(186d) Quenched Periodic Extension for Interpolation Using Radial Basis Functions

Interpolation of regular data is important in solution of partial differential equations, computer simulation of fluid flow and heat/mass transfer, digital image processing and medical imaging, among other applications. Radial basis functions (RBF) enable conceptually straightforward interpolation of functions with (1) prescribed smoothness (i.e., mathematical regularity), (2) arbitrary dimensionality, and (3) satisfaction of physically motivated constraints, such as vanishing divergence of an incompressible fluid velocity field. RBF have advantages over traditional methods such as bilinear or bicubic interpolation kernels. Compactly supported RBF enable fast interpolation and sparse matrices.

In numerical practice, however, RBF are bedeviled by severe ill conditioning of the matrices. In the case of periodic functions, the translational nature of the RBF kernels allows the coefficients to be obtained by fast Fourier transforms (FFT) and the convolution theorem to avoid the poor condition number, as well as lowering computing time. However, functions on finite domains require careful handling to extend periodically, especially in three dimensions. Gibb’s phenomenon and spurious oscillations are typical problems.

Here we report on a versatile quenched periodic extension (QPE) method that has been developed for interpolating functions on finite domains. The QPE method guarantees removal of the discontinuity across the boundaries for any type of function. The QPE method can easily be extended to any number of dimensions. Additionally, the QPE method includes ellipsoidal support for the basis function, which can be advantageous for high-aspect-ratio grids. The presented RBF interpolation method can be used in a wide range of applications such as medical imaging and particle computer simulations among others. The latter will be illustrated with computer simulations of low-Reynolds-number drop deformations.