From 25f45eb1817cc0fde14227d6c977f476c34ad480 Mon Sep 17 00:00:00 2001 From: Peter Schaefer Date: Mon, 11 Feb 2013 17:19:56 +0100 Subject: [PATCH] =?utf8?q?[doc]=20Bib=20aktualisiert=20[doc]=20referenzen?= =?utf8?q?=20eingef=C3=BCgt=20[doc]=20Interpolation,=20Volle=20Quadratur?= =?utf8?q?=20=C3=BCberarbeitet=20[res]=20h-matrix=20und=20numerik=20Skript?= =?utf8?q?=20hinzugef=C3=BCgt?= MIME-Version: 1.0 Content-Type: text/plain; charset=utf8 Content-Transfer-Encoding: 8bit --- doc/doc.bib | 121 +++++++++++++++++------- doc/doc.pdf | Bin 336179 -> 340348 bytes doc/doc.tex | 210 ++++++++++++++++++++--------------------- res/hmatrix2010.pdf | Bin 0 -> 546631 bytes res/numerik2010x11.pdf | Bin 0 -> 821601 bytes 5 files changed, 189 insertions(+), 142 deletions(-) create mode 100644 res/hmatrix2010.pdf create mode 100644 res/numerik2010x11.pdf diff --git a/doc/doc.bib b/doc/doc.bib index a4b22bb..c8ee552 100644 --- a/doc/doc.bib +++ b/doc/doc.bib @@ -1,40 +1,95 @@ -@comment{x-kbibtex-personnameformatting=<%l><, %f>} +% This file was created with JabRef 2.7b. +% Encoding: UTF-8 -@article{dor:adapt, - __markedentry = "[treecity:6]", - abstract = "We construct a converging adaptive algorithm for linear elements applied to Poisson's equation in two space dimensions. Starting from a macro triangulation, we describe how to construct an initial triangulation from a priori information. Then we use a posteriori error estimators to get a sequence of refined triangulations and approximate solutions. It is proved that the error, measured in the energy norm, decreases at a constant rate in each step until a prescribed error bound is reached. Extensions to higher-order elements in two space dimensions and numerical results are included.", - author = "Dörfler, Willy", - copyright = "Copyright © 1996 Society for Industrial and Applied Mathematics", - issn = "00361429", - journal = "SIAM Journal on Numerical Analysis", - jstor_articletype = "research-article", - jstor_formatteddate = "Jun., 1996", - number = "3", - owner = "treecity", - pages = "pp. 1106--1124", - publisher = "Society for Industrial and Applied Mathematics", - timestamp = "2012.10.04", - title = "{A Convergent Adaptive Algorithm for Poisson's Equation}", - url = "http://www.jstor.org/stable/2158497", - volume = "33", - year = "1996" +@ARTICLE{bor:errbox, + author = {Börm, Steffen and Grasedyck, Lars}, + title = {Low-Rank Approximation of Integral Operators by Interpolation}, + journal = {Computing}, + year = {2004}, + volume = {72}, + pages = {325-332}, + abstract = {A central component of the analysis of panel clustering techniques + for the approximation of integral operators is the so-called η -admissibility + condition “ min {diam(τ),diam(σ)} ≤ 2ηdist(τ,σ)” that ensures that + the kernel function is approximated only on those parts of the domain + that are far from the singularity. Typical techniques based on a + Taylor expansion of the kernel function require a subdomain to be + “far enough” from the singularity such that the parameter η has to + be smaller than a given constant depending on properties of the kernel + function. In this paper, we demonstrate that any η is sufficient + if interpolation instead of Taylor expansion␣is␣used for the kernel + approximation, which paves the way for grey-box panel clustering + algorithms.