## 数学代写|凸优化作业代写Convex Optimization代考|CS168

statistics-lab™ 为您的留学生涯保驾护航 在代写凸优化Convex Optimization方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写凸优化Convex Optimization代写方面经验极为丰富，各种代写凸优化Convex Optimization相关的作业也就用不着说。

## 数学代写|凸优化作业代写Convex Optimization代考|Rate of Convergence

The following proposition describes how the convergence rate of the proximal algorithm depends on the magnitude of $c_k$ and on the order of growth of $f$ near the optimal solution set (see also Fig. 5.1.3).

Proposition 5.1.4: (Rate of Convergence) Assume that $X^$ is nonempty and that for some scalars $\beta>0, \delta>0$, and $\gamma \geq 1$, we have $$f^+\beta(d(x))^\gamma \leq f(x), \quad \forall x \in \Re^n \text { with } d(x) \leq \delta,$$
where
$$d(x)=\min {x^* \in X^}\left|x-x^\right|$$
Let also
$$\sum {k=0}^{\infty} c_k=\infty,$$
so that the sequence $\left{x_k\right}$ generated by the proximal algorithm (5.1) converges to some point in $X^*$ by Prop. 5.1.3. Then:
(a) For all $k$ sufficiently large, we have
$$d\left(x_{k+1}\right)+\beta c_k\left(d\left(x_{k+1}\right)\right)^{\gamma-1} \leq d\left(x_k\right)$$
if $\gamma>1$, and

$$d\left(x_{k+1}\right)+\beta c_k \leq d\left(x_k\right),$$
if $\gamma=1$ and $x_{k+1} \notin X^$. (b) (Superlinear Convergence) Let $1<\gamma<2$ and $x_k \notin X^$ for all $k$. Then if $\inf {k \geq 0} c_k>0$, $$\limsup {k \rightarrow \infty} \frac{d\left(x_{k+1}\right)}{\left(d\left(x_k\right)\right)^{1 /(\gamma-1)}}<\infty .$$ (c) (Linear Convergence) Let $\gamma=2$ and $x_k \notin X *$ for all $k$. Then if $\lim {k \rightarrow \infty} c_k=\bar{c}$ with $\bar{c} \in(0, \infty)$, $$\limsup {k \rightarrow \infty} \frac{d\left(x_{k+1}\right)}{d\left(x_k\right)} \leq \frac{1}{1+\beta \bar{c}},$$ while if $\lim {k \rightarrow \infty} c_k=\infty$, $$\lim {k \rightarrow \infty} \frac{d\left(x_{k+1}\right)}{d\left(x_k\right)}=0 .$$ (d) (Sublinear Convergence) Let $\gamma>2$. Then
$$\limsup {k \rightarrow \infty} \frac{d\left(x{k+1}\right)}{d\left(x_k\right)^{2 / \gamma}}<\infty .$$

An interesting interpretation of the proximal iteration is obtained by considering the function
$$\phi_c(z)=\inf {x \in \Re^n}\left{f(x)+\frac{1}{2 c}|x-z|^2\right}$$ for a fixed positive value of $c$. It can be seen that $$\inf {x \in \Re^n} f(x) \leq \phi_c(z) \leq f(z), \quad \forall z \in \Re^n,$$
from which it follows that the set of minima of $f$ and $\phi_c$ coincide (this is also evident from the geometric view of the proximal minimization given in Fig. 5.1.7). The following proposition shows that $\phi_c$ is a convex differentiable function, and derives its gradient.

Proposition 5.1.7: The funetion $\phi_c$ of Eq. (5.14) is convex and differentiable, and we have
$$\nabla \phi_c(z)=\frac{z-x_c(z)}{c} \quad \forall z \in \Re^n,$$
where $x_c(z)$ is the unique minimizer in Eq. (5.14). Moreover
$$\nabla \phi_c(z) \in \partial f\left(x_c(z)\right), \quad \forall z \in \Re^n$$
Proof: We first note that $\phi_c$ is convex, since it is obtained by partial minimization of $f(x)+\frac{1}{2 c}|x-z|^2$, which is convex as a function of $(x, z)$ (cf. Prop. 3.3.1 in Appendix B). Furthermore, $\phi_c$ is real-valued, since the infimum in Eq. (5.14) is attained.

