Data Interpolation by Using Two-Parameter Boundary Shape Functions

Abstract

This paper addresses the interpolation of noisy data, a problem in which high-order polynomial methods frequently exhibit ill-conditioning and the Runge phenomenon. To mitigate these issues, a hybrid mixed-type method (MTM) is first developed by augmenting classical collocation equations with Galerkin-derived moment equations and optimizing a characteristic length parameter within the polynomial representation. Subsequently, the study develops a boundary shape function interpolation method (BSFIM) for two-dimensional interpolation over rectangular and arbitrary planar domains. In the BSFIM, a family of two-parameter boundary shape functions is systematically derived such that each basis function automatically and exactly satisfies the measured boundary data. This boundary-conforming and term-wise separable basis function design yields a low-dimensional, and well-conditioned approximation space that allows the extraction of highly accurate interpolants from substantially fewer interior data points than those required in traditional polynomial or radial basis function (RBF) methods. Extensive numerical examples, including problems on irregular domains, discontinuous scattered data, and Stokes-flow-based pressure profile construction, demonstrate that the BSFIM can achieve interpolation errors below the noise level and exhibit robust performance under conditions involving high measurement noise.