}, + doi = {10.1007/s00607-003-0036-0}, + issn = {0010-485X}, + issue = {3-4}, + owner = {treecity}, + publisher = {Springer-Verlag}, + timestamp = {2013.02.11}, + url = {http://dx.doi.org/10.1007/s00607-003-0036-0} } -@article{fer:errbem, - author = "Ferraz-Leite, Samuel and Praetorius, Dirk", - journal = "Computing", - number = "4", - pages = "135--162", - title = "{Simple a posteriori error estimators fot the h-version of the boundary element method}", - volume = "83", - year = "2008" +@ARTICLE{dor:adapt, + author = {Dörfler, Willy}, + title = {{A Convergent Adaptive Algorithm for Poisson's Equation}}, + journal = {SIAM Journal on Numerical Analysis}, + year = {1996}, + volume = {33}, + pages = {pp. 1106--1124}, + number = {3}, + abstract = {We construct a converging adaptive algorithm for linear elements applied + to Poisson's equation in two space dimensions. Starting from a macro + triangulation, we describe how to construct an initial triangulation + from a priori information. Then we use a posteriori error estimators + to get a sequence of refined triangulations and approximate solutions. + It is proved that the error, measured in the energy norm, decreases + at a constant rate in each step until a prescribed error bound is + reached. Extensions to higher-order elements in two space dimensions + and numerical results are included.}, + copyright = {Copyright © 1996 Society for Industrial and Applied Mathematics}, + issn = {00361429}, + jstor_articletype = {research-article}, + jstor_formatteddate = {Jun., 1996}, + owner = {treecity}, + publisher = {Society for Industrial and Applied Mathematics}, + timestamp = {2012.10.04}, + url = {http://www.jstor.org/stable/2158497} } -@techreport{mai:3dbem, - author = "Maischak, Matthias", - institution = "Institut f{\"u}r Angewandte Mthematik, University of Hannover", - month = jul, - title = "{The analytical computation of the Galerkin elements for the Laplace, Lam{\'e} and Helmholtz equation in 3D-BEM}", - year = "2000" +@ARTICLE{fer:errbem, + author = {Ferraz-Leite, Samuel and Praetorius, Dirk}, + title = {{Simple a posteriori error estimators fot the h-version of the boundary + element method}}, + journal = {Computing}, + year = {2008}, + volume = {83}, + pages = {135--162}, + number = {4} +} + +@TECHREPORT{mai:3dbem, + author = {Maischak, Matthias}, + title = {{The analytical computation of the Galerkin elements for the Laplace, + Lam{\'e} and Helmholtz equation in 3D-BEM}}, + institution = {Institut f{\"u}r Angewandte Mthematik, University of Hannover}, + year = {2000}, + month = jul +} + +@BOOK{pla:nummat, + title = {{Numerische Mathematik kompakt}}, + publisher = {Vieweg Verlag}, + year = {2006}, + author = {Plato, Robert}, + owner = {treecity}, + timestamp = {2013.02.11} +} + +@UNPUBLISHED{pra:hmat, + author = {Praetorius, Dirk}, + title = {Hierarchische Matrizen und Fast Multipole Method}, + year = {2009}, + owner = {treecity}, + timestamp = {2013.02.11} } diff --git a/doc/doc.pdf b/doc/doc.pdf index 065fa5a1c030e0edefe0dd14fe49f8a166f6d306..dbab7fc6fda8453510ae755344a5b7b1c0e04775 100644 GIT binary patch delta 233426 zcmV(&K;gf$!W8`C6p%@OZrwHz-QTa!QYs*p91dU0q)mzxP0*x9H$@kT-PlHKr?Hd1 zHvRRU(Tlt??Ko`!U)Y5g=9MUNhI8i3LweU}Wg0Di)BpX`&6h7?b9k}QDrz0#c=`Nr 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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} \noindent -Wie wir im Folgenden