A partial differential equation approach to multidimensional extrapolation

In this short note, a general methodology for multidimensional extrapolation is presented. The approach assumes a level set function exists which separates the region of known values from the region to be extrapolated. It is shown that arbitrary orders of polynomial extrapolation can be formulated by simply solving a series of linear partial differential equations (PDEs). Examples of constant, linear and quadratic extrapolation are given.