Interpolation

Lets consider a cell defined by its corner nodes : $$\begin{aligned}{} g_{0000...0} & = (s_0^1,s_0^2,s_0^3,s_0^4,\cdots,s_0^D) \\ g_{1000...0} & = (s_1^1,s_0^2,s_0^3,s_0^4,\cdots,s_0^D) \\ g_{0100...0} & = (s_0^1,s_1^2,s_0^3,s_0^4,\cdots,s_0^D) \\ g_{1100...0} & = (s_1^1,s_1^2,s_0^3,s_0^4,\cdots,s_0^D) \\ g_{0010...0} & = (s_0^1,s_0^2,s_1^3,s_0^4,\cdots,s_0^D) \\ g_{1010...0} & = (s_1^1,s_0^2,s_1^3,s_0^4,\cdots,s_0^D) \\ g_{0110...0} & = (s_0^1,s_1^2,s_1^3,s_0^4,\cdots,s_0^D) \\ g_{1110...0} & = (s_1^1,s_1^2,s_1^3,s_0^4,\cdots,s_0^D) \\ & \\ g_{1111...1} & = (s_1^1,s_1^2,s_1^3,s_1^4,\cdots,s_1^D) \end{aligned}$$

Lets consider a point $X$ in that cell. This point partition the cell into $2^D$ sub-cells : $$X = (x_1,x_2,x_3,\cdots,x_D) ~~,~~ x_j \in [s_0^j,s_1^j] ~~,~~ j \in [1,D]$$

Lets define the volume of the cell as : $$V = \prod_{j\in[1,D]} (s_1^j-s_0^j)$$

Lets define the volume of a sub-cell as : $$V_{\sigma_1\sigma_2\sigma_3\cdots\sigma_{D}}(X) = \prod_{j\in[1,D]} \small \begin{cases} s_1^j-x_j,& \sigma_j=0 \\ x_j-s_0^j,& \sigma_j=1 \end{cases}$$

Linear interpolation factors are given by : $$L_{\sigma_1\sigma_2\sigma_3\cdots\sigma_{D}}(X) = \frac{1}{V} \cdot V_{\sigma_1\sigma_2\sigma_3\cdots\sigma_{D}}(X)$$

A scalar field that is known at each node can thus be interpolated : $$f(X) = \prod_{\sigma_1\sigma_2\sigma_3\cdots\sigma_{D}} f_{\sigma_1\sigma_2\sigma_3\cdots\sigma_{D}} \cdot V_{\sigma_1\sigma_2\sigma_3\cdots\sigma_{D}}(X)$$

Where : $$f_{\sigma_1\sigma_2\sigma_3\cdots\sigma_{D}} = F(g_{\sigma_1\sigma_2\sigma_3\cdots\sigma_{D}})$$