Moving least squares

<h2 id="definition">Definition</h2>

<p>Consider a function 
  
    
      
        f
        :
        
          
            R
          
          
            n
          
        
        →
        
          R
        
      
    
    {\displaystyle f:\mathbb {R} ^{n}\to \mathbb {R} }
  
 and a set of sample points 
  
    
      
        S
        =
        {
        (
        
          x
          
            i
          
        
        ,
        
          f
          
            i
          
        
        )
        
          |
        
        f
        (
        
          x
          
            i
          
        
        )
        =
        
          f
          
            i
          
        
        }
      
    
    {\displaystyle S=\{(x_{i},f_{i})|f(x_{i})=f_{i}\}}
  
. Then, the moving least square approximation of degree 
  
    
      
        m
      
    
    {\displaystyle m}
  
 at the point 
  
    
      
        x
      
    
    {\displaystyle x}
  
 is 
  
    
      
        
          
            
              p
              ~
            
          
        
        (
        x
        )
      
    
    {\displaystyle {\tilde {p}}(x)}
  
 where 
  
    
      
        
          
            
              p
              ~
            
          
        
      
    
    {\displaystyle {\tilde {p}}}
  
 minimizes the weighted least-square error 
</p>

∑
          
            i
            ∈
            I
          
        
        (
        p
        (
        
          x
          
            i
          
        
        )
        −
        
          f
          
            i
          
        
        
          )
          
            2
          
        
        θ
        (
        ‖
        x
        −
        
          x
          
            i
          
        
        ‖
        )
      
    
    {\displaystyle \sum _{i\in I}(p(x_{i})-f_{i})^{2}\theta (\|x-x_{i}\|)}

<p>over all polynomials 
  
    
      
        p
      
    
    {\displaystyle p}
  
 of degree 
  
    
      
        m
      
    
    {\displaystyle m}
  
 in 
  
    
      
        
          
            R
          
          
            n
          
        
      
    
    {\displaystyle \mathbb {R} ^{n}}
  
. 
  
    
      
        θ
        (
        s
        )
      
    
    {\displaystyle \theta (s)}
  
 is the weight and it tends to zero as 
  
    
      
        s
        →
        ∞
      
    
    {\displaystyle s\to \infty }
  
.
</p><p>In the example 
  
    
      
        θ
        (
        s
        )
        =
        
          e
          
            −
            
              s
              
                2
              
            
          
        
      
    
    {\displaystyle \theta (s)=e^{-s^{2}}}
  
.  The smooth interpolator of "order 3" is a quadratic interpolator.
</p>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Local_regression/0kvMY73g">Local regression</a></li>
<li><a href="/facts/Diffuse_element_method/AuCcII2n">Diffuse element method</a></li>
<li><a href="/facts/Moving_average/tWY74500">Moving average</a></li></ul>

