Skip to main content

version 9 - Interpolation on an unstructured (mostly-structured) grid in 4D+


I frequently need to perform interpolations on an unstructured grid. Mostly, with the purpose of filling grid holes and then later continue with regular grid interpolations.



Mathematica V9 and V10 supports linear Interpolation on an unstructured grid up to 3D:


In[4]:= if = 
 Interpolation[RandomReal[1, {2000, 2}], InterpolationOrder -> 1]; if[.5]

Out[4]= 0.61608

In[5]:= if = 
 Interpolation[RandomReal[1, {2000, 3}], InterpolationOrder -> 1]; if[.5, .5]

Out[5]= 0.41285


In[6]:= if = 
 Interpolation[RandomReal[1, {2000, 4}], InterpolationOrder -> 1]; if[.5, .5, .5]

Out[6]= 0.496552

In V8 however, it was possible to apply the same to even higher dimensions. On my present machine I can get up to 5D (higher dimensions apparently require more than the 16GB RAM that I have):


In[2]:= if = 
 Interpolation[RandomReal[1, {2000, 5}], InterpolationOrder -> 1]; if[.5, .5, .5, .3]


Out[2]= 4.82823

In[3]:= if = 
 Interpolation[RandomReal[1, {2000, 6}], InterpolationOrder -> 1]; if[.5, .5, .5, .3, .6]


Out[3]= 0.119046

To compensate I used the Imtek Mathematica Supplement. However, this accumulated some incompatibilities with the recent Versions of Mathematica and the performance for higher dimensions is not sufficient for my applications. I am about to switch to Ingolf Dahl's Obtuse package which adds five interpolation methods to the Mathematica Interpolation command. Alternatively, there in this Mathematica Stackexchange post one can find a Radial Basis Function based Interpolation command.


Follow Up: I ended up using a RBF Interpolation code from this article



Now my question(s) are:




  1. Why has this functionality been removed for V8+ and is there are work-around to this new limitations?




  2. What other possibilities are there to perform unstructured grid interpolations in 4D+ with reasonable Performance and some control over the applied methods?




  3. Unfortunately I often need to interpolate missing grid points at the edges of the N-dim grid. For this reason I need to prevent/suppress divergent behavior, similar to the divergent extrapolation Problems when using higher order spline/polynomial inter/extrapolation methods. I know that the data cube is reasonably well behaved and would be content with linear Interpolation/Extrapolation behavior - any tips for that?







Comments

Post a Comment

Popular posts from this blog

plotting - How to draw lines between specified dots on ListPlot?

I would like to create a plot where I have unconnected dots and some connected. So far, I have figured out how to draw the dots. My code is the following: ListPlot[{{1, 1}, {2, 2}, {3, 3}, {4, 4}, {1, 4}, {2, 5}, {3, 6}, {4, 7}, {1, 7}, {2, 8}, {3, 9}, {4, 10}, {1, 10}, {2, 11}, {3, 12}, {4,13}, {2.5, 7}}, Ticks -> {{1, 2, 3, 4}, None}, AxesStyle -> Thin, TicksStyle -> Directive[Black, Bold, 12], Mesh -> Full] I have thought using ListLinePlot command, but I don't know how to specify to the command to draw only selected lines between the dots. Do have any suggestions/hints on how to do that? Thank you. Answer One possibility would be to use Epilog with Line : ListPlot[ {{1, 1}, {2, 2}, {3, 3}, {4, 4}, {1, 4}, {2, 5}, {3, 6}, {4, 7}, {1, 7}, {2, 8}, {3, 9}, {4, 10}, {1, 10}, {2, 11}, {3, 12}, {4, 13}, {2.5, 7}}, Ticks -> {{1, 2, 3, 4}, None}, AxesStyle -> Thin, TicksStyle -> Directive[Black, Bold, 12], Mesh -> Full, Epilog -> { Line[ ...
Post a Comment
Read more

equation solving - Invert and fit implicitly defined curve

I need to fit an implicitly defined curve. I thought I could get some data out of Solve , and then using FindFit . Therefore, I would like to find the relation the parametric curve defined by $F(x,y)=0$: Solve[-(1/2) + 1/2 (0.41202 BesselK[0, 0.1 Sqrt[x^2 + y^2]] + (0.101483 x BesselK[1, 0.1 Sqrt[x^2 + y^2]])/Sqrt[x^2 + y^2]) == 0, y] But I can't get an output: Solve was unable to solve the system with inexact coefficients or the system obtained by direct rationalization of inexact numbers present in the system. Since many of the methods used by Solve require exact input, providing Solve with an exact version of the system may help. >> Edit: In particular, I would like to fit the data coming from the curve with the expression of another curve, and not with a function $f(x)$. In particular, since this clearly looks like a cardioid , I would like it to fit to something like it. What other strategies could I try?
Post a Comment
Read more

dynamic - How can I make a clickable ArrayPlot that returns input?

I would like to create a dynamic ArrayPlot so that the rectangles, when clicked, provide the input. Can I use ArrayPlot for this? Or is there something else I should have to use? Answer ArrayPlot is much more than just a simple array like Grid : it represents a ranged 2D dataset, and its visualization can be finetuned by options like DataReversed and DataRange . These features make it quite complicated to reproduce the same layout and order with Grid . Here I offer AnnotatedArrayPlot which comes in handy when your dataset is more than just a flat 2D array. The dynamic interface allows highlighting individual cells and possibly interacting with them. AnnotatedArrayPlot works the same way as ArrayPlot and accepts the same options plus Enabled , HighlightCoordinates , HighlightStyle and HighlightElementFunction . data = {{Missing["HasSomeMoreData"], GrayLevel[ 1], {RGBColor[0, 1, 1], RGBColor[0, 0, 1], GrayLevel[1]}, RGBColor[0, 1, 0]}, {GrayLevel[0], GrayLevel...
Post a Comment
Read more