### 数学代写|优化算法代写optimization algorithms代考|Improvement of the Lower Estimate of the Accuracy

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## 数学代写|优化算法代写optimization algorithms代考|Approximate Solving Problem by the Choice

Let $\widetilde{I}$ be any class of informational operators [285]. Assume that the class creates the informational operators of one type with different sets of functionals. For example, if a set of values of function is used, then their number or set of nodes can change their number or even the value of function is computed within constant $N$ (or both). Informational operators of different types (the value of the function and its derivatives, the coefficient of the factorize by certain basis, etc.) create different classes. It is possible to introduce the characteristics:
\begin{aligned} &\rho(\Pi, A, \widetilde{I})=\inf {I{N}(f) \in I} \rho\left(\Pi, A, I_{N}(P)\right)\left(\rho\left(\Pi, A, I_{N}(P)\right) \equiv \rho(\Pi, A)\right) \ &\rho(\Pi, \Lambda, \widetilde{I})=\inf _{A \in \Lambda} \rho(\Pi, A, \widetilde{I}) \end{aligned}
where $\rho=(\Pi, A, \tilde{I})$ is a lower boundary of the error of the algorithm $A \in \Lambda$ in the problem class $\Pi$ using information from class $\tilde{I}$, and $\rho(\Pi, \Lambda, \tilde{I})$ is a lower bound of the error of algorithms in the computing model $(\Pi, \Lambda, \widehat{I})$.

Information $I_{N}^{0}(P) \in \widetilde{I}$, for which the condition $\rho\left(\Pi, A, I_{N}^{0}(P)\right)=\rho(\Pi, A, \widetilde{I})$ is performed, is called an optimal in classes $\Pi, \tilde{I}$ by using the algorithm $A \in \Lambda$. If $\rho\left(\Pi, A^{0}, I_{N}^{0}(P)\right)=\rho(\Pi, \Lambda, \widetilde{I})$, then the algorithm $A^{0} \in \Lambda$ and the information $I_{N}^{0}(P) \in \widetilde{I}$ are called optimal in this computational model $(\Pi, \Lambda, \widetilde{I})$.

Likewise, it is possible to introduce the definition of complexity for the problem $P$ and the problem of class $\Pi$ and their characteristics:

• $T(\Pi, A, \widetilde{I}, \varepsilon)=\inf {I{N}(P) \in \widetilde{I}} T\left(\Pi, A, I_{N}(P), \varepsilon\right)$ is $\varepsilon$-complexity of the algorithm $A \in \Lambda(\varepsilon)$ in the problem of class $\Pi$ within the use of information $\tilde{I}$.
• $T(\Pi, \Lambda(\varepsilon), \tilde{I})=\inf _{A \in \Lambda(\varepsilon)} T(\Pi, A, \tilde{I}, \varepsilon)$ is $\varepsilon$-complexity of the problem in this computation model $(\Pi, \Lambda(\varepsilon), \widetilde{I})$.
• $T(P, A, \widetilde{I}, \varepsilon)$ is the $\varepsilon$-complexity of the algorithm $A \in \Lambda(\varepsilon)$ when the problem $P \in \Pi$ is solved using information $\tilde{I}$.
• $T(P, \Lambda(\varepsilon), \widetilde{I})$ is the $\varepsilon$-complexity of the problem $P$ by using the algorithms $\Lambda(\varepsilon)$ and information $\tilde{I}$, as well as the definition of complexity optimal algorithm and optimal information.

It is possible to introduce an optimization of nodes in numerical integrating as an example of such optimization: by optimization with accuracy for a fixed $N$ by computing $\varepsilon$-solution with $N=O\left(\varepsilon^{-1 / q}\right), q$ is the index of the smoothness of the subintegral function.

This case is about the optimization of choosing the functionals within the constrain of the same type of informational operator that is a set of values of the subintegral function.

Examples of the value optimization of the characteristics $E(\rho(\cdot))$ and $T$ moving to another class of informational operators are contained in [298].

