Mesh quality error with interpolation of table

Question

I am trying to interpolate a table of the following form.

{{x1,y1,z1},{x2,y2,z2}...}

Where z depends on x and y, and x ranges from 0.01 to 1 in steps of 0.01, and y ranges from 0.01 to 15 in steps of 0.1 . It is a huge table so I cannot post the elements.

This is what the surface looks like.

I want to interpolate the table, however I am getting the following error regarding the mesh,

Interpolation::fememtlq: The quality 0.` of the underlying mesh is too low. 
The quality needs to be larger than 0.`.

I read that this error can be caused by collinear points, so I performed the following perturbation. I tried this at multiple values of epsilon, and also without adding a perturbation to z and the results are more or less the same.

  epsilon = 10^-5;
  perturbData = {#1 + RandomReal[epsilon {-1, 1}], #2 + 
  RandomReal[epsilon {-1, 1}], #3 + 
  RandomReal[epsilon {-1, 1}]} & @@@ intrct;
  reg = ConvexHullMesh[perturbData[[All, 1 ;; 2]]];
  f = Interpolation[perturbData, InterpolationOrder -> 1]

However when I plot this to check how it looks I get this surface which does not seem right at all.

I am trying to find the roots of this function, but I cannot do this properly with this setup.

So I would like to know if there are any alternative ways to solve the interpolation error, or to refine the one I have.

Answer 1

One option is to use every 5th data point:

Needs["NDSolve`FEM`"]
mesh = ToElementMesh[wtable[[All, 1 ;; 2]][[1 ;; -1 ;; 5]]]


mesh["Wireframe"]

if = ElementMeshInterpolation[{mesh}, 
  wtable[[All, 3]][[1 ;; -1 ;; 5]]]

Plot3D[if[x, y], {x, y} \[Element] mesh]

Or a bit simpler:

if = Interpolation[wtable[[1 ;; -1 ;; 5]], "InterpolationOrder" -> 1];

A second option is to look for the bad data points and slightly perturb them:

mesh = ToElementMesh[wtable[[All, 1 ;; 2]]]
ToElementMesh::femimq: The element mesh has insufficient quality of 0.`. A quality estimate below 0. may be caused by a wrong ordering of element incidents or self-intersecting elements.

pos = Position[mesh["Quality"], _?(# <= 0. &)]
{{1, 10791}}

badInci = Extract[ElementIncidents[mesh["MeshElements"]], pos]
{{5611, 6823, 6232}}

badCoords = mesh["Coordinates"][[#]] & /@ badInci
{{{0.46`, 2.4000000000000004`}, {0.56`, 3.6`}, {0.51`, 3.`}}}

This is not a good polygon:

Area[Polygon[#]] & /@ badCoords
{Undefined}

We make a new mesh:

newCoords = mesh["Coordinates"];
epsilon = 10^-3;
(newCoords[[#]] += epsilon) & /@ badInci[[All, 2]];
mesh2 = ToElementMesh[newCoords]

And a new interpolating function:

if2 = ElementMeshInterpolation[mesh2, wtable[[All, 3]]]

Plot3D[if2[x, y], {x, y} \[Element] mesh2]