Talk:Kansa method

A need for corrections

Wikimedia received a communication from Dr. Kansa. I working to sort out his inability to create an account but while that's being resolved, I offered to post some comments on the talk page.--S Philbrick(Talk) 14:27, 2 January 2018 (UTC) His comments:

The term. "Ill-conditioned system of equations" means that the condition number of the equation system exceeds the inverse of the machine epsilon. The machine epsilon can be reduced by hardware methods such as using computer chips with more bits per word or by computer science methods that increases the number of bits (digits of precision) per word. There is currently a debate whether compactly supported methods that converge at a polynomial rate, h(K+1), are always faster than global C methods that yield full systems of equations that converge exponentially, (c/h) , <1, c is the shape parameter, and h is the fill distance of discretization for the same target accuracy. On average, full systems of equations require O(NG3) to find the expansion coefficients whereas compactly supported methods require O(MCS(3/2) ) operations to find the expansion coefficients. Obviously, if both methods require the same number of discretization points, the compactly supported method is faster.

However, if extended precision is used so that the shape parameter is increased, there is no need to decrease h which in turn increases the size of the matrix. This approach has been verified by Huang,Yen, and Cheng (2008) that showed quadruple precision calculations of the inverse multiquadric method Although extended precision requires more CPU time than single or double precision, outperforms linear finite element compactly supported methods on the total execution time. Kansa and Holoborodko (2017) and (2018) showed software based extended precision methods outperform double precision methods in obtaining meaningful solutions not previously possible and execution time because the total number of discretization points, NG << NCS. Also, in treating multi-dimensional problems, the curse of dimensionality is a severe problem; minimizing the total number of discretization points is a major concern. By increasing c using extended precision, the total number of discretization points is minimized to only discretizing the local and global extrema, and reducing the total execution time. Compactly supported radial basis function methods have their place when problems have a large amount of extremely fine-scale spatial structure.

References Huang, CS; Yen, HD; Cheng, AHD "On the Increasingly Flat Radial Basis Function and Optimal Shape Parameter for the Solution of Elliptic PDEs", Eng. Anal. Bound. Elem. 2010,vol.34,:pp. 802-809.