Interpolationbased

Interpolation-based refers to methods that rely on constructing an interpolant, a function that exactly matches a set of known data points, to estimate unknown values or to guide optimization. The interpolant is chosen from families such as polynomials, splines, or radial basis functions. The conventional spelling is interpolation-based; some sources may use the concatenated form interpolationbased.

Common interpolation schemes include polynomial interpolation (Lagrange or Newton form), spline interpolation (notably cubic splines), piecewise-polynomial interpolants, and localized basis methods like radial basis functions. In statistics and geostatistics, methods such as kriging and Gaussian process regression are also interpolation-based modeling approaches that provide estimates with associated uncertainties.

Applications span reconstructing missing data, resampling or rescaling signals and images, and enabling numerical differentiation or integration from sample values. In the numerical solution of differential equations, interpolation-based schemes supply function values at non-grid points and help construct accurate approximations of derivatives or fluxes.

Advantages of interpolation-based methods include exact reproduction of known data values and the ability to impose smoothness or localness through the choice of interpolant. Limitations include sensitivity to data quality, potential oscillations with high-degree global polynomials (the Runge phenomenon), and unreliable extrapolation beyond the observed range. The accuracy depends strongly on the interpolant type and the distribution of data points.

In optimization and machine learning, interpolation-based models serve as surrogate functions that approximate an objective using observed samples, guiding search steps or providing predictions with quantified uncertainty. These methods balance fidelity to data with computational efficiency and are widely used in derivative-free optimization and surrogate modeling.