Approximate Information There is known information (approximated) $I_{N \sigma}(P)$ instead of information (exact) $I_{N}(P)$ where $\sigma \geq 0$ characterizes the deviation of the approximate information from the exact one. It is possible to consider the characteristics for the approximate information $I_{N \sigma}(P)$ that are similar to those that were given above for $I_{N}(P)$ assuming that information $I_{N \sigma}(P)$ can be adjusted considering $I_{0}$-information about the problem of the class $\Pi$. Thus, the central algorithm [270] in this case decreases the effect of error of the information $I_{N o}(P)$ on the approximate solution. Examples of constructing these algorithms are given in $[33,106]$.

## 数学代写|优化算法代写optimization algorithms代考|Basic Approaches to Constructing the Accuracy

Consider the problem of the computation of the integral that looks
\begin{aligned} &I_{1}(\omega)=\int_{a}^{b} f(x) e^{-i \omega x} d x \ &I_{2}(\omega)=\int_{a}^{b} f(x) \sin \omega x d x \ &I_{3}(\omega)=\int_{a}^{b} f(x) \cos \omega x d x \end{aligned}
assuming that $f(x) \in F(F)$ is a certain class of functions, and $\omega$ is a certain real number $(\omega \mid \geq 2 \pi(b-a))$.

Let the information about $f(x)$ be given by $N$ values at nodes $\left{x_{i}\right}_{0}^{N-1}$ from its definition domain: $\left{f_{i}\right}_{0}^{N-1}=\left{f\left(x_{i}\right)\right}_{0}^{N-1}$, $\varepsilon_{i}$ characterizes the accuracy of the problem $f\left(x_{i}\right)=f_{i}:\left|\tilde{f}{i}-f{i}\right| \leq \varepsilon_{i}, i=\overline{0, N-1}$.

We concretize the general definition of the accuracy optimal algorithm that is given in the par. $1.4$ for the problem of the approximate computation $I(\omega)$ (we will understand one of the integrals $(1.20,1.21$, and $1.22)$ under $I(\omega))$ ).

Mark $R=R\left(f, A,\left{x_{i}\right}_{0}^{N-1},\left{\varepsilon_{i}\right}_{0}^{N-1}, \omega\right)$ as the result of the approximate computation $I(\omega)$ with quadrature formula $A$.
Introduce the characteristics
\begin{aligned} &V\left(f, A,\left{x_{i}\right}_{0}^{N-1},\left{\varepsilon_{i}\right}_{0}^{N-1}, \omega\right)=\rho(I(\omega), R) \ &V\left(F, A,\left{\varepsilon_{i}\right}_{0}^{N-1}, \omega\right)=\sup {f \in F} V\left(f, A,\left{x{i}\right}_{0}^{N-1},\left{\varepsilon_{i}\right}_{0}^{N-1}, \omega\right) \ &V=V\left(F,\left{\varepsilon_{i}\right}_{0}^{N-1}, \omega\right)=\inf {A} V\left(F, A,\left{\varepsilon{i}\right}_{0}^{N-1}, \omega\right) \ &V(F, \omega)=V(F, 0, \omega) \end{aligned}

## 数学代写|优化算法代写optimization algorithms代考|The function f(x)

Definition 1.1 The function $f^{\pm}(x)$ is called majorizing (minorant) class of functions $F_{N}$ that are defined in some domain $D$ if:

