Polyharmonic spline
In applied mathematics, polyharmonic splines are used for function approximation and data interpolation.
), the polyharmonic RBFs with the natural logarithm might be implemented as: or, more simply adding a continuity extension in
is full rank, the system of equations (2) always has a unique solution and it can be solved using a linear solver specialised for symmetric matrices.
Many practical details of implementing and using polyharmonic splines are explained in Fasshauer.
[4] In Iske[5] polyharmonic splines are treated as special cases of other multiresolution methods in scattered data modelling.
Main disadvantages are: One straightforward approach to speeding up model construction and evaluation is to use a subset of
nearest interpolation nodes to build a local model every time we evaluate the spline.
As a result, the total time needed for model construction and evaluation at
The main drawback is that it introduces small discontinuities in the spline and requires problem-specific tuning: a proper choice of the neighbors count,
Recently, methods have been developed to overcome the aforementioned difficulties without sacrificing main advantages of polyharmonic splines.
evaluation were proposed: Second, an accelerated model construction by applying an iterative solver to an ACBF-preconditioned linear system was proposed by Brown et al.[8] This approach reduces running time from
The approaches above are often employed by commercial geospatial data analysis libraries and by some open source implementations (e.g. ALGLIB).
Sometimes domain decomposition methods are used to improve asymptotic behavior, reducing memory requirements from
For example, the thin plate radial basis function is a solution of the modified 2-dimensional biharmonic equation.
A computer algebra system will show that So the thin plate radial basis function is a solution of the equation
again indicates that the right hand side of the PDEs for the biharmonic and triharmonic RBFs are Dirac delta functions.
{\displaystyle \nabla ^{2}f=(f_{xx}\ f_{xy}\ f_{xz}\ f_{yx}\ f_{yy}\ f_{yz}\ f_{zx}\ f_{zy}\ f_{zz})}
making the integral the simplified thin plate energy functional.
To show that polyharmonic splines minimize equation (3), the fitting term must be transformed into an integral using the definition of the Dirac delta function: So equation (3) can be written as the functional where
A weak solution of equation (4) will still minimize (3) while getting rid of the delta function through integration.
divided by the square root of a total degree 8 polynomial.
Just as in the previous smoothing spline coefficient derivation, the top half of (2) becomes
This derivation of the polyharmonic smoothing spline equation system did not assume the constraints necessary to guarantee that
formed from the solution of the polyharmonic smoothing spline equation system.
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.
[9] The next figure shows the interpolation through four points (marked by "circles") using different types of polyharmonic splines.
The "curvature" of the interpolated curves grows with the order of the spline and the extrapolation at the left boundary (x < 0) is reasonable.
The figure also includes the radial basis functions φ = exp(−r2) which gives a good interpolation as well.
Finally, the figure includes also the non-polyharmonic spline phi = r2 to demonstrate, that this radial basis function is not able to pass through the predefined points (the linear equation has no solution and is solved in a least squares sense).
Note, for other radial basis functions, such as φ = exp(−kr2) with k = 1, the interpolation is no longer reasonable and it would be necessary to adapt k. The next figure shows the same interpolation as in the first figure, with the only exception that the polynomial term of the function is not taken into account (and the case phi = r2 is no longer included).
