Abstract
Abstract The aim of this paper is to present a new stable method for smoothing and differentiating noisy data defined on a bounded domain Ω ⊂ R N \Omega \subset {{\mathbb{R}}}^{N} with N ≥ 1 N\ge 1 . The proposed method stems from the smoothing properties of the classical diffusion equation; the smoothed data are obtained by solving a diffusion equation with the noisy data imposed as the initial condition. We analyze the stability and convergence of the proposed method and we give optimal convergence rates. One of the main advantages of this method lies in multivariable problems, where some of the other approaches are not easily generalized. Moreover, this approach does not require strong smoothness assumptions on the underlying data, which makes it appealing for detecting data corners or edges. Numerical examples demonstrate the feasibility and robustness of the method even with the presence of a large amounts of noise.
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