数学代写|优化算法代写optimization algorithms代考| Complexity of Real Computation Processes

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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|On the Computer Constructing Technology

Scheme of constructing (choice) of $T$-effective computational algorithm depends on many factors (class problems, input data, dimension and characteristics of the problems, computational resources that are available to the user, constrains (2.1), (2.2), and (2.3)); therefore, in the class problem $F$, it is advisable to distinguish multitude (subclasses) of problems that have common features in the context of computing [14]:

• One-off problems with a small amount of computing and moderate constraints on process time
• Problems (or series of problems) that are needed to be solved in real time
• Problems with a very large amount of computations that are needed to be solved in a practically reasonable amount of time (that cannot be achieved on traditional computing machines)

The performance of the conditions (2.1), (2.2), and (2.3) depending upon the statement of the problem can be achieved by choosing one of the following combinations of computing resources: $X,\left(X, I_{n}\right),(X, Y),\left(X, Y, I_{n}\right)$. In the first two situations, the possibilities of the computer are fixed. In the first situation, the information $I_{n}$ is also fixed; conditions (2.1), (2.2), and (2.3) are satisfied by the choice of the algorithm and its parameters; in the second one, it is still possible to select the set $I_{n}$ for this type of information operator. In the third situation, the information is fixed, and the parameters of the computer can be chosen besides the algorithm. In the fourth situation, all computing resources are used.

The first group of problems can be solved by the choice $X$ or $\left(X, I_{n}\right)$ of a regular sequential computer. Herewith, it is possible to devote three levels of detalization of the computing model. At the first level, there are algorithms that are focused on class $F$ solving problems using the information $I_{n}$. Herewith, there is support: approximation, stability, convergence of the approximate solution, the possibility to achieve a certain accuracy for the given input information, and the volume of computations as a function of the problem size (volume of input data). At this level, there is a possibility to discover the impossibility of computation of the $\varepsilon$-solution using specific input data, and there might be a possibility to clarify the class of problems and the requirements for the input information to provide a certain accuracy of the approximate solution, and it is possible (in this regard) to choose a new algorithm.
The second level (detalization) is related to the use of elements of the multitude $Y$ (machine word length, rounding rule) to compute the error estimate of rounding. Herewith, a multitude $A(\varepsilon)$ can be defined as conclusions in the case of the advisability of certain algorithms using from the multitude $A(\varepsilon)$ to save process time.
At the third level, where computational algorithm is a program for computing the $\varepsilon$-solution on a certain computer, time $T(\varepsilon)$ and memory $M(\varepsilon)[114]$ are estimated.
The variants $(X, Y)$ and $\left(X, Y, I_{n}\right)$ are specific to the second group of problems, for example, for digital signals processing and digital images processing using specialized computers. To achieve high rapid rates, the computer architecture is coherent with the computational algorithm $[131,277]$.

It is possible to use the third or fourth variants of the organization of computing to solve the problems of the third group. Herewith, the one purpose high-end computers [220] and computers of all purposes can be used [199].

数学代写|优化算法代写optimization algorithms代考|Specificity of Using Characteristic Estimates

In constructing real computational processes of computations, $\varepsilon$-solution is often used by some estimates of global error, its component and process time. Herewith, they distinguish estimates in the following way: a priori and a posteriori, majorizing and asymptotic, and determinate and stochastic. The possibility and advisability of these estimates using and the methods of their construction depend on the type, structure, and accuracy of a priori data, the problem, and the CA from that why the estimate is computed, and it also depends on the computational resources [114,238].

Majorizing a priori estimate guarantees the upper bound of the estimated deriv atives, and they are performed through known derivatives. Their computation does not require some significant computational expenses, but the value of estimates are often overrated; therefore, the conclusions based on them as for the possibility of computing of the solution under the conditions $(2.1)$ and $(2.2)$ may be false.

Asymptotic estimates approximate the estimated derivative. The variability of the parameter can be achieved by the desirable estimate proximity to the estimated derivative, but the computation of such estimates is related to significant computational expenses, and these estimates are usually a posteriori.

In the algorithmic support of solving problems under the conditions (2.1) and (2.2), given the properties of the estimates, it must be expected the possibility of computing of the various types of estimates of characteristics $E\left(E_{\mathrm{u}}, E_{\mu}, E_{\tau}\right)$ [238]. By the relaxed constraints (2.1) and (2.2), less precise and less complex (computational) estimates may be sufficient. By the tighten constrains (2.1) and (2.2), asymptotic (a posteriori) estimates are used. For example, the condition (2.2) may apply strict requirements to the accuracy of estimates of computational process parameters that are computed on the basis of errors estimate of the solution.

数学代写|优化算法代写optimization algorithms代考|Classes of Computational Problems, Informational

In the given technology of constructing problems solution per time that does not exceed the given $T$, available information plays a great significance. The more a priori information of different principles is known on the problem and algorithm uses it, the more accuracy effective and time it can be solved.

Note that the effectiveness of the algorithms is determined by the estimate of their characteristics so that the estimates should be of high quality (constants that are included in majorizing estimates of errors, accurate, estimates, unimprovable, etc.). And yet even high-quality estimates are constructed on a class of problems. And the wider this class is, the less suitable this estimate may be for a particular problem. Therefore, it is important to have a classification of problems that considers the additional a priori information. This will make a possibility to “select” such a class for a solved problem that is most likely to be used to obtain the required solution of a certain quality.

Consequently, the improvement of the quality of solving problems depends on the “narrowing” of the class of problems to which the solved problem belongs and the building of algorithms of such solving problems and the most accurate estimates of their characteristics.

However, it is not always possible to obtain $\varepsilon$-solution of some problems (although the total input information may be enough for this) using the given technology, or it cannot be checked that the solution was achieved. In these cases, it is important to have algorithms that are accuracy optimal (all available information on the problem is used as much as possible to improve accuracy) and a posteriori error estimates (that are more accurate next to a priori ones).

On the back of the accuracy optimal algorithm of this solving problem and a posteriori estimate of the error, it is often possible to obtain a solution that satisfies the user or draw a conclusion that it was not possible to obtain such a solution. We consider key principles of the problems classification and algorithms through the examples of some specific classes of problems of computational and applied mathematics.

数学代写|优化算法代写optimization algorithms代考|On the Computer Constructing Technology

• 计算量小、处理时间适度限制的一次性问题
• 需要实时解决的问题（或一系列问题）
• 需要在实际合理的时间内解决的大量计算问题（传统计算机无法实现）

有限元方法代写

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MATLAB代写

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