Fast Solution of the Radial Basis Function Interpolation Equations: Domain Decomposition Methods

Abstract

Abstract. In this paper we consider domain decomposition methods for solving the radial basis function interpolation equations. There are three interwoven threads to the paper. The first thread provides good ways of setting up and solving small-to medium-sized radial basis function interpolation problems. These may occur as subproblems in a domain decomposition solution of a larger interpolation problem. The usual formulation of such a problem can suffer from an unfortunate scale dependence not intrinsic in the problem itself. This scale dependence occurs, for instance, when fitting polyharmonic splines in even dimensions. We present and analyze an alternative formulation, available for all strictly conditionally positive definite basic functions, which does not suffer from this drawback, at least for the very important example previously mentioned. This formulation changes the problem into one involving a strictly positive definite symmetric system, which can be easily and efficiently solved by Cholesky factorization. The second section considers a natural domain decomposition method for the interpolation equations and views it as an instance of von Neumann's alternating projection algorithm. Here the underlying Hilbert space is the reproducing kernel Hilbert space induced by the strictly conditionally positive definite basic function. We show that the domain decomposition method presented converges linearly under very weak nondegeneracy conditions on the possibly overlapping subdomains. The last section presents some algorithmic details and numerical results of a domain decomposition interpolatory code for polyharmonic splines in 2 and 3 dimensions. This code has solved problems with 5 million centers and can fit splines with 10,000 centers in approximately 7 seconds on very modest hardware.

Overlapping additive Schwarz preconditioners for elliptic PDEs on the unit sphere

“…As has been pointed out in [1], the essential ingredients for a domain decomposition algorithm are: (i) A method for subdividing the physical space.…”

“…Based on those overlapping subsets of scattered data, we define an additive Schwarz operator for solving (1). We prove a theorem which gives a bound on the condition number of the Schwarz operator.…”

Symmetric boundary knot method

“…Therefore, developing the corresponding fast algorithms is of vital importance in solving large size problems. In this regard, the fast multipole and domain decomposition techniques [17] seem very promising and are now under study. …”

Preconditioning for Radial Basis Function Partition of Unity Methods

“…In our case, we are starting from a local approximation, so the coefficient matrices are already sparse. Preconditioning approaches based on domain decomposition such as those derived in [5] and [13] could however be effective in combination with the partition of unity method that in itself introduces a subdomain structure.…”