Which one of the following techniques would you suggest?
- Linear interpolation
- Least square
- Weighted Least square
- Moving Least square
- RBF
- Any other ?
It all depends on what you mean by arbitrarily distributed data set?
If for example you use a (pseudo) random numbers generator, then any interpolation method would produce irrelevant data, since there is no rule to generate a new value between any two existing ones.
If, on the other hand, you measure what is called an ergodic system (http://en.wikipedia.org/wiki/Ergodicity) then you try to fit your data into some theoretically proven curve. E.g. an oscillator will move close to a harmonic motion, and the interpolation curve should be a sine & cosine, using for example the least square method, to obtain the amplitudes.
In a similar way, a linear interpolation is correct if the expected process is linear, etc.
Best Technique in terms of general applicability, easy coding, easy application and good accuracy is the Volume Spline Technique, which can be considered as RBF.
See my papers IFASD 1995 Machester, and IFASD 2003 Amsterdam.
Please find attached a generic interpolation implementation of the V-spline in FORTRAN
It depends on the dimension, data and additional properties you would like the interpolant to have. Please be more specific.
@ Michael - Thank you !
@Yulia -I am mainly concerned about interpolation techniques used in shape function construction for meshfree methods.
One good technique to interpolate in that condition, non regular distribution, and no structure, is the Krigging method. This techniqueis well developed even in comercial software packages like Sufer.
Main problem with Krigging is estimating the variogram. There are ways to do this, but like all interpolation methods, you need to have data at a scale smaller than the spatial scale of variations in the true surface being estimated. This is generally assumed, but not always correct. A good example is the problem with interpolating rainfall data. At long time scales (e.g. annual), interpolation of rainfall data is fairly easy (but often needs additional information like a DEM) as the variations in the spatial distribution of rainfall tend to be at a large scale. However, as we progress to finer temporal resolutions, the scale of the spatial variability in the rainfall decreases, reaching a point where there is insufficient data points to have confidence in the interpolated values.
You need to think about the density of data points, and how this compares with the surface you are trying to interpolate. You will then need to consider the uncertainty in the interpolation and what impact this will have on the output of the finite element model.
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