Let us fix $z$, and for notational simplicity, denote $\bar{z}=x_c(z)$. To show that $\phi_c$ is differentiable with the given form of gradient, we note that by the optimality condition of Prop. 3.1.4, we have $v \in \partial \phi_c(z)$, or equivalently $0 \in \partial \phi_c(z)-v$, if and only if $z$ attains the minimum over $y \in \Re^n$ of
$$\phi_c(y)-v^{\prime} y=\inf _{x \in \Re^n}\left{f(x)+\frac{1}{2 c}|x-y|^2\right}-v^{\prime} y$$

# 凸优化代写

## 数学代写|凸优化作业代写Convex Optimization代考|Rate of Convergence

$$d(x)=\min {x^* \in X^}\left|x-x^\right|$$

$$\sum{k=0}^{\infty} c_k=\infty,$$

(a)对于所有$k$足够大的，我们有
$$d\left(x_{k+1}\right)+\beta c_k\left(d\left(x_{k+1}\right)\right)^{\gamma-1} \leq d\left(x_k\right)$$

$$d\left(x_{k+1}\right)+\beta c_k \leq d\left(x_k\right),$$

$$\limsup {k \rightarrow \infty} \frac{d\left(x{k+1}\right)}{d\left(x_k\right)^{2 / \gamma}}<\infty .$$

$$\phi_c(z)=\inf {x \in \Re^n}\left{f(x)+\frac{1}{2 c}|x-z|^2\right}$$为固定正值$c$。可以看出$$\inf {x \in \Re^n} f(x) \leq \phi_c(z) \leq f(z), \quad \forall z \in \Re^n,$$

$$\nabla \phi_c(z)=\frac{z-x_c(z)}{c} \quad \forall z \in \Re^n,$$

$$\nabla \phi_c(z) \in \partial f\left(x_c(z)\right), \quad \forall z \in \Re^n$$

$$\phi_c(y)-v^{\prime} y=\inf _{x \in \Re^n}\left{f(x)+\frac{1}{2 c}|x-y|^2\right}-v^{\prime} y$$

## Matlab代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

## 数学代写|凸优化作业代写Convex Optimization代考|ESE6050

statistics-lab™ 为您的留学生涯保驾护航 在代写凸优化Convex Optimization方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写凸优化Convex Optimization代写方面经验极为丰富，各种代写凸优化Convex Optimization相关的作业也就用不着说。

## 数学代写|凸优化作业代写Convex Optimization代考|GENERALIZED SIMPLICIAL DECOMPOSITION

In this section we will aim to highlight some of the applications and the fine points of the general algorithm of the preceding section. As vehicle we will use the simplicial decomposition approach, and the problem
\begin{aligned} & \text { minimize } f(x)+c(x) \ & \text { subject to } x \in \Re^n, \end{aligned}
where $f: \Re^n \mapsto(-\infty, \infty]$ and $c: \Re^n \mapsto(-\infty, \infty]$ are closed proper convex functions. This is the Fenchel duality context, and it contains as a special case the problem to which the ordinary simplicial decomposition method of Section 4.2 applies (where $f$ is differentiable, and $c$ is the indicator function of a bounded polyhedral set). Here we will mainly focus on the case where $f$ is nondifferentiable and possibly extended real-valued.

We apply the polyhedral approximation scheme of the preceding section to the equivalent EMP
\begin{aligned} & \operatorname{minimize} f_1\left(x_1\right)+f_2\left(x_2\right) \ & \text { subject to }\left(x_1, x_2\right) \in S, \end{aligned}

where
$$f_1\left(x_1\right)=f\left(x_1\right), \quad f_2\left(x_2\right)=c\left(x_2\right), \quad S=\left{\left(x_1, x_2\right) \mid x_1=x_2\right} .$$
Note that the orthogonal subspace has the form
$$S^{\perp}=\left{\left(\lambda_1, \lambda_2\right) \mid \lambda_1=-\lambda_2\right}=\left{(\lambda,-\lambda) \mid \lambda \in \Re^n\right} .$$
Optimal primal and dual solutions of this EMP problem are of the form $\left(x^{o p t}, x^{o p t}\right)$ and $\left(\lambda^{o p t},-\lambda^{o p t}\right)$, with
$$\lambda^{o p t} \in \partial f\left(x^{o p t}\right), \quad-\lambda^{o p t} \in \partial c\left(x^{o p t}\right),$$
consistently with the optimality conditions of Prop. 4.4.1. A pair of such optimal solutions $\left(x^{o p t}, \lambda^{o p t}\right)$ satisfies the necessary and sufficient optimality conditions of the Fenchel Duality Theorem [Prop. 1.2.1(c)] for the original problem.

