### 数学代写|数值分析代写numerical analysis代考|MATH2722

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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

## 数学代写|数值分析代写numerical analysis代考|Solving Fredholm Integral Equation via Tight Framelets

Many methods have been presented to find exact and approximate solutions of different integral equations. In this work, we introduce a new method for solving the above-mentioned class of equations. We use quasi-affine tight framelets systems generated by the UEP and OEP for solving some types of integral equations. Consider the second-kind linear Fredholm integral equation of the form:
$$u(x)=f(x)+\lambda \int_a^b \mathcal{K}(x, t) u(t) d t,-\infty<a \leq x \leq b<\infty,$$
where $\lambda$ is a real number, $f$ and $\mathcal{K}$ are given functions and $u$ is an unknown function to be determined. $\mathcal{K}$ is called the kernel of the integral Equation (10). A function $u(x)$ defined over $[a, b]$ can be expressed by quasi-affine tight framelets as Equation (5). To find an approximate solution $u_n$ of (10), we will truncate the quasi-affine framelet representation of $u$ as in Equation (6). Then,
$$u(x) \approx u_n(x)=\sum_{\ell=1}^r \sum_{j<n} \sum_{k \in \mathbb{Z}} c_{j, k}^{\ell} \psi_{j, k}^{\ell}(x),$$
where
$$c_{j, k}^{\ell}=\int_{\mathbb{R}} u_n(x) \psi_{j, k}^{\ell}(x) d x .$$
Substituting (11) into (10) yields
$$\sum_{\ell=1}^r \sum_{j<n} \sum_{k \in \mathbb{Z}} c_{j, k}^{\ell} \psi_{j, k}^{\ell}(x)=f(x)+\lambda \sum_{\ell=1}^r \sum_{j<n} \sum_{k \in \mathbb{Z}} c_{j, k}^{\ell} \int_a^b \mathcal{K}(x, t) \psi_{j, k}^{\ell}(t) d t$$
Multiply Equation (12) by $\sum_{s=1}^r \psi_{p, q}^s(x)$ and integrate both sides from $a$ to $b$. This can be a generalization of Galerkin method used in Reference [29,30]. Then, with a few algebra, Equation (12) can be simplified to a system of linear equations with the unknown coefficients $c_{j, k}^{\ell}$ (to be determined) given by
$$\sum_{s, \ell=1}^r \sum_{j<n} \sum_{k \in \mathbb{Z}} c_{j, k}^{\ell} m_{j, k, p, q}^{\ell, s}=g_{p, q \prime} \quad p, q \in \mathbb{Z},$$
where
$$m_{j, k, p, q}^{\ell, s}=\int_a^b \psi_{j, k}^{\ell}(x) \psi_{p, q}^s(x) d x-\lambda \int_a^b \int_a^b \mathcal{K}(x, t) \psi_{j, k}^{\ell}(t) \psi_{p, q}^s(x) d x d t, \quad p, q \in \mathbb{Z}$$
and
$$g_{p, q}=\sum_{s=1}^r \int_a^b f(x) \psi_{p, q}^s(x) d x, \quad p, q \in \mathbb{Z} .$$

## 数学代写|数值分析代写numerical analysis代考|Error Analysis

In this section, we get an upper bound for the error of our method. Let $\phi$ be as in Equation (1) and $W_2^m(\mathbb{R})$ is the Sobolev space consists of all square integrable functions $f$ such that $\left{f^{(k)}\right}_{k=0}^m \in L^2(\mathbb{R})$. Then, $X^0(\Psi)$ provides approximation order $m$, if
$$\left|f-S_n f\right|_2 \leq C 2^{-n m} \mid f^{(m)} |_2, \quad \forall f \in W_2^m(\mathbb{R}), n \in \mathbb{N} .$$
The approximation order of the truncated function $S_n$ was studied in References [20,31]. It is well known in the literature that the vanishing moments of the framelets can be determined by its low and high pass filters $\hat{h}_{\ell} \ell=0, \ldots, r$. Also, if the quasi-affine framelet system has vanishing moments of order say $m_1$ and the low pass filter of the system satisfy the following,
$$1-\left|\hat{h}_0(\xi)\right|^2=\mathcal{O}\left(|\cdot|^{2 m}\right),$$
at the origin, then the approximation order of $X^0(\Psi)$ is equal to $\min \left{m_1, m\right}$. Therefore, as the OEP increases the vanishing moments of the quasi-affine framelet system, the accuracy order of the truncated framelet representation, will increase as well.