sehen werden, lassen sich asymptotisch glatte Kernfunktionen gut durch Polynome interpolieren. Hierzu benötigen wir das folgende Lemma: - +Wie wir im Folgenden sehen werden, lassen sich asymptotisch glatte Kernfunktionen gut durch Polynome interpolieren. Hierzu benötigen wir aus \cite[Theorem 3.2]{bor:errbox} das folgende Lemma: \begin{lem}\label{thm:sem:ipolnD} %Interpol über Glatten KERN nD -Die Funktion $u \in \C^{\infty}([0,1])^d$ erfüllt für Konstanten $C_u,\rho_u > 0$ -\begin{align} +Die Funktion $u \in \C^{\infty}([0,1])^d$ erfülle für Konstanten $C_u,\rho_u > 0$ +\begin{align*} \norm{\partial_j^nu}_{\infty,{[0,1]^d}} &\leq C_u \rho_u^nn!\quad\text{~für alle~} j \in \{1,\ldots,d\} \text{~und~} n \in \N_0. -\end{align} +\end{align*} 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_p^d p\left(1+\frac{2}{\sqrt d\rho_u}\right)^{-p}. + \norm{u-\I_p^d u}_{\infty,{[0,1]^d}} &\leq C_u 8e(1+\rho_u\sqrt d)\Lambda_p^d (p+1)\left(1+\frac{2}{\rho_u\sqrt d}\right)^{-(p+1)}. \end{align} \end{lem} % Quadratur ----------------------------------------- @@ -528,7 +527,7 @@ Dann gilt für alle $p\in \N_0$ Weiterhin wollen wir uns kurz die Ableitung der asymptotisch glatten Kernfunktion mit einer Parametrisierung anschauen. \begin{lem} \label{thm:sem:kett} - 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 $T_k$ + Sei $\kappa(\cdot , \cdot) : \R^3\times\R^3 \rightarrow \R$ eine asymptotisch glatte Kernfunktion und sei weiterhin $ g: \R^4 \rightarrow \R^6$ die Parametrisierung zweier achsenorientierten Rechtecke $T_j$ und $T_k$, mit $T_j \cap T_k = \emptyset$ \begin{align} g(\lambda) = (\gamma_j(\lambda_1,\lambda_2),\gamma_k(\lambda_3,\lambda_4)), \end{align} @@ -544,9 +543,9 @@ Weiterhin wollen wir uns kurz die Ableitung der asymptotisch glatten Kernfunktio wobei $\bs a_j,\bs b_j \in \R^3$ die paarweise verschiedenen Einheitsvektoren zur Parametrisierung $\gamma_j$ sind und für den Index $k$ analog, gilt dann für die Verknüpfung $\kappa \circ g$ die Kettenregel \begin{align*} \abs{\partial^{\alpha}(\kappa \circ g)(\lambda)} - &= \diam(T_j)_{\bs a}^{\alpha_1}\diam(T_j)_{\bs b}^{\alpha_2} \diam(T_k)_{\bs a}^{\alpha_3}\diam(T_k)_{\bs b}^{\alpha_4} \abs{\partial^{t_{jk}(\alpha)}(\kappa)(g(\lambda))} + &= \diam(T_j)_{\bs a}^{\alpha_1}\diam(T_j)_{\bs b}^{\alpha_2} \diam(T_k)_{\bs a}^{\alpha_3}\diam(T_k)_{\bs b}^{\alpha_4} \abs{\partial^{t_{jk}(\alpha)} \kappa (g(\lambda))} \end{align*} - mit dem Multiindex $\alpha = (\alpha_1, \alpha_2, \alpha_3, \alpha_4)$, wobei $\abs{\alpha} \geq 1$ gilt. + für jeden Multiindex $\alpha = (\alpha_1, \alpha_2, \alpha_3, \alpha_4)$, mit $\abs{\alpha} \geq 1$. \end{lem} @@ -554,12 +553,7 @@ Weiterhin wollen wir uns kurz die Ableitung der asymptotisch glatten Kernfunktio \begin{align*} D(\kappa \circ g)(\lambda) = D\kappa(g(\lambda)) \circ D g(\lambda) \qquad \in \R^{1\times 4}. \end{align*} -Mithilfe der Jacobimatrizen $A \in \R^{1\times 6}$, $B \in\R^{6\times 4}$ und $m,\ell \in \N$ -\begin{align*} - A_{1\ell} &= \partial_{\ell} \kappa(g(\lambda)) \\ - B_{\ell m} &= \partial_m g_{\ell}(\lambda) -\end{align*} -untersuchen wir zunächst die Ableitungen +Mithilfe der Jacobimatrizen $A := \partial \kappa(g(\lambda)) \in \R^{1\times 6}$, $B := \partial g(\lambda) \in\R^{6\times 4}$ untersuchen wir zunächst die partiellen Ableitungen \begin{align} \partial_m (\kappa \circ g)(\lambda) &= (AB)_{1m} = \sum_{\ell=1}^6 