1. $f^{+}(x) \geq f(x)\left(f^{-}(x) \leq f(x)\right)$ for all $m$,
2. $f^{+}(x) \in F_{N}\left(f^{-}(x) \in F_{N}\right)$.
The Chebyshev center $\left(y_{1}, \ldots, y_{N}\right)$ and the Chebyshev radius $\rho^{}(\omega)$ of domain of uncertainty of solving the problem $(1.20,1.21$, and $1.22$ ) can be defined as follows [102]: $$\left(y_{1}, \ldots, y_{m}\right),\left(y_{1}, \ldots, y_{m}\right)=F\left(x_{1}, \ldots, x_{n}\right) \ldots$$ The quadrature formula that computes $I^{}(\omega)$ will be called accuracy optimal, and $\rho^{}(\omega)$ is the error of introduction of the value domain of the integral $I(\omega)$ using $I^{}(\omega)$ or the optimal estimate of the error of numerical integration $I(\omega)$ on the class $F_{N}\left(\delta=\rho^{}(\omega)\right)$. The quadrature formula $R(\omega)$ of the computation $I(\omega)$ for which $$\sup {f \in F{N}}|R(\omega)-I(\omega)| \leq \rho^{}(\omega)+\eta, \eta>0 \text { and } \eta=o\left(\rho^{}\right), O\left(\rho^{}\right)$$
$\left(y_{1}, \ldots, y_{N}\right)$ is called asymptotically optimal or accuracy order optimal.
Within given information about the problem, any quadrature formula can’t give an accuracy less than $\rho^{}(\omega)$. For interpolation classes $\left(y_{1}, \ldots, y_{m}\right)=F\left(x_{1}, \ldots, x_{n}\right)$, the Chebyshev radius $\rho^{}(\omega)$ ) coincides with an optimal estimate $V_{1}$.

The use of the limiting function method for the estimate $V$ is based on the following statement [293].

Theorem $1.3$ Let $f(x) \in F$ ( $F$ is a class of limiting functions) on $f(x)$ the information about its value in $N$ nodes of a random grid, and there is at least one quadrature formula $A \in M$ such as that $I^{+}(\omega) \leq I(\omega) \leq \Gamma(\omega)$. Then the next estimate is valid for $V_{1}$ :

$$V_{1} \geq \sup {F{N} \in F} \rho^{}(\omega)$$ It follows from the definition of the estimates $V$ and $V_{1}$ : $$V \geq V_{1}$$ In the case of $F \equiv F_{N}$, we have $V=\rho^{}(\omega)$.
Remark 1.1 Similar statements are colligated on n-dimensional case [293, 298], and they are used to construct optimal error estimates and prove some optimal cubature formulae of computation of multidimensional integrals from highoscillating functions of the form
\begin{aligned} I_{1}^{n}(\omega) &=\underbrace{\int_{0}^{1} \ldots \int_{0}^{1}}{n} f\left(x{1}, \ldots, x_{n}\right) \sin \omega x_{1} \cdot \ldots \cdot \sin \omega x_{n} d x_{1} \ldots d x_{n}, \ I_{2}^{n}(\omega) &=\underbrace{\int_{0}^{1} \ldots \int_{0}^{1}}{n} f\left(x{1}, \ldots, x_{n}\right) \cos \omega x_{1} \ldots \ldots \cos \omega x_{n} d x_{1} \ldots d x_{n} \end{aligned}
in the case when $n>1, f(X)$ is a known function, $f(X)=f\left(x_{1}, \ldots, x_{n}\right) \in F(F$ is a certain class of functions $X=\left{x_{1}, \ldots, x_{n}\right}, \omega$ is a certain real number $(|\omega| \geq 2 \pi)$, and information about $f(X)$ is given by $N$ values in node points $\left{X_{i}\right}_{0}^{N-1}$ from its domain of definition: $\left{f_{i}\right}_{0}^{N-1}=\left{f\left(X_{i}\right)\right}_{0}^{N-1}$.

