### 分类： 自适应算法代写

• 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

## 计算机代写|自适应算法代写Adaptive algorithm代考|Control of Time Steps

Step size control is an important and necessary means to increase the efficiency of a time integration method. In fact, a constant time step is often not adequate to reach a given accuracy, since it would require a huge number of small steps. The discretization sequence, first time then space, permits us to consider naturally the spatial discretization as a perturbation of the time integration process. We assume for the moment that the spatial perturbation is always kept below a certain level and does not affect mainly the step size selection procedure. Thus, the generated time step sequence $\left{\tau_j\right}_{j=0,1, \ldots}$ is supposed to be nearly the same as in the case of no perturbation.
The local truncation error $\delta_\tau(t)$ is defined as the error after a single step of length $\tau$ starting with the exact local solution $u(t)$. Using the short notation $u_{n+1}=\Phi\left(u_n\right)$ for the Rosenbrock method (II.18)-(II.19), we have at $t=t_n$ with time step $\tau_n=t_{n+1}-t_n$
$$\delta_{\tau_n}\left(t_n\right)=\Phi\left(u\left(t_n\right)\right)-u\left(t_n+\tau_n\right) .$$
The asymptotic behaviour of the local error for an order $p$ method can be described by
$$\delta_{\tau_n}\left(t_n\right)=\phi\left(t_n\right) \tau_n^{p+1}+o\left(\tau_n^{p+2}\right), \quad \tau_n \rightarrow 0 .$$
Assuming appropriate temporal regularity of the mapping $F(t, u(t))$ in (II.4), the coefficient vector $\phi(t)$ is a smooth function of $t$.

