Weighted Least Squares (WLS)

One of the most important building blocks of the meshless methods is the Moving Least Squares approximation, which is implemented in the EngineMLS class.

Figure 1: Example of 1D MLS approximation

In general, approximation function can be written as \[\hat u({\bf{p}}) = \sum\limits_i^m {{\alpha _i}{b_i}({\bf{p}})} = {{\bf{b}}^{\rm{T}}}{\bf{\alpha }}\] where $\hat u,\,{B_i}\,and\,{\alpha _i}$ stand for approx. function, coefficients and basis function, respectively. We minimize the sum of residuum squares, i.e., the sum of squares of difference between the approx. function and target function, in addition we can also add weight function that controls the significance of nodes, i.e., [{_ <math display='block' xmlns='http://www.w3.org/1998/Math/MathML'>

<semantics>
 <mrow>
  <msup>
   <mi>r</mi>
   <mn>2</mn>
  </msup>
  <mo>=</mo><mstyle displaystyle='true'>
   <munderover>
    <mo>∑</mo>
    <mi>j</mi>
    <mi>n</mi>
   </munderover>
   <mrow>
    <mi>W</mi><mrow><mo>(</mo>
     <mrow>
      <msub>
       <mstyle mathvariant='bold' mathsize='normal'><mi>p</mi></mstyle>
       <mi>j</mi>
      </msub>
      </mrow>
    <mo>)</mo></mrow><msup>
     <mrow>
      <mrow><mo>(</mo>
       <mrow>
        <mi>u</mi><mo stretchy='false'>(</mo><msub>
         <mstyle mathvariant='bold' mathsize='normal'><mi>p</mi></mstyle>
         <mi>j</mi>
        </msub>
        <mo stretchy='false'>)</mo><mo>−</mo><mover accent='true'>
         <mi>u</mi>
         <mo>^</mo>
        </mover>
        <mo stretchy='false'>(</mo><msub>
         <mstyle mathvariant='bold' mathsize='normal'><mi>p</mi></mstyle>
         <mi>j</mi>
        </msub>
        <mo stretchy='false'>)</mo></mrow>
      <mo>)</mo></mrow></mrow>
     <mn>2</mn>
    </msup>
    </mrow>
  </mstyle><mo>=</mo><msup>
   <mrow>
    <mrow><mo>(</mo>
     <mrow>
      <mstyle mathvariant='bold' mathsize='normal'><mi>B</mi><mi>α</mi></mstyle><mo>−</mo><mstyle mathvariant='bold' mathsize='normal'><mi>u</mi></mstyle></mrow>
    <mo>)</mo></mrow></mrow>
   <mtext>T</mtext>
  </msup>
  <mstyle mathvariant='bold' mathsize='normal'><mi>W</mi></mstyle><mrow><mo>(</mo>
   <mrow>
    <mstyle mathvariant='bold' mathsize='normal'><mi>B</mi><mi>α</mi></mstyle><mo>−</mo><mstyle mathvariant='bold' mathsize='normal'><mi>u</mi></mstyle></mrow>
  <mo>)</mo></mrow></mrow>
 <annotation encoding='MathType-MTEF'>MathType@MTEF@5@5@+=
 feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
 hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
 fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOCamaaCa
 aaleqabaGaaGOmaaaakiabg2da9maaqahabaGaam4vamaabmaabaGa
 aCiCamaaBaaaleaacaWGQbaabeaaaOGaayjkaiaawMcaamaabmaaba
 GaamyDaiaacIcacaWHWbWaaSbaaSqaaiaadQgaaeqaaOGaaiykaiab
 gkHiTiqadwhagaqcaiaacIcacaWHWbWaaSbaaSqaaiaadQgaaeqaaO
 GaaiykaaGaayjkaiaawMcaamaaCaaaleqabaGaaGOmaaaaaeaacaWG
 QbaabaGaamOBaaqdcqGHris5aOGaeyypa0ZaaeWaaeaacaWHcbGaaC
 ySdiabgkHiTiaahwhaaiaawIcacaGLPaaadaahaaWcbeqaaiaabsfa
 aaGccaWHxbWaaeWaaeaacaWHcbGaaCySdiabgkHiTiaahwhaaiaawI
 cacaGLPaaaaaa@5BA9@
 </annotation>
</semantics>

</math>}]