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# 数学代写|优化理论代写Optimization Theory代考|MATH414 Input Information, Algorithms, and Complexity of Computations

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## 数学代写|优化理论代写Optimization Theory代考|Input Information, Algorithms, and Complexity of Computations

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 $a^\gamma\left(I_n(f)\right)=\gamma(f)$. 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
$$\rho_\mu\left(F_n(f), a\right) \rightarrow 0 \text { при } n \rightarrow \infty$$

## 数学代写|优化理论代写Optimization Theory代考|Asymptotic Model of Analytic Complexity

Asymptotic Model of Analytic Complexity In the practice of numerical analysis (quadrature formulae, numerical integration of ODE, etc.), an asymptotic model is often used which differs from the given model (the “worst” case) because it uses not only one informational operator but an asymptotically congruent sequence of the linear informational operator $\left{I_{n_k}(f)\right}$ of cardinality $n_k$ and the suitable sequence of asymptotically congruent algorithms $\left{a_k\right}: \lim {k \rightarrow \infty} \rho\left(E\mu\left(I_{n_k}(f)\right), a_k\right)=0, \forall f \in F$.
A special feature of the asymptotic model is that the algorithm is asymptotically “tunable” on a particular class problem that provides a more accurate approximation. We give the computation of integral as an example of using an asymptotic model:
$$\Phi(f)=\int_0^1 f(t) d t, f(t) \in C^p[0,1], \quad p \geq 1$$
with quadrature formulae of order $r(1 \leq r \leq p)$ using the sequence of informational operators:

\begin{aligned} I_{n_k}(f) &=\left{f\left(t_i\right), i=\overline{1, n_k}\right}, \quad t_i=(i-1) h_k, \quad h_k=2^{-k}, \quad n_k=2^k+1, \quad k \ &=0,1,2, \ldots \end{aligned}
(step of discretization is given in this form only for convenience).
It is obvious
$$E_\mu\left(f, a_k, h_k\right)=C_r f^{(r)}(\xi) h_k^r \rightarrow 0, \quad k \rightarrow \infty, \quad \xi \in(0,1)$$

## 数学代写|优化理论代写Optimization Theory代考|Input Information, Algorithms, and Complexity of Computations

$\rho_\mu(F, a)=\sup f \in F \rho\left(E \mu\left(I_n(f)\right), a\right)$ 是算法错误 $a \in A$ 关于问题的类别 $F$ 使用信息 $I_n(f)$ (全同错误 [270])。

$\rho_\mu(F, A)=\inf a \in A \rho \mu(F, a)$ 是类算法的误差下界 $A$ 在问题类别中 $F$ 使用信自 $I_n(f)$ (信息半径 $\left.[270]\right)$ 如果有算法 $a_0 \in A$ 为此 $\rho_\mu\left(F, a_0\right)=\rho_\mu(F, A)$ ，则称其为类中精度最优 $F$ 使用信息 $I_n(f)$.

$$\inf a \in A \sup \varphi \in F_n(f) \rho_\mu\left(I_n(f), a\right)=\rho_\mu\left(I_n(f), a^\gamma\right)=\operatorname{rad} U(f) .$$

$$\rho_\mu\left(F_n(f), a\right) \rightarrow 0 \mathrm{pp} \text { 和 } n \rightarrow \infty$$

## 数学代写|优化理论代写Optimization Theory代考|Asymptotic Model of Analytic Complexity

$$\Phi(f)=\int_0^1 f(t) d t, f(t) \in C^p[0,1], \quad p \geq 1$$

\left 的分隔符缺失或无法识别
(为了方使起见，以这种形式給出离散化步骤)。

$$E_\mu\left(f, a_k, h_k\right)=C_r f^{(r)}(\xi) h_k^r \rightarrow 0, \quad k \rightarrow \infty, \quad \xi \in(0,1)$$

## MATLAB代写

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