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zO<^ibyn=b$?rkb_W1CnTK*RiL_1O%LaG-fW_FMB%|L^tXb?yl%+Os0V?8_-AOIGDq z>^in?gdP!m$7iyz`iDDh$J|{R8oii`i4KAh%(+3w>$gzx;z)C9n`227={qO4MqoPD zq{Jb&dxEC>Hoy(LyhQBT=hE)TfS!3->WTpHhcw?ZZ-V)CjWZg7n zQ%k#B5wOz-4+RpvhV;UFGEkC+r5)2uG=A#pX6=tA0Dd_3D)9 z+~s(Gtk2EbHgZu-GoSzQ;aZ+sLIpg;Uia$;ZD*dosFy|`Tf!jrdl3!hsUSh~#~j2I zo4PO;rQJT0U!oNI*C->hjoK_@8h(2}B4SRvjg4;hVq0Fp^xONt(Fr7`1nRseI#;xt ZH1NixG)WLGE}x(X7>AizQC$h=zW`3@E`tC7 diff --git a/semester3/numcs/parts/01_interpolation/02_piece-wise/02_splines.tex b/semester3/numcs/parts/01_interpolation/02_piece-wise/02_splines.tex index bc0191b..134da3d 100644 --- a/semester3/numcs/parts/01_interpolation/02_piece-wise/02_splines.tex +++ b/semester3/numcs/parts/01_interpolation/02_piece-wise/02_splines.tex @@ -3,6 +3,7 @@ \begin{definition}[]{Raum der Splines} Sei $[a, b] \subseteq \R$ ein Intervall, sei $\mathcal{G} = \{ a = x_0 < x_1 < \ldots < x_N = b \}$ und sei $d \geq 1 \in \N$. Die Menge + % FIXME: What's up with the |[x_{j - 1}, x_j] Notation? Isn't there a | missing? And what does it even mean? \begin{align*} \mathcal{S}_{d, \mathcal{G}} = \{ s \in C^{d - 1}[a, b], \smallhspace s_j := s_{|[x_{j - 1}, x_j]|} \text{ ist ein polynom von Grad höchstens } d \} \end{align*} @@ -13,4 +14,91 @@ \inlineremark Obige Definition ist undefiniert für $d = 0$, aber $\mathcal{S}_{d, \mathcal{G}}$ kann als die Menge der stückweise Konstanten Funktionen betrachtet werden. Im Vergleich zu den Kubischen Hermite-Interpolanten sind die Kubischen-Splines (für $d = 3$) \textit{zweimal} Ableitbar statt nur \textit{einmal} -\inlineremark $\dim(\mathcal{S}_{d, \mathcal{G}}) = N + d$. Es werden oft kubische Splines in Anwendungen verwendet +\inlineremark $\dim(\mathcal{S}_{d, \mathcal{G}}) = N + d$. Es werden oft kubische Splines in Anwendungen verwendet, also ist\\ +$\dim(\mathcal{S}_{d, \mathcal{G}}) = N + 3$, wir haben aber nur $N + 1$ Funktionswerte, also beleiben noch zwei Freiheitsgrade übrig. + +Dies bedeutet, dass wir ein underdeterminiertes lineares Gleichungssystem haben für $h_j = x_j - x_{j - 1}$: +\rmvspace +\begin{align*} + \begin{bmatrix} + b_0 & a_1 & b_1 & 0 & \dots & & \dots & 0 \\ + 0 & b_1 & a_2 & b_2 & & & & \\ + & 0 & \ddots & \ddots & \ddots & & & \vdots \\ + \vdots & & & \ddots & \ddots & \ddots & & \\ + & & & & \ddots & a_{N - 2} & b_{N - 2} & 0 \\ + 0 & \dots & & \dots & 0 & b_{N - 2} & a_{N - 1} & b_{N - 1} + \end{bmatrix} + \begin{bmatrix} + c_0 \\ c_1\\ \\[0.2cm] \vdots \\ \\[0.2cm] c_{N - 1} \\ c_N + \end{bmatrix} + = + \begin{bmatrix} + 3 \left( \frac{y_1 - y_0}{h_1^2} + \frac{y_2 - y_1}{h_2^2} \right) \\ + \\ \vdots \\ \\ + 3 \left( \frac{y_{N - 1} - y_{N - 2}}{h_{N - 1}^2} + \frac{y_N - y_{N - 1}}{h_N^2} \right) \\ + \end{bmatrix} +\end{align*} + +\rmvspace +Wobei im Resultatvektor Einträge der Form +\rmvspace +\begin{align*} + 3 \left( \frac{y_j - y_{j - 1}}{h_j^2} + \frac{y_{j + 1} - y_j}{h^2_{j + 1}} \right) +\end{align*} +enthalten sind und mit $a_j := \frac{2}{h_j} + \frac{2}{h_{j + 1}}$ und $b_j := \frac{1}{h_{j + 1}}$ für $j = 0, 1, \ldots, N - 1$ + +Wir müssen also zwei weitere Gleichungen