Interpolation
Estimating values between known data points by fitting a function that passes through them.
Definition
Interpolation constructs a function that passes exactly through a set of known data points, allowing estimation of values in between. It differs from regression, which fits an approximate trend without passing through every point.
Interpolation should not be confused with extrapolation: estimating beyond the range of known data is far riskier, since the fitted function has no data to constrain it there. Many notorious modeling errors stem from trusting an interpolant pushed outside its support.
The reliability of an interpolant depends on staying within the range of the data; extrapolation beyond it is far riskier because nothing constrains the fitted function there, and many modeling failures trace to trusting a curve pushed outside its support. High-degree polynomial interpolation can also oscillate wildly between points, which is why piecewise splines, smooth and locally controlled, are the usual choice for dependable interpolation.
Common schemes
- Linear: connect points with straight segments.
- Polynomial: a single curve through all points (risky at high degree).
- Spline: piecewise low-degree polynomials, smooth and stable.
- Multidimensional interpolation on grids.
Why it matters
Interpolation is used to resample data, build lookup tables, and evaluate functions known only at discrete points. High-degree polynomial interpolation can oscillate wildly (Runge's phenomenon), which is why splines are usually preferred for smoothness.
Fusion connection
Interpolation evaluates tabulated physical data, such as reaction rates and material properties, at the precise conditions a Hyperion simulation requires between tabulated entries.