Characterization and construction of radial basis functions

By Robert Schaback, Holger Wendland. 2001

147 citations recorded by the source corpus.

We review characterizations of (conditional) positive definiteness and show how they apply to the theory of radial basis functions. We then give complete proofs for the (conditional) positive definiteness of all practically relevant basis functions. Furthermore, we show how some of these characterizations may lead to construction tools for positive definite functions. Finally, we give new construction techniques based on discrete methods which lead to non-radial, even nontranslation invariant, local basis functions.

Subfields: Atomic and Molecular Physics, and Optics, Computational Mechanics, Mechanics of Materials

View Paper: https://www.semanticscholar.org/paper/1a9d530618a144cec37ac3fbce69561b8de59204

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Atomic and Molecular Physics, and Optics

In Computational Mechanics

In Mechanics of Materials

Citations in the corpus

Toward the next generation of recommender systems: a survey of the state-of-the-art and possible extensions (2005, background): “One example of an approxima-tion-based approach to defining function u ij in (16) constitutes radial basis functions [16], [30], [92] that are defined as follows: Given a set of points X ¼ f x 1 ; . . . ; x m g (where x i 2 IR N ) and the values of an unknown function f (e.g., the rating function)…”
An enriched radial point interpolation method (e-RPIM) for analysis of crack tip fields (2010, background): “These three types of RBFs and their shape parameters used in this paper can be found in Table 1.”
On the Approximation Order and Numerical Stability of Local Lagrange Interpolation by Polyharmonic Splines (2003, background): “For more details concerning conditionally positive definite functions, see [19, 24].”
A level set method for structural shape and topology optimization using radial basis functions (2009, background): “Hence, to overcome the unfavorable features related to the global support, the CS-RBF is well accepted due to its strictly positive definiteness and the sparsity of collection matrices [33–34,39].”
A meshfree radial point interpolation method for analysis of functionally graded material (FGM) plates (2004, background): “Mathematicians have given complete proofs for the (conditional) positive definiteness of some practically relevant radial basis functions, including the present MQ (Schaback and Wendland 2002; Powell 1992).”
Structural Shape and Topology Optimization Using an Implicit Free Boundary Parametrization Method (2006, methodology): “In the present method, since the IMQs are positive definite RBFs [Schaback and Wendland (2001)] and boundarycondition free (radially unbounded), according to a recent result in [Platte and Driscoll (2005)], the present implementation of the method of lines is time-stable for all knot distributions…”
A level set‐based parameterization method for structural shape and topology optimization (2008, background): “The particular attractiveness of CS-RBFs is that the interplant can naturally inherit the continuity of the RBFs [36], which would enable us to select appropriate shape functions with favorable continuity [43, 49].”
A meshfree method: meshfree weak–strong (MWS) form method, for 2-D solids (2003, methodology): “To ensure invertible interpolation matrix of RBF, the polynomial that is added into RBF cannot be arbitrary ( Schaback and Wendland 2000; Cheng et al. 2003).”
Multivariate Interpolation by Polynomials and Radial Basis Functions (2005, unknown): “For all α ∈ ZZd0 let wα be a positive real number, and consider the inner product (p, q)w := ∑ α∈ZZd0 1 wα (Dαp)(0)(Dαq)(0) on the space IP d∞ of all d–variate polynomials.”