4 Classical methods for numerical solution of PDEs such as finite difference, finite elements, finite volume, pseudo-spectral methods are base on polynomial interpolation. Local polynomial based methods (finite difference, finite elements and finite volume) are limited by their algebraic convergence rate. MQ collection methods in comparison to finite element method have superior accuracy.
5 Global polynomial methods such as spectral methods have exponential convergence rate but are limited by being tied to a fixed grid. Standard" multivariate approximation methods (splines or finite elements) require an underlying mesh (e.g., a triangulation) for the definition of basis functions or elements. This is usually rather difficult to accomplish in space dimensions > 2. RBF method are not tied to a grid but to a category of methods called meshless methods. The global non polynomial RBF methods are successfully applied to achieve exponential accuracy where traditional methods either have difficulties or fail. RBF methods are generalization of Multi Quadric RBF, MQ RBF have a rich theoretical development.
2. Literature Review RBF developed by Iowa State university, Rolland Hardy in 1968 for scattered data be easily used in computations which polynomial interpolation has failed in some cases. RBF present a topological surface as well as other three dimensional shapes. In 1979 at Naval post graduate school Richard Franke compared different methods to solve scattered data interpolation problem and he applied Hardy's MQ method and shows it is the best approximation & also the matrix is invertible. 6
7 In 1986 Charles Micchelli a mathematician with IBM proved the system matrix for MQ is invertible and the theoretical basis began to develop. His approach is based on conditionally positive definite functions. In 1990 the first use of MQ to solve PDE was presented by Edward Kansa. In 1992 spectral convergence rate of MQ interpolation presented by Nelson &Madych.
8 Later, applications were found in many areas such as in the numerical solution of PDEs, artificial intelligence, learning theory, neural networks, signal processing, sampling theory, statistics,finance, and optimization. Originally, the motivation for two of the most common basis meshfree approximation methods (radial basis functions and moving least squares methods) came from applications in geodesy, geophysics, mapping, or meteorology.
10 It should be pointed out that meshfree local regression methods have been used independently in statistics for more than 100 years. Radial Basis Function "RBF" interpolate a multi dimensional scattered data which easily generalized to several space dimension & provide spectral accuracy. So it is so popular
3. RBF Method 3.1 Radial Function 11 In many applications it is desirable to have invariance not only under translation, but also under rotation & reflection. This leads to positive definite functions which are also radial. Radial functions are invariant under all Euclidean transformations (translations, reflections & rotations)
3.2 Positive Definite Matrices & Functions 13 If the only vector c that turns (1) into an equality is the zero vector, then A is called positive definite. An important property of positive definite matrices is that all their eigenvalues are positive, and therefore a positive definite matrix is non-singular (but certainly not vice versa).
3.5 RBF categories: We have two kinds of RBF methods 1.Basis RBF 2.Augmented RBF 19 Micchelli gave sufficient conditions for φ(r) to guarantee that the matrix A is unconditionally nonsingular. RBF method is uniquely solvable.