The global error $e_{n+1}:=u_{n+1}-u\left(t_{n+1}\right)$ at the forward time level $t=t_{n+1}$ can be seen to satisfy
$$e_{n+1}=\Phi\left(e_n+u\left(t_n\right)\right)-\Phi\left(u\left(t_n\right)\right)+\delta_{\tau_n}\left(t_n\right) .$$
Consequently, this error is the sum of the local error and the difference of the actual Rosenbrock step $\Phi\left(u_n\right)$ and the hypothetical step $\Phi\left(u\left(t_n\right)\right)$ taken from the exact solution $u\left(t_n\right)$. To measure the errors we introduce an appropriate norm III $\cdot$ II which is often a mixed absolute-relative norm in practical computations for reason of robustness (see Chapter V.\$4). It is now a fundamental property of a stable one-step integration method that $$||\left|e_{n+1}\right|\left|\leq I|| e_n\left|\left|+||\left|\delta_{\tau_n}\left(t_n\right)\right|\right| \leq\right||| e_0||\left|+\sum_{j=0}^n\right||| \delta_{\tau_j}\left(t_j\right) \mid\right|,$$ showing that the global error consists of propagated and accumulated local truncation errors. Estimating and controlling the latter within an automatic step size selection procedure ensure that the step sizes$\tau_j$are chosen sufficiently small to have the desired precision, but they have to be also sufficiently large to avoid unnecessary computational work. ## 计算机代写|自适应算法代写Adaptive algorithm代考|Estimation of Spatial Errors Since the pioneering works of BABUS̆KA and RHEINBOLDT$[11,12]$quite a lot of a posteriori error estimates have been developed for mastering finite element calculations. Now they are widely used in the mesh-controlled solution of partial differential equations. A good survey is given in [10] and more recently in [161], where also a substantial bibliography on the subject can be found. We deal with a posteriori error estimators based on the use of hierarchical basis functions. Such estimators have been accepted to provide efficient and reliable assessment of spatial errors and to form a basis of adaptive local mesh refinement$[173,54,19,36]$. Our aim is the extension of the hierarchical bases technique to time-dependent nonlinear problems within the setting of linearly implicit time integrators. The crucial point herein is the construction of robust estimators for the fully discretized equations (III.16) which are singularly perturbed by the presence of the (variable) time step. A robust estimator has to yield upper and lower bounds on the error uniformly in the time step$\tau \geq 0$. In general, a straightforward application of standard adaptive finite element solvers runs into troubles in the limit case$\tau \rightarrow 0$. For selfadjoint scalar problems robust estimators were constructed in$[33,162]$. Our analysis takes into account the abstract framework of [19]. Analogously to Chapter IV.§1 we are interested in analyzing the local error behaviour. The spatial discretization is considered as a perturbation of the time integration process. Starting with the Rosenbrock solution$u_n$at$t=t_n$, we will estimate the error$u_{n+1}-u_{h, n+1}$caused by the spatial approximation of all stage values$K_{h, n i}^{\prime} \in \mathcal{V}h$. Although system (III.16) is linear the nonlinearity on the right-hand side gives rise to a nonlinear spatial error transport. Let us consider a hierarchical decomposition $$\overline{\mathcal{V}}_h=\mathcal{V}_h \oplus \mathcal{Z}_h,$$ where$\mathcal{Z}_h$is the subspace that corresponds to the span of all additional basis functions needed to enrich the space$\mathcal{V}_h$to higher order. Consequently, any function$\bar{v} \in \overline{\mathcal{V}}_h$has the unique decomposition$\bar{v}=v+z$, where$v \in \mathcal{V}_h$and$z \in \mathcal{Z}_h$. The hierarchical basis error estimator tries to bound the spatial error by evaluating its components in the space$\mathcal{Z}_h$, i.e., $$C_1||\left|E{h, n+1}\right|\left||\leq|\left|u_{n+1}-u_{h, n+1}\right|\right| \leq C_2||\left|E_{h, n+1} |\right|,$$ where$E_{h, n+1} \in \mathcal{Z}_h$is the computed a posteriori error estimate. # 自适应算法代考 ## 计算机代写|自适应算法代写Adaptive algorithm代考|Control of Time Steps 步长控制是提高时间积分方法效率的重要且必要的手段。事实上，恒定的时间步长通常不足以达到给 定的精度，因为它需要大量的小步长。离散化序列，先是时间，然后是空间，使我们可以自然地将空 间离散化视为时间积分过程的扰动。我们暂时假设空间扰动始终保持在一定水平以下，并且主要不影 相同。 局部截断错误$\delta_\tau(t)$定义为单步长度后的误差$\tau$从精确的本地解决方案开始$u(t)$. 使用短符号$u_{n+1}=\Phi\left(u_n\right)$对于 Rosenbrock 方法(II.18)-(II.19)，我们有$t=t_n$随看时间步长$\tau_n=t_{n+1}-t_n$$$\delta_{\tau_n}\left(t_n\right)=\Phi\left(u\left(t_n\right)\right)-u\left(t_n+\tau_n\right) .$$ 订单局部误差的渐近行为$p$方法可以描述为 $$\delta_{\tau_n}\left(t_n\right)=\phi\left(t_n\right) \tau_n^{p+1}+o\left(\tau_n^{p+2}\right), \quad \tau_n \rightarrow 0 .$$ 假设映射具有适当的时间规律性$F(t, u(t))$在 (II.4) 中，系数向量$\phi(t)$是一个光滑的函数$t$. 全局错误$e_{n+1}:=u_{n+1}-u\left(t_{n+1}\right)$在前向时间水平$t=t_{n+1}$可以看出满足 $$e_{n+1}=\Phi\left(e_n+u\left(t_n\right)\right)-\Phi\left(u\left(t_n\right)\right)+\delta_{\tau_n}\left(t_n\right) .$$ 因此，该误差是局部误差与实际 Rosenbrock 步长之差的总和$\Phi\left(u_n\right)$和假设的步㡜$\Phi\left(u\left(t_n\right)\right)$取自精 确解$u\left(t_n\right)$. 为了衡量错误，我们引入了一个适当的标准 III·|I 出于稳健性的原因，在实际计算中通常是 混合的绝对-相对范数（参见第$\mathrm{V}$章$\$4$ ）。现在，稳定的一步积分方法的一个基本属性是
$$\left|\left|e_{n+1}\right|\left|\leq I\left|e_n\right|+\right||| \delta_{\tau_n}\left(t_n\right)\right| \leq|| e_0\left|\left|+\sum_{j=0}^n\right|\right| \delta_{\tau_j}\left(t_j\right)||,$$

## 计算机代写|自适应算法代写Adaptive algorithm代考|Rosenbrock Methods and Basic Results

We are interested in approximating the nonlinear Cauchy problem (II.4) in time. For this, we use linearly implicit one-step methods proposed by RoSENBROCK [134] to achieve higher order methods for stiff problems by working the Jacobian matrix into the integration formula. Applied to the initial-value problem (II.4) with step size $\tau>0$ a so-called s-stage Rosenbrock method has the recursive form
\begin{aligned} u_{n+1}= & u_n+\tau \sum_{i=1}^s b_i K_{n i}^{\prime} \ K_{n i}^{\prime}= & F\left(t_n+\alpha_i \tau, K_{n i}\right)-\tau A\left(t_n, u_n\right) \sum_{j=1}^i \gamma_{i j} K_{n j}^{\prime} \ & +\tau \gamma_i F_t\left(t_n, u_n\right) \end{aligned}
with the intermediate values
$$K_{n i}=u_n+\tau \sum_{j=1}^{i-1} \alpha_{i j} K_{n j}^{\prime}, \quad 1 \leq i \leq s$$
and
$$\alpha_i=\sum_{j=1}^{i-1} \alpha_{i j}, \quad \gamma_i=\sum_{j=1}^i \gamma_{i j} .$$

