Gaussian extended cubature formulae for polyharmonic functions (2000)
This fact enables transforming the polyharmonic smoothing spline equation system to a symmetric positive definite system of equations that can be solved twice as fast using the Cholesky decomposition. The next figure shows the interpolation through four points marked by "circles" using different types of polyharmonic splines.
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- Polyharmonic spline?
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This is different for the logarithmic form of the spline, although the scaling has not much influence. This analysis is reflected in the figure, where the interpolation shows not much differences.
A Goursat Decomposition for Polyharmonic Functions in Euclidean Space
This indicates, that the polynomial term is useful if extrapolation occurs. The main advantage of polyharmonic spline interpolation is that usually very good interpolation results are obtained for scattered data without performing any "tuning", so automatic interpolation is feasible. This is not the case for other radial basis functions.
Recently, methods have been developed to overcome the aforementioned difficulties. For example Beatson et al.
From Wikipedia, the free encyclopedia. The same interpolation as in the first figure, but the points to be interpolated are scaled by The same interpolation as in the first figure, but without the polynomial term. Harder and R. Desmarais: Interpolation using surface splines. Journal of Aircraft, , Issue 2, pp.
A generalized mean value property for polyharmonic functions
Duchon: Splines minimizing rotation-invariant semi-norms in Sobolev spaces. Schempp and K. Zeller eds , Springer, Berlin, pp.
Scattered Data Approximation. Cambridge University Press. Fasshauer G.
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Cambridge University Dept. However these algorithms can not be developed for the estimation of parameters of a polyharmonic signal.
A Goursat Decomposition for Polyharmonic Functions in Euclidean Space - Semantic Scholar
A solution of considered problem can find wide practical application in self-learning of robotic systems and adaptive noise damping systems. For example, the problem of overcoming of function uncertainties in control laws arises from the design of mobile robot control systems for a motion along physically detectable, but analytically unknown paths border of physical object. Solution of this problem is based an using a strategy of self-learning and an algorithm of the approximation of the unknown periodic functional dependencies.
Analytical conditions of existence of the solution are presented for the case of on-line continuous-time estimation of the parameters of a polyharmonic function. The design procedure of the estimation algorithm is proposed.
Published in: 2nd International Conference. Control of Oscillations and Chaos. Proceedings Cat. Article :. DOI: