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index 9a090d5..2b76df5 100644 --- a/doc/doc.tex +++ b/doc/doc.tex @@ -383,7 +383,7 @@ unter bestimmten Voraussetzungen an die affinen Randstücke $T_j,T_k$ und den as An dieser stelle werden wir zunächst den Interpolationsoperator auf dem Intervall $[0,1]$ definieren. \begin{defi} -Für einen festen Grad $p \in \N$ und paarweise verschiedene Knoten $x_j \in [-1,1]$ lautet das Lagrange'sche Interpolations Problem: +Für einen festen Grad $p \in \N$ und paarweise verschiedene Knoten $x_j \in [0,1]$ lautet das Lagrange'sche Interpolations Problem: Zu gegebenen Funktionswerten $y_j \in \R$ finde ein Polynom $q \in \P^{p-1}$ so, dass \begin{align*} q(x_j) &=y_j \quad \text{ für alle } j=0,\ldots,p. @@ -397,64 +397,26 @@ wissen wir aus \todo{cite}, dass das Lagrange'sche Interpolations Problem eine e \begin{align*} q&=\sum_{j=0}^py_jL_j. \end{align*} -Mit diesem Wissen definieren wir uns nun den Interpolationsoperator $\I_p u : \C[-1,1]\to\P^{p-1}$ +Mit diesem Wissen definieren wir uns nun den Interpolationsoperator $\I_p u : \C[0,1]\to\P^{p-1}$ \begin{align} \I_pu &:= \sum_{j=0}^p u(x_j)L_j. \end{align} -Ferner definieren wir noch die Lebesgue Konstante -\begin{align} -\Lambda_p &:= \max_{x\in[-1,1]} \sum_{j=0}^p\abs{L_j(x)}. -\end{align} - -\todo{ -\subsection{Interpolation} -Zunächst wollen wir die Chebyshev-Interpolation auf dem Intervall $[0,1]$ definieren. Hierzu sei $\P_p$ die Menge aller Polynome vom Grad $p$. -% \begin{defi} -% Seien $x_0,\ldots,x_p \in [0,1]$ paarweise verschiedene Knoten und $y_0,\ldots,y_p$ die zugehörigen Funktionswerte. Die Lagrange'sche Interpolationsaufgabe lautet dann folgendermaßen: Finde ein Polynom $q$ vom Grad $p-1$, sodass -% \begin{align*} -% q(x_i) &=y_i \quad \text{ für alle } i=0,\ldots,p. -% \end{align*} -% \end{defi} -% Ferner definieren wir die zu den Knoten $x_0,\ldots,x_p$ zugehörigen Lagrange-Polynome durch -% \begin{align} -% L_j(x) &= \prod_{i=1,i\neq j}^p \frac{x-x_i}{x_j-xi}. -% \end{align} -% \todo{cite} zeigt, dass die Lagrange'sche Interpolationsaufgabe eine eindeutige Lösung besitzt, die gegeben ist durch -% \begin{align} -% q&=\sum_{j=0}^py_jL_j. -% \end{align} -% Wir definieren den Interpolationsoperator $\I_pu$ durch -% \begin{align} -% \I_pu &:= \sum_{j=0}^p u(x_j)L_j. -% \end{align} -% Außerdem definieren wir die Lebesgue-Konstante durch +% Ferner definieren wir noch die Lebesgue Konstante % \begin{align} % \Lambda_p &:= \max_{x\in[0,1]} \sum_{j=0}^p\abs{L_j(x)}. % \end{align} -\todo{cite} stellt einen Bezug zwischen der Bestapproximation und der Polynominterpolation her und ist später von Bedeutung. -\begin{sat}\label{math:ipol} -Sei $\Lambda_p$ die Lebesgue-Konstante zu $p$ paarweise verschiedener Knoten $x_1,\ldots,x_p$, die alle im Intervall $[0,1]$ liegen. Dann gilt für jedes $u \in \C[0,1]$ +Eine einfache Fehlerabschätzung für den Interpolationsoperator ist \begin{align} - \norm{\I_pu-u}_{\infty} &\leq (1 + \Lambda_p) \min_{v\in \P_p} \norm{u-v}_{\infty}, +\norm{u-\I_pu}_{\infty,[0,1]} &\leq \frac{\norm{u^{(p)}}_{\infty,[0,1]}}{p!} \max_{x\in[0,1]} \prod_{j=0}^n \abs{x-x_j} +\quad \text{ für alle }u \in \C^p([0,1]). \end{align} -wobei $\P_p$ die Menge aller Polynome vom Grad $p$ bezeichnet. -\hfill $\square$ -\end{sat} -Eine einfache Fehlerabschätzung für den Interpolationsfehler liefert \todo{cite}. Unter der Voraussetzung $u \in \C^{p+1}([0,1])$ gilt: -\begin{align} -\norm{u-\I_pu}_{\infty,[0,1]} &\leq \max_{x\in[0,1]} \left| \prod_{j=0}^n (x-x_j) \right| \frac{1}{(p+1)!