Here, $u_n$ denotes an approximation of $u\left(t_n\right)$ at $t_n=n \tau$. The coefficients $b_i, \alpha_{i j}$, and $\gamma_{i j}$ are suitable chosen to obtain a desired order of consistency and stability for stiff problems (see also Appendix B§1). We always assume that $\gamma_{i i}=\gamma>0$, and $\alpha_i \in[0,1]$ for all $i$.
It was the fundamental idea of ROSENBROCK that only linear systems with the operators $I+\tau \gamma_{i i} A\left(t_n, u_n\right)$ have to be solved successively one after the other. An iterative Newton method as known from implicit Runge-Kutta methods is no longer required.
For convenience, we set $\alpha_{i j}=0$ for $j \geq i, \gamma_{i j}=0$ for $j>i$, and use the notation
$$\begin{gathered} \beta_{i j}=\alpha_{i j}+\gamma_{i j}, \quad c_i=\alpha_i+\gamma_i, \quad \mathcal{B}=\left(\beta_{i j}\right)_{i, j=1}^s, \ b=\left(b_1, \ldots, b_s\right)^T, \quad \alpha^k=\left(\alpha_1^k, \ldots, \alpha_s^k\right)^T, \quad \mathbf{1}=(1, \ldots, 1)^T \in \mathbb{R}^s . \end{gathered}$$
The Rosenbrock method applied to ordinary differential equations (ODEs) with sufficiently differentiable right-hand side has (classical) order $p$ if the global error satisfies
$$\epsilon_n:=u_n-u\left(t_n\right)=O\left(\tau^p\right) \quad \text { as } \tau \rightarrow 0,$$
uniformly on bounded time intervals. The method is called $\mathcal{A}(\Theta)$-stable, if its stability function
$$R(z)=1+z b^T(I-z \mathcal{B})^{-1} \mathbf{1}$$
is bounded in modulus by 1 for $|\arg (z)| \geq \pi-\Theta$. If additionally the absolute limit of the stability function at infinity $|R(\infty)|$ is strictly smaller than 1, we call the method strongly $\mathcal{A}(\Theta)$-stable.

# 自适应算法代考

$$\partial_t u(x, t)=F(x, t, u(x, t)) \quad \text { in } \Omega \times(0, T], B(x, t, u(x, t)) u(x, t) \quad=g(x, t, u(x, t))$$

$$\mathcal{V} \hookrightarrow^{d s} \mathcal{H} \hookrightarrow^{d s} \mathcal{V}^{\prime}$$

u_{n+1}=u_n+\tau \sum_{i=1}^s b_i K_{n i}^{\prime} K_{n i}^{\prime}=\quad F\left(t_n+\alpha_i \tau, K_{n i}\right)-\tau A\left(t_n, u_n\right) \sum_{j=1}^i \gamma_{i j} K_{n j}^{\prime}+\tau \gamma_i F_t\left(t_n\right.
$$与中间值$$
K_{n i}=u_n+\tau \sum_{j=1}^{i-1} \alpha_{i j} K_{n j}^{\prime}, \quad 1 \leq i \leq s

\alpha_i=\sum_{j=1}^{i-1} \alpha_{i j}, \quad \gamma_i=\sum_{j=1}^i \gamma_{i j}
$$这里， u_n 表示近似值 u\left(t_n\right) 在 t_n=n \tau. 系数 b_i, \alpha_{i j} ，和 \gamma_{i j} 适合选择以获得所需的刚性问题的一致性 和稳定性顺序 (另请参见附录 B§1）。我们总是假设 \gamma_{i i}=\gamma>0 ，和 \alpha_i \in[0,1] 对全部 i. ROSENBROCK 的基本思想是只有线性系统与操作符 I+\tau \gamma_{i i} A\left(t_n, u_n\right) 必须一个接一个地解决。不 再需要从隐式 Runge-Kutta 方法中得知的迭代牛顿法。 为了方便，我们设 \alpha_{i j}=0 为了 j \geq i, \gamma_{i j}=0 为了 j>i ，并使用符号$$
\beta_{i j}=\alpha_{i j}+\gamma_{i j}, \quad c_i=\alpha_i+\gamma_i, \quad \mathcal{B}=\left(\beta_{i j}\right)_{i, j=1}^s, b=\left(b_1, \ldots, b_s\right)^T, \quad \alpha^k=\left(\alpha_1^k, \ldots, \alpha_s^k\right)^T
$$应用于右手边充分可微的常微分方程 (ODE) 的 Rosenbrock 方法具有 (经典) 阶 p 如果全局误差满足$$
\epsilon_n:=u_n-u\left(t_n\right)=O\left(\tau^p\right) \quad \text { as } \tau \rightarrow 0
$$在有限的时间间隔内均匀分布。该方法称为 \mathcal{A}(\Theta)-稳定，如果它的稳定性函数$$
R(z)=1+z b^T(I-z \mathcal{B})^{-1} \mathbf{1}


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## MATLAB代写

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