On unsymmetric collocation by radial basis functions

Solving partial diierential equations by collocation with radial basis functions can be eeciently done by a technique rst proposed by E. Kansa in 1990. It rewrites the problem as a generalized interpolation problem, and the solution is obtained by solving a (possibly large) linear system. The method has been used successfully in a variety of applications, but a proof of nonsingularity of the linear system still was missing. This paper shows that a general proof of this fact is impossible. However, numerical evidence shows that cases of singularity are rare and have to be constructed with quite some eeort.