<ul><li><a href="http://dl.acm.org/citation.cfm?id=301704">The approximation power of moving least squares</a> David Levin, Mathematics of Computation, Volume 67, 1517-1531, 1998 <a href="https://www.ams.org/mcom/1998-67-224/S0025-5718-98-00974-0/S0025-5718-98-00974-0.pdf">[1]</a></li>
<li><a href="http://www.sciencedirect.com/science/article/pii/S0045794905000726/">Moving least squares response surface approximation: Formulation and metal forming applications</a> Piotr Breitkopf; Hakim Naceur; Alain Rassineux; Pierre Villon, Computers and Structures, Volume 83, 17-18, 2005.</li>
<li><a href="https://doi.org/10.1007%2FBF00364252">Generalizing the finite element method: diffuse approximation and diffuse elements</a>, B Nayroles, G Touzot. Pierre Villon, P, Computational Mechanics Volume 10, pp 307-318, 1992</li></ul>
<h2 id="external-links">External links</h2>
<ul><li><a href="http://www.nealen.net/projects/mls/asapmls.pdf">An As-Short-As-Possible Introduction to the Least Squares, Weighted Least Squares and Moving Least Squares Methods for Scattered Data Approximation and Interpolation</a></li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1"><p>Liang, Jian; Zhao, Hongkai (January 2013). "Solving Partial Differential Equations on Point Clouds". SIAM Journal on Scientific Computing. 35 (3): A1461 – A1486. Bibcode:2013SJSC...35A1461L. doi:10.1137/120869730. S2CID 9984491. <a href="/wiki/Bibcode_(identifier)" target="_blank">/wiki/Bibcode_(identifier)</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>Gross, B. J.; Trask, N.; Kuberry, P.; Atzberger, P. J. (15 May 2020). "Meshfree methods on manifolds for hydrodynamic flows on curved surfaces: A Generalized Moving Least-Squares (GMLS) approach". Journal of Computational Physics. 409: 109340. arXiv:1905.10469. Bibcode:2020JCoPh.40909340G. doi:10.1016/j.jcp.2020.109340. S2CID 166228451. <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
<li id="fn:3"><p>Gross, B. J.; Kuberry, P.; Atzberger, P. J. (15 March 2022). "First-passage time statistics on surfaces of general shape: Surface PDE solvers using Generalized Moving Least Squares (GMLS)". Journal of Computational Physics. 453: 110932. arXiv:2102.02421. Bibcode:2022JCoPh.45310932G. doi:10.1016/j.jcp.2021.110932. ISSN 0021-9991. S2CID 231802303.
 <a href="https://doi.org/10.1016/j.jcp.2021.110932" target="_blank">https://doi.org/10.1016/j.jcp.2021.110932</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li>
<li id="fn:4"><p>Liang, Jian; Zhao, Hongkai (January 2013). "Solving Partial Differential Equations on Point Clouds". SIAM Journal on Scientific Computing. 35 (3): A1461 – A1486. Bibcode:2013SJSC...35A1461L. doi:10.1137/120869730. S2CID 9984491. <a href="/wiki/Bibcode_(identifier)" target="_blank">/wiki/Bibcode_(identifier)</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li>
<li id="fn:5"><p>Gross, B. J.; Kuberry, P.; Atzberger, P. J. (15 March 2022). "First-passage time statistics on surfaces of general shape: Surface PDE solvers using Generalized Moving Least Squares (GMLS)". Journal of Computational Physics. 453: 110932. arXiv:2102.02421. Bibcode:2022JCoPh.45310932G. doi:10.1016/j.jcp.2021.110932. ISSN 0021-9991. S2CID 231802303.
 <a href="https://doi.org/10.1016/j.jcp.2021.110932" target="_blank">https://doi.org/10.1016/j.jcp.2021.110932</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li>
<li id="fn:6"><p>Gross, B. J.; Trask, N.; Kuberry, P.; Atzberger, P. J. (15 May 2020). "Meshfree methods on manifolds for hydrodynamic flows on curved surfaces: A Generalized Moving Least-Squares (GMLS) approach". Journal of Computational Physics. 409: 109340. arXiv:1905.10469. Bibcode:2020JCoPh.40909340G. doi:10.1016/j.jcp.2020.109340. S2CID 166228451. <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li>
<li id="fn:7"><p>Wang, Hong-Yan; Xiang, Dao-Hong; Zhou, Ding-Xuan (1 March 2010). "Moving least-square method in learning theory". Journal of Approximation Theory. 162 (3): 599–614. doi:10.1016/j.jat.2009.12.002. ISSN 0021-9045. <a href="https://doi.org/10.1016/j.jat.2009.12.002" target="_blank">https://doi.org/10.1016/j.jat.2009.12.002</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></p></li>
<li id="fn:8"><p>Trask, Nathaniel; Patel, Ravi G.; Gross, Ben J.; Atzberger, Paul J. (13 September 2019). "GMLS-Nets: A framework for learning from unstructured data". arXiv:1909.05371 [cs.LG]. <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></p></li>
</ol>

Moving least squares open-in-new

Moving least squares