finden (oder zwei Freiheitsgrade eliminieren). + +\fancydef{Vollständige kubische Spline-Interpolation} Falls wir die zusätzlichen Bedingungen $s'(x_0) = c_0$ und $s'(x_N) = c_N$ mit gegebenen $c_0$ und $c_N$ haben. +Sie ist auch bekannt als \textit{clamped cubic spline}. +In der obigen Matrix können dann die erste und letzte Spalte weggelassen werden. + +\fancydef{Natürliche kubische Spline-Interpolation} Falls wir die zusätzlichen Bedingungen $s''(x_0) = 0$ und $s''(x_N) = 0$ haben. +Dann fügen wir obigem SLE zwei Zeilen hinzu (1. und $(N + 1)$-te), die $2, 1, 0, 0, \ldots = \frac{y_1 - y_0}{h_1}$ und $0, \ldots, 0, 1, 2 = \frac{y_N - y_{N - 1}}{h_N}$. +Die Matrix ist nun also positive-definite und symmetrisch + +\vspace{-1.25pc} +{ + \begin{wrapfigure}[6]{l}{0.5\textwidth} + \fancydef{Periodische kubische Spline-Interpolation} Falls wir die zusätzlichen Bedingungen $s'(x_0) = s'(x_N)$ und $s''(x_0) = s''(x_N)$ haben. + Dies macht nur Sinn, wenn $y_0 = y_N$, also nehmen wir das an und wir haben eine Spalte weniger und eine Reihe mehr, also ist die Systemmatrix rechts + \end{wrapfigure} + \begin{align*} + A := + \begin{bmatrix} + a_1 & b_1 & 0 & \dots & 0 & b_0 \\ + b_1 & a_2 & b_2 & & & 0 \\ + 0 & \ddots & \ddots & \ddots & & \vdots \\ + 0 & & & \ddots & a_{N - 1} & b_{N - 1} \\ + b_0 & 0 & \dots & 0 & b_{N - 1} & a_0 + \end{bmatrix} + \end{align*} +} + +\inlineremark Die SLE können in $\tco{n}$ gelöst werden. + +\inlineremark Mit der ``not-a-knot''-Bedingung $s'''$ ist stetig in $x_1$ und $x_{N - 1}$ braucht man mindestens $4$ Knoten. +Da wir kubische Splines betrachten erzwing die Bedingung dass ein Polynom nur in den ersten beiden und ein anderes in den letzten beiden Subintervallen erscheint, +also gilt $s_1 = s_2$ und $s_{N - 1} = s_N$ + +\inlineremark Der natürliche Spline minimiert die Gesamtkrümmung des Funktionsgraphen: +\begin{align*} + \int_{a}^{b} |s''(x)|^2 \dx x \leq \int_{a}^{b} |g''(x)|^2 \dx x +\end{align*} +für alle Funktionen zweimal stetig differenzierbaren Funktionen $g$, für welche $g(x_j) = y_j$ gilt für jedes $j = 0, \ldots, N$ + +\begin{theorem}[]{Interpolationsfehler vollständiger kubischer Splines} + Wenn $f \in C^4[a, b]$ und $s$ der vollständige kubische Spline-Interpolation von $f$ auf einem äquidistantem Gitter mit Gitterweite $h$ ist, + dann ist der Fehler für $k = 0, 1, 2, 3$: + \begin{align*} + ||f^{(k)} - s^{(k)}||_{L^\infty} \leq \frac{5}{384} h^{4 - k} ||f^{(4)}||_{L^\infty} + \end{align*} +\end{theorem} + +\innumpy verwendet \texttt{scipy.interpolate.CubicSpline} aktuell die ``not-a-knot''-Bedingung. +Es ist möglich mithilfe von \texttt{bc\_type} beim Instanziieren der Klasse die Art des Splines zu ändern. +Folgende (relevante) Optionen stehen laut +\color{NavyBlue}\href{https://docs.scipy.org/doc/scipy/reference/generated/scipy.interpolate.CubicSpline.html}{Dokumentation}\color{black}\smallhspace +zur Verfügung: \texttt{"not-a-knot"} (was der Default ist), \texttt{"periodic"}, \texttt{"clamped"} und \texttt{"natural"} + +Auf Seite 114-115 im Skript finden sich einige Abbildungen zur Konvergenz der verschiedenen Varianten des \texttt{CubicSplices}