A_{1\ell} B_{\ell m} = \sum_{\ell=1}^6 \partial_{\ell} \kappa(g(\lambda)) \partial_m g_{\ell}(\lambda). \end{align} @@ -568,18 +562,18 @@ Für die ersten drei Komponenten von $g$ gilt \begin{align*} g_{1,2,3}(\lambda) &= \gamma_j(\lambda_1,\lambda_2) = \bs v + \lambda_1 a {\bs a} + \lambda_2 b {\bs b}, \end{align*} -mit $\bs a, \bs b \in \{(1,0,0)^T,(0,1,0)^T,(0,0,1)^T\}$, $\bs a \neq \bs b$ und $a,b \in \R$ mit $a,b > 0$, +mit den Variablen $\bs a, \bs b \in \{(1,0,0)^T,(0,1,0)^T,(0,0,1)^T\}$, $\bs a \neq \bs b$ und $a,b \in \R$ mit $a,b > 0$ der Parametrisierung zum Rechteck $T_j$, wodurch sich die folgenden Ableitungen ergeben. \begin{align*} \partial_1 g_{1,2,3}(\lambda) &= a {\bs a} & \partial_2 g_{1,2,3}(\lambda) &= b {\bs b} & \partial_{3,4} g_{1,2,3}(\lambda) &= 0 \end{align*} -Für die letzten drei Komponenten, wobei $\tilde{ \bs a}, \tilde{\bs b}$ analog definiert seien, +Für die letzten drei Komponenten, wobei für die Parametrisierung des Rechtecks $T_k$ die Variablen $\tilde{ \bs a}, \tilde{\bs b}$ analog definiert seien, \begin{align*} g_{4,5,6}(\lambda) &= \gamma_k(\lambda_3,\lambda_4) = \tilde {\bs v} + \lambda_3 \tilde a \tilde{ \bs a} + \lambda_4 \tilde b \tilde{ \bs b} \end{align*} -können wir die Ableitungen analog anschreiben. Hierbei müssen wir nur beachten, dass sich die Indizierung von $ \tilde {\bs a}, \tilde {\bs b}$ ändert, weshalb wir hier die Komponenten +können wir die Ableitungen ebenfalls anschreiben. Hierbei müssen wir nur beachten, dass sich die Indizierung von $ \tilde {\bs a}, \tilde {\bs b}$ ändert, weshalb wir hier die Komponenten \begin{align*} \partial_{1,2} g_{\ell}(\lambda) &= 0 & \partial_3 g_{\ell}(\lambda) &= \tilde a \tilde{ \bs a}_{\ell -3} & @@ -608,10 +602,10 @@ für die Einheitsvektoren mit dem Skalarprodukt $\cdot$, können wir den Gradien % \begin{equation} \begin{align*} % \begin{split} - \partial_1 (\kappa \circ g)(\lambda) &= a \cdot \partial_{ind(\bs a)}\kappa(g(\lambda))& - \partial_3 (\kappa \circ g)(\lambda) &= \tilde a \cdot \partial_{ind(\tilde{\bs a})+3}\kappa(g(\lambda))\\ - \partial_2 (\kappa \circ g)(\lambda) &= b \cdot \partial_{ind(\bs b)}\kappa(g(\lambda))& - \partial_4 (\kappa \circ g)(\lambda) &= \tilde b \cdot \partial_{ind(\tilde{\bs b})+3}\kappa(g(\lambda)) + \partial_1 (\kappa \circ g)(\lambda) &= a \partial_{ind(\bs a)}\kappa(g(\lambda))& + \partial_3 (\kappa \circ g)(\lambda) &= \tilde a \partial_{ind(\tilde{\bs a})+3}\kappa(g(\lambda))\\ + \partial_2 (\kappa \circ g)(\lambda) &= b \partial_{ind(\bs b)}\kappa(g(\lambda))& + \partial_4 (\kappa \circ g)(\lambda) &= \tilde b \partial_{ind(\tilde{\bs b})+3}\kappa(g(\lambda)) % \end{split} \end{align*} % \end{equation} @@ -619,7 +613,7 @@ genau anschreiben, wobei auch hier wieder auf die richtige Indizierung geachtet Des Weiteren bestehen die ersten Ableitungen nur aus einem Skalar und dem asymptotisch glatten Kern $\partial_{\ell}\kappa(g(\lambda))$, weshalb wir die Kettenregel beliebig oft anwenden können. Mithilfe der Funktion $t_{jk} : \N^4 \rightarrow \N^6$ können wir abschließend für Multiindex $\alpha \in \N_0^4$ die partiellen Ableitungen $\partial^{\alpha} (\kappa \circ g)(\lambda)$ anschreiben \begin{align*} - \partial^\alpha (\kappa \circ g)(\lambda) &= a^{\alpha_1} b^{\alpha_2} \tilde a^{\alpha_3} \tilde b^{\alpha_4} \cdot \partial^{t_{jk}(\alpha)}\kappa(g(\lambda)). + \partial^\alpha (\kappa \circ g)(\lambda) &= a^{\alpha_1} b^{\alpha_2} \tilde a^{\alpha_3} \tilde b^{\alpha_4} \partial^{t_{jk}(\alpha)}\kappa(g(\lambda)). \end{align*} \end{beweis} @@ -639,19 +633,19 @@ Mit diesen Vorüberlegungen können wir nun den folgenden Satz über die Interpo \begin{sat} \label{thm:sem:pol:V} Seien $T_j,T_k \subseteq \R^3$ zwei $\zeta_Q$-zulässige Rechtecke. Sei $\kappa(\cdot,\cdot)$ eine asymptotisch glatte Kernfunktion mit Konstanten $c_1,c_2>0$ und Singularitätsordung $s\geq0$. Dann gilt mit \begin{align*} - C_{\zeta_Q,j,k}&:=8e\frac{c_1}{(c_2\dist(T_j,T_k))^s} \left(1+\frac{2\zeta_Q}{c_2} \right) + C_{\zeta_Q,j,k}&:=8e\frac{c_1}{(c_2\dist(T_j,T_k))^s} \left(1+\frac{2}{c_2\zeta_Q} \right) \end{align*} die Abschätzung \begin{align*} \norm{\kappa(\gamma_j(\cdot),\gamma_k(\cdot))&-\I_p^4\kappa(\gamma_j(\cdot),\gamma_k(\cdot))}_{\infty,[0,1]^4} - \leq C_{\zeta_Q,j,k}\Lambda_p^4 p\left(1+\frac{c_2}{\zeta_Q}\right)^{-p}. + \leq C_{\zeta_Q,j,k}\Lambda_p^4 (p+1)\left(1+c_2\zeta_Q\right)^{-(p+1)}. \end{align*} \end{sat} \begin{beweis} Zunächst definieren wir die Konstanten \begin{align*} - C_{\kappa} &=\frac{c_1}{(c_2\dist(T_j,T_k))^s} \quad \text{und} \quad \rho_{\kappa} = \frac{\zeta_Q}{c_2} + C_{\kappa} &=\frac{c_1}{(c_2\dist(T_j,T_k))^s} \quad \text{und} \quad \rho_{\kappa} = \frac{1}{c_2\zeta_Q} \end{align*} und können dann die Konstante $C_{\zeta_Q,j,k}$ kurz \begin{align*} @@ -662,30 +656,30 @@ Bezeichne $g_{jk} : [0,1]^4 \rightarrow (T_j \times T_k) \subset \R^6$ die Param \begin{align} g_{jk}(\lambda) = (\gamma_j(\lambda_1,\lambda_2),\gamma_k(\lambda_3,\lambda_4)) \end{align} - mit der jeweiligen Parametrisierung $\gamma_j, \gamma_k$ zu $T_j, T_k$ aus Definition \ref{math:def:T}. So gilt Aufgrund von Lemma \ref{thm:sem:kett} + mit der jeweiligen Parametrisierung $\gamma_j, \gamma_k$ zu $T_j, T_k$ aus Definition \ref{math:def:T}. So gilt aufgrund von Lemma \ref{thm:sem:kett} \begin{align*} % \Abs{\partial_x^{\alpha}\partial_y^{\beta}\kappa} &= - \Abs{\partial_\lambda^{\alpha}\kappa(\gamma_j(\lambda_1,\lambda_2),\gamma_k(\lambda_1,\lambda_2))} + \Abs{\partial_\lambda^{\alpha}\kappa(\gamma_j(\lambda_1,\lambda_2),\gamma_k(\lambda_3,\lambda_4))} &=\Abs{\partial_\lambda^{\alpha}\kappa(g_{jk}(\lambda))}\\ - &= \diam(T_j)_{\bs a}^{\alpha_1}\diam(T_j)_{\bs b}^{\alpha_2} \diam(T_k)_{\bs a}^{\alpha_3}\diam(T_k)_{\bs b}^{\alpha_4} \Abs{\partial^{t_{jk}(\alpha)}(\kappa)(g_{jk}(\lambda))}\\ - &\leq\diam(T_j)^{\alpha_1 + \alpha_2}\diam(T_k)^{\alpha_3 + \alpha_4} \Abs{\partial^{t_{jk}(\alpha)}(\kappa)(g_{jk}(\lambda))}\\ - &\leq \max\{\diam(T_j),\diam(T_k)\}^{\abs{\alpha}} \Abs{\partial^{t_{jk}(\alpha)}(\kappa)(g_{jk}(\lambda))}, + &= \diam(T_j)_{\bs a}^{\alpha_1}\diam(T_j)_{\bs b}^{\alpha_2} \diam(T_k)_{\bs a}^{\alpha_3}\diam(T_k)_{\bs b}^{\alpha_4} \Abs{\partial^{t_{jk}(\alpha)}\kappa(g_{jk}(\lambda))}\\ + &\leq\diam(T_j)^{\alpha_1 + \alpha_2}\diam(T_k)^{\alpha_3 + \alpha_4} \Abs{\partial^{t_{jk}(\alpha)}\kappa(g_{jk}(\lambda))}\\ + &\leq \max\{\diam(T_j),\diam(T_k)\}^{\abs{\alpha}} \Abs{\partial^{t_{jk}(\alpha)}\kappa(g_{jk}(\lambda))}, \end{align*} mit Multiindex $\alpha = (\alpha_1, \ldots , \alpha_4)$, wobei $\abs{\alpha}>0$ sei. Ferner gilt mit Definition \ref{thm:sem:glatt} und den Konstanten $C_{\kappa},\rho_{\kappa}$ die Abschätzung \begin{align*} -\max\{\diam(T_j),\diam(T_k)\}^{\abs{\alpha}} &\Abs{\partial^{t_{jk}(\alpha)}(\kappa)(g_{jk}(\lambda))}\\ +\max\{\diam(T_j),\diam(T_k)\}^{\abs{\alpha}} &\Abs{\partial^{t_{jk}(\alpha)}\kappa(g_{jk}(\lambda))}\\ \leq& \max\{\diam(T_j),\diam(T_k)\}^{\abs{\alpha}} c_1 (c_2 \abs{\gamma_j(\lambda_1,\lambda_2) - \gamma_k(\lambda_3,\lambda_4)})^{-(\abs{\alpha}+s)}\abs{\alpha}!