### 数学代写|优化算法代写optimization algorithms代考|Analytic Computational Complexity

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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|优化算法代写optimization algorithms代考|Input Information, Algorithms, and Complexity

Consider the idealized computation model: Information $I_{n}(f)$ is given accurately, and the model $c$ is fixed. Here are some characteristics that are related to the lower and upper estimates of the error on the example of the passive pure minimax strategy $[102,253]$ (see Chap. 1 for more details):

• $\rho_{\mu}(F, a)=\sup {f \in F} \rho\left(E{\mu}\left(I_{n}(f)\right), a\right)$ is an error of algorithm $a \in A$ on the class of the problems $F$ using information $I_{n}(f)$ (global error [270]).
• $\rho_{\mu}(F, A)=\inf {a \in A} \rho{\mu}(F, a)$ is a lower boundary of error of algorithms of class $A$ in the class of problems $F$ using information $I_{n}(f)$ (radius of information [270])
If there is an algorithm $a_{0} \in A$ for which $\rho_{\mu}\left(F, a_{0}\right)=\rho_{\mu}(F, A)$, then it is called accuracy optimal in class $F$ using information $I_{n}(f)$.

The narrowing of the $F$ class that is provided by the incompleteness of information (in relation to $f \in F) F_{n}(f)=\left{\varphi: I_{n}(\varphi)=I_{n}(f), \varphi, f \in F\right}$ allows to introduce the characteristics that are equivalent to the mentioned one above: $\rho_{\mu}\left(F_{n}(f), a\right)$ and $\rho_{\mu}\left(F_{n}(f), A\right)$, which are also called the local error and the local radius of information [270], respectively.

Let $U(f)$ be the multitude of solving problems from $F_{n}(f)$, and $\gamma(f)$ is the center of this multitude (the Chebyshev center). The algorithm $a^{\gamma} \in A$ is called a central one if $a^{y}\left(I_{n}(f)\right)=\gamma(f)$. These algorithms are accuracy optimal. Their important quality is that they minimize the local error of the algorithm:
$$\inf {a \in A} \sup {\varphi \in F_{n}(f)} \rho_{\mu}\left(I_{n}(f), a\right)=\rho_{\mu}\left(I_{n}(f), a^{\gamma}\right)=\operatorname{rad} U(f) .$$
Note that $A(\varepsilon) \neq \varnothing$ only when $\rho_{\mu}\left(F_{n}(f), A\right)<\varepsilon$.
Consider that class $A$ contains stable congruent algorithms and
$$\rho_{\mu}\left(F_{n}(f), a\right) \rightarrow 0 \text { при } n \rightarrow \infty$$
The algorithm $a$ is called asymptotically accuracy optimal (accuracy order optimal) if the condition is performed:
$$\rho_{\mu}\left(F_{n}(f), A\right) / \rho_{\mu}\left(F_{n}(f), a\right) \rightarrow 1(\leq \text { const }), \quad n \rightarrow \infty$$
Let $A(\varepsilon) \neq \varnothing$. The value $T(F, a, \varepsilon)=\sup {f \in F} T\left(I{n}(f), a, \varepsilon\right)$ is called $\varepsilon$-complexity of the algorithm $a \in A(\varepsilon)$ on the class of problems $F$ using information $I_{n}(f)$; $T(F, A, \varepsilon)=\inf {a \in A(\varepsilon)} T(F, a, \varepsilon)$ is $\varepsilon$-complexity of class $F$ problems using algorithms $A(\varepsilon)$ and information $I{n}(f)$. If there is an algorithm $a^{0} \in A(\varepsilon)$ for which

$T\left(F, a^{0}, \varepsilon\right)=T(F, A, \varepsilon)$, then it is called complexity optimal algorithm in this computational model. As in the case of error, it is possible to enter characteristics $T\left(F_{n}(f), a, \varepsilon\right)$ and $T\left(F_{n}(f), A, \varepsilon\right)$ on the class $F_{n}(f)$ and definitions of the optimal, asymptotically optimal, and complexity order optimal of the algorithm.

