The paper shows the adaptation of the cross validation approach, known from interpolation problems, for estimating the value of a shape parameter for radial basis functions. The latter are involved in two collocation techniques used on an unstructured grid to find approximate solution of differential equations. To obtain accurate results, the shape parameter should be chosen as a result of a trade-off between accuracy and conditioning of the system. The cross validation approach called “leave one out” takes these aspects into consideration. The numerical examples that summarize the investigations show the usefulness of the approach.
If the inline PDF is not rendering correctly, you can download the PDF file here.
 Belytschko T., Krongauz Y., Organ D., Flrming M., Krysl P., Meshless methods: an overview and recent developments, Computer Methods in Applied Mechanics and Engineering, vol. 139, 1996, 3-47.
 Liu G.R., Meshlees Methods – Moving beyond the Finite Element Method, CRC Press, Boca Raton, Florida 2003.
 Kansa E., Multiquadrics – A scattered data approximation scheme with applications to computational fluid dynamics I: Surface approximations and partial derivative estimates, Computers and Mathematics with Applications, vol. 19, 1990, 127-145.
 Kansa E., Multiquadrics – A scattered data approximation scheme with applications to computational fluid dynamics I: Solutions to parabolic, hyperbolic, and elliptic partial differential equations, Computers and Mathematics with Applications, vol. 19, 1990, 147-161.
 Fasshauer G.E., Meshfree Approximation Methods with Matlab, World Scientific Publishing, Singapore, 2007.
 Cheng A.H.D., Multiquadrics and its shape parameter – a numerical investigation of error estimate, condition number and round-off error by arbitrary precision computation, Engineering analysis with boundary elements, vol. 36, 2012, 220-239.
 Ferreira A.J.M, A formulation of the multiquadric radial basis function method for the analysis of laminated composite plates, Composit Structures, vol. 59, 2003, 385-92.
 Schaback R., Error estimates and condition numbers for radial basis function interpolation, Advances in Computational Mathematics, vol. 3, 1995, 251-264.
 Krowiak A., Radial basis function-based pseudospectral method for static analysis of thin plates, Engineering Analysis with Boundary Elements, vol. 71, 2016, 50-58.
 Krowiak A., On choosing a value of shape parameter in Radial Basis Function collocation methods, Numerical Methods for Partial Differential Equations, submitted for publication.
 Hon Y.C., Schaback R., On nonsymmetric collocation by radial basis functions, Appl. Math. Comput., vol. 119, 2001, 177-186.
 Chen W., Fu Z.J., Chen C.S., Recent Advances in Radial Basis Function Collocation Methods, Springer, 2014.
 Rippa S., An algorithm for selecting a good value for the parameter c in radial basis function interpolation, Adv. in Comput. Math., vol. 11, 1999, 193-210.