Presmoothing

Presmoothing is a preprocessing smoothing step applied before a primary computation, typically to reduce high-frequency noise or to stabilize a numerical method. The term is used in several disciplines, most notably numerical linear algebra and signal processing.

In numerical linear algebra and multigrid methods, presmoothing refers to applying a smoothing operator to the current approximation on the fine grid before transferring information to a coarser grid. The goal is to attenuate high-frequency error components that are poorly represented on coarse grids and can slow or destabilize convergence. A subsequent post-smoothing step may follow after coarse-grid correction. Common smoothers include Gauss-Seidel, Jacobi (often with relaxation), damped polynomial smoothers, or other relaxation schemes. The number of pre-smoothing steps and the choice of smoother influence the overall efficiency and robustness of the multigrid cycle.

In data analysis and signal processing, presmoothing describes applying a smoothing filter to data prior to further processing, such as differentiation, de-noising, or feature extraction. The intent is to suppress noise that would otherwise be amplified by subsequent operations. While presmoothing can improve stability and reduce spurious artifacts, it can also bias or blur important features if the smoothing is too aggressive or poorly tuned. Typical approaches include moving-average filters, Gaussian smoothing, and Savitzky–Golay filtering, chosen to balance noise reduction with preservation of signal structure.

Overall, presmoothing is about applying a smoothing operation early in a workflow to improve accuracy, stability, or convergence in the presence of noise or high-frequency error, with the specific technique selected to suit the application and data characteristics.