The mentioned characteristics are about the so-called “worst”-case model. The estimates are focused on the “worst” (the most complex) problem of the class. Of course, these estimates are guaranteed, and they are achieved on any problem of class $\Pi$, but this problem can be atypical for a given class. Therefore, there are some possible situations where the $\varepsilon$-solution can be obtained at lower expenses $(T)$. Consequently, in order to minimize the computational complexity of the $\varepsilon$-solution construction, the current question of the problems of classes narrowing, the ways where input data is presented, and the use of a priori information on a problem is relevant.

## 数学代写|优化算法代写optimization algorithms代考|Computer Architecture and the Complexity

There is an opinion (see, for example, [218]) that the optimization of the mathematical support of applied problems and the progress of computing techniques make equal contributions to the increasing possibilities of complex solving problems, in a point of fact in decreasing the computational complexity.

Consider the effect of rounding of the numbers on the computational complexity. Hypothesis [22] on that it is enough to compute the estimate of function $f$ for obtaining the solution with accuracy $O(\varepsilon)$ and perform intermediate computations in implemented CA with $O\left(\ln \varepsilon^{-1}\right)$ binary digit bits found confirmation in solving many problems $[42,106,114]$.

Thus, in the building of the $\varepsilon$-solution, the program uses numeric arrays with a total volume of $N$ numbers, and then memory $O\left(N \ln \varepsilon^{-1}\right)$ is required to store them.
Next, there is a possibility to consider the example of separate classes of problems and how the rounding error affects the possibility of $\varepsilon$-solution computation and the complexity of CP.
Let
$$\begin{gathered} \rho\left(E_{\mathrm{H}}\left(I_{n}(f), a, c\right) \leq \varepsilon_{1}<\varepsilon\right. \ \rho\left(E_{\mu \tau}\left(I_{n}(f), a, c\right)>\varepsilon_{2}, \quad \varepsilon_{2}=\varepsilon-\varepsilon_{1}\right. \end{gathered}$$
where $E_{\mu \tau}=E_{\mu}+E_{\tau}, a \in A$, and the relations are performed (in a point of fact in the numerical integration of $\mathrm{ODE}$, the computation of integrals and other classes of problems) $[106,114]$;

$$E_{\mu}=O\left(n^{-p}\right), \quad E_{\tau}=O\left(n 2^{-\tau}\right),$$
where $p$ is the order of the numerical method accuracy, and $\tau$ is the length of the mantissa in binary number notation in the floating-point mode. Herewith,
$$\varepsilon_{\mu \tau}^{0}(\tau)=\min {n} \rho\left(E{\mu z}\left(I_{n}(f), a, c\right)=O\left(n_{0}^{-p}(\tau)\right),\right.$$
where $n_{0}(\tau)=O\left(2^{\tau /(p+1)}\right)$, and $E_{\mu}\left(n_{0}(\tau)\right)=O\left(n_{0}^{-p}(\tau)\right), E_{\tau}\left(n_{0}(\tau)\right)=O\left(n_{0}^{-p}(\tau)\right)$.
Method error predominates when $n \ll n_{0}$. It can be decreased using optimal sets $I_{n}$, increasing $n$ (considering (2.10)), using accuracy optimal and close to them CA, moving to another class of input data $I_{n}$ (to increase the order of accuracy) and relevant CA.