} \norm{u^{(p+1)}}_{\infty,[0,1]} -\end{align} -Bei der Chebyshev-Interpolation wählt man die Knoten in Bezug auf diese Abschätzung optimal, indem man den ausschließlich durch die Knotenwahl bestimmten Term -\begin{align*} - \max_{x\in[0,1]} \left| \prod_{j=0}^n (x-x_j) \right| -\end{align*} -minimiert. Die Lösung dieses Minimierungsproblems sind die Nullstellen der Chebyshev-Polynome. Diese sind gegeben durch +\todo{ +Chebyshev-Polynome: \begin{align} - x_j &= \frac12 \cos \left( \frac{2j-1}{p} \frac12 \right) + \frac12 \quad \text{für} j=1,\ldots,p. + x_j &= \cos \left( \frac{2j-1}{p} \frac12 \right) \quad \text{ für } j=1,\ldots,p. \end{align} -Sieh dazu auch \todo{cite}.\\ Im folgenden bezeichne $\I_p$ immer den Chebyshev-Interpolationsoperator. -\todo{multi Interpolation!} -} +Multi-Interpol} + \subsection{Gauss-Quadratur} Im Folgenden wollen wir die klassische Gauss-Quadratur definieren.\\ Unter einer Quadratur verstehen wir die approximative Berechnung eines Integrals der Form: @@ -473,45 +435,73 @@ Eine Quadratur hießt interpolatorisch (vom Grad $n$), falls für jede integrier \Q_n(f) &:= \sum_{k=0}^n w_kp(x_k) \end{align} gilt, wobei $p$ das Interpolationspolynom von $f$ vom Grad $n$ zu den Knoten $x_0,\ldots,x_n$ ist. -\todo{\subsection{Bedingung} +\todo{\subsection{---} } \begin{align} \dist (T_j, T_k)&\geq \zeta \min\{ \diam(T_j) , \diam(T_k)\}\\ \dist (T_j,T_k) &\geq \zeta_1 \max\{ \diam_{|\bs a}(T_j) , \diam_{|\bs a}(T_k)\} \end{align} - -\subsection{Semianalytisch} -% Das Integral über $T_j$ kann durch die analytische Doppelintegral Funktion ersetzt werden. Die Quadratur über $T_k$ wird nun mittels einer doppelten Gauss-Quadratur berechnet. \begin{align} \int_{T_j} \int_{T_k} &\approx \q_{T_j} \int_{T_k}\\ \int_{T_j} \int_{T_k} &\approx \q_{I_1} \q_{J_1} \int_{I_2}\int_{J_2}\\ \int_{T_j} \int_{T_k} &\approx \q_{I_1} \int_{I_2} \int_{J_1} \int_{J_2}\\ \int_{T_j} \int_{T_k} &\approx \q_{T_j} \q_{J_1} \int_{J_2} \end{align} + +\subsection{Glatter Kern} +% Das Integral über $T_j$ kann durch die analytische Doppelintegral Funktion ersetzt werden. Die Quadratur über $T_k$ wird nun mittels einer doppelten Gauss-Quadratur berechnet. + \begin{defi} Die Kern-Funktion $\kappa(\bs x,\bs y)$ heißt asymptotisch glatt, falls sie glatt ist für $\bs x \neq \bs y$ und Konstanten $c_1,c_2>0$ und eine Ordnung der Singularität $s \in \R$ existieren, sodass -% \begin{align} -% \abs{\partial_{\bs x}^{\alpha}\partial_{\bs y}^{\beta}\kappa(\bs x, \bs y)} -% &\leq c_1(c_2(\abs{\bs x - \bs y})^{-(\abs{\alpha}-\abs{\beta}+s)}(\alpha + \beta)! -% \end{align} - \begin{align}\label{math:sem:glatt} + \begin{align*}\label{math:sem:glatt} % Glatter KERN \abs{\partial_{\bs x}^{\alpha}\partial_{\bs y}^{\beta}\kappa(\bs x, \bs y)} &\leq c_1(c_2(\abs{\bs x - \bs y})^{-(\abs{\alpha}+\abs{\beta}+s)}(\abs{\alpha} + \abs{\beta})! - \end{align} + \end{align*} für alle Multiindizes $\alpha,\beta \in \N_0^d$ mit $\abs{\alpha}+\abs{\beta}\geq 1$ gilt. \end{defi} Wie wir im Folgenden sehen werden, lassen sich asymptotisch glatte Kernfunktionen besonders gut durch Polynome interpolieren. Hierzu benötigen wir das Lemma \todo{cite}: -\begin{lem} +\begin{lem}\label{math:sem:ipol} %Interpol über Glatten KERN 1D Sei $J \subseteq \R$ ein abgeschlossenes und beschränktes Intervall und angenommen die Funktion $u$ erfüllt für Konstanten $C_u,\rho_u > 0$ \begin{align*} \norm{u^{(n)}}_{\infty,J} &\leq C_u \rho_u^nn!