A two-stage filter for smoothing multivariate noisy data on unstructured grids

Experimental data as well as numerical simulations are very often affected by “noise”, random fluctuations that distort the final output or the intermediate products of a numerical process. In this paper, a new method for smoothing data given on nonstructured grids is proposed. It takes advantage of the smoothing properties of kernel-weighted averaging and least-squares techniques. The weighted averaging is performed following a Shepard-like procedure, where Gaussian kernels are employed, while least-squares fitting is reduced to the use of very few basis functions so as to improve smoothness, though at the price of interpolation accuracy. Once we have defined an n-point grid and want to make a smoothed fit at a given grid point, this method reduces to consideration of all m-point stencils (for a given value of m where m < n) that include that point, and to make a least-squares fitting, for that particular point, in each of those stencils; finally, the various results, thus obtained, are weight-averaged, the weights being inversely proportional to the distance of the point to the middle of the stencil. Then this process is repeated for all the grid points so as to obtain the smoothing of the input function. Though this method is generalized for the multivariate case, one-, two-, and five-dimensional test cases are shown as examples of the performance of this method.