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.
Basics
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., \[{r^2} = \sum\limits_j^n {W\left( {{{\bf{p}}_j}} \right){{\left( {u({{\bf{p}}_j}) - \hat u({{\bf{p}}_j})} \right)}^2}} = {\left( {{\bf{B\alpha }} - {\bf{u}}} \right)^{\rm{T}}}{\bf{W}}\left( {{\bf{B\alpha }} - {\bf{u}}} \right)\] Where $\bf{B}$ is non-square matrix of dimension $m \times n$ and $\bf{W}$ diagonal matrix of weights. The least squares problem can be solved with three approaches
- Normal equation (fastest – less accurate) – using Cholesky decomposition
- QR decomposition of $\bf{B}$ ($rank(\bf{B})=m$ – number of basis functions)
- SVD decomposition of $\bf{B}$ (more expensive – more reliable, no rank demand)
In MM we use SVD.
Normal equations
We seek the minimum of \[{\bf{r}} = {\bf{u}} - {\bf{B\alpha }}\], \[{r^2} = {\left( {{\bf{u}} - {\bf{B\alpha }}} \right)^{\rm{T}}}\left( {{\bf{u}} - {\bf{B\alpha }}} \right)\] By seeking the zero gradient in terms of coefficient \[\frac{\partial }{{\partial {\alpha _i}}}\left( {{{\left( {{\bf{u}} - {\bf{B\alpha }}} \right)}^{\rm{T}}}\left( {{\bf{u}} - {\bf{B\alpha }}} \right)} \right) = 0\] resulting in \[{\bf{B}}{{\bf{B}}^{\rm{T}}}{\bf{\alpha }} = {{\bf{B}}^{\rm{T}}}{\bf{u}}\] The coefficient matrix $\bf{B}^T\bf{B}$ is symmetric and positive definite. However, solving above problem directly is poorly behaved with respect to round-off effects since the condition number of $\bf{B}^T\bf{B}$is the square of that of $\bf{B}$. Or in case of weighted LS \[{\bf{WB}}{{\bf{B}}^{\rm{T}}}{\bf{\alpha }} = {\bf{W}}{{\bf{B}}^{\rm{T}}}{\bf{u}}\]
QR Decomposition
\[{\bf{B}} = {\bf{QR}} = \left[ {{{\bf{Q}}_1},{{\bf{Q}}_2}} \right]\left[ {\begin{array}{*{20}{c}} {{{\bf{R}}_1}}\\ 0 \end{array}} \right]\] \[{\bf{B}} = {{\bf{Q}}_1}{{\bf{R}}_1}\] $\bf{Q}$ is unitary matrix ($\bf{Q}^{-1}=\bf{Q}^T$). Useful property of a unitary matrices is that multiplying with them does not alter the (Euclidean) norm of a vector, i.e., \[\left\| {{\bf{Qx}}} \right\|{\bf{ = }}\left\| {\bf{x}} \right\|\] And $\bf{R}$ is upper diagonal matrix \[{\bf{R = (}}{{\bf{R}}_{\bf{1}}}{\bf{,}}0{\bf{)}}\] therefore we can say \[\begin{array}{l} \left\| {{\bf{B\alpha }} - {\bf{u}}} \right\| = \left\| {{{\bf{Q}}^{\rm{T}}}\left( {{\bf{B\alpha }} - {\bf{u}}} \right)} \right\| = \left\| {{{\bf{Q}}^{\rm{T}}}{\bf{B\alpha }} - {{\bf{Q}}^{\rm{T}}}{\bf{u}}} \right\|\\ = \left\| {{{\bf{Q}}^{\rm{T}}}\left( {{\bf{QR}}} \right){\bf{\alpha }} - {{\bf{Q}}^{\rm{T}}}{\bf{u}}} \right\| = \left\| {\left( {{{\bf{R}}_1},0} \right){\bf{\alpha }} - {{\left( {{{\bf{Q}}_1},{{\bf{Q}}_{\bf{2}}}} \right)}^{\rm{T}}}{\bf{u}}} \right\|\\ = \left\| {{{\bf{R}}_{\bf{1}}}{\bf{\alpha }} - {{\bf{Q}}_{\bf{1}}}{\bf{u}}} \right\| + \left\| {{\bf{Q}}_2^{\rm{T}}{\bf{u}}} \right\| \end{array}\] Of the two terms on the right we have no control over the second, and we can render the first one zero by solving \[{{\bf{R}}_{\bf{1}}}{\bf{\alpha }} = {\bf{Q}}_{_{\bf{1}}}^{\rm{T}}{\bf{u}}\] Which results in a minimum. We could also compute it with pseudo inverse \[\mathbf{\alpha }={{\mathbf{B}}^{-1}}\mathbf{u}\] Where pseudo inverse is simply \[{{\mathbf{B}}^{-1}}=\mathbf{R}_{\text{1}}^{\text{-1}}{{\mathbf{Q}}^{\text{T}}}\] (once again, R is upper diagonal matrix, and Q is unitary matrix). And for weighted case \[\mathbf{\alpha }={{\left( {{\mathbf{W}}^{0.5}}\mathbf{B} \right)}^{-1}}\left( {{\mathbf{W}}^{0.5}}\mathbf{u} \right)\]
