---
tag: BM703GV34G
kind: concept
name: Multivariate interpolation
msc: [41A63, 41A05, 65D05]
sources:
  - url: https://en.wikipedia.org/wiki/Multivariate_interpolation
same_as:
  wikidata: Q6334079
  wikipedia: Multivariate interpolation
---

## Summary

Multivariate interpolation is the construction of a function of several variables that takes given values at given points of $\R^d$ with $d \ge 2$.
In one variable, polynomials of degree at most $n$ always interpolate uniquely at $n + 1$ distinct points.
In several variables, this depends on the position of the points, see [[unisolvent point set|5276MDSDQV]].
For gridded data one uses tensor products of one-dimensional methods, such as bilinear and bicubic interpolation.
For scattered data one uses methods adapted to the points, such as [[radial basis function interpolation|XZ33P3HBB9]], [[inverse distance weighting|QHN67MSH8S]], [[moving least squares|T0JZN6WP5X]] and [[kriging|VP9S3PNJQ0]].

## Explanations

### L1 human

Imagine temperature readings taken at weather stations scattered over a country.
Multivariate interpolation fills in a temperature map that agrees with the readings at the stations.
On a regular grid one can interpolate first along rows and then along columns.
For example, bilinear interpolation on the unit square with corner values $f_{00}, f_{10}, f_{01}, f_{11}$ gives $f_{00}(1 - x)(1 - y) + f_{10} x (1 - y) + f_{01} (1 - x) y + f_{11} x y$.
For scattered stations other methods are needed.
The [[definition|C4B5FB6A34]] states the problem.