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Problems wollen wir nun einige Begriffe definieren. Zunächst wollen wir ein achsenorientiertes Rechteck beschreiben, dessen Seiten parallel zu den Achsen des kartesischen Koordinatensystems liegen. -\begin{defi}\label{math:def:T} Sei $\bs v \in \R^3$, $\bs a,\bs b \in \{(1,0,0)^T,(0,1,0)^T,(0,0,1)^T\}$ mit $\bs a\neq \bs b$ und $a,b \in \R$ mit $a,b > 0$. Dann heißt +\begin{defi}\label{math:def:T} Sei $\bs v \in \R^3$, $\bs a,\bs b \in \{(1,0,0)^T,(0,1,0)^T,(0,0,1)^T\}$ mit $\bs a\neq \bs b$ und $\alpha, \beta \in \R$ mit $\alpha, \beta > 0$. Dann heißt \begin{align*} -T := \{\bs v + \lambda_1 a {\bs a} + \lambda_2 b {\bs b} ~|~ \lambda_1,\lambda_2 \in[0,1]\} +T := \{\bs v + \lambda_1 \alpha {\bs a} + \lambda_2 \beta {\bs b} ~|~ \lambda_1,\lambda_2 \in[0,1]\} \end{align*} achsenorientiertes Rechteck. Ferner heißt \begin{align*} - \gamma_T := [0,1]^2 \to T: \lambda_1,\lambda_2 \mapsto \bs v + \lambda_1 a {\bs a} + \lambda_2 b {\bs b} + \gamma_T := [0,1]^2 \to T: \lambda_1,\lambda_2 \mapsto \bs v + \lambda_1 \alpha {\bs a} + \lambda_2 \beta {\bs b} \end{align*} die zu $T$ zugehörige Parametrisierung. \end{defi} @@ -492,18 +492,6 @@ gilt, wobei $p$ das Interpolationspolynom von $f$ vom Grad $n$ zu den Knoten $t_ 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 gut durch Polynome interpolieren. Hierzu benötigen wir das folgende Lemma: -% \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{\partial^n u}_{\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*} -% \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{lem} -% \hfill$\square$ - \begin{lem}[BG, Theorem 3.2]\label{thm:sem:ipolnD} %Interpol über Glatten KERN nD Sei ${[0,1]^d} \subseteq \R^d$ eine achsenorientierte Box wobei die Intervalle $[0,1]$ abgeschlossen sind, und vorausgesetzt die Funktion $u$ erfüllt für Konstanten $C_u,\rho_u > 0$ \begin{align} @@ -512,9 +500,123 @@ Sei ${[0,1]^d} \subseteq \R^d$ eine achsenorientierte Box wobei die Intervalle $ Dann gilt für alle $p\in \N_0$ \begin{align}\label{math:sem:ipolnD} \norm{u-\I_p^d u}_{\infty,{[0,1]^d}} &\leq C_u 8e(1+\sqrt d\rho_u)\Lambda_k^d(p+1)\left(1+\frac{2}{\sqrt d\rho_u}\right)^{-(p+1)}. -\end{align} -\end{lem} \hfill$\square$ +\end{align}\hfill$\square$ +\end{lem} % Quadratur ----------------------------------------- + +\begin{lem} + Sei $\kappa(\cdot , \cdot) : \R^6 \rightarrow \R$ eine asymptotisch glatte Kernfunktion und sei weiterhin $ g: \R^4 \rightarrow \R^6$ die Parametrisierung der achsenorientierten Rechtecke $T_j$ und $T_k$ + \begin{align} + g(\lambda) = (\gamma_j(\lambda_1,\lambda_2),\gamma_k(\lambda_3,\lambda_4)), + \end{align} + wobei $\gamma_j$ die Parametrisierung des achsenorientierten Rechtecks $T_j$ aus Definition \ref{math:def:T} ist. + Mit der Hilfsfunktion $t_{jk} : \N^4 \rightarrow \N^6$ + \begin{align} + t_{jk}(\bs z) = + \begin{pmatrix} + \bs z_1 \cdot \bs a_j + \bs z_2 \cdot \bs b_j \\ + \bs z_3 \cdot {\bs a_k} + \bs z_4 \cdot {\bs b_k} + \end{pmatrix}, + \end{align} + wobei $\bs a_j,\bs b_j \in \R^3$ die paarweise verschiedenen Einheitsvektoren zur Parametrisierung aus Definition \ref{math:def:T} sind, gilt dann für die partiellen Ableitungen der Verknüpfung $\kappa \circ g$ die Kettenregel + \begin{align*} + \abs{\partial^z(\kappa \circ g)(x)} + &= \diam(T_j)_{\bs a}^{z_1}\diam(T_j)_{\bs b}^{z_2} \diam(T_k)_{\bs a}^{z_3}\diam(T_k)_{\bs