matrix problem in the polynomial point interpolation method. The value of terms in the diagonal line of moment matrix is changed with different offsets. The effects of the offsets on the Kronecker delta function and partition of unity properties are investigated and an optimum offset range is proposed. The algorithm is validated solving some patch tests and various elasticity problems in 2-D domain. Their results demonstrate that the proposed algorithm is very easy to implement and completely solves the singular moment matrix problem. The accuracy of the method with proposed algorithm is investigated for regular and irregular local domains.

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Numerical modelling of a peripheral arterial stenosis using dimensionally reduced models and kernel methods

Köppl

Santin

Haasdonk

et al. 2018

Numer Methods Biomed Eng

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In this work, we consider 2 kinds of model reduction techniques to simulate blood flow through the largest systemic arteries, where a stenosis is located in a peripheral artery, i.e., in an artery that is located far away from the heart. For our simulations, we place the stenosis in one of the tibial arteries belonging to the right lower leg (right posterior tibial artery). The model reduction techniques that are used are on the one hand dimensionally reduced models (1-D and 0-D models, the so-called mixed-dimension model) and on the other hand surrogate models produced by kernel methods. Both methods are combined in such a way that the mixed-dimension models yield training data for the surrogate model, where the surrogate model is parametrised by the degree of narrowing of the peripheral stenosis. By means of a well-trained surrogate model, we show that simulation data can be reproduced with a satisfactory accuracy and that parameter optimisation or state estimation problems can be solved in a very efficient way. Furthermore, it is demonstrated that a surrogate model enables us to present after a very short simulation time the impact of a varying degree of stenosis on blood flow, obtaining a speedup of several orders over the full model.

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Near-optimal data-independent point locations for radial basis function interpolation

Cited by 112 publications

References 5 publications

Interpolation of spatial data – A stochastic or a deterministic problem?

Interpolation of spatial data – A stochastic or a deterministic problem?

A diagonal offset algorithm for the polynomial point interpolation method

Numerical modelling of a peripheral arterial stenosis using dimensionally reduced models and kernel methods

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