\\ - = & \max\{\diam(T_j),\diam(T_k)\}^{\abs{\alpha}} c_1 (c_2 \abs{\bs x - \bs y})^{-(\abs{\alpha}+s)}\abs{\alpha}!\\ - = & \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)^{\abs{\alpha}}\abs{\alpha}!\\ + = & \max\{\diam(T_j),\diam(T_k)\}^{\abs{\alpha}} c_1 (c_2 \dist(T_j,T_k))^{-(\abs{\alpha}+s)}\abs{\alpha}!\\ + = & \frac{c_1}{(c_2 \dist(T_j,T_k))^s} \left( \frac{\max(\diam(T_j),\diam(T_k))}{c_2 \dist(T_j,T_k)} \right)^{\abs{\alpha}}\abs{\alpha}!\\ \leq& C_{\kappa}\rho_{\kappa}^{\abs{\alpha}}\abs{\alpha}!, \end{align*} -wobei $\bs x \in T_j$ und $\bs y \in T_k$ sei, mit $\zeta_Q$-zulässigen $T_j, T_k$ . Daher sind die Voraussetzungen für Lemma \ref{thm:sem:ipolnD} erfüllt, und es gilt die Abschätzung +wobei $\gamma_j(\lambda_1,\lambda_2) \in T_j$ und $\gamma_k(\lambda_3,\lambda_4) \in T_k$ sei. Daher sind die Voraussetzungen für Lemma \ref{thm:sem:ipolnD} erfüllt, und es gilt die Abschätzung \begin{align*} \norm{\kappa(\gamma_j(\cdot),\gamma_k(\cdot))&-\I_p^4\kappa(\gamma_j(\cdot),\gamma_k(\cdot))}_{\infty,[0,1]^4}\\ - &\leq C_{\kappa} 8e(1+2\rho_{\kappa})\Lambda_p^4p\left(1+\frac{1}{\rho_{\kappa}}\right)^{-p}\\ + &\leq C_{\kappa} 8e(1+2\rho_{\kappa})\Lambda_p^4(p+1)\left(1+\frac{1}{\rho_{\kappa}}\right)^{-(p+1)}\\ % &= C_{\zeta,j,k}\Lambda_p^4p\left(1+\frac{2}{\rho_{\kappa}}\right)^{-p}\\ - &= C_{\zeta_Q,j,k}\Lambda_p^4p\left(1+\frac{c_2}{\zeta_Q}\right)^{-p}. + &= C_{\zeta_Q,j,k}\Lambda_p^4(p+1)\left(1+c_2\zeta_Q\right)^{-(p+1)}. \end{align*} Damit ist der Beweis abgeschlossen. \end{beweis} @@ -699,7 +693,7 @@ Da sich wie gezeigt die Kernfunktion besonders gut durch Polynome Interpolieren \begin{sat}\label{thm:sem:quad:V} Seien $T_j,T_k$ zwei $\zeta_Q$-zulässige Rechtecke mit zugehörigen Parametrisierungen $\gamma_j,\gamma_k$. 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} \label{math:sem:zetaQ:c} - \tilde C_{\zeta_Q,j,k}&:=2^4e\frac{c_1\abs{T_j}\abs{T_k}}{(c_2\dist(T_j,T_k))^s} \left(1+\frac{2\zeta_Q}{c_2} \right) + \tilde C_{\zeta_Q,j,k}&:=2^4e\frac{c_1\abs{T_j}\abs{T_k}}{(c_2\dist(T_j,T_k))^s} \left(1+\frac{2}{c_2\zeta_Q} \right) \end{align} Dann gilt mit $\bs \lambda_j = (\lambda_1,\lambda_2)$ und $\bs \lambda_k = (\lambda_3,\lambda_4)$ für das Integral \begin{align} @@ -714,12 +708,12 @@ und für den durch die Gauss-Quadratur von $A_{jk}$ zum Grad $p$ entstehenden Te \end{align*} die Abschätzung \begin{align} - \abs{A_{jk} - (A_p)_{jk}}&\leq \tilde C_{\zeta_Q,j,k}\Lambda_p^4p\left(1+\frac{c_2}{\zeta_Q}\right)^{-2p} + \abs{A_{jk} - (A_p)_{jk}}&\leq \tilde C_{\zeta_Q,j,k}\Lambda_{2p+1}^4(p+1)\left(1+c_2\zeta_Q\right)^{-2(p+1)} \end{align} \end{sat} -\begin{beweis} Da die Gauss-Quadratur interpolatorisch mit dem Exaktheitsgrad $2p+1$ ist, gilt: +\begin{beweis}Wir wissen aus \cite[Korollar 6.38]{pla:nummat}, dass die Gauss-Quadratur interpolatorisch mit dem Exaktheitsgrad $2p+1$ ist, unter Berücksichtigung, der hier verwendeten Indizierung von 0 statt 1. Deshalb gilt \begin{align*} (A_p)_{jk} &=\abs{T_j}\abs{T_k}\sum_{a=0}^p w_a \sum_{b=0}^p w_b \sum_{c=0}^p w_c \sum_{d=0}^p w_d \kappa(\gamma_j( \lambda_{a},\lambda_{b}),\gamma_k(\lambda_{c},\lambda_{d}))\\ @@ -732,14 +726,26 @@ wobei $\I_{2p+1}$ den Chebyshev-Interpolationsoperator vom Grad $2p+1$ bezeichne &\leq \abs{T_j}\abs{T_k}\int_{[0,1]^2}\int_{[0,1]^2} \Abs{\kappa(\gamma_j(\bs \lambda_j),\gamma_k(\bs \lambda_k)) - \I_{2p+1}^4 \kappa(\gamma_j(\bs \lambda_j),\gamma_k(\bs \lambda_k))} d{\bs \lambda_k} d{\bs \lambda_j}\\ &\leq \abs{T_j}\abs{T_k} \norm{\kappa(\gamma_j(\cdot),\gamma_k(\cdot))-\I_{2p+1}^4\kappa(\gamma_j(\cdot),\gamma_k(\cdot))}_{\infty,[0,1]^4} \end{align*} -Mithilfe von Satz \ref{thm:sem:pol:V} erhalten wir die Behauptung +Mithilfe von Satz \ref{thm:sem:pol:V} für den Grad $2p+1$ erhalten wir die Behauptung \begin{align*} \abs{A_{jk} - (A_p)_{jk}} - &\leq \abs{T_j}\abs{T_k} C_{\zeta_Q,j,k}\Lambda_p^42p\left(1+\frac{c_2}{\zeta_Q}\right)^{-2p}\\ - &= \tilde C_{\zeta_Q,j,k}\Lambda_p^4p\left(1+\frac{c_2}{\zeta_Q}\right)^{-2p}. + &\leq \abs{T_j}\abs{T_k} C_{\zeta_Q,j,k}\Lambda_{2p+1}^42(p+1)\left(1+c_2\zeta_Q\right)^{-2(p+1)}\\ + &= \tilde C_{\zeta_Q,j,k}\Lambda_{2p+1}^4(p+1)\left(1+c_2\zeta_Q\right)^{-2(p+1)}. \end{align*} \end{beweis} +\noindent + Da die Konstante $\tilde C_{\zeta_Q,j,k}$ durch die Netzverfeinerung, aufgrund der Distanz sehr groß werden könnte, untersuchen wir sie an dieser Stelle noch einmal etwas genauer. + Mitithilfe der folgenden Abschätzung + \begin{align*} + \tilde C_{\zeta_Q,j,k} & = 2^4 e \frac{c_1 \abs{T_j}\abs{T_k}}{(c_2 \dist(T_j,T_k))^s}\left( 1 + \frac{2}{c_2\zeta_Q} \right)\\ + & \leq 2^4 e \frac{c_1 \abs{T_j}\abs{T_k}}{(c_2 \zeta_Q \max\{ \diam(T_j) , \diam(T_k)\})^s} \left( 1 + \frac{2}{c_2\zeta_Q} \right)\\ + & \leq 2^4 e \frac{c_1 \max\{ \diam(T_j) , \diam(T_k)\}^4}{(c_2 \zeta_Q \max\{ \diam(T_j) , \diam(T_k)\})^s} \left( 1 + \frac{2}{c_2\zeta_Q} \right)\\ + & = 2^4 e \frac{c_1}{(c_2 \zeta_Q)^s} \max\{ \diam(T_j) , \diam(T_k)\}^{4-s} \left( 1 + \frac{2}{c_2\zeta_Q} \right), + \end{align*} + welche unabhängig von der im Netz auftretenden Distanzen ist, können wir also eine gute Aussage über die Stabilität der Zulässigkeitsbedingung im Zuge der Netzverfeinerung treffen. + + \begin{bem} $\Lambda_p$ wächst für Chebyshev-Polynome im wesentlichen logarithmisch in $p$. Daher konvergiert $(A_p)_{jk}$ für wachsenden Quadraturgrad $p$ exponentiell schnell gegen $A_{jk}$. @@ -757,7 +763,7 @@ Im Folgenden wollen wir nun annehmen, dass $T_1,T_2,\ldots,T_n$ Randelemente sin \end{align*} \item Sind $T_j$ und $T_k$ $\zeta_Q$-zulässig, so ist \begin{align*} - (A_p)_{jk} := (A_{\zeta_Q})_{jk}. + (A_p)_{jk} := \abs{T_j}\abs{T_k}\sum_{a=0}^p w_a \sum_{b=0}^p w_b \sum_{c=0}^p w_c \sum_{d=0}^p w_d \kappa(\gamma_j( \lambda_{a},\lambda_{b}),\gamma_k(\lambda_{c},\lambda_{d})). \end{align*} \end{itemize} \end{defi} @@ -771,42 +777,28 @@ Wir wollen nun zeigen, dass die approximierende Matrix bezüglich der Frobenius- \begin{sat} - Seien $T_1,T_2,\ldots,T_n$ Rechtecke. Sei $\kappa : \R^2\times \R^2 \to \R$ eine asymptotisch glatte Kernfunktion mit Konstanten $c_1,c_2$ und Singularitätsordung $s \geq 0$. Sei $\tilde C_{\zeta_Q,j,k}$ wie in \eqref{math:sem:zetaQ:c}. Seien $A\in \R^{n \times n}$ eine Matrix, deren Einträge gegeben sind durch + Seien $T_1,T_2,\ldots,T_n$ Rechtecke. Sei $\kappa : \R^3\times \R^3 \to \R$ eine asymptotisch glatte Kernfunktion mit Konstanten $c_1,c_2$ und Singularitätsordung $s \geq 0$. Sei $\tilde C_{\zeta_Q,j,k}$ wie in \eqref{math:sem:zetaQ:c}. Sei $A\in \R^{n \times n}$ eine Matrix, deren Einträge gegeben sind durch \begin{align*} A_{jk} &= \int_{T_j} \int_{T_k} \kappa(\bs x,\bs y) ds_{\bs y} ds_{\bs x}. \end{align*} und sei $A_p$ die $A$ approximierende Matrix gemäß Definition \ref{thm:sem:quad:Ap}. Dann gilt \begin{align*} - \norm{A-A_p}_F \leq n \max_{j=1,\ldots,n} \max_{k=1,\ldots,n} \tilde C_{\zeta_Q,j,k}\Lambda_p^4p\left(1+\frac{c_2}{\zeta_Q}\right)^{-2p}. + \norm{A-A_p}_F \leq n \max_{j=1,\ldots,n} \max_{k=1,\ldots,n} \tilde C_{\zeta_Q,j,k}\Lambda_{2p+1}^4(p+1)\left(1+c_2\zeta_Q\right)^{-2(p+1)}. \end{align*} \end{sat} -\begin{beweis} Betrachten wir zunächst die Differenz zwischen $A$ und $A_p$ in einem festen Eintrag $(A-A_p)_{jk}$. Sind $T_j$ und $T_k$ unzulässig ist die Differenz laut Definition $0$. Sind $T_j$ und $T_k$ hingegen zulässig, so gilt nach Satz \ref{thm:sem:quad:V} -% \begin{align} -% \abs{A_{jk} - (A_p)_{jk}}&= \tilde C_{\zeta_Q,j,k}\Lambda_p^4p\left(1+\frac{2c_2}{\zeta_Q}\right)^{-2p} -% \end{align} +\begin{beweis} Betrachten wir zunächst die Differenz zwischen $A$ und $A_p$ in einem festen Eintrag $(A-A_p)_{jk}$. Sind $T_j$ und $T_k$ unzulässig, ist die Differenz laut Definition $0$. Sind $T_j$ und $T_k$ hingegen zulässig, können wir Satz \ref{thm:sem:quad:V} anwenden. Damit erhalten wir \begin{align*} \norm{A-A_p}_F^2 &= \sum_{j,k=1}^n (A_{jk} - (A_p)_{jk})^2\\ - &\leq \sum_{j,k=1}^n \left(\tilde C_{\zeta_Q,j,k}\Lambda_p^4p\left(1+\frac{2c_2}{\zeta_Q}\right)^{-2p}\right)^2\\ - &\leq \sum_{j,k=1}^n \max_{j=1,\ldots,n} \max_{k=1,\ldots,n} \left(\tilde C_{\zeta_Q,j,k}\Lambda_p^4p\left(1+\frac{c_2}{\zeta_Q}\right)^{-2p}\right)^2\\ - &= n^2 \max_{j=1,\ldots,n} \max_{k=1,\ldots,n} \left(\tilde C_{\zeta_Q,j,k}\Lambda_p^4p\left(1+\frac{c_2}{\zeta_Q}\right)^{-2p}\right)^2. + &\leq \sum_{j,k=1}^n \left(\tilde C_{\zeta_Q,j,k}\Lambda_{2p+1}^4(p+1)\left(1+c_2\zeta_Q\right)^{-2(p+1)}\right)^2\\ + &\leq \sum_{j,k=1}^n \max_{j=1,\ldots,n} \max_{k=1,\ldots,n} \left(\tilde C_{\zeta_Q,j,k}\Lambda_{2p+1}^4(p+1)\left(1+c_2\zeta_Q\right)^{-2(p+1)}\right)^2\\ + &= n^2 \max_{j=1,\ldots,n} \max_{k=1,\ldots,n} \left(\tilde C_{\zeta_Q,j,k}\Lambda_{2p+1}^4(p+1)\left(1+c_2\zeta_Q\right)^{-2(p+1)}\right)^2. \end{align*} Durch ziehen der Wurzel auf beiden Seiten, folgt dann die Behauptung. \end{beweis} -\begin{bem} - Da die Konstante $C_{\zeta_Q,j,k}$ durch die Netzverfeinerung, aufgrund der Distanz sehr groß werden könnte, untersuchen wir sie noch einmal etwas genauer. - Die Konstante können wir nach oben durch die Zulässigkeitsbedingung abschätzen. - \begin{align*} - C_{\zeta_Q,j,k} & = 2^4 e \frac{c_1 \abs{T_j}\abs{T_k}}{(c_2 dist(T_j,T_k))^s}\left( 1 + \frac{2\zeta_Q}{c_2} \right)\\ - & \leq 2^4 e \frac{c_1 \abs{T_j}\abs{T_k}}{(c_2 \zeta_Q \max\{ \diam(T_j) , \diam(T_k)\})^s} \left( 1 + \frac{2\zeta_Q}{c_2} \right)\\ - & \leq 2^4 e \frac{c_1 \max\{ \diam(T_j) , \diam(T_k)\}^4}{(c_2 \zeta_Q \max\{ \diam(T_j) , \diam(T_k)\})^s} \left( 1 + \frac{2\zeta_Q}{c_2} \right)\\ - & = 2^4 e \frac{c_1}{(c_2 \zeta_Q)^s} \max\{ \diam(T_j) , \diam(T_k)\}^{4-s} \left( 1 + \frac{2\zeta_Q}{c_2} \right) - \end{align*} -\end{bem} - \noindent \todo{ \begin{itemize} diff --git a/res/hmatrix2010.pdf b/res/hmatrix2010.pdf new file mode 100644 index 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