## 数学代写|凸优化作业代写Convex Optimization代考|Dual/Cutting Plane Implementation

Let us also provide a dual implementation, which is an equivalent outer linearization/cutting plane-type of method. The Fenchel dual of the minimization of $f+c$ [cf. Eq. (4.31)] is
\begin{aligned} & \text { minimize } f^{\star}(\lambda)+c^{\star}(-\lambda) \ & \text { subject to } \lambda \in \Re^n, \end{aligned}
where $f^{\star}$ and $c^{\star}$ are the conjugates of $f$ and $c$, respectively. According to the theory of the preceding section, the generalized simplicial decomposition algorithm (4.32)-(4.34) can alternatively be implemented by replacing $c^$ by a piecewise linear/cutting plane outer linearization, while leaving $f^$ unchanged, i.e., by solving at iteration $k$ the problem
\begin{aligned} & \text { minimize } f^{\star}(\lambda)+C_k^{\star}(-\lambda) \ & \text { subject to } \lambda \in \Re^n, \end{aligned}
where $C_k^{\star}$ is an outer linearization of $c^{\star}$ (the conjugate of $C_k$ ). This problem is the (Fenchel) dual of problem (4.32) [or equivalently, the low-dimensional problem (4.36)].

Note that solutions of problem (4.37) are the subgradients $\lambda_k$ satisfying $\lambda_k \in \partial f\left(x_k\right)$ and $-\lambda_k \in \partial C_k\left(x_k\right)$, where $x_k$ is the solution of the problem (4.32) [cf. Eq. (4.33)], while the associated subgradient of $c^*$ at $-\lambda_k$ is the vector $\tilde{x}k$ generated by Eq. (4.34), as shown in Fig. 4.5.1. In fact, the function $C_k^{\star}$ has the form $$C_k^{\star}(-\lambda)=\max {j \in J_k}\left{c\left(-\lambda_j\right)-\tilde{x}_j^{\prime}\left(\lambda-\lambda_j\right)\right}$$
where $\lambda_j$ and $\tilde{x}_j$ are vectors that can be obtained either by using the generalized simplicial decomposition method (4.32)-(4.34), or by using its dual, the cutting plane method based on solving the outer approximation problems (4.37). The ordinary cutting plane method, described in the beginning of Section 4.1, is obtained as the special case where $f^{\star}(\lambda) \equiv 0$ [or equivalently, $f(x)=\infty$ if $x \neq 0$, and $f(0)=0$ ].

# 凸优化代写

## 数学代写|凸优化作业代写Convex Optimization代考|GENERALIZED SIMPLICIAL DECOMPOSITION

\begin{aligned} & \text { minimize } f(x)+c(x) \ & \text { subject to } x \in \Re^n, \end{aligned}

\begin{aligned} & \operatorname{minimize} f_1\left(x_1\right)+f_2\left(x_2\right) \ & \text { subject to }\left(x_1, x_2\right) \in S, \end{aligned}

$$f_1\left(x_1\right)=f\left(x_1\right), \quad f_2\left(x_2\right)=c\left(x_2\right), \quad S=\left{\left(x_1, x_2\right) \mid x_1=x_2\right} .$$

$$S^{\perp}=\left{\left(\lambda_1, \lambda_2\right) \mid \lambda_1=-\lambda_2\right}=\left{(\lambda,-\lambda) \mid \lambda \in \Re^n\right} .$$

$$\lambda^{o p t} \in \partial f\left(x^{o p t}\right), \quad-\lambda^{o p t} \in \partial c\left(x^{o p t}\right),$$

## 数学代写|凸优化作业代写Convex Optimization代考|Dual/Cutting Plane Implementation

\begin{aligned} & \text { minimize } f^{\star}(\lambda)+c^{\star}(-\lambda) \ & \text { subject to } \lambda \in \Re^n, \end{aligned}

\begin{aligned} & \text { minimize } f^{\star}(\lambda)+C_k^{\star}(-\lambda) \ & \text { subject to } \lambda \in \Re^n, \end{aligned}

## Matlab代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

## 数学代写|凸优化作业代写Convex Optimization代考|EE364a

statistics-lab™ 为您的留学生涯保驾护航 在代写凸优化Convex Optimization方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写凸优化Convex Optimization代写方面经验极为丰富，各种代写凸优化Convex Optimization相关的作业也就用不着说。

## 数学代写|凸优化作业代写Convex Optimization代考|DUALITY OF INNER AND OUTER LINEARIZATION

We have considered so far cutting plane and simplicial decomposition methods, and we will now aim to connect them via duality. To this end, we define in this section outer and inner linearizations, and we formalize their conjugacy relation and other related properties. An outer linearization of a closed proper convex function $f: \Re^n \mapsto(-\infty, \infty]$ is defined by a finite set of vectors $\left{y_1, \ldots, y_{\ell}\right}$ such that for every $j=1, \ldots, \ell$, we have $y_j \in \partial f\left(x_j\right)$ for some $x_j \in \Re^n$. It is given by
$$F(x)=\max _{j=1, \ldots, \ell}\left{f\left(x_j\right)+\left(x-x_j\right)^{\prime} y_j\right}, \quad x \in \Re^n,$$

and it is illustrated in the left side of Fig. 4.3.1. The choices of $x_j$ such that $y_j \in \partial f\left(x_j\right)$ may not be unique, but result in the same function $F(x)$ : the epigraph of $F$ is determined by the supporting hyperplanes to the epigraph of $f$ with normals defined by $y_j$, and the points of support $x_j$ are immaterial. In particular, the definition (4.14) can be equivalently written in terms of the conjugate $f^{\star}$ of $f$ as
$$F(x)=\max _{j=1, \ldots, \ell}\left{x^{\prime} y_j-f^{\star}\left(y_j\right)\right},$$
using the relation $x_j^{\prime} y_j=f\left(x_j\right)+f^{\star}\left(y_j\right)$, which is implied by $y_j \in \partial f\left(x_j\right)$ (the Conjugate Subgradient Theorem, Prop. 5.4.3 in Appendix B).

Note that $F(x) \leq f(x)$ for all $x$, so as is true for any outer approximation of $f$, the conjugate $F^{\star}$ satisfies $F^{\star}(y) \geq f^{\star}(y)$ for all $y$. Moreover, it can be shown that $F^{\star}$ is an inner linearization of the conjugate $f^{\star}$, as illustrated in the right side of Fig. 4.3.1. Indeed we have, using Eq. (4.15),
\begin{aligned} F^{\star}(y)= & \sup {x \in \Re^n}\left{y^{\prime} x-F(x)\right} \ & =\sup {x \in \Re^n}\left{y^{\prime} x-\max {j=1, \ldots, \ell}\left{y_j^{\prime} x-f^{\star}\left(y_j\right)\right}\right}, \ & =\sup {\substack{x \in \Re^n, \xi \in \Re \ y_j^{\prime} x-f^{\star}\left(y_j\right) \leq \xi, j=1, \ldots, \ell}}\left{y^{\prime} x-\xi\right} . \end{aligned}

## 数学代写|凸优化作业代写Convex Optimization代考|GENERALIZED POLYHEDRAL APPROXIMATION

We will now consider a unified framework for polyhedral approximation, which combines the cutting plane and simplicial decomposition methods. We consider the problem
$$\begin{array}{ll} \operatorname{minimize} & \sum_{i=1}^m f_i\left(x_i\right) \ \text { subject to } & x \in S, \end{array}$$
where
$$x \stackrel{\text { def }}{=}\left(x_1, \ldots, x_m\right),$$

is a vector in $\Re^{n_1+\cdots+n_m}$, with components $x_i \in \Re^{n_i}, i=1, \ldots, m$, and
$f_i: \Re^{n_i} \mapsto(-\infty, \infty]$ is a closed proper convex function for each $i$, $S$ is a subspace of $\Re^{n_1+\cdots+n_m}$.

We refer to this as an extended monotropic program (EMP for short). $\dagger$
A classical example of EMP is a single commodity network optimization problem, where $x_i$ represents the (scalar) flow of an arc of a directed graph and $S$ is the circulation subspace of the graph (see e.g., [Ber98]). Also problems involving general linear constraints and an additive extended realvalued convex cost function can be converted to EMP. In particular, the problem
$$\begin{array}{ll} \text { minimize } & \sum_{i=1}^m f_i\left(x_i\right) \ \text { subject to } & A x=b, \end{array}$$
where $A$ is a given matrix and $b$ is a given vector, is equivalent to
$$\begin{array}{ll} \text { minimize } & \sum_{i=1}^m f_i\left(x_i\right)+\delta_Z(z) \ \text { subject to } & A x-z=0, \end{array}$$
where $z$ is a vector of artificial variables, and $\delta_Z$ is the indicator function of the set $Z={z \mid z=b}$. This is an EMP with constraint subspace
$$S={(x, z) \mid A x-z=0} .$$

# 凸优化代写

## 数学代写|凸优化作业代写Convex Optimization代考|DUALITY OF INNER AND OUTER LINEARIZATION

$$F(x)=\max _{j=1, \ldots, \ell}\left{f\left(x_j\right)+\left(x-x_j\right)^{\prime} y_j\right}, \quad x \in \Re^n,$$

$$F(x)=\max _{j=1, \ldots, \ell}\left{x^{\prime} y_j-f^{\star}\left(y_j\right)\right},$$

\begin{aligned} F^{\star}(y)= & \sup {x \in \Re^n}\left{y^{\prime} x-F(x)\right} \ & =\sup {x \in \Re^n}\left{y^{\prime} x-\max {j=1, \ldots, \ell}\left{y_j^{\prime} x-f^{\star}\left(y_j\right)\right}\right}, \ & =\sup {\substack{x \in \Re^n, \xi \in \Re \ y_j^{\prime} x-f^{\star}\left(y_j\right) \leq \xi, j=1, \ldots, \ell}}\left{y^{\prime} x-\xi\right} . \end{aligned}

## 数学代写|凸优化作业代写Convex Optimization代考|GENERALIZED POLYHEDRAL APPROXIMATION

$$\begin{array}{ll} \operatorname{minimize} & \sum_{i=1}^m f_i\left(x_i\right) \ \text { subject to } & x \in S, \end{array}$$

$$x \stackrel{\text { def }}{=}\left(x_1, \ldots, x_m\right),$$

$f_i: \Re^{n_i} \mapsto(-\infty, \infty]$是每个$i$的闭固有凸函数，$S$是$\Re^{n_1+\cdots+n_m}$的一个子空间。

EMP的一个经典例子是单个商品网络优化问题，其中$x_i$表示有向图的弧线的(标量)流，$S$是图的循环子空间(参见示例[Ber98])。此外，涉及一般线性约束和可加扩展重值凸代价函数的问题也可以转换为EMP
$$\begin{array}{ll} \text { minimize } & \sum_{i=1}^m f_i\left(x_i\right) \ \text { subject to } & A x=b, \end{array}$$

$$\begin{array}{ll} \text { minimize } & \sum_{i=1}^m f_i\left(x_i\right)+\delta_Z(z) \ \text { subject to } & A x-z=0, \end{array}$$

$$S={(x, z) \mid A x-z=0} .$$

## Matlab代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

## 数学代写|有限元方法代写Finite Element Method代考|EG55M1

statistics-lab™ 为您的留学生涯保驾护航 在代写有限元方法Finite Element Method方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写有限元方法Finite Element Method代写方面经验极为丰富，各种代写有限元方法Finite Element Method相关的作业也就用不着说。

## 数学代写|有限元方法代写Finite Element Method代考|Triangular element

An examination of the weak form in Eq. (9.2.10) and the finite element matrices in Eq. (9.2.19b) shows that $\psi_i^e$ should be at least linear functions of $x$ and $y$. The complete linear polynomial in $x$ and $y$ in $\Omega_e$ is of the form
$$u_h^e(x, y)=c_1^e+c_2^e x+c_3^e y$$
where $c_i^e$ are constants. The set ${1, x, y}$ is linearly independent and complete. Equation (9.2.21) defines a unique plane for fixed $c_i^e$. Thus, if $u(x, y)$ is a curved surface, $u_h^e(x, y)$ approximates the surface by a plane (see Fig. 9.2.2). In particular, $u_h^e(x, y)$ is uniquely defined on a triangle by the three nodal values of $u_h^e(x, y)$; three nodes are placed at the vertices of the triangle so that the geometry of the triangle is uniquely defined, and the nodes are numbered in counterclockwise direction, as shown in Fig. 9.2.2, so that the unit normal always points upward from the domain. Let
$$u_h^e\left(x_1, y_1\right)=u_1^e, \quad u_h^e\left(x_2, y_2\right)=u_2^e, \quad u_h^e\left(x_3, y_3\right)=u_3^e$$
where $\left(x_i, y_i\right)$ denote the coordinates of the $i$ th vertex of the triangle. Note that the triangle is uniquely defined by the three pairs of coordinates $\left(x_i, y_i\right)$.
The three constants $c_i^e(i=1,2,3)$ in Eq. (9.2.21) can be expressed in terms of three nodal values $u_i^e(i=1,2,3)$. Thus, the polynomial in Eq. (9.2.21) is associated with a triangular element and there are three nodes identified, namely, the vertices of the triangle. Equations in (9.2.22) have the explicit form
\begin{aligned} & u_1 \equiv u_h\left(x_1, y_1\right)=c_1+c_2 x_1+c_3 y_1 \ & u_2 \equiv u_h\left(x_2, y_2\right)=c_1+c_2 x_2+c_3 y_2 \ & u_3 \equiv u_h\left(x_3, y_3\right)=c_1+c_2 x_3+c_3 y_3 \end{aligned}
where the element label $e$ is omitted for simplicity. In matrix form, we have
$$\left{\begin{array}{l} u_1 \ u_2 \ u_3 \end{array}\right}=\left[\begin{array}{lll} 1 & x_1 & y_1 \ 1 & x_2 & y_2 \ 1 & x_3 & y_3 \end{array}\right]\left{\begin{array}{l} c_1 \ c_2 \ c_3 \end{array}\right} \text { or } \mathbf{u}=\mathbf{A c}$$

## 数学代写|有限元方法代写Finite Element Method代考|Linear rectangular element

Next, consider the complete polynomial
$$u_h^e(x, y)=c_1^e+c_2^e x+c_3^e y+c_4^e x y$$
which contains four linearly independent terms, and is linear in $x$ and $y$, with a bilinear term in $x$ and $y$. This polynomial requires an element with four nodes. There are two possible geometric shapes: a triangle with the fourth node at the center (or centroid) of the triangle, or a rectangle with the nodes at the vertices. A triangle with a fourth node at the center does not provide a single-valued variation of $u$ at interelement boundaries, resulting in incompatible variation of $u$ at interelement boundaries, and is therefore not admissible (see Fig. 9.2.7). The linear rectangular element is a compatible element because on any side $u_h^e$ varies only linearly and there are two nodes to uniquely define it. Here we consider an approximation of the form Eq. (9.2.27) and use a rectangular element with sides $a$ and $b$ [see Fig. 9.2.8(a)]. For the sake of convenience, we choose a local coordinate system $(\bar{x}, \bar{y})$ to derive the interpolation functions. We assume that (element label is omitted)

$$u_h(\bar{x}, \bar{y})=c_1+c_2 \bar{x}+c_3 \bar{y}+c_4 \bar{x} \bar{y}$$
and require
\begin{aligned} & u_1=u_h(0,0)=c_1 \ & u_2=u_h(a, 0)=c_1+c_2 a \ & u_3=u_h(a, b)=c_1+c_2 a+c_3 b+c_4 a b \ & u_4=u_h(0, b)=c_1+c_3 b \end{aligned}

## 数学代写|有限元方法代写Finite Element Method代考|Triangular element

$$u_h^e(x, y)=c_1^e+c_2^e x+c_3^e y$$

$$u_h^e\left(x_1, y_1\right)=u_1^e, \quad u_h^e\left(x_2, y_2\right)=u_2^e, \quad u_h^e\left(x_3, y_3\right)=u_3^e$$

\begin{aligned} & u_1 \equiv u_h\left(x_1, y_1\right)=c_1+c_2 x_1+c_3 y_1 \ & u_2 \equiv u_h\left(x_2, y_2\right)=c_1+c_2 x_2+c_3 y_2 \ & u_3 \equiv u_h\left(x_3, y_3\right)=c_1+c_2 x_3+c_3 y_3 \end{aligned}

$$\left{\begin{array}{l} u_1 \ u_2 \ u_3 \end{array}\right}=\left[\begin{array}{lll} 1 & x_1 & y_1 \ 1 & x_2 & y_2 \ 1 & x_3 & y_3 \end{array}\right]\left{\begin{array}{l} c_1 \ c_2 \ c_3 \end{array}\right} \text { or } \mathbf{u}=\mathbf{A c}$$

## 数学代写|有限元方法代写Finite Element Method代考|Linear rectangular element

$$u_h^e(x, y)=c_1^e+c_2^e x+c_3^e y+c_4^e x y$$

$$u_h(\bar{x}, \bar{y})=c_1+c_2 \bar{x}+c_3 \bar{y}+c_4 \bar{x} \bar{y}$$

\begin{aligned} & u_1=u_h(0,0)=c_1 \ & u_2=u_h(a, 0)=c_1+c_2 a \ & u_3=u_h(a, b)=c_1+c_2 a+c_3 b+c_4 a b \ & u_4=u_h(0, b)=c_1+c_3 b \end{aligned}

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

## 数学代写|有限元方法代写Finite Element Method代考|AMCS329

statistics-lab™ 为您的留学生涯保驾护航 在代写有限元方法Finite Element Method方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写有限元方法Finite Element Method代写方面经验极为丰富，各种代写有限元方法Finite Element Method相关的作业也就用不着说。

## 数学代写|有限元方法代写Finite Element Method代考|Finite Element Discretization

In two dimensions there is more than one simple geometric shape that can be used as a finite element (see Fig. 9.2.1). As we shall see shortly, the interpolation functions depend not only on the number of nodes in the element and the number of unknowns per node, but also on the shape of the element. The shape of the element must be such that its geometry is uniquely defined by a set of points, which serve as the element nodes in the development of the interpolation functions. As will be discussed later in this section, a triangle is the simplest geometric shape, followed by a rectangle.

The representation of a given region by a set of elements (i.e., discretization or mesh generation) is an important step in finite element analysis. The choice of element type, number of elements, and density of elements depends on the geometry of the domain, the problem to be analyzed, and the degree of accuracy desired. Of course, there are no specific formulae to obtain this information. In general, the analyst is guided by his or her technical background, insight into the physics of the problem being modeled (e.g., a qualitative understanding of the solution), and experience with finite element modeling. The general rules of mesh generation for finite element formulations include:

1. Select elements that characterize the governing equations of the problem.
2. The number, shape, and type (i.e., linear or quadratic) of elements should be such that the geometry of the domain is represented as accurately as desired.
3. The density of elements should be such that regions of large gradients of the solution are adequately modeled (i.e., use more elements or higher-order elements in regions of large gradients).
4. Mesh refinements should vary gradually from high-density regions to low-density regions. If transition elements are used, they should be used away from critical regions (i.e., regions of large gradients). Transition elements are those which connect lower-order elements to higher-order elements (e.g., linear to quadratic).

## 数学代写|有限元方法代写Finite Element Method代考|Weak Form

In the development of the weak form we need only consider a typical element. We assume that $\Omega_e$ is a typical element, whether triangular or quadrilateral, of the finite element mesh, and we develop the finite element model of Eq. (9.2.1) over $\Omega_e$. Various two-dimensional elements will be discussed in the sequel.

Following the three-step procedure presented in Chapters 2 and 3, we develop the weak form of Eq. (9.2.1) over the typical element $\Omega_e$. The first step is to multiply Eq. (9.2.1) with a weight function $w$, which is assumed to be differentiable once with respect to $x$ and $y$, and then integrate the equation over the element domain $\Omega_e$ :
$$0=\int_{\Omega_{\varepsilon}} w\left[-\frac{\partial}{\partial x}\left(F_1\right)-\frac{\partial}{\partial y}\left(F_2\right)+a_{00} u-f\right] d x d y$$
where
$$F_1=a_{11} \frac{\partial u}{\partial x}+a_{12} \frac{\partial u}{\partial y}, \quad F_2=a_{21} \frac{\partial u}{\partial x}+a_{22} \frac{\partial u}{\partial y}$$
In the second step we distribute the differentiation among $u$ and $w$ equally. To achieve this we integrate the first two terms in (9.2.4a) by parts. First we note the identities
\begin{aligned} & \frac{\partial}{\partial x}\left(w F_1\right)=\frac{\partial w}{\partial x} F_1+w \frac{\partial F_1}{\partial x} \quad \text { or } \quad-w \frac{\partial F_1}{\partial x}=\frac{\partial w}{\partial x} F_1-\frac{\partial}{\partial x}\left(w F_1\right) \ & \frac{\partial}{\partial y}\left(w F_2\right)=\frac{\partial w}{\partial y} F_2+w \frac{\partial F_2}{\partial y} \quad \text { or } \quad-w \frac{\partial F_2}{\partial y}=\frac{\partial w}{\partial y} F_2-\frac{\partial}{\partial y}\left(w F_2\right) \end{aligned}

## 数学代写|有限元方法代写Finite Element Method代考|Weak Form

$$0=\int_{\Omega_{\varepsilon}} w\left[-\frac{\partial}{\partial x}\left(F_1\right)-\frac{\partial}{\partial y}\left(F_2\right)+a_{00} u-f\right] d x d y$$

$$F_1=a_{11} \frac{\partial u}{\partial x}+a_{12} \frac{\partial u}{\partial y}, \quad F_2=a_{21} \frac{\partial u}{\partial x}+a_{22} \frac{\partial u}{\partial y}$$

\begin{aligned} & \frac{\partial}{\partial x}\left(w F_1\right)=\frac{\partial w}{\partial x} F_1+w \frac{\partial F_1}{\partial x} \quad \text { or } \quad-w \frac{\partial F_1}{\partial x}=\frac{\partial w}{\partial x} F_1-\frac{\partial}{\partial x}\left(w F_1\right) \ & \frac{\partial}{\partial y}\left(w F_2\right)=\frac{\partial w}{\partial y} F_2+w \frac{\partial F_2}{\partial y} \quad \text { or } \quad-w \frac{\partial F_2}{\partial y}=\frac{\partial w}{\partial y} F_2-\frac{\partial}{\partial y}\left(w F_2\right) \end{aligned}

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

## 数学代写|有限元方法代写Finite Element Method代考|ME672

statistics-lab™ 为您的留学生涯保驾护航 在代写有限元方法Finite Element Method方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写有限元方法Finite Element Method代写方面经验极为丰富，各种代写有限元方法Finite Element Method相关的作业也就用不着说。

## 数学代写|有限元方法代写Finite Element Method代考|General Outline

A typical finite element program consists of three basic units (see Fig. 8.3.1):

1. Preprocessor
2. Processor
3. Postprocessor
In the preprocessor part of the program, the input data of the problem are read in and/or generated. This includes the geometry (e.g., length of the domain and boundary conditions), the data of the problem (e.g., coefficients in the differential equation), finite element mesh information (e.g., element type, number of elements, element length, coordinates of the nodes, and connectivity matrix), and indicators for various options (e.g., print, no print, type of field problem analyzed, static analysis, eigenvalue analysis, transient analysis, and degree of interpolation).

In the processor part, all steps of the finite element analysis discussed in the preceding chapters, except for postprocessing, are performed. The major steps of the processor are:

1. Generation of the element matrices using numerical integration.
2. Assembly of element equations.
3. Imposition of the boundary conditions.
4. Solution of the algebraic equations for the nodal values of the primary variables.

## 数学代写|有限元方法代写Finite Element Method代考|Preprocessor

The preprocessor unit consists of reading input data and generating finite element mesh, and printing the data and mesh information. The input data to a finite element program consist of element type, IELEM (i.e., Lagrange element or Hermite element), number of elements in the mesh (NEM), specified boundary conditions on primary and secondary variables (number of boundary conditions, global node number and degree of freedom, and specified values of the degrees of freedom), the global coordinates of global nodes, and element properties [e.g., coefficients $a(x), b(x), c(x), f(x)$, etc.] If a uniform mesh is used, the length of the domain should be read in, and global coordinates of the nodes can be generated in the program.

The preprocessor portion that deals with the generation of finite element mesh information (when not supplied by the user) can be separated into a subroutine (MESH1D), depending on the convenience and complexity of the program. Mesh generation includes computation of the global coordinates $X_I$ and the connectivity array NOD $\left(=B_{i j}\right)$. Recall that the connectivity matrix describes the relationship between element nodes to global nodes:
$\operatorname{NOD}(n, j)=$ Global node number corresponding to the $j$ th (local) node of element $n$
This array is used in the assembly procedure as well as to transfer information from element to the global system and vice versa. For example, to extract the vector ELX of global coordinates of element nodes from the vector GLX of global coordinates of global nodes, we can use the matrix NOD as follows. The global coordinate $x_i^{(n)}$ of the $i$ th node of the $n$th element is the same as the global coordinate $X_I$ of the global node $I$, where $I=$ $\operatorname{NOD}(n, i)$ :
$$\left{x_i^{(n)}\right}=\left{X_l\right}, \quad I=\operatorname{NOD}(n, i) \rightarrow \operatorname{ELX}(i)=\operatorname{GLX}(\operatorname{NOD}(n, i))$$

## 数学代写|有限元方法代写Finite Element Method代考|Preprocessor

$\operatorname{NOD}(n, j)=$元素$n$的第$j$个(本地)节点对应的全局节点号

## Matlab代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。