### 统计代写|随机控制代写Stochastic Control代考|ELEC6410

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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

## 统计代写|随机控制代写Stochastic Control代考|Particle model due to Brownian motion force

The position of particles in water at time $t$, is designated by $(X(t), Y(t))$. Different random locations of the particle are described with the aid of stochastic differential equation. The integration of the movements of the particle in water is done in two steps. A deterministic step consisting of velocity field of water and a random step known as diffusion modelled by the stochastic process A. W. Heemink (1990);
\begin{aligned} d X(t) & \stackrel{\text { It̂̂ }}{=}\left[U+\frac{D}{H} \frac{\partial H}{\partial x}+\frac{\partial D}{\partial x}\right] d t+\sqrt{2 D} d W_{1}(t), X(0)=x_{0} \ d Y(t) \stackrel{\text { Itô }}{=}\left[V+\frac{D}{H} \frac{\partial H}{\partial y}+\frac{\partial D}{\partial y}\right] d t+\sqrt{2 D} d W_{2}(t), Y(0)=y_{0} . \end{aligned}
Here $D$ is the dispersion coefficient in $m^{2} / s ; U(x, y, t), V(x, y, t)$ are the averaged flow velocities $(m / s)$ in respectively $x, y$ directions; $H(x, y, t)$ is the total depth in $m$ at location $(x, y)$, and $d W(t)$ is a Brownian motion with mean $(0,0)^{T}$ and $\mathbb{E}\left[d W_{1}(t) d W_{2}(t)^{\mathrm{T}}\right]=I d t$ where $I$ is a $2 \times 2$ identity matrixP.E. Kloeden et al. (2003). Note that the advective part of the particle model eqns.(25)-(26) is not only containing the averaged water flow velocities but also spatial variations of the diffusion coefficient and the averaged depth. This correction term makes sure that particles are not allowed to be accumulated in regions of low diffusivity as demonstrated by (see e.g., J. R. Hunter et al. (1993); R.W.Barber et al. (2005)). At closed boundaries particle bending is done by halving the time step sizes until the particle no longer crosses closed boundary. As a result there is no loss of mass through such boundaries. The position $(X(t), Y(t))$ process is Markovian and the evolution of its probability density function $(p(x, y, t))$, is described by an advection-diffusion type of the partial differential equation known as the Fokker-Planck equation (see e.g.,A.H. Jazwinski (1970))

## 统计代写|随机控制代写Stochastic Control代考|Boundaries

Numerical schemes such as the Euler scheme often show very poor convergence behaviour G.N. Milstein (1995); P.E. Kloeden et al. (2003). This implies that, in order to get accurate results, small time steps are needed thus requiring much computation. Another problem with the Euler (or any other numerical scheme) is its undesirable behaviour in the vicinity of boundaries; a time step that is too large may result in particles unintentionally crossing boundaries. To tackle this problem two types of boundaries are prescribed. Closed boundaries representing boundaries intrinsic to the domain, and open boundaries which arise from the modeller’s decision to artificially limit the domain at that location. Besides these boundary types, the is of what what happens if, during integration, a particle crosses one of these two borders is also considered as in J.W. Stijnen et al. (2003); W. M. Charles et al. (2009);

• In case an open boundary is crossed by a particle, the particle remains in the sea but is now outside the scope of the model and is therefore removed;
• In case a closed boundary is crossed by a particle during the advective step of integration, the step taken is cancelled and the time step halved until the boundary is no longer crossed. However, because of the halving, say $n$ times, the integration time is reduced to $2^{-n} \Delta t$, leaving a remaining $\left(1-2^{-n}\right) \Delta t$ integration time. This means at least another $2^{n}-1$ steps need to be taken at the new integration step in order to complete the full time-step $\Delta t$. This way, shear along the coastline is modelled;
• If a closed boundary is crossed during the diffusive part of integration, the step size halving procedure described above is maintained with the modification that in addition to the position, the white noise process is also restored to its state prior to the abandoned integration step. Again the process of halving the time step and continuing integration is repeated until no boundaries are crossed and the full $\Delta t$ time step has been integrated.

## 统计代写|随机控制代写Stochastic Control代考|Particle model due to Brownian motion force

$$d X(t) \stackrel{\text { It }}{=}\left[U+\frac{D}{H} \frac{\partial H}{\partial x}+\frac{\partial D}{\partial x}\right] d t+\sqrt{2 D} d W_{1}(t), X(0)=x_{0} d Y(t) \stackrel{\text { It. }}{=}\left[V+\frac{D}{H} \frac{\partial H}{\partial y}+\frac{\partial D}{\partial y}\right] d t$$

## 统计代写|随机控制代写Stochastic Control代考|Boundaries

• 如果粒子穿过开放边界，粒子仍留在海中，但现在不在模型范围内，因此被移除；
• 如果在积分平流步骤期间粒子穿过封闭边界，则取消所采取的步骤并将时间步长减半，直到不再越过边界。但是，由于减半，说n次，积分时间减少到2−nD吨, 剩下一个(1−2−n)D吨整合时间。这至少意味着另一个2n−1需要在新的集成步骤中采取步骤以完成完整的时间步骤D吨. 这样，沿海岸线的切变就被建模了；
• 如果在积分的扩散部分期间越过闭合边界，则保持上述步长减半过程，修改为除了位置之外，白噪声过程也恢复到放弃积分步骤之前的状态。再次重复将时间步长减半并继续积分的过程，直到没有跨越边界并且完全D吨时间步已被整合。

## 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。