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数学代写|应用数学代考APPLIED MATHEMATICS代写|Case study

Setup. To illustrate the preceding results on dimensional methods, and the process of modelling a simple mechanical system, we study the motion of a pendulum released from rest. Figure $1.1$ illustrates the system, which consists of a string of length $\ell$, with one end attached to a fixed support point, and the other end attached to a ball of mass $m$. We assume the string is always in tension and hence straight, and we let $\theta$ denote the angle between the string and a vertical line through the support point, and arbitrarily take the positive direction to be counter-clockwise. We assume that gravitational acceleration $g$ is directed in the downward, vertical direction. When the ball is raised and released from the rest conditions $\theta=\theta_0$ and $\frac{d \theta}{d t}=0$ at time $t=0$, the ball will swing back-and-forth in a periodic motion. We seek to understand various aspects of this motion; for example, how the period depends on the parameters $m, g, \ell$, and $\theta_0$.
Outline of model. We assume that the motion occurs in a plane and introduce an origin and $x, y$ coordinates as shown. The standard unit vectors in the positive $x$ and $y$ directions are denoted by $\vec{i}$ and $\vec{j}$, and the position vector for the ball is denoted by $\vec{r}$. It will be convenient to introduce unit vectors $\vec{e}r$ and $\vec{e}\theta$ that are parallel and perpendicular to $\vec{r}$. For any angle $\theta$, the components of these vectors are $\vec{r}=\ell \sin \theta \vec{i}+$ $\ell \cos \theta \vec{j}, \vec{e}r=\sin \theta \vec{i}+\cos \theta \vec{j}$, and $\vec{e}\theta=\cos \theta \vec{i}-\sin \theta \vec{j}$. By differentiating the position with respect to time, we obtain the velocity and acceleration vectors
$$\begin{gathered} \frac{d \vec{r}}{d t}=\ell \cos \theta \frac{d \theta}{d t} \vec{i}-\ell \sin \theta \frac{d \theta}{d t} \vec{j} \ \frac{d^2 \vec{r}}{d t^2}=\left[\ell \cos \theta \frac{d^2 \theta}{d t^2}-\ell \sin \theta\left(\frac{d \theta}{d t}\right)^2\right] \vec{i}-\left[\ell \sin \theta \frac{d^2 \theta}{d t^2}+\ell \cos \theta\left(\frac{d \theta}{d t}\right)^2\right] \vec{j} \end{gathered}$$

数学代写|应用数学代考APPLIED MATHEMATICS代写|Reduced equation for period

Reduced equation for period. The quantities $P, g, \ell, \theta_0$ have dimensions $[P]=$ $T,[g]=L / T^2,[\ell]=L$ and $\left[\theta_0\right]=1$. A dimensional basis is ${T, L}$, and the dimensional exponent matrix in this basis is
$$A=\left(\Delta_P, \Delta_g, \Delta_{\ell}, \Delta_{\theta_0}\right)=\left(\begin{array}{rrrr} 1 & -2 & 0 & 0 \ 0 & 1 & 1 & 0 \end{array}\right) .$$
An arbitrary power product has the form $\pi=P^{b_1} g^{b_2} \ell^{b_3} \theta_0^{b_4}$. The equation $A v=0$, where $v=\left(b_1, \ldots, b_4\right)$, has two free variables, and the general solution is
$$b_1=-2 b_3, \quad b_2=-b_3, \quad b_3 \text { and } b_4 \text { free. }$$
Since there are two free variables, there are two independent solutions. For the first solution, we choose $b_3=-1 / 2$ and $b_4=0$, which gives $\pi_1=P \sqrt{g / \ell}$. For the second solution, we choose $b_3=0$ and $b_4=1$, which gives $\pi_2=\theta_0$. This is a full set of independent dimensionless power products, and is normalized with respect to $P$.
By the $\pi$-theorem, the period equation $P=F\left(g, \ell, \theta_0\right)$ must be equivalent to
$$\pi_1=\phi\left(\pi_2\right) \quad \text { or } \quad P=\sqrt{\frac{\ell}{g}} \phi\left(\theta_0\right),$$
for some function $\phi$. Thus the relation between the quantities $P, g, \ell, \theta_0$ is not characterized by an unknown function of three quantities $F\left(g, \ell, \theta_0\right)$, but is instead characterized by an unknown function of one quantity $\phi\left(\theta_0\right)$. Equivalently, the dependence of $F\left(g, \ell, \theta_0\right)$ on the quantities $g$ and $\ell$ is completely dictated by dimensional considerations.