As mentioned earlier, integral equations describe many different events in applications such as image processing and data reconstructions, for which the regularity of the function $f$ is low and does not meet the required order of smoothness. This makes the determination of the approximation order difficult from the functional analysis side. Instead, it is assumed that the solution function to satisfy a decay condition with a wavelet characterization of Besov space $B_{2,2}^s$. We refer the reader to Reference [32] for more details. Hence, we impose the following decay condition such that
$$N_f=\sum_{\ell=1}^r \sum_{j \geq 0} \sum_{k \in \mathbb{Z}} 2^{s j}\left|\left\langle f, \psi_{j, k}^{\ell}\right\rangle\right|<\infty,$$
where $s \geq-1$.

# 数值分析代考

## 数学代写|数值分析代写numerical analysis代考|Solving Fredholm Integral Equation via Tight Framelets

$$u(x)=f(x)+\lambda \int_a^b \mathcal{K}(x, t) u(t) d t,-\infty<a \leq x \leq b<\infty,$$

$$u(x) \approx u_n(x)=\sum_{\ell=1}^r \sum_{j<n} \sum_{k \in \mathbb{Z}} c_{j, k}^{\ell} \psi_{j, k}^{\ell}(x),$$

$$c_{j, k}^{\ell}=\int_{\mathbb{R}} u_n(x) \psi_{j, k}^{\ell}(x) d x .$$

$$\sum_{\ell=1}^r \sum_{j<n} \sum_{k \in \mathbb{Z}} c_{j, k}^{\ell} \psi_{j, k}^{\ell}(x)=f(x)+\lambda \sum_{\ell=1}^r \sum_{j<n} \sum_{k \in \mathbb{Z}} c_{j, k}^{\ell} \int_a^b \mathcal{K}(x, t) \psi_{j, k}^{\ell}(t) d t$$

$$\sum_{s, \ell=1}^r \sum_{j<n} \sum_{k \in \mathbb{Z}} c_{j, k}^{\ell} m_{j, k, p, q}^{\ell, s}=g_{p, q^{\prime}} \quad p, q \in \mathbb{Z},$$

$$m_{j, k, p, q}^{\ell, s}=\int_a^b \psi_{j, k}^{\ell}(x) \psi_{p, q}^s(x) d x-\lambda \int_a^b \int_a^b \mathcal{K}(x, t) \psi_{j, k}^{\ell}(t) \psi_{p, q}^s(x) d x d t, \quad p, q \in \mathbb{Z}$$
$$g_{p, q}=\sum_{s=1}^r \int_a^b f(x) \psi_{p, q}^s(x) d x, \quad p, q \in \mathbb{Z}$$

## 数学代写|数值分析代写numerical analysis代考|Error Analysis

$$\left|f-S_n f\right|2 \leq C 2^{-n m}\left|f^{(m)}\right|_2, \quad \forall f \in W_2^m(\mathbb{R}), n \in \mathbb{N} .$$ 截断函数的逼近阶数 $S_n$ 在参考文献 [20,31] 中进行了研究。在文献中众所周知，小框架的消失时刻可以通 过其低通和高通滤波器来确定 $\hat{h}{\ell} \ell=0, \ldots, r$. 此外，如果准仿射框架系统具有消失的秩序时刻说 $m_1$ 并 且系统的低通滤波器满足以下条件，
$$1-\left|\hat{h}0(\xi)\right|^2=\mathcal{O}\left(|\cdot|^{2 m}\right),$$ 在原点，那么近似阶 $X^0(\Psi)$ 等于 \min \eft{m_1, m\right } } \text { . 因此，随着 OEP 增加准仿射小框架系统的消失 } 矩，截断小框架表示的精度阶数也将增加。 如前所述，积分方程描述了图像处理和数据重建等应用中的许多不同事件，其中函数的正则性 $f$ 低且不满 足所需的平滑度顺序。这使得从泛函分析方面难以确定近似阶数。相反，假设解函数满足具有 Besov 空 间小波特征的衰减条件 $B{2,2}^s$. 我们建议读者参阅参考文献 [32] 了解更多详细信息。因此，我们施加以下衰 减条件，使得
$$N_f=\sum_{\ell=1}^r \sum_{j \geq 0} \sum_{k \in \mathbb{Z}} 2^{s j}\left|\left\langle f, \psi_{j, k}^{\ell}\right\rangle\right|<\infty$$

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## MATLAB代写

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