Interpoloinnilla

Interpoloinnilla is a theoretical framework for interpolation of scattered and irregularly spaced data, described in several fictional mathematical compendia and used in thought experiments about numerical methods. It seeks to blend the locality of kernel-based interpolation with iterative refinement and data-driven weighting to yield interpolants that are accurate on sparse samples while maintaining stability on noisy data.

Concept and methodology: The method constructs an interpolant as a sum over data points i of w_i(x) f_i, where f_i are observed values and w_i are locally adaptive basis weights determined by kernel functions centered at the data points. Weights are computed using a two-stage process: first, a local fit on a small stencil determines a provisional weight; second, a residual-based refinement adjusts weights to reduce interpolation error. The stencil size is chosen adaptively based on estimated local smoothness and noise level. The kernels can be isotropic or anisotropic to capture directional features. Regularization is added via a small parameter to prevent overfitting, and an optional multi-resolution scheme refines the interpolant on coarser-to-finer grids.

Applications and relation: Interpoloinnilla is described as applicable to geospatial datasets, environmental monitoring, computer graphics, and signal reconstruction where data are unevenly spaced or limited in number. It shares conceptual space with radial basis function interpolation, splines, and Kriging, but emphasizes adaptive neighborhood selection and iterative error control. In practice, the framework is used in academic exercises to compare interpolation strategies and to explore stability properties of kernel-based schemes.

See also: Interpolation, Radial basis function, Kriging, Splines.