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How to deal with noisy data in a mesh free Galerkin method

I'm currently working with a particle tracking velocimetry method, specifically trying to extract shear and vorticity data from the non-gridded velocity results. Of several methods used, the most efficient and accurate was based on a non-Sibsonian element free method. Unlike a more typical Galerkin problem which finds displacements by solving a PDE, this method uses the displacements of natural neighbors to find local flow gradients. 
But here's the rub: this method is very sensitive to noisy data. Small errors in the displacement data cause large errors in the calculated gradients. Thus far smoothing the displacement data has been unsuccessful in correcting this problem. 
Has anyone worked on a similar problem or had to deal with noisy inputs into a meshfree method like this? I'm open to any suggestions.

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