##  [Moving Least Squares](/moving-least-squares-0) 

 Definition

A meshless approximation technique that constructs local polynomial approximants by solving a weighted least-squares problem whose weights depend on the evaluation point, producing smooth reconstructions from scattered data.

 

 

 

 

 

 





## Principle

Principle

Fit local polynomial models to data in a neighborhood of each target point using distance-based weights so that the approximation varies smoothly with the evaluation location and adapts to point distribution.

 

 

 

 

 





## Demonstration

Demonstration

Given scattered displacement samples from a deforming surface, compute at each query point a local quadratic fit using nearby samples weighted by a Gaussian kernel centered at the query, yielding a continuous reconstructed displacement field.

 

 

 

 

## Misapplication

Misapplication

Using MLS with an excessively large support radius destroys locality and can smear sharp features; conversely, too small support yields ill-conditioned least-squares and noisy reconstructions.

 

 

 

 

 





## Consequence

Consequence

Properly tuned, MLS yields smooth, differentiable approximations useful for gradient recovery, surface reconstruction, and meshless PDE discretizations with controllable regularity.

 

 

 

 

## Reversal

Reversal

The reverse is global least squares or interpolation that uses a single fit for the whole domain: that approach loses local adaptivity and can suffer from oscillation or poor conditioning for large data sets.

 

 

 

 

 





## Boundary

Boundary

Effective for scattered data and meshless discretizations where locality and smoothness are desired; less suitable when exact interpolation at nodes is mandatory or when data are extremely noisy without preprocessing.

 

 

 

 

 





## Semantic Tension

Semantic Tension

Competes with radial basis function interpolation and kernel regression: MLS emphasizes local polynomial reproduction and moving evaluation, whereas RBFs provide global smooth interpolants that may require different conditioning strategies.

 

 

 

 

 





## Synthesis

Synthesis

Moving Least Squares: a meshless local fitting method that produces smooth field reconstructions by solving weighted least-squares problems whose weights move with the evaluation point, balancing locality and smoothness via support choice.