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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 电子工程代写|并行计算代写Parallel Computing代考|Time Discretisation: Running a Movie

With given positions $p_{n}(t=0), i \in{1,2, \ldots, N}$, we can throw all of our analysis skills onto the Eqs. (3.1)-(3.3), and hope that we can find an expression for $p_{n}(t)$. This is laborious. Bad news: Without further assumption it is impossible to solve this problem.

Let’s assume that we live in a movie universe where the whole universe runs at 60 frames per second. In each frame, we know all objects $p_{n}$ as well as their velocities $v_{n}$. We can compute the forces at this very frame. They push the particles, i.e. they alter their velocities. Due to the (changed) velocity, all objects end up at a different place in the next frame (Fig. 3.1). The longer the time in-between two frames, the more the particles can change their position (the velocity acts longer). To make the impact of our force pushes realistic, the forces are the bigger the bigger the time in-between two frames, too. The more frames per second we use, the more accurate our universe. With the time step size going to zero, we eventually end up with the real thing.

The construction of the movie universe from the real one is called discretisation. We know that time is continuous – at least at our scale – and computers struggle to hold things that are infinitely small and detailed. After all, we have finite memory, finite compute power and finite compute time. So we sample or discretise the time and track the solution in time steps. Let dt be the time step size, i.e. the time in-between two snapshots. ${ }^{3}$ Our computer program simulation the space bodies then reads as follows.

## 电子工程代写|并行计算代写Parallel Computing代考|Numerical Versus Analytical Solutions

Computers are calculators. If we pass them a certain problem like “here are two bodies interacting through gravity”, they yield values as solution: “the bodies end up at position $\mathrm{x}, \mathrm{y}, \mathrm{z}$ “. They yield numerical solutions. This is different to quite a lot of maths we do in school. There, we manipulate formulae and compute expressions like $F(a, b)=\int_{a}^{b} 4 x d x=2 b^{2}-2 a^{2}$. Indeed, many teachers save us till the very last minute (or until we have pocket calculators) from inserting actual numbers for $b$ and $a$.

In programming languages, we often speak of variables. But these variables still contain data at any point of the program run. They are fundamentally different to variables in a mathematical formula which might or might not hold specific values. We conclude: There are two different ways to handle equations: We can try to solve them for arbitrary input, i.e. find expressions like $F(a, b)=2 b^{2}-2 a^{2}$. Once we have such a solution, we can insert different $a$ and $b$ values. The solution is an analytical solution. ${ }^{4}$ On a computer, we typically work numerically. We hand in numbers, and we get answers for these particular numbers. But we do not get any universal solution.

Definition $3.2$ (Analytical versus numerical solution) If we solve an equation via formula rewrites such as integration rules, we obtain an analytical solution over the variables. Analytical solutions describe a generic system behaviour. If we solve it for one particular set of initial values right from the start, we strive for a numerical solution.

Computers yield numerical solutions. This statement is not $100 \%$ correct. There are computer programs which yield symbolic solutions. They are called computer algebra systems. While they are very powerful, they cannot find an analytical solution always and obviously do not yield a result if there is no analytical solution. We walk down the numerics route in this course.

Analogous to this distinction of numerical and analytical solutions, we can also distinguish how we manipulate formulae. If you want to determine the derivative $\partial_{t} y(t)$ of $y(t)=t^{2}$, you know $\partial_{t} y(t)=2 t$ from school. You manipulate the formulae symbolically. In a computer, you can also evaluate $\partial_{t} y(t)$ only for a given input. This often comprises some algorithmic approximations. In this case, you again tackle the expression numerically.

# 并行计算代考

## 有限元方法代写

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## 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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