From de3f8901fe4f769beb4901d8892dd58db40cae75 Mon Sep 17 00:00:00 2001 From: jowl Date: Sat, 29 Nov 2025 03:06:29 -0600 Subject: [PATCH] test --- zsplins.aux | 30 +++++ zsplins.fdb_latexmk | 82 ++++++++++++++ zsplins.fls | 269 ++++++++++++++++++++++++++++++++++++++++++++ zsplins.log | 234 ++++++++++++++++++++++++++++++++++++++ zsplins.pdf | Bin 0 -> 210763 bytes zsplins.synctex.gz | Bin 0 -> 21865 bytes zsplins.tex | 213 +++++++++++++++++++++++++++++++++++ 7 files changed, 828 insertions(+) create mode 100644 zsplins.aux create mode 100644 zsplins.fdb_latexmk create mode 100644 zsplins.fls create mode 100644 zsplins.log create mode 100644 zsplins.pdf create mode 100644 zsplins.synctex.gz create mode 100644 zsplins.tex diff --git a/zsplins.aux b/zsplins.aux new file mode 100644 index 0000000..db7865a --- /dev/null +++ b/zsplins.aux @@ -0,0 +1,30 @@ +\relax +\providecommand\babel@aux[2]{} +\@nameuse{bbl@beforestart} +\catcode `"\active +\catcode `<\active +\catcode `>\active 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iF0@q}43p}DcFKpH;&;!_P`HKjD=>%?`cn|uKmP}avbh8R literal 0 HcmV?d00001 diff --git a/zsplins.tex b/zsplins.tex new file mode 100644 index 0000000..6bb9c2d --- /dev/null +++ b/zsplins.tex @@ -0,0 +1,213 @@ +\documentclass{article} + +% --- PAQUETES ESENCIALES --- +\usepackage[utf8]{inputenc} % Para reconocer tildes y ñ directamente +\usepackage[T1]{fontenc} +\usepackage[spanish]{babel} % Configuración para español (fechas, silabeo) +\usepackage{amsmath} % Matemáticas avanzadas (\text, \operatorname, etc.) +\usepackage{amssymb} % Símbolos matemáticos (\mathbb) +\usepackage{geometry} % (Opcional) Para mejorar los márgenes +% --------------------------- + +\title{Proyecto 1: Métodos numéricos del Z-spline cúbico} +\author{Reyes Garcia Joel} +\date{\today} + +\begin{document} + +\maketitle + +\begin{abstract} +En este trabajo se presentan los métodos numéricos asociados a... +\end{abstract} + +\section{Los Z-splines} + +\subsection{Introducción} + +Los Z-splines constituyen una familia de splines cardinales de soporte +compacto que preservan momentos discretos y poseen alto orden de +regularidad. La idea central es construir una base spline +cuya definición incorpora explícitamente operadores de diferencias +finitas derivados de desarrollos de Taylor. Su creador, +Julián T. Becerra-Sagredo, desarrolló esta teoría en dos trabajos clave +(2003 y 2020), en los cuales se presentan tanto la construcción como +sus propiedades analíticas y aplicaciones. + +\subsection{Definición cardinal y construcción básica} + +Sea $\{x_j\} = \{j\}$ una malla equiespaciada. +Los Z-splines se definen mediante un núcleo cardinal $Z_m(x)$ tal que la +interpolación de un conjunto de datos $\{y_j\}$ está dada por + +\[ +f_m(x) = \sum_{j\in\mathbb{Z}} y_j \, Z_m(x - j). +\] + +Cada función base $Z_m$ es un polinomio por tramos de grado $2m-1$ y con +regularidad $C^{m-1}$. El parámetro $m$ determina el orden del spline. + +La construcción impone condiciones de interpolación entre nodos y +conservación de momentos discretos. Para cada intervalo +$[j,j+1]$, el polinomio está determinado por un conjunto de condiciones +de tipo Hermite–Birkhoff derivadas de operadores de diferencias finitas. + +\subsection{Conservación de momentos} + +Una de las propiedades fundamentales es que las Z-splines preservan los +primeros $2m-1$ momentos discretos. Sea + +\[ +M_k = \sum_j y_j j^k, \qquad k=0,\dots,2m-2, +\] + +entonces la interpolante satisface + +\[ +\int_{-\infty}^{\infty} x^k f_m(x) \, dx = M_k, +\] + +lo que asegura que la interpolación no altera estos invariantes +(discretos o físicos). + +\subsection{Reproducción polinómica} + +El artículo demuestra que las Z-splines reproducen exactamente +polinomios hasta grado $2m-2$: + +\[ +p(x) = x^n, \qquad n \le 2m-2 \quad \Rightarrow \quad +f_m(x) = p(x). +\] + +Esto implica un orden de aproximación + +\[ +\| f - f_m \| = \mathcal{O}(h^{\,2m-1}), +\] + +cuando $f$ es suficientemente regular. + +\subsection{Soporte compacto y regularidad} + +Cada $Z_m$ tiene soporte compacto mínimo: + +\[ +Z_m(x) = 0 \qquad \text{si } |x| > m, +\] + +y regularidad + +\[ +Z_m \in C^{m-1}. +\] + +Esto permite su uso eficiente en contextos numéricos, especialmente en +métodos de remapeo o mallas adaptativas. + +\subsection{Relación con diferencias finitas} + +Una contribución teórica clave es que los coeficientes de los polinomios +que forman $Z_m$ coinciden con los coeficientes de operadores de +diferencias finitas de orden elevado. En particular, se demuestra que +para derivadas, + +\[ +f_m^{(k)}(x_j) = \sum_{\ell=-m}^m c^{(k)}_{\ell} \, y_{j+\ell}, +\] + +donde los $c^{(k)}_\ell$ son coeficientes de diferencias finitas +centradas obtenidos por expansión de Taylor. + +\subsection{Límite hacia la función sinc} + +A medida que $m \to \infty$, las Z-splines convergen hacia la función +ideal de reconstrucción cardinal: + +\[ +Z_m(x) \longrightarrow \operatorname{sinc}(x) += \frac{\sin(\pi x)}{\pi x}. +\] + +Esto establece un vínculo entre splines compactos y teoría de muestreo. + +\subsection{Z-splines en mallas no uniformes} + +Los artículos extienden la teoría a mallas arbitrarias +$\{x_j\}$ no equidistantes. En ese caso, el núcleo cardinal ya no es +trasladable, y se construyen polinomios locales mediante matrices de +diferencias finitas adaptadas a la geometría de los nodos. + +En regiones de frontera se emplean splines ``one-sided'', lo cual permite +resolver la falta de simetría. + +\subsection{Aplicaciones numéricas} + +Las Z-splines son especialmente útiles en contextos donde es esencial la +conservación de cantidades físicas discretas: + +\begin{itemize} + \item métodos de remapeo ALE y semi-Lagrangianos, + \item interpolaciones conservativas en dinámica de fluidos, + \item regularización de funciones delta, + \item remallado adaptativo, + \item filtrado numérico (denoising) preservando momentos. +\end{itemize} + +En estas aplicaciones, la preservación de momentos juega un papel +fundamental para mantener estabilidad física del sistema. + +\subsection{Conclusión} + +Los Z-splines constituyen una herramienta moderna de +interpolación de gran interés tanto teórico como práctico, especialmente +en contextos de simulación numérica donde la preservación de invariantes +es esencial. + + +\section{El Z-spline cúbico} + +La función base Z-spline cúbica para datos a intervalos arbitrarios $a_{1}, a_{2}, a_{3}$ y $a_{4}$ +está dada por: + +\begin{equation} \label{eq:aiz1} +\widetilde{Z}_{1} (x) = \left\{ +\begin{array}{lcl} +0 & & x < -a_{1}-a_{2} \\ [0.3cm] + +\left( \frac{a_{2}+a_{1}}{a_{1}} \right) + \left( +\frac{3a_{2}+a_{1}}{a_{2}a_{1}} \right) x & & \\ [0.1cm] +\quad + \frac{3a_{2}+2a_{1}}{a_{2}a_{1}(a_{2}+a_{1})} +x^2 + \frac{1}{a_{2}a_{1}(a_{2}+a_{1})}x^{3} & & -a_{1}-a_{2} \leq x \leq -a_{2}, \\ [0.3cm] + +1- \left( \frac{1}{a_{3}}-\frac{1}{a_{2}} \right) x & & \\ [0.1cm] +\quad - \frac{a_{3}+2(a_{2}+a_{1})}{a_{3}a_{2}(a_{2}+a_{1})} x^2 +- \frac{a_{3}+a_{2}+a_{1}}{a_{3}a_{2}^2(a_{2}+a_{1})} x^3 & & -a_{2} \leq x \leq 0, \\ [0.3cm] + +1+ \left( \frac{1}{a_{2}}-\frac{1}{a_{3}} \right) x & & \\ [0.1cm] +\quad - \frac{a_{2}+2(a_{3}+a_{4})}{a_{2}a_{3}(a_{3}+a_{4})} x^2 ++ \frac{a_{2}+a_{3}+a_{4}}{a_{2}a_{3}^2(a_{3}+a_{4})} x^3 & & 0 \leq x \leq a_{3}, \\ [0.3cm] + +\left( \frac{a_{3}+a_{4}}{a_{4}} \right) - \left( +\frac{3a_{3}+a_{4}}{a_{3}a_{4}} \right) x & & \\ [0.1cm] +\quad + \frac{3a_{3}+2a_{4}}{a_{3}a_{4}(a_{3}+a_{4})} x^2 +- \frac{1}{a_{3}a_{4}(a_{3}+a_{4})}x^{3} & & a_{3} \leq x \leq a_{3}+a_{4}, \\ [0.3cm] + +0 & & x > a_{3}+a_{4}. +\end{array} +\right. +\end{equation} + + +\subsection{Z-spline cúbico en los extremos de un intervalo} + +\section{Fórmula de integración numérica} + +\section{Fórmula de derivación numérica} +\section{Convergencia} +\subsection{Integración} +\subsection{Derivación} +\subsection{Interpolación equidistante en 2D} +\subsection{Interpolación en 1D con puntos arbitrarios} + +\end{document} \ No newline at end of file