Let $x_1,\dots,x_n\in[0,1]^d$ and $y_1,\dots,y_n\in[0,1]$ satisfy: $$\forall i,j\in {1,\dots,n},\hspace{10pt} x_i \le x_j \rightarrow y_i \le y_j.$$

I seek to find a continuous function $f:[0,1]^d\rightarrow[0,1]$ such that:

  1. $\forall i\in {1,\dots,n},\hspace{10pt} f(x_i)=y_i$.
  2. $\forall k\in {1,\dots,d},\hspace{10pt} \forall z=[z_1,\dots,z_d]\in[0,1]^d,\hspace{10pt} \frac{\partial f(z)}{\partial z_k} \ge 0$.
  3. $\forall z\in[0,1]^d,\hspace{10pt} f(z)=\sum_{i=1}^n{w_i(z) y_i}$ and $\sum_{i=1}^n{w_i(z)}=1$, where $w_1,\dots,w_n:[0,1]^d\rightarrow[0,1]$ are continuous functions independent of $y_1,\dots,y_n$ (i.e. they can be calculated without knowing $y_1,\dots,y_n$).

Does such an interpolation method exist? The usual approach for linear interpolation on scattered data via Delaunay triangulation satisfies my first and third conditions, but not the second (monotonicity) constraint. (For example, consider $d=2,n=5$, $x_1=[0,0],x_2=[1,0],x_3=[0,1],x_4=[1,1],x_5=[0.1,0.1]$, $y_1=0,y_2=0,y_3=1,y_4=1,y_5=1$.)

Edit: It seems 2. and 3. above are mutually inconsistent, except in the special case of a complete grid. For example, suppose $d=2,n=4$, $x_1=[0,0],x_2=[1,1],x_3=[0.1,0.2],x_4=[0.2,0.1]$, $y_1=0,y_2=1$ and consider $f([0.3,0.3])$. Monotonicity implies $f([0.3,0.3])\ge y_3$ and $f([0.3,0.3])\ge y_4$. But if $y_3=1$, the former inequality can only hold for all values of $y_4$ if $w_3([0.3,0.3])=1$, and if $y_4=1$, the latter inequality can only hold for all values of $y_3$ if $w_4([0.3,0.3])=1$.

Given this example, I do not understand the monotonicity claim of this paper: https://www.nas.nasa.gov/assets/pdf/staff/Murman_S_apnum_jun13.pdf


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.