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# 数学代写|Example Polynomial approximation 数值分析代写

## 数值分析代写

Determine the input and output variables $X$ and $Y$

Collect training data set containing observations $\left(x_{i}, y_{i}\right)$ where $i=1, \ldots, N$.

Choose a model to use for relating $X$ to $Y$
(e.g. linear model)

Choose a loss function to minimize
(e.g. residual sum of squares)

Solve the minimization problem to find the parameters
(e.g. normal equations)

Consider our inverse problem involving the PDE
$$\nabla \cdot K \nabla u=0 \quad \text { in }(-1,1)^{2}$$

Our input $X$ will be set as an observation of the measurement operator (Dirichlet-to-Neumann map $\Lambda$ ) on the boundary

For example let $\left(x_{i_{r}}, y_{j_{r}}\right), r=1, \ldots, R$ denote points on the boundary of the domain.

Then for each Dirichlet boundary condition $f_{\el l, r}$ on the boundary, we would have the Neumann data $g_{\ell, r}$.
Input and output for the inverse conductivity problem

Then each observation of the Dirichlet-to-Neumann map will be given as
$$\left[\begin{array}{c} f_{1, \ell} \ f_{2, \ell} \ \vdots \ f_{R, \ell} \ g_{1, \ell} \ g_{2, \ell} \ \vdots \ g_{R, \ell} \end{array}\right], \quad \ell=1, \ldots, L .$$

Collecting all such measurements $\ell=1, \ldots, L$, with some choice of $f_{\ell}$, forms one observation of our input $x_{i}$
$$x_{i}=\left[\begin{array}{cccc} f_{1,1} & f_{1,2} & \cdots & f_{1, L} \ \vdots & \vdots & \vdots & \vdots \ f_{R, 1} & f_{R, 1} & \cdots & f_{R, L} \ g_{1,1} & g_{1,2} & \cdots & g_{1, L} \ \vdots & \vdots & \vdots & \vdots \ g_{R, 1} & g_{R, 2} & \cdots & g_{R, L} \end{array}\right]$$

## 数值分析代考

（例如线性模型）

（例如残差平方和）

（例如正规方程）

$$\nabla \cdot K \nabla u=0 \quad \text { in }(-1,1)^{2}$$

$$\left[\begin{数组}{c} f_{1, \ell} \ f_{2, \ell} \ \vdots \ f_{R, \ell} \ g_{1, \ell} \ g_{2, \ell} \ \vdots \ g_{R, \ell} \end{array}\right], \quad \ell=1, \ldots, L 。$$

$$x_{i}=\left[\begin{array}{cccc} f_{1,1} & f_{1,2} & \cdots & f_{1, L} \ \vdots & \vdots & \vdots & \vdots \ f_{R, 1} & f_{R, 1} & \cdots & f_{R, L} \ g_{1,1} & g_{1,2} & \cdots & g_{1, L} \ \vdots & \vdots & \vdots & \vdots \ g_{R, 1} & g_{R, 2} & \cdots & g_{R, L} \end{数组}\right]$$

Matlab代写