## 数学代写|优化算法代写optimization algorithms代考|Approximate Solving Problem by the Choice

\begin{aligned} &\rho(\Pi, A, \widetilde{I})=\inf {I {N}(f) \in I} \rho\left( \Pi, A, I_{N}(P)\right)\left(\rho\left(\Pi, A, I_{N}(P)\right) \equiv \rho(\Pi, A)\right ) \ &\rho(\Pi, \Lambda, \widetilde{I})=\inf _{A \in \Lambda} \rho(\Pi, A, \widetilde{I}) \end{aligned}

• $T(\Pi, A, \widetilde{I}, \varepsilon)=\inf {I {N}(P) \in \widetilde{I}} T\left(\Pi, A, I_{N}( P), \varrepsilon\right)一世s\伐普西隆−C这米pl和X一世吨是这F吨H和一种lG这r一世吨H米一个 \in \Lambda(\varepsilon)一世n吨H和pr这bl和米这FCl一种ss\π在一世吨H一世n吨H和在s和这F一世nF这r米一种吨一世这n\波浪号{I}$。
• 吨(圆周率,Λ(e),一世~)=信息一种∈Λ(e)吨(圆周率,一种,一世~,e)是e-此计算模型中问题的复杂性(圆周率,Λ(e),一世~).
• 吨(磷,一种,一世~,e)是个e- 算法的复杂性一种∈Λ(e)当问题磷∈圆周率使用信息解决一世~.
• 吨(磷,Λ(e),一世~)是个e- 问题的复杂性磷通过使用算法Λ(e)和信息一世~，以及复杂度最优算法和最优信息的定义。

## 数学代写|优化算法代写optimization algorithms代考|Basic Approaches to Constructing the Accuracy

\begin{aligned} &V\left(f, A,\left{x_{i}\right}_{0}^{N-1},\left{\varepsilon_{i}\right} _{0}^{N-1}, \omega\right)=\rho(I(\omega), R) \ &V\left(F, A,\left{\varepsilon_{i}\right}_{ 0}^{N-1}, \omega\right)=\sup {f \in F} V\left(f, A,\left{x {i}\right}_{0}^{N-1 },\left{\varepsilon_{i}\right}_{0}^{N-1}, \omega\right) \ &V=V\left(F,\left{\varepsilon_{i}\right}_ {0}^{N-1}, \omega\right)=\inf {A} V\left(F, A,\left{\varepsilon {i}\right}_{0}^{N-1} , \omega\right) \ &V(F, \omega)=V(F, 0, \omega) \end{aligned}

## 数学代写|优化算法代写optimization algorithms代考|The function f(x)

1. F+(X)≥F(X)(F−(X)≤F(X))对全部米,
2. F+(X)∈Fñ(F−(X)∈Fñ).
切比雪夫中心(是1,…,是ñ)和切比雪夫半径ρ(ω)解决问题的不确定性域(1.20,1.21， 和1.22) 可以定义如下[102]：(是1,…,是米),(是1,…,是米)=F(X1,…,Xn)…计算的求积公式一世(ω)将被称为精度最优，并且ρ(ω)是积分值域引入的误差一世(ω)使用一世(ω)或数值积分误差的最优估计一世(ω)在课堂上Fñ(d=ρ(ω)). 求积公式R(ω)计算的一世(ω)为此支持F∈Fñ|R(ω)−一世(ω)|≤ρ(ω)+这,这>0 和 这=这(ρ),这(ρ)
(是1,…,是ñ)称为渐近最优或精度阶最优。
在有关问题的给定信息内，任何求积公式的准确度都不能低于ρ(ω). 对于插值类(是1,…,是米)=F(X1,…,Xn), 切比雪夫半径ρ(ω)) 与最优估计一致在1.

\begin{aligned}形式的高振荡函数计算多维积分的一些最优容积公式 I_{1}^{n}(\omega) &=\underbrace{\int_{0}^{1} \ldots \int_{0}^{1}} {n} f\left(x {1}, \ldots, x_{n}\right) \sin \omega x_{1} \cdot \ldots \cdot \sin \omega x_{n} d x_{1} \ldots d x_{n}, \ I_{2} ^{n}(\omega) &=\underbrace{\int_{0}^{1} \ldots \int_{0}^{1}} {n} f\left(x {1}, \ldots, x_ {n}\right) \cos \omega x_{1} \ldots \ldots \cos \omega x_{n} d x_{1} \ldots d x_{n} \end{aligned}

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## MATLAB代写

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