### 数学代写|优化算法代写optimization algorithms代考|Identification and Clarification of A Priori Information

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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|Identification and Clarification of A Priori Information

Input data about the problem and its quality is very important in many aspects. Mention some of them:

• The more qualitative information on the problem is, the more qualitative approximate solution we can compute.
• Maximum use of all available information on the problem allows to limit the class of solved problems, and thus, it increases the “potential ability” of the NM; the more accurate input data is, the more accurate estimates of error and the less uncertainly range of the approximate solving problem.
• The computer technology of solving problem with the fixed values of quality with accuracy and fast speed is based on the analysis of error estimates.

We stop on some aspects of identification and clarification of a priori information on a problem.

An appropriate a priori information on the problem is required for obtaining a problem solution of a high quality, for example, the order of the derivative, constants that constrain it, the Hölder constant, and the corresponding mark-for the problems of function recovery and functionals. Useful information may also be about geometric properties-convexity, monotonies, number of extremums, etc. Such information is necessary to obtain an error estimate of the finding solution. If this information is given with sufficient low accuracy, then the conclusions on the quality of solving the problem will be inaccurate.

Consequently, obtaining qualitative a priori information is important in solving applied problems. Such information can be obtained from specialists who have a good knowledge of the physical phenomenon that we are studying. This information can also be obtained by using algorithms for identifying and clarifying a priori information.

For example, if the function is approximated from Lipschitz interpolational class, $F \equiv C_{L, N,} \varepsilon$ [287], and not only $L$ and $\varepsilon$ are known, but an only approximation to them is known. In such cases, it is advisable to use methods of residual and quasisolutions for approximating function [203].

For the class $F \equiv C_{L, N, e}$ the approximating function is the solution of the problem:
$$\min {f \in F} \max {i} \varepsilon_{i^{*}}$$
Otherwise, the method of quasisolutions involves finding a function that deviates less from the given set of points $\left(x_{i}, \tilde{f}{i}\right), \tilde{f}{i}=f_{i}+\varepsilon_{i}, i=\overline{0, N-1}$.

The solving problem $(1.15)$ is a line spline $S(x, L)$ in which the maximal deviation from the given points $\left(x_{i}, \tilde{f}{i}\right), i=\overline{0, N-1}$ is the minimum [203]: \begin{aligned} &S(x, L)=\widehat{f}{i}+\frac{x-x_{i}}{x_{i+1}-x_{i}}\left(\widetilde{f}{i+1}-\tilde{f}{i}\right), x \in\left[x_{i}, x_{i+1}\right], i=\overline{0, N-1}, \ &\widehat{f}{i}=\frac{\tilde{f}{i}-\tilde{f}{i}}{2}, \tilde{f}{i}^{\pm}=\max {1 \leq j \leq N}\left[\pm\left(\tilde{f}{j} \mp L\left|x_{j}-x_{i}\right|\right)\right], i=\overline{0, N-1} \end{aligned}
It often happens that the quantitative a priori information that is used to define a class $F$ is given in the form of constraints on some functional. A uniform norm of the derivative is used a functional $\Phi(f)$ itself for classes $C_{L, N}$ and $C_{L, N,}$. We will approximate the function $f(x)$ by a function that is the solution of the following problem:
$$\min _{f \in F} \Phi(f)$$

## 数学代写|优化算法代写optimization algorithms代考|Accuracy Optimal Computational Algorithms

In the due form, the concept of the optimality of the solution or the algorithm is determined by some criteria. Such criteria can be a requirement of the solution to have a given error of a method or the algorithm to have the highest possible rate of convergence. Specific content of criteria has an important value for its use.

One of the main criteria for the optimality of approximate solving problem can be the requirement of its maximum accuracy (or minimum error) by the given resources that can be used in the solution process. The concept of the resource includes the amount and accuracy of input data of the problem, free use of computer memory, limit the time of computing on this computing machinery, the available supply of mathematical software of computing machinery, etc.