应用数学代考

数学代写|应用数学代考APPLIED MATHEMATICS代写|Case study

$$\frac{d \vec{r}}{d t}=\ell \cos \theta \frac{d \theta}{d t} \vec{i}-\ell \sin \theta \frac{d \theta}{d t} \vec{j} \frac{d^2 \vec{r}}{d t^2}=\left[\ell \cos \theta \frac{d^2 \theta}{d t^2}-\ell \sin \theta\left(\frac{d \theta}{d t}\right)^2\right] \vec{i}-\left[\ell \sin \theta \frac{d^2 \theta}{d t^2}+\ell \cos \theta\left(\frac{d \theta}{d t}\right)^2\right] \vec{j}$$

数学代写|应用数学代考APPLIED MATHEMATICS代写|Reduced equation for period

$$A=\left(\Delta_P, \Delta_g, \Delta_{\ell}, \Delta_{\theta_0}\right)=\left(\begin{array}{llllllll} 1 & -2 & 0 & 0 & 0 & 1 & 1 & 0 \end{array}\right) .$$

$b_1=-2 b_3, \quad b_2=-b_3, \quad b_3$ and $b_4$ free.

$$\pi_1=\phi\left(\pi_2\right) \quad \text { or } \quad P=\sqrt{\frac{\ell}{g}} \phi\left(\theta_0\right),$$

avatest.org 为您提供可靠及专业的论文代写服务以便帮助您完成您学术上的需求，让您重新掌握您的人生。我们将尽力给您提供完美的论文，并且保证质量以及准时交稿。除了承诺的奉献精神，我们的专业写手、研究人员和校对员都经过非常严格的招聘流程。所有写手都必须证明自己的分析和沟通能力以及英文水平，并通过由我们的资深研究人员和校对员组织的面试。

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。

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数学代写|应用数学代考APPLIED MATHEMATICS代写|Natural Boundary Condition

Finally, let us address natural boundary conditions instead of fixed boundary conditions, which we hav considered so far. These can be motivated, for instance, by the following problem.
Example 7.5.3: Motivation for natural boundary conditions
A river with parallel straight banks $b$ units apart has a stream velocity given by
$$\boldsymbol{v}(x, y)=\left(\begin{array}{c} 0 \ v(x) \end{array}\right) .$$
Assuming that one of the banks is the $y$-axis and that the point $(0,0)$ is the point of departure, what route should a boat take to reach the opposite bank in the shortest possible time? Assume that the speed of the boat in still water is $c \in \mathbb{R}^{+}$with $c>v(x)$ for all $x$. This problem differs from those in earlier sections in that right-hand endpoint, the point of arrival on the line $x=b$, is not specified. Instead, it must be determined as part of the solution. It can be shown that the time required for the boat to cross the river along a given path $y=y(x)$ is
$$F(y)=\int_a^b \frac{\sqrt{c^2\left[1+y^{\prime}(x)^2\right]-v(x)^2}-v(x) y^{\prime}(x)}{c^2-v(x)^2} \mathrm{~d} x .$$
Thus, the variational problem is to minimize $F$ subject to the conditions
$$y(0)=0, y(b) \text { free. }$$

数学代写|应用数学代考APPLIED MATHEMATICS代写|Best Approximations in Inner Product Spaces

The potentially excellent approximation properties of the Fourier series might be best understood by noting that the (truncated) Fourier series can be characterized as the best approximation in certain inner product spaces. Getting to the heart of the underlying theory – presented in this and the next section – will also allow us to formulate generalized Fourier series, therefore extending the concept of classical Fourier series to a much broader class of function spaces.

Definition 8.1.1: Best approximations
Let $(X,|\cdot|)$ be a normed linear space, $f \in X$, and $V \subset X$ be a linear subspace. An element $v^* \in V$ is called best approximation of $f$ from $V$ with respect to $|\cdot|$ if
$$\left|f-v^\right| \leq|f-v|$$ holds for all $v \in V$. This means that there is no element $v \in V$ which is closer to $f$ that $v^$.
Example 8.1.2: $\left(\mathbb{R}^2,|\cdot|_{\infty}\right)$
Given is the linear space $\left(\mathbb{R}^2,|\cdot|_{\infty}\right)$ with $\left|(x, y)^{\top}\right|_{\infty}=\max {|x|,|y|}$, the element $f=(0,1)^{\top}$, and the linear subspace $V=\mathbb{R} \times{0}$ which corresponds to the $x$-axis of the $x y$-plane. Then, every element
$$v_r^=(r, 0)^{\boldsymbol{\top}} \in V, \quad r \in[-1,1],$$ is a best approximation of $f=(0,1)^{\top}$ from $V$ with respect to $|\cdot|_{\infty}$. This can be noted by observing that $$\left|f-v_r^\right|_{\infty}=\left|(-r, 1)^{\top}\right|_{\infty}=\max {|r|, 1}=1$$
for $r \in[-1,1]$ and
$$|f-v|_{\infty}>1$$
for all other elements from $V$.

应用数学代考

数学代写|应用数学代考APPLIED MATHEMATICS代写|Natural Boundary Condition

$$\boldsymbol{v}(x, y)=(0 v(x))$$

$$F(y)=\int_a^b \frac{\sqrt{c^2\left[1+y^{\prime}(x)^2\right]-v(x)^2}-v(x) y^{\prime}(x)}{c^2-v(x)^2} \mathrm{~d} x .$$

$$y(0)=0, y(b) \text { free. }$$

数学代写|应用数学代考APPLIED MATHEMATICS代写|Best Approximations in Inner Product Spaces

$$|f-v|_{\infty}>1$$

avatest.org 为您提供可靠及专业的论文代写服务以便帮助您完成您学术上的需求，让您重新掌握您的人生。我们将尽力给您提供完美的论文，并且保证质量以及准时交稿。除了承诺的奉献精神，我们的专业写手、研究人员和校对员都经过非常严格的招聘流程。所有写手都必须证明自己的分析和沟通能力以及英文水平，并通过由我们的资深研究人员和校对员组织的面试。

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。

Posted on Categories:优化理论代写, 变方法代写Calculus of Variations, 数学代写, 机器学习代考

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数学代写|应用数学代考APPLIED MATHEMATICS代写|Necessary conditions for extrema

Considering the functions $f: \mathbb{R} \rightarrow \mathbb{R}$ we know from real analysis that $f^{\prime}(x)=0$ is a necessary condition for $x$ to be an extrema. In fact, this idea can be generalized for functionals $F: A \rightarrow \mathbb{R}$, at least in a certain sense. At the same time we should remember from multidimensional calculus that there can already be different concepts of derivatives for functions $f: \mathbb{R}^n \rightarrow \mathbb{R}^m$. Usually, we distinguish between

the total derivative (best linear approximation) and

the directional derivative (rate of change in a certain direction, partial derivative).
Indeed, both concepts can be generalized for functionals $F: A \rightarrow \mathbb{R}$ on general normed vector spaces. The resulting generalization of the total derivative is referred to as Fréchet derivative while the generalization of the directional derivative is referred to as Gâteaux derivative. In the context of calculus of variations, however, we are only interested in the Gâteaux derivative.

数学代写|应用数学代考APPLIED MATHEMATICS代写|Asymptotic Expansion of Integrals

Many physical quantities can be described by integrals. In particular, the solution of differential equations often yield formulas involving integrals. Unfortunately, in many cases, these integrals cannot be evaluated in closed form. For example, the initial value problem
$$y^{\prime \prime}+2 \lambda+y^{\prime}=0, \quad y(0)=0, \quad y^{\prime}(0)=1,$$
has the solution
$$y_\lambda(t)=\int_0^t e^{-\lambda s^2} \mathrm{~d} s .$$
Yet, the solution cannot be calculated, because there is no antiderivative in terms of simple functions, for the integral on the right hand side. For some problems, however, we may want to at least know the behavior for fixed $\lambda$ and large $t$ or, conversely, for fixed $t$ and large $\lambda$. Problems like this are common in applied mathematics and in this chapter we will address some standard techniques to solve them for certain types of integrals.

应用数学代考

数学代写|应用数学代考APPLIED MATHEMATICS代写|Asymptotic Expansion of Integrals

$$y^{\prime \prime}+2 \lambda+y^{\prime}=0, \quad y(0)=0, \quad y^{\prime}(0)=1$$

$$y_\lambda(t)=\int_0^t e^{-\lambda s^2} \mathrm{~d} s .$$

avatest.org 为您提供可靠及专业的论文代写服务以便帮助您完成您学术上的需求，让您重新掌握您的人生。我们将尽力给您提供完美的论文，并且保证质量以及准时交稿。除了承诺的奉献精神，我们的专业写手、研究人员和校对员都经过非常严格的招聘流程。所有写手都必须证明自己的分析和沟通能力以及英文水平，并通过由我们的资深研究人员和校对员组织的面试。

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。

Posted on Categories:优化理论代写, 变方法代写Calculus of Variations, 数学代写, 机器学习代考

avatest™帮您通过考试

avatest™的各个学科专家已帮了学生顺利通过达上千场考试。我们保证您快速准时完成各时长和类型的考试，包括in class、take home、online、proctor。写手整理各样的资源来或按照您学校的资料教您，创造模拟试题，提供所有的问题例子，以保证您在真实考试中取得的通过率是85%以上。如果您有即将到来的每周、季考、期中或期末考试，我们都能帮助您！

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数学代写|应用数学代考APPLIED MATHEMATICS代写|Pitfalls of Regular Perturbation

At the end of the last chapter we already observed problems for the regular perturbation method when secular terms arise in the perturbation approximation. In what follows, we demonstrate some more pitfalls of regular perturbation. By doing so we hope to sharpen our understanding of when we can the expect regular perturbation method to succeed and – most notably- when we cannot.
Example $4.2 .1$
$$\varepsilon x^2+2 x+1=0$$
with $0<\varepsilon \ll 1$. Of course, it is not hard to solve this equation exactly and its solutions are given by
$$x_{1,2}=-\frac{1}{\varepsilon} \pm \sqrt{\frac{1}{\varepsilon^2}-\frac{1}{\varepsilon}}=\frac{1}{\varepsilon}(-1 \pm \sqrt{1-\varepsilon}) .$$
Note that
$$\lim {\varepsilon \rightarrow 0} x_1=-\frac{1}{2}, \quad \lim {\varepsilon \rightarrow 0} x_2=-\infty .$$
Yet, our goal is to illustrate the failure of the regular perturbation method for this example. If we attempt regular perturbation by substituting the perturbation series
$$p_{\varepsilon}=x_0+\varepsilon x_1+\varepsilon^2 x_2+\ldots$$
into the perturbed problem (4.8), we get, after comparing the coefficients of the different orders of $\varepsilon$, the following sequence of equations:
\begin{aligned} 2 x_0+1 &=0, \ x_0^2+2 x_1 &=0, \ 2 x_1 x_0+2 x_2 &=0, \quad \ldots \end{aligned}

数学代写|应用数学代考APPLIED MATHEMATICS代写|Asymptotic Expansion of Integrals

Many physical quantities can be described by integrals. In particular, the solution of differential equations often yield formulas involving integrals. Unfortunately, in many cases, these integrals cannot be evaluated in closed form. For example, the initial value problem
$$y^{\prime \prime}+2 \lambda+y^{\prime}=0, \quad y(0)=0, \quad y^{\prime}(0)=1,$$
has the solution
$$y_\lambda(t)=\int_0^t e^{-\lambda s^2} \mathrm{~d} s .$$
Yet, the solution cannot be calculated, because there is no antiderivative in terms of simple functions, for the integral on the right hand side. For some problems, however, we may want to at least know the behavior for fixed $\lambda$ and large $t$ or, conversely, for fixed $t$ and large $\lambda$. Problems like this are common in applied mathematics and in this chapter we will address some standard techniques to solve them for certain types of integrals.

应用数学代考

数学代写|应用数学代考APPLIED MATHEMATICS代写|Pitfalls of Regular

Perturbation 过这样做，我们㣇望加深我们对何时可以预期常规扰动方法成功以及 – 最值得注意的是 – 何时不能成功的理解。

$$\varepsilon x^2+2 x+1=0$$

$$x_{1,2}=-\frac{1}{\varepsilon} \pm \sqrt{\frac{1}{\varepsilon^2}-\frac{1}{\varepsilon}}=\frac{1}{\varepsilon}(-1 \pm \sqrt{1-\varepsilon}) .$$

$$\lim \varepsilon \rightarrow 0 x_1=-\frac{1}{2}, \quad \lim \varepsilon \rightarrow 0 x_2=-\infty .$$

$$p_{\varepsilon}=x_0+\varepsilon x_1+\varepsilon^2 x_2+\ldots$$

$$2 x_0+1=0, x_0^2+2 x_1 \quad=0,2 x_1 x_0+2 x_2=0, \quad \ldots$$

数学代写|应用数学代考APPLIED MATHEMATICS代写|Asymptotic Expansion of Integrals

$$y^{\prime \prime}+2 \lambda+y^{\prime}=0, \quad y(0)=0, \quad y^{\prime}(0)=1,$$

$$y_\lambda(t)=\int_0^t e^{-\lambda s^2} \mathrm{~d} s .$$

avatest.org 为您提供可靠及专业的论文代写服务以便帮助您完成您学术上的需求，让您重新掌握您的人生。我们将尽力给您提供完美的论文，并且保证质量以及准时交稿。除了承诺的奉献精神，我们的专业写手、研究人员和校对员都经过非常严格的招聘流程。所有写手都必须证明自己的分析和沟通能力以及英文水平，并通过由我们的资深研究人员和校对员组织的面试。

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。

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数学代写|应用数学代考APPLIED MATHEMATICS代写|Representation of quantities

We say that the quantity $q$ can be represented with respect to the vector of dimension symbols $\left(\mathbf{E}_1, \ldots, \mathbf{E}_m\right)$ if
$$[q]=\mathbf{E}_1^{\alpha_1} \cdots \mathbf{E}_m^{\alpha_m}$$
for some vector $\boldsymbol{\alpha}=\left(\alpha_1, \ldots, \alpha_m\right) \in \mathbb{R}^m$. The vector $\boldsymbol{\alpha}$ is often called dimension vector of $q$ with respect to $\left(\mathbf{E}_1, \ldots, \mathbf{E}_m\right)$

We note that there can be different representations of quantities. Let us consider the quantity $q$ measuring energy, i. e., $[q]=\mathbf{E}$. Consulting Table $2.2, q$ can be represented with respect to $(\mathbf{F}, \mathbf{L})$, since
$$[q]=\mathbf{E}=\mathbf{F} \cdot \mathbf{L} \quad \text { (force times length). }$$
The corresponding dimension vector is given by $\alpha=(1,1)$. Yet, at the same time $q$ can be represented with respect to $(\mathbf{m}, \mathbf{t}, \mathbf{L})$. Note that force can be derived by
$$\mathbf{F}=\mathbf{m} \cdot \mathbf{L} \cdot \mathbf{t}^{-2}$$
and we can therefore rewrite (2.4) as
$$[q]=\mathbf{E}=\mathbf{m} \cdot \mathbf{t}^{-2} \cdot \mathbf{L}^2 .$$
This time, the corresponding dimension vector is given by $\boldsymbol{\alpha}=(1,-2,2)$.

数学代写|应用数学代考APPLIED MATHEMATICS代写|Uniqueness of the dimension vector

Let $q$ be a quantity which can be represented with respect to the vector of dimension symbols $\left(\mathbf{E}_1, \ldots, \mathbf{E}_m\right)$. The dimension $\boldsymbol{\alpha}=\left(\alpha_1, \ldots, \alpha_m\right)$ vector is unique if and only if $\left(\mathbf{E}_1, \ldots, \mathbf{E}_m\right)$ is independent.

Definition 2.2.7: Dimension matrix
Let $\left(q_1, \ldots, q_n\right)$ be a vector of quantities which can be represented with respect to the vector of independent dimension symbols $\left(\mathbf{E}1, \ldots, \mathbf{E}_m\right)$. That is, there are unique dimension vectors such that $$\left[q_i\right]=\mathbf{E}_1^{\alpha{i, 1}} \cdots \mathbf{E}m^{\alpha{i, m}}$$
for $i=1, \ldots, n$. The matrix containing the dimensional vectors
$$A:=\left(\begin{array}{ccc} \alpha_{1,1} & \cdots & \alpha_{1, m} \ \vdots & & \vdots \ \alpha_{n, 1} & \cdots & \alpha_{n, m} \end{array}\right) \in \mathbb{R}^{n \times m}$$
is called the dimension matrix of $\left(q_1, \ldots, q_n\right)$ with respect to $\left(\mathbf{E}_1, \ldots, \mathbf{E}_m\right)$. The rank of $A$, $\operatorname{rank}(A)=$ maximal number of independent columns, is referred to as the dimension rank.

应用数学代考

数学代写|应用数学代考APPLIED MATHEMATICS代写|Representation of quantities

$$[q]=\mathbf{E}_1^{\alpha_1} \cdots \mathbf{E}_m^{\alpha_m}$$

$$[q]=\mathbf{E}=\mathbf{F} \cdot \mathbf{L} \quad \text { (force times length). }$$

$$\mathbf{F}=\mathbf{m} \cdot \mathbf{L} \cdot \mathbf{t}^{-2}$$

$$[q]=\mathbf{E}=\mathbf{m} \cdot \mathbf{t}^{-2} \cdot \mathbf{L}^2 .$$

数学代写|应用数学代考APPLIED MATHEMATICS代写|Uniqueness of the dimension vector

$$\left[q_i\right]=\mathbf{E}_1^{\alpha i, 1} \cdots \mathbf{E} m^{\alpha i, m}$$

avatest.org 为您提供可靠及专业的论文代写服务以便帮助您完成您学术上的需求，让您重新掌握您的人生。我们将尽力给您提供完美的论文，并且保证质量以及准时交稿。除了承诺的奉献精神，我们的专业写手、研究人员和校对员都经过非常严格的招聘流程。所有写手都必须证明自己的分析和沟通能力以及英文水平，并通过由我们的资深研究人员和校对员组织的面试。

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。

Posted on Categories:优化理论代写, 变方法代写Calculus of Variations, 数学代写, 机器学习代考

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数学代写|应用数学代考APPLIED MATHEMATICS代写|Introduction – What is Applied Mathematics?

Applied mathematics is a broad subject area dealing with the description, analysis, and prediction of real-world phenomena. It is more than a set of methods that can be used to solve equations that come from physics, engineering, and other applied sciences. In fact, applied mathematics also is about mathematical modeling and an entire process that intertwines with the physical reality. In this course, by a mathematical model (sometimes just called a model) we usually refer to an equation that describes some physical problem or phenomenon. Moreover, by mathematical modeling we mean the process of formulating, analyzing, and potentially refining a mathematical model. In particular, this process includes the following steps:

1. Introducing the relevant quantities or variables in the model
2. Solving the model by some (analytical or numerical) method
3. Comparing the solution of the model to real-world data and interpreting the results
4. If necessary, revising the model until it describes the underlying physical problem sufficiently accurate

Hence, mathematical modeling involves physical intuition, formulation of equations, solution methods, and analysis. Please note that solution methods can be both, of analytical or numerical nature. In this course, however, we will restrict ourselves to analytical methods (the ones involving the use of pen and paper) rather than using numerical methods (the ones involving the use of computers). Moreover, you (the student) are expected to already be familiar with basic solution methods for ordinary differential equations, which will be the class of mathematical models we will focus on in this course. Finally, let us – at least loosely – agree on what we demand from a ‘good’ model:

• It should be as simple as possible and as complex as necessary ${ }^1$
• It should apply to many situations
• It should be predictive

数学代写|应用数学代考APPLIED MATHEMATICS代写|Physical Dimensions and Units

Almost all physical quantities we will deal with in this course have a physical (or some other) dimension. Furthermore, this dimension is usually measured by using certain units. It is worth listing and getting familiar some of the more important dimensions and their corresponding units right from the start of this course. This will save us some trouble later on. Let us start by noting that there are two different classes of physical dimensions:

1. Basis dimensions (also called primary dimensions)
2. Derived dimensions (also called secondary dimensions)
As the name indicates, basis dimensions are defined independent or fundamental dimensions, from which other dimensions can be obtained. Table $2.1$ lists the seven basis dimensions as well as their corresponding symbols and units. We should note that there can be many possible units for measuring a certain dimension. For instance, length can be measured not just by the unit meter but also by millimeter, kilometer, inches, foot, mile, and many more. In this course, however, we will only use SI units (more commonly referred to as metric units) from the International System of Units (also Le Systeme International d’Unites) [oWMTT01]. The SI units corresponding to the seven basis dimensions can be found in Table $2.1$ as well.

应用数学代考

数学代写|应用数学代考APPLIED MATHEMATICS代写|Introduction – What is Applied Mathematics?