\text{~~~für alle} n \in \N_0. \end{align*} Dann gilt für alle $k\in \N_0$ -\begin{align}\label{math:sem:ipol} +\begin{align*} \min_{v \in \P_k} \norm{u-v}_{\infty,J} &\leq C_u 4e(1+\rho_u\abs{J})(k+1)\left(1+\frac{2}{\rho_u\abs{J}}\right)^{-(k+1)}. -\end{align} +\end{align*} +\end{lem} +\hfill$\square$ + +\begin{lem}\label{math:sem:ipolnD} %Interpol über Glatten KERN nD +Sei $B \subseteq \R^d$ ein abgeschlossenes und beschränktes Intervall und angenommen die Funktion $u$ erfüllt für Konstanten $C_u,\rho_u > 0$ +\begin{align*} + \norm{\partial_j^nu}_{\infty,B} &\leq C_u \rho_u^nn!\text{~~~für alle} n \in \N_0. +\end{align*} +Dann gilt für alle $k\in \N_0$ +\begin{align*} +\min_{v \in \P_k} \norm{u-\I_k^Bu}_{\infty,B} &\leq C_u 8e\Lambda_k^d(1+\rho_u\diam (B))(k+1)\left(1+\frac{2}{\rho_u\diam(B)}\right)^{-(k+1)}. +\end{align*} \end{lem} \hfill$\square$ + +\begin{align*} + \Abs{\partial_x^n\partial_y^m\kappa}&=\Abs{\partial_x^n\partial_y^m\kappa(\gamma_j(x),\gamma_k(y))}\\ + &=\diam(T_j)^n \diam(T_k)^m \Abs{(\partial_x^n\partial_y^m\kappa)(\gamma_j(x),\gamma_k(y))} +\end{align*} + +\begin{align*} + \diam(T_j)^n& \diam(T_k)^m \Abs{(\partial_x^n\partial_y^m\kappa)(\gamma_j(x),\gamma_k(y))}\\ + \leq& \diam(T_j)^n \diam(T_k)^m c_1 (c_2 \abs{\gamma_j(x) - \gamma_k(y)})^{-(\abs{n}+\abs{m}+s)}(\abs{n}+\abs{m})!\\ + \leq & \max(\diam(T_j),\diam(T_k))^{n+m} c_1 (c_2 \abs{\bs x - \bs y})^{-(n+m+s)}(n+m)!\\ + = & \frac{c_1}{c_2 \abs{(\bs x - \bs y})^s} \left( \frac{\max(\diam(T_j),\diam(T_k))}{c_2 \abs{\bs x - \bs y}} \right)^{m+n}(\abs{n}+\abs{m})!\\ + \leq& C_{\kappa}\rho_{\kappa}^nn! +\end{align*} + +\begin{align*} + \min_{v \in \P_k} \norm{u-\I_k^Bu}_{\infty,B} &\leq C_k 8e\Lambda_k^d(1+\rho_u\diam (B))(k+1)\left(1+\frac{2}{\rho_u\diam(B)}\right)^{-(k+1)}. +\end{align*} + +\subsection{1D Integral} \begin{lem} Seien $T_j,T_k$ zwei $\zeta_1$-zulässige, affine Randstücke mit $\diam(T_j) \leq \diam(T_k)$. Weiterhin sei $\kappa$ eine asymptotisch glatte Kernfunktion mit den Konstanten $c_1,c_2>0$ und Singularitätsordung $s\geq0$. Dann gilt mit @@ -564,7 +554,10 @@ und mithilfe von Satz \ref{math:ipol} gilt \norm{\kappa(\cdot,\bs y)-\I_p\kappa(\cdot,\bs y)}_{\infty,T_{j| x_{\bs b}}} &\leq C_{\zeta,j,\kappa}(1+\Lambda_p)(p+1)\left( 1+ \frac{2c_1}{\zeta} \right)^{-(p+1)}. \end{align*} -Da $\bs y$ und $x_{\bs b}$ beliebig war und die rechte Seite unabhängig von $\bs y,x_{\bs b}$ ist, folgt die Behauptung für das Supremum.\hfill$\square$ +Da $\bs y$ und $x_{\bs b}$ beliebig war und die rechte Seite unabhängig von $\bs y,x_{\bs b}$ ist, folgt die Behauptung für das Supremum. +\hfill$\square$ + +\subsection{Matrix} \begin{sat} Seien $Tj,Tk$ zwei $\zeta_1$-zulässige affine Randstücke mit zugehörigen Parametrisierungen $\gamma_j,\gamma_k$ und $\diam(T_j) \leq \diam(T_k)$. Sei weiterhin $\kappa : \R^3\times\R^3\to \R : (\bs x, \bs y) \mapsto \kappa(\bs x,\bs y)$ eine asymptotisch glatte Kernfunktion mit Konstanten $c_1,c_2,s$. Sei \begin{align} -- 2.47.3