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1.
Mira Bozzini Licia Lenarduzzi Robert Schaback 《Advances in Computational Mathematics》2002,16(4):375-387
We present an adaptive method to extract shape-preserving information from a univariate data sample. The behavior of the signal is obtained by interpolating at adaptively selected few data points by a linear combination of multiquadrics with variable scaling parameters. On the theoretical side, we give a sufficient condition for existence of the scaled multiquadric interpolant. On the practical side, we give various examples to show the applicability of the method. 相似文献
2.
This paper describes a numerical realization of an extended continuous Newton method defined by Diener. It traces a connected set of locally one-dimensional trajectories which contains all critical points of a smooth functionf:
n
. The results show that the method is effectively applicable.The authors would like to thank L. C. W. Dixon for pointing out some errors in the original version of this paper and for several suggestions of improvements. 相似文献
3.
Robert Schaback 《Constructive Approximation》1993,9(4):373-389
Five points in general position inR 2 always lie on a unique conic, and three points plus two tangents also have a unique interpolating conic, the type of which depends on the data. These well-known facts from projective geometry are generalized: an odd number 2n+1≥5 of points inR 2, if they can be interpolated at all by a smooth curve with nonvanishing curvature, will have a uniqueGC 2 interpolant consisting of pieces of conics of varying type. This interpolation process reproduces conics of arbitrary type and preserves strict convexity. Under weak additional assumptions its approximation order is ?(h 5), whereh is the maximal distance of adjacent data pointsf(t i ) sampled from a smooth and regular planar curvef with nonvanishing curvature. Two algorithms for the construction of the interpolant are suggested, and some examples are presented. 相似文献
4.
Fahimeh Saberi Zafarghandi Maryam Mohammadi Robert Schaback 《Mathematical Methods in the Applied Sciences》2019,42(11):3877-3899
The paper provides the fractional integrals and derivatives of the Riemann‐Liouville and Caputo type for the five kinds of radial basis functions, including the Powers, Gaussian, Multiquadric, Matérn, and Thin‐plate splines, in one dimension. It allows to use high‐order numerical methods for solving fractional differential equations. The results are tested by solving two test problems. The first test case focuses on the discretization of the fractional differential operator while the second considers the solution of a fractional order differential equation. 相似文献
5.
Doz. Dr. Robert Schaback 《Numerische Mathematik》1973,21(2):166-180
Summary This paper generates interpolatingM-splines in the sense of Lucas [J. of Approx. Th. 5, 1–14 (1972)] by a simple algebraic construction. The method yieldsM-spline interpolants for every finite family of functionals commuting with the remainder term of a generalized Taylor formula. These assumptions are fulfilled for a large class of spline interpolation problems (e.g. splines generated by certain singular differential operators and splines of several variables) without any further requirements about the geometrical distribution or denseness of the interpolation points. A generalization ofB-splines is used to improve the numerical behaviour of the interpolation process. 相似文献
6.
This paper applies difference operators to conditionally positive definite kernels in order to generate kernel
-splines that have fast decay towards infinity. Interpolation by these new kernels provides better condition of the linear system,
while the kernel -spline inherits the approximation orders from its native kernel. We proceed in two different ways: either the kernel -spline is constructed adaptively on the data knot set , or we use a fixed difference scheme and shift its associated kernel -spline around. In the latter case, the kernel -spline so obtained is strictly positive in general. Furthermore, special kernel -splines obtained by hexagonal second finite differences of multiquadrics are studied in more detail. We give suggestions
in order to get a consistent improvement of the condition of the interpolation matrix in applications. 相似文献
7.
It is often observed that interpolation based on translates of radial basis functions or non-radial kernels is numerically
unstable due to exceedingly large condition of the kernel matrix. But if stability is assessed in function space without considering
special bases, this paper proves that kernel-based interpolation is stable. Provided that the data are not too wildly scattered,
the L
2 or L
∞ norms of interpolants can be bounded above by discrete ℓ2 and ℓ ∞ norms of the data. Furthermore, Lagrange basis functions are uniformly bounded and Lebesgue constants grow at most like the
square root of the number of data points. However, this analysis applies only to kernels of limited smoothness. Numerical
examples support our bounds, but also show that the case of infinitely smooth kernels must lead to worse bounds in future
work, while the observed Lebesgue constants for kernels with limited smoothness even seem to be independent of the sample
size and the fill distance. 相似文献
8.
9.
Under mild additional assumptions this paper constructs quasi-interpolants in the form
with approximation order ℓ−1, whereh(x) is a linear combination of translatesψ(x−jh) of a functionψinCℓ(
). Thus the order of convergence of such operators can be pushed up to a limit that only depends on the smoothness of the functionψ. This approach can be generalized to the multivariate setting by using discrete convolutions with tensor products of odd-degreeB-splines. 相似文献
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10.
J. Beeckmann R. Hesse J. Schaback H. Pitsch E. Varea N. Chaumeix 《Proceedings of the Combustion Institute》2019,37(2):1521-1528
Laminar burning velocities are of great importance in many combustion models as well as for validation and improvement of chemical kinetic schemes. Determining laminar burning velocities with high accuracy is quite challenging and different approaches exist. Hence, a comparison of existing methods measuring and evaluating laminar burning velocities is of interest. Here, two optical diagnostics, high speed tomography and Schlieren cinematography, are simultaneously set up to investigate methods for evaluating laminar flame speed in a spherical flame configuration. The hypothesis to obtain the same flame propagation radii over time with the two different techniques is addressed. Another important aspect is the estimation of flame properties, such as the unstretched flame propagation speed and Markstein length in the burnt gas phase and if these are estimated satisfactorily by common experimental approaches. Thorough evaluation of the data with several extrapolation techniques is undertaken. A systematic extrapolation approach is presented to give more confidence into results generated experimentally. The significance of the linear extrapolation routine is highlighted in this context. Measurements of spherically expanding flames are carried out in two high-pressure, high-temperature, constant-volume vessels at RWTH in Aachen, Germany and at ICARE in Orleans, France. For the discussion of the systematic extrapolation approach, flame speed measurements of methane / air mixtures with mixture Lewis numbers moderately away from unity are used. Conditions were varied from lean to rich mixtures, at temperatures of 298–373 K, and pressures of 1 atm and 5 bar. 相似文献