The rounding error predominates when $n \gg n_{0}$. The decreasing $\rho\left(E_{\mu \mathrm{r}}\right)$ can be achieved by immediate increasing $n$ (considering $(2.10)$ ) or using the same capabilities as when $n \gg n_{0}$ (except increasing), as well as by increasing of $\tau$. From the relations (2.9) and (2.10), it follows that the performance of the constrain $\varepsilon_{\mu \tau}^{0} \leq \varepsilon_{2}$ is related to the conditions:
$$n=O\left(\varepsilon_{2}^{-1 / p}\right), \quad \tau=O\left(\log \varepsilon_{2}^{-1}\right), \quad \varepsilon_{2} \rightarrow 0$$
Consider the case (2.6). Let $\varepsilon$-solution be computed on the one processor using a linear computational algorithm per hour:
$$T(\varepsilon)=T_{l}(\varepsilon)+T_{a}(\varepsilon),$$
where $T_{I}(\varepsilon)$ is the process time of computation of the set of functionals $I_{n}(f)$ (information complexity), and $T_{a}(\varepsilon)$ is the process time of CA implementation for given information $I_{n}(f)$ (combinatorial complexity). Thus,
$$\begin{gathered} T_{l}(\varepsilon)=n(\varepsilon) \beta_{f}(\varepsilon) \alpha(\varepsilon), \ T_{a}(\varepsilon)=n(\varepsilon) \beta_{a} \alpha(\varepsilon), \end{gathered}$$
where $\alpha(\varepsilon)$ is the time performance of the “mean” operation in the computation of $\varepsilon$ solution; $\beta_{f}(\varepsilon)$ is an average number of computation operations of the functional $i_{j}$; and $\beta_{a}$ is an average number of operations that are related to the use of one function in implementation of CA.

Note that $\beta_{f}(\varepsilon)$ does not depend on $\varepsilon$ if the functionals in $I_{n}(f)$ can be computed closely to the accurate arithmetical operations. Then,
$$T(\varepsilon)=n(\varepsilon) \beta \alpha(\varepsilon), \quad \beta=\beta_{f}+\beta_{a} .$$

## 数学代写|优化算法代写optimization algorithms代考|Optical Models of Computations

Optical Models of Computations Dynamic holography [102] is a promising way of implementing a variety of optical converters. For instance, the effect of the energy transfer of a beam of light into another coherent light of beam that goes in the other direction (courtesy of their transverse in a dynamic environment) is an optical analogue of the transistor. It is possible to control the temporal variations in its intensity by changing the intensity of the amplifying beam. In another variant of the optical analogue of the electronic transistor, such control is achieved by changing not the intensity but the phase of the intensity beam.

Another example can be an optical switching device that is similar to a high-speed electronic commutator that is an inherent part of the most important devices of the computing techniques.

The advantage of holography is the possibility of a single transformation of the most complex images and not just the primary plane or spherical waves.

For today, the experiments have been performed on the creation of optical of bistable devices that switch $10^{-12}$ with elements of optical fiber communication lines, the information of which is moved with the help of optical solutions with the duration of reaching $10^{-13}$. With such a switching time, the productivity of a digital optical processor $10^{5}-10^{6}$ has simultaneous channels that would be equal to $10^{18}$ operations per second; in other words, it is on six order higher than the productive potential of electronic schemes. Examples of primitive actions for an optical computer include addition and subtraction of mappings, computation of Fourier transform, mapping identification, and so on.

## 数学代写|优化算法代写optimization algorithms代考|Input Information, Algorithms, and Complexity

• ρμ(F,一种)=支持F∈Fρ(和μ(一世n(F)),一种)是算法错误一种∈一种关于问题的类别F使用信息一世n(F)（全局错误 [270]）。
• ρμ(F,一种)=信息一种∈一种ρμ(F,一种)是类算法的误差下界一种在问题类别中F使用信息一世n(F)（信息半径[270]）
如果有算法一种0∈一种为此ρμ(F,一种0)=ρμ(F,一种)，则称其为类中精度最优F使用信息一世n(F).

приρμ(Fn(F),一种)→0 在 n→∞

ρμ(Fn(F),一种)/ρμ(Fn(F),一种)→1(≤ 常量 ),n→∞

## 数学代写|优化算法代写optimization algorithms代考|Computer Architecture and the Complexity

ρ(和H(一世n(F),一种,C)≤e1<e ρ(和μτ(一世n(F),一种,C)>e2,e2=e−e1

eμτ0(τ)=分钟nρ(和μ和(一世n(F),一种,C)=这(n0−p(τ)),

n=这(e2−1/p),τ=这(日志⁡e2−1),e2→0

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