In such a statement, it is natural to consider the question of the “potential ability” of NM at the beginning of the study, in other words, on that maximum accuracy of the solution that can be achieved for this given input information on the problem.
Every CA of solving a certain problem uses only a finite number of input data on the problem, and thus, it automatically is a CA of solving the class of all those problems that have the same input data. On this multitude of problems, there are always two problems in the solution of which the worst and best limits of the optimized characteristics are achieved. Therefore, every, including the optimal one, CA of solving a problem that concerns us will have some “potential ability.” If, for example, there are two problems with the same input data, accurate solutions of which $x_{1}$ and $x_{2}$ are the elements of the metric space, moreover, the distance between them are
$$\rho\left(x_{1}, x_{2}\right) \geq d>0$$
Then a solution $x$ obtains for each CA their solving that have a property
$$\max {i=1,2} \rho\left(x, x{i}\right) \geq \frac{d}{2}$$
This means that there is no CA that would give a solution to the considered problem with a guaranteed accuracy of less than $d / 2$. If there is a need to improve the accuracy of solving the problem, it should be included by some additional information about it. Then the problem will belong to a new more “narrow” class of problems, and the CA of which solution will have a new more powerful “potential ability.” Similar considerations are valid for any other index (characteristics) of CA and problems.

## 数学代写|优化算法代写optimization algorithms代考|Time Optimal Computational Algorithms

The computational complexity of the problem essentially depends on the computing model (computer architecture). Assume that it is possible to use a computing model for the building $E$-solution that is based on the use of input information of $I$, informational operator of $L_{M}(I)$ that can be introduced, for example, in the form of some multitude of functionals, as well as on a multitude of parameters $X, Y$.

Define the time-optimal algorithms and similar to them algorithms. Let the problem $P(I) L_{N}(I)=I=\left(i_{1}, i_{2}, \ldots, i_{N}\right), I \in$ 田 be input data, and $M$ is a multitude of algorithms $A$ of solving problem with a given accuracy $\varepsilon, \varepsilon>0$, on fixed computing machinery, and $Q(A, I, \varepsilon)$ is a number of arithmetic operations that are required for this. Consider the characteristics
\begin{aligned} &Q_{N}(A, \varepsilon)=\sup {I \in \mathcal{J}} Q(A, I, \varepsilon) \ &Q{N}(\varepsilon)=\inf {A \in M} Q{N}(A, \varepsilon) \end{aligned}
Call an algorithm a time-optimal one on which $Q_{N}(\varepsilon)$ is achieved. If $Q_{N}\left(A^{}, \varepsilon\right)=Q_{N}(\varepsilon)+\xi, \xi>0$, then $A^{}$ is called time-optimal with accuracy within $\xi$. If $\xi=o\left[Q_{N}(\varepsilon)\right]$ or $\xi=O\left[Q_{N}(\varepsilon)\right]$, then it is asymptotically optimal or time-optimal in order algorithm, correspondingly.

The purpose of constructing a lower complexity estimate $Q_{N}(\varepsilon)$ is to prove that none of the algorithms in this computational model has a less complexity of computation than the current function $Q_{N}(\varepsilon)$. Unfortunately, the well-known “high” (nontrivial) lower estimates are most likely an exception to the rule.

The scheme of upper estimates of complexity constructing is like this. CA $A^{}$ is built based on a certain method of solving the problem in the current computing model, and it is proved that the computational complexity does not exceed $Q_{N}\left(A^{}, \varepsilon\right)$ within the input data from the class. $Q_{N}\left(A^{}, \varepsilon\right)$ is called the upper estimate of the computational complexity of CA $A^{}$ of solving problem obtaining.

## 数学代写|优化算法代写optimization algorithms代考|Identification and Clarification of A Priori Information

• 关于问题的定性信息越多，我们可以计算的定性近似解就越多。
• 最大限度地利用有关问题的所有可用信息可以限制已解决问题的类别，从而增加 NM 的“潜在能力”；输入数据越准确，误差估计越准确，近似求解问题的不确定性范围越小。
• 准确、快速地解决质量固定值问题的计算机技术是基于对误差估计的分析。

$$\min {f \in F} \max {i} \varepsilon_{i^{*}}$$

ρ(X1,X2)≥d>0

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## MATLAB代写

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