1. 在模型中引入相关量或变量
2. 通过某种（分析或数值）方法求解模型
3. 将模型的解决方案与真实世界的数据进行比较并解释结果
4. 如有必要，修改模型直到它足够准确地描述潜在的物理问题

• 它应该尽可能简单，必要时尽可能复杂1
• 它应该适用于许多情况
• 应该是有预见性的

数学代写|应用数学代考APPLIED MATHEMATICS代写|Physical Dimensions and Units

1. 基本尺寸（也称为主要尺寸）
2. 派生维度（也称为次级维度）
顾名思义，基础维度是定义独立的或基本的维度，从中可以得到其他维度。桌子2.1列出了七个基本维度及其相应的符号和单位。我们应该注意到，可以有许多可能的单位来衡量某个维度。例如，长度不仅可以用单位米来衡量，还可以用毫米、千米、英寸、英尺、英里等等来衡量。然而，在本课程中，我们将仅使用国际单位制（也称为 Le Systeme International d’Unites）[oWMTT01] 中的 SI 单位（通常称为公制单位）。七个基本维度对应的SI单位见表2.1以及。

avatest.org 为您提供可靠及专业的论文代写服务以便帮助您完成您学术上的需求，让您重新掌握您的人生。我们将尽力给您提供完美的论文，并且保证质量以及准时交稿。除了承诺的奉献精神，我们的专业写手、研究人员和校对员都经过非常严格的招聘流程。所有写手都必须证明自己的分析和沟通能力以及英文水平，并通过由我们的资深研究人员和校对员组织的面试。

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。

Posted on Categories:Optimization Theory, 优化理论, 优化理论代写, 数学代写, 机器学习代写, 机器学习代考

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数学代写|机器学习中的优化理论代写Optimization for Machine Learning代考|Steepest Descent Direction

The Taylor expansion (7) computes an affine approximation of the function $f$ near $x$, since it can be written as
$$f(z)=T_x(z)+o(|x-z|) \quad \text { where } \quad T_x(z) \stackrel{\text { def. }}{=} f(x)+\langle\nabla f(x), z-x\rangle,$$
see Fig. 8. First order methods operate by locally replacing $f$ by $T_x$.
The gradient $\nabla f(x)$ should be understood as a direction along which the function increases. This means that to improve the value of the function, one should move in the direction $-\nabla f(x)$. Given some fixed $x$, let us look as the function $f$ along the 1-D half line
$$\tau \in \mathbb{R}^{+}=[0,+\infty[\longmapsto f(x-\tau \nabla f(x)) \in \mathbb{R}$$

If $f$ is differentiable at $x$, one has
$$f(x-\tau \nabla f(x))=f(x)-\tau\langle\nabla f(x), \nabla f(x)\rangle+o(\tau)=f(x)-\tau|\nabla f(x)|^2+o(\tau) .$$
So there are two possibility: either $\nabla f(x)=0$, in which case we are already at a minimum (possibly a local minimizer if the function is non-convex) or if $\tau$ is chosen small enough,
$$f(x-\tau \nabla f(x))<f(x)$$
which means that moving from $x$ to $x-\tau \nabla f(x)$ has improved the objective function.
Remark 2 (Orthogonality to level sets). The level sets of $f$ are the sets of point sharing the same value of $f$, i.e. for any $s \in \mathbb{R}$
$$\mathcal{L}_s \stackrel{\text { def. }}{=}{x ; f(x)=s} .$$

The gradient descent algorithm reads, starting with some $x_0 \in \mathbb{R}^p$
$$x_{k+1} \stackrel{\text { def. }}{=} x_k-\tau_k \nabla f\left(x_k\right)$$
where $\tau_k>0$ is the step size (also called learning rate). For a small enough $\tau_k$, the previous discussion shows that the function $f$ is decaying through the iteration. So intuitively, to ensure convergence, $\tau_k$ should be chosen small enough, but not too small so that the algorithm is as fast as possible. In general, one use a fix step size $\tau_k=\tau$, or try to adapt $\tau_k$ at each iteration (see Fig. 9).

Remark 4 (Greedy choice). Although this is in general too costly to perform exactly, one can use a “greedy” choice, where the step size is optimal at each iteration, i.e.
$$\tau_k \stackrel{\text { def. }}{=} \underset{\tau}{\operatorname{argmin}} h(\tau) \stackrel{\text { def. }}{=} f\left(x_k-\tau \nabla f\left(x_k\right)\right) .$$
Here $h(\tau)$ is a function of a single variable. One can compute the derivative of $h$ as
$$h(\tau+\delta)=f\left(x_k-\tau \nabla f\left(x_k\right)-\delta \nabla f\left(x_k\right)\right)=f\left(x_k-\tau \nabla f\left(x_k\right)\right)-\left\langle\nabla f\left(x_k-\tau \nabla f\left(x_k\right)\right), \nabla f\left(x_k\right)\right\rangle+o(\delta) .$$
One note that at $\tau=\tau_k, \nabla f\left(x_k-\tau \nabla f\left(x_k\right)\right)=\nabla f\left(x_{k+1}\right)$ by definition of $x_{k+1}$ in (13). Such an optimal $\tau=\tau_k$ is thus characterized by
$$h^{\prime}\left(\tau_k\right)=-\left\langle\nabla f\left(x_k\right), \nabla f\left(x_{k+1}\right)\right\rangle=0 .$$

数学代写|机器学习中的优化理论代写Optimization for Machine Learning代 考|Steepest Descent Direction

$$f(z)=T_x(z)+o(|x-z|) \quad \text { where } \quad T_x(z) \stackrel{\text { def. }}{=} f(x)+\langle\nabla f(x), z-x\rangle,$$

$$\tau \in \mathbb{R}^{+}=[0,+\infty[\longmapsto f(x-\tau \nabla f(x)) \in \mathbb{R}$$

$$f(x-\tau \nabla f(x))=f(x)-\tau\langle\nabla f(x), \nabla f(x)\rangle+o(\tau)=f(x)-\tau|\nabla f(x)|^2+o(\tau) .$$

$$f(x-\tau \nabla f(x))s \stackrel{\text { def. }}{=} x ; f(x)=s .$$ # to |Gradient Descent 梯度下降算法读取，从一些开始 $x_0 \in \mathbb{R}^p$ $$x{k+1} \stackrel{\text { def. }}{=} x_k-\tau_k \nabla f\left(x_k\right)$$ 在哪里 $\tau_k>0$ 是步长 (也称为学习率) 。对于足够小的 $\tau_k$ ，前面的讨论表明函数 $f$ 通过迭代詚减。所以直觉上，为了确保收敛， $\tau_k$ 应该选择足够小，但又不能太小，以便算法层可能仜。一般来说，一个人使用固定的步长 $\tau_k=\tau$ ，或尝试适应 $\tau_k$ 在每次迭代中 (见图 9)。

$$\tau_k \stackrel{\text { def. }}{=} \underset{\tau}{\operatorname{argmin}} h(\tau) \stackrel{\text { def. }}{=} f\left(x_k-\tau \nabla f\left(x_k\right)\right) \text {. }$$

$$h(\tau+\delta)=f\left(x_k-\tau \nabla f\left(x_k\right)-\delta \nabla f\left(x_k\right)\right)=f\left(x_k-\tau \nabla f\left(x_k\right)\right)-\left\langle\nabla f\left(x_k-\tau \nabla f\left(x_k\right)\right), \nabla f\left(x_k\right)\right\rangle+o(\delta) .$$

$$h^{\prime}\left(\tau_k\right)=-\left\langle\nabla f\left(x_k\right), \nabla f\left(x_{k+1}\right)\right\rangle=0 .$$

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。

Posted on Categories:Optimization Theory, 优化理论, 优化理论代写, 数学代写, 机器学习代写, 机器学习代考

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数学代写|机器学习中的优化理论代写Optimization for Machine Learning代考|Convexity

Convex functions define the main class of functions which are somehow “simple” to optimize, in the sense that all minimizers are global minimizers, and that there are often efficient methods to find these minimizers (at least for smooth convex functions). A convex function is such that for any pair of point $(x, y) \in\left(\mathbb{R}^p\right)^2$,
$$\forall t \in[0,1], \quad f((1-t) x+t y) \leqslant(1-t) f(x)+t f(y)$$
which means that the function is below its secant (and actually also above its tangent when this is well defined), see Fig. 4. If $x^{\star}$ is a local minimizer of a convex $f$, then $x^{\star}$ is a global minimizer, i.e. $x^{\star} \in$ argmin $f$. Convex function are very convenient because they are stable under lots of transformation. In particular, if $f, g$ are convex and $a, b$ are positive, $a f+b g$ is convex (the set of convex function is itself an infinite dimensional convex cone!) and so is $\max (f, g)$. If $g: \mathbb{R}^q \rightarrow \mathbb{R}$ is convex and $B \in \mathbb{R}^{q \times p}, b \in \mathbb{R}^q$ then $f(x)=g(B x+b)$ is convex. This shows immediately that the square loss appearing in (3) is convex, since $|\cdot|^2 / 2$ is convex (as a sum of squares). Also, similarly, if $\ell$ and hence $L$ is convex, then the classification loss function (4) is itself convex.

Strict convexity. When $f$ is convex, one can strengthen the condition (5) and impose that the inequality is strict for $t \in] 0,1[$ (see Fig. 4, right), i.e.
$$\forall t \in] 0,1[, \quad f((1-t) x+t y)<(1-t) f(x)+t f(y) .$$
In this case, if a minimum $x^{\star}$ exists, then it is unique. Indeed, if $x_1^{\star} \neq x_2^{\star}$ were two different minimizer, one would have by strict convexity $f\left(\frac{x_1^{\star}+x_2^{\star}}{2}\right)<f\left(x_1^{\star}\right)$ which is impossible.
Example 2 (Least squares). For the quadratic loss function $f(x)=\frac{1}{2}|A x-y|^2$, strict convexity is equivalent to $\operatorname{ker}(A)={0}$. Indeed, we see later that its second derivative is $\partial^2 f(x)=A^{\top} A$ and that strict convexity is implied by the eigenvalues of $A^{\top} A$ being strictly positive. The eigenvalues of $A^{\top} A$ being positive, it is equivalent to $\operatorname{ker}\left(A^{\top} A\right)={0}$ (no vanishing eigenvalue), and $A^{\top} A z=0$ implies $\left\langle A^{\top} A z, z\right\rangle=|A z|^2=0$ i.e. $z \in \operatorname{ker}(A)$

数学代写|机器学习中的优化理论代写Optimization for Machine Learning代考|Convex Sets

A set $\Omega \subset \mathbb{R}^p$ is said to be convex if for any $(x, y) \in \Omega^2,(1-t) x+t y \in \Omega$ for $t \in[0,1]$. The connexion between convex function and convex sets is that a function $f$ is convex if and only if its epigraph $\operatorname{epi}(f) \stackrel{\text { def. }}{=}\left{(x, t) \in \mathbb{R}^{p+1} ; t \geqslant f(x)\right}$ is a convex set.
Remark 1 (Convexity of the set of minimizers). In general, minimizers $x^{\star}$ might be non-unique, as shown on Figure 3. When $f$ is convex, the set argmin $(f)$ of minimizers is itself a convex set. Indeed, if $x_1^{\star}$ and $x_2^{\star}$ are minimizers, so that in particular $f\left(x_1^{\star}\right)=f\left(x_2^{\star}\right)=\min (f)$, then $f\left((1-t) x_1^{\star}+t x_2^{\star}\right) \leqslant(1-t) f\left(x_1^{\star}\right)+t f\left(x_2^{\star}\right)=$ $f\left(x_1^{\star}\right)=\min (f)$, so that $(1-t) x_1^{\star}+t x_2^{\star}$ is itself a minimizer. Figure 5 shows convex and non-convex sets.

数学代写|机器学习中的优化理论代写Optimization for Machine Learning代 考|Convexity

$$\forall t \in[0,1], \quad f((1-t) x+t y) \leqslant(1-t) f(x)+t f(y)$$

$$\forall t \in] 0,1[, \quad f((1-t) x+t y)<(1-t) f(x)+t f(y) .$$

数学代写|机器学习中的优化理论代写Optimization for Machine Learning代写|Convex Sets

$f\left((1-t) x_1^{\star}+t x_2^{\star}\right) \leqslant(1-t) f\left(x_1^{\star}\right)+t f\left(x_2^{\star}\right)=f\left(x_1^{\star}\right)=\min (f)$ ，以便 $(1-t) x_1^{\star}+t x_2^{\star}$ 本身就是一个最小化器。图 5

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。

Posted on Categories:Optimization Theory, 优化理论, 优化理论代写, 数学代写, 机器学习代写, 机器学习代考

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No general prescriptions for selecting appropriate learning rate; typically no fixed learning rate appropriate for entire learning period

“Bold driver” heuristic: monitor error after each epoch (sweep through entire training set)

1. If error decreases, increase learning rate: $\epsilon=\epsilon * \rho$
2. If error increases, decrease rate, reset parameters:
$$\epsilon=\epsilon * \sigma ; \quad \mathbf{w}^t=\mathbf{w}^{t-1}$$

Sensible choices for parameters: $\rho=1.1, \quad \sigma=0.5$

Momentum

If the error surface is a long and narrow valley, gradient descent goes quickly down the valley walls, but very slowly along the valley floor

We can alleviate this problem by updating parameters using a combination of the previous update and the gradient update:
$$\Delta w_j^t=\beta \Delta w^{t-1}+(1-\beta)\left(-\epsilon \partial E / \partial w_j\left(\mathbf{w}^t\right)\right)$$

Usually $\beta$ is set quite high, about $0.95$.

This is like giving momentum to the weights

数学代写|机器学习中的优化理论代写Optimization for Machine Learning代考|Mini-Batch and Online Optimization

When the dataset is large, computing the exact gradient is expensive

This seems wasteful since the only thing we use the gradient for is to compute a small change in the weights, then throw this out and recompute the gradient all over again

An approximate gradient is useful as long as it points in roughly the same direction as the true gradient

One easy way to do this is to divide the dataset into small batches of examples, compute the gradient using a single batch, make an update, then move to the next batch of examples: mini-batch optimization

In the limit, if each batch contains just one example, then this is the ‘online’ learning, or stochastic gradient descent mentioned in Lecture 2.

These methods are much faster than exact gradient descent, and are very effective when combined with momentum, but care must be taken to ensure convergence

Rather than take a fixed step in the direction of the negative gradient or the momentum-smoothed negative gradient, it is possible to do a search along that direction to find the minimum of the function

Usually the search is a bisection, which bounds the nearest local minimum along the line between any two points such that there is a third point $\mathbf{w}_3$ with $E\left(\mathbf{w}_3\right)<E\left(\mathbf{w}_1\right)$ and $E\left(\mathbf{w}_3\right)<E\left(\mathbf{w}_2\right)$

数学代写机器学习中的优化理论代写Optimization for Machine Learning代 考|Adaptive Stepsize

“Bold driver”启发式: 在每个 epoch 之后监控错误（扫描整个训线集）

$$\epsilon=\epsilon * \sigma ; \quad \mathbf{w}^t=\mathbf{w}^{t-1}$$

$$\Delta w_j^t=\beta \Delta w^{t-1}+(1-\beta)\left(-\epsilon \partial E / \partial w_j\left(\mathbf{w}^t\right)\right)$$

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。