MATH 590: Meshfree Methods - Chapter 6 - Applied Mathematics

Example: Reproduction of Linear Functions using Gaussian RBFsThere is a simple remedy for this problem.Just add the polynomial functionsto the Gaussian basisProblemx ↦→ 1, x ↦→ x, x ↦→ y{e −ε2 ‖·−x 1 ‖ 2 , . . . , e −ε2 ‖·−x N ‖ 2 }.Now we have N + 3 unknowns, namely the coefficients c k ,k = 1, . . . , N + 3, in the expansionP f (x) =N∑c k e −ε2 ‖x−x k ‖ 2 +c N+1 +c N+2 x +c N+3 y, x = (x, y) ∈ R 2 ,k=1but we have only N conditions to determine them, namely theinterpolation conditionsP f (x j ) = f (x j ) = (x j + y j )/2, j = 1, . . . , N.fasshauer@iit.edu MATH 590 – Chapter 6 16