b}^{z_4} \abs{\partial^{t_{jk}(z)}(\kappa)(g(x))}. + \end{align*} +\end{lem} +\beweis + +\begin{align} + \kappa \circ g : \R^4 \to \R^6 \to \R +\end{align} + Kettenregel +\begin{align} + D(\kappa \circ g)(x) = D\kappa(g(x)) \circ D g(x) \quad \in \R^{1\times 4} +\end{align} + +\begin{align} + A_{1\ell} &= \partial_{\ell} \kappa(g(x))\\ + B_{\ell m} &= \partial_m g_{\ell}(x) +\end{align} + +\begin{align} + AB_{1m} & = \sum_{\ell=1}^6 A_{1\ell} B_{\ell m} = \sum_{\ell=1}^6 \underbrace{\partial_{\ell} \kappa(g(x))}_{\text{Asymp Glatt}} \cdot \underbrace{ \partial_m g_{\ell}(x)}_{\text{?}} +\end{align} + +\begin{align} + \partial_m (\kappa \circ g)(x) = \sum_{\ell=1}^6 \partial_{\ell} \kappa(g(x)) \cdot \partial_m g_{\ell}(x) +\end{align} + + +\begin{align} + g(\lambda) = (\gamma_j(\lambda_1,\lambda_2),\gamma_k(\lambda_3,\lambda_4)) +\end{align} + +Für $\ell \in \{1,2,3\}$ +\begin{align} + g_{\ell}(\lambda) &= \gamma_j(\lambda_1,\lambda_2) = \bs v + \lambda_1 \alpha {\bs a} + \lambda_2 \beta {\bs b}\\ + \partial_{34} g_{\ell}(\lambda) &= 0\\ + \partial_1 g_{\ell}(\lambda) &= \alpha {\bs a}_{\ell}\\ + \partial_2 g_{\ell}(\lambda) &= \beta {\bs b}_{\ell} +\end{align} + +Für $\ell \in \{4,5,6\}$ +\begin{align} + g_{\ell}(\lambda) &= \gamma_k(\lambda_3,\lambda_4) = \tilde {\bs v} + \lambda_3 \tilde \alpha \tilde{ \bs a} + \lambda_4 \tilde \beta \tilde{ \bs b}\\ + \partial_{12} g_{\ell}(\lambda) &= 0\\ + \partial_3 g_{\ell}(\lambda) &= \tilde \alpha \tilde{ \bs a}_{\ell -3}\\ + \partial_4 g_{\ell}(\lambda) &= \tilde \beta \tilde{ \bs b}_{\ell -3} +\end{align} + + + +{% +\newcommand{\mc}[3]{\multicolumn{#1}{#2}{#3}} +\begin{align} + \partial_m (\kappa \circ g)(x) = + \begin{pmatrix} + \partial_1 \kappa(g(x)) & \cdots & \partial_6 \kappa(g(x)) + \end{pmatrix} + \cdot + \begin{pmatrix} + \alpha {\bs a} & \beta {\bs b} & \mc{2}{c}{\bs 0}\\ + \mc{2}{c}{\bs 0} & \tilde \alpha \tilde{ \bs a} &\tilde \beta \tilde{ \bs b} + \end{pmatrix} +\end{align} + +}% + +\begin{align} + ind : \{(1,0,0)^T,(0,1,0)^T,(0,0,1)^T\} &\rightarrow \{ 1 , 2 , 3 \}\\ + x &\mapsto (1, 2, 3) \cdot x +\end{align} + +\begin{align} + \partial_1 (\kappa \circ g)(x) &= \alpha \cdot \partial_{ind(\bs a)}\kappa(g(x))\\ + \partial_2 (\kappa \circ g)(x) &= \beta \cdot \partial_{ind(\bs b)}\kappa(g(x))\\ + \partial_3 (\kappa \circ g)(x) &= \tilde \alpha \cdot \partial_{ind(\tilde{\bs a})+3}\kappa(g(x))\\ + \partial_4 (\kappa \circ g)(x) &= \tilde \beta \cdot \partial_{ind(\tilde{\bs b})+3}\kappa(g(x)) +\end{align} + + \begin{align} + t_{jk} : \N^4 &\rightarrow \N^6\\ + \bs z &\mapsto + \begin{pmatrix} + z_1 \cdot \bs a + z_2 \cdot \bs b \\ + z_3 \cdot \tilde{\bs a} + z_4 \cdot \tilde{\bs b} + \end{pmatrix} +\end{align} + +\begin{align} + \partial^z (\kappa \circ g)(x) &= \alpha^{z_1} \beta^{z_2} \tilde \alpha^{z_3} \tilde \beta^{z_4} \cdot \partial^{t_{jk}(z)}\kappa(g(x)) + \quad z \in \N^4 +\end{align} + + + + +\hfill$\square$ + \begin{defi} Seien $T_j,T_k \in \R^3$ achsenorientierte Rechtecke und sei $\zeta_Q > 0$ fest. Dann heißt das Paar $(T_j,T_k)$ $\zeta_Q$-zulässig, genau dann wenn \begin{align}\label{math:sem:zetaQ} \dist (T_j, T_k)&\geq \zeta_Q \max\{ \diam(T_j) , \diam(T_k)\}. @@ -1032,7 +1134,7 @@ dy_2 dy_1 dx_2 dx_1\\ \clearpage -\section{fasträguläre Partionierung in \Matlab} +\section{Fastreguläre Partionierung in \Matlab} \todo{ \scriptsize \begin{itemize} \item Datenstruktur -- 2.47.3