Weighted Least Squares (WLS) - Revision history

AndrejKP: Typos

2022-06-16T12:57:59Z

Typos

2022-06-16T12:55:27Z

b_i (s_j - p) -> b_i (s_j) in notation cheat sheet (seems like what is used from then on)

2019-09-08T16:26:58Z

‎MLS

2019-09-08T16:20:26Z

‎Weighted Least Squares

2019-09-08T16:20:14Z

‎Weighted Least Squares

2019-09-08T16:19:26Z

‎Definition of local approximation

2019-09-08T16:19:10Z

‎Definition of local approximation

2019-09-08T16:16:52Z

‎Definition of local approximation

2019-09-08T16:16:01Z

‎Notation cheat sheet

2019-09-08T16:14:51Z

‎Notation Cheat sheet

@@ Line 103: / Line 103: @@
 the weighted least squares problem whenever we move the point of approximation.
-Note that if out weight is constant or if $n = m$, when approximation reduces to interpolation, the weights do not play
+Note that if our weight is constant or if $n = m$, when approximation reduces to interpolation, the weights do not play
 any role and this method is redundant. In fact, its benefits arise when supports are rather large.
 See <xr id="fig:comparisonMLSandWLS"/> for comparison between MLS and WLS approximations. MLS approximation remains close to
-actual function while still inside the support domain, while WLS approximation becomes bad when
+the actual function while still inside the support domain, while WLS approximation becomes bad when
 we come out of the reach of the weight function.
 {{reflist}}

@@ Line 11: / Line 11: @@
    \vec p \in \R^k           & \dots \text{center point of approximation} \\
    b_i\colon \R^k \to \R     & \dots \text{basis functions } \quad i=1,\dots,m \\
-   B_{j, i} \in \R           & \dots \text{value of basis functions in support points } b_i(s_j-p) \quad j=1,\dots,n, \quad i=1,\dots,m\\
+   B_{j, i} \in \R           & \dots \text{value of basis functions in support points } b_i(\vec{s}_j) \quad j=1,\dots,n, \quad i=1,\dots,m\\
    \omega \colon \R^k \to \R & \dots \text{weight function} \\
    w_j \in \R                & \dots \text{weights } \omega(\vec{s}_j-\vec{p})  \quad j=1,\dots,n \\

@@ Line 85: / Line 85: @@
 \]
-= MLS =
+= Moving Least Squares =
 <figure id="fig:comparisonMLSandWLS">

@@ Line 82: / Line 82: @@
 constant and applied the linearity.
 \[
-  \widehat{\mathcal L u}(\vec x) = \sum_{i=1}^m \alpha_i (\mathcal L b_i)(\vec x) = (\mathcal L \b b(\vec x)^\T \b \alpha.
+  \widehat{\mathcal L u}(\vec x) = \sum_{i=1}^m \alpha_i (\mathcal L b_i)(\vec x) = (\mathcal L \b b(\vec x))^\T \b \alpha.
 \]

@@ Line 82: / Line 82: @@
 constant and applied the linearity.
 \[
-  \widehat{\mathcal L u}(\vec x) = \sum_{i=1}^m \alpha_i (\mathcal L b_i)(\vec x).
+  \widehat{\mathcal L u}(\vec x) = \sum_{i=1}^m \alpha_i (\mathcal L b_i)(\vec x) = (\mathcal L \b b(\vec x)^\T \b \alpha.
 \]

← Older revision		Revision as of 16:19, 8 September 2019
Line 65:		Line 65:

	The problem of finding the coefficients $\b{\alpha}$ that minimize the error $\b{e}$ numerically is described in [[Solving overdetermined systems]].		The problem of finding the coefficients $\b{\alpha}$ that minimize the error $\b{e}$ numerically is described in [[Solving overdetermined systems]].
		+
	Here, we simply write the solution $\b \alpha = (WB)^+ Wu = (B^\T W^2 B)^{-1} B^\T W^2 u$.		Here, we simply write the solution $\b \alpha = (WB)^+ Wu = (B^\T W^2 B)^{-1} B^\T W^2 u$.

@@ Line 66: / Line 66: @@
 The problem of finding the coefficients $\b{\alpha}$ that minimize the error $\b{e}$ numerically is described in [[Solving overdetermined systems]].
-In our MLS engine we use SVD with regularization by default.
+In our WLS engine we use SVD with regularization by default.
 = Weighted Least Squares =

@@ Line 20: / Line 20: @@
 We will also use \(\b{s}, \b{u}, \b{b}, \b{\alpha}, \b{\chi} \) to denote a column of corresponding values,
-$W$ as a $n\times n$ diagonal matrix filled with $w_j$ on the diagonal and $B$ as a $n\times m$ matrix filled with $B_{j, i} = b_i(\vec p_j)$.
+$W$ as a $n\times n$ diagonal matrix filled with $w_j$ on the diagonal and $B$ as a $n\times m$ matrix filled with $B_{j, i} = b_i(\vec s_j)$.
 = Definition of local approximation =

@@ Line 2: / Line 2: @@
 It is also often called GFDM (generalized finite differences method).
-= Notation Cheat sheet =
+= Notation cheat sheet =
 \begin{align*}
    m \in \N                  & \dots \text{number of basis functions} \\
@@ Line 19: / Line 19: @@
 \end{align*}
-We will also use \(\b{s}, \b{u}, \b{b}, \b{\alpha}, \b{\chi} \) to annotate a column of corresponding values,
+We will also use \(\b{s}, \b{u}, \b{b}, \b{\alpha}, \b{\chi} \) to denote a column of corresponding values,
-$W$ as a $n\times n$ diagonal matrix filled with $w_j$ on the diagonal and $B$ as a $n\times m$ matrix filled with $B_{j, i}$.
+$W$ as a $n\times n$ diagonal matrix filled with $w_j$ on the diagonal and $B$ as a $n\times m$ matrix filled with $B_{j, i} = b_i(\vec p_j)$.
 = Definition of local approximation =