Notes for "A-optimal encoding we

2019-07-11 本文已影响0人 jjx323

Benjamin Crestel, Alen Alexanderian, Georg Stadler and Omar Ghattas, A-optimal encoding weights for nonlinear inverse problems, with application to Helmholtz inverse problem, Inverse Problems 33(2017) 074008.

Formula (21)

Define
$\begin{align} \mathcal{L}(u_i, m, p_i) = & \frac{1}{2N_w}\sum_{i=1}^{N_w}\|Bu_i - d(w^i)\|_{\Gamma_{noise}^{-1}}^{2} + \frac{1}{2}\|m-m_0\|_{\mathcal{E}}^{2} \\ & -\frac{1}{N_w}\sum_{i=1}^{N_w}\left\{\int \nabla u_i \cdot\nabla p_i dx - \int \kappa^2 m u_i p_i dx - \int f(w^i)p_i dx\right\}, \end{align}$
and
$\begin{align} \langle e(u_i, m), p_i\rangle = \int \nabla u_i \cdot\nabla p_i dx - \int \kappa^2 m u_i p_i dx - \int f(w^i)p_i dx \end{align}$
Then the first-order necessary optimiality conditions are
$\begin{align} & \langle e(u_i, m), p_i\rangle = 0,\quad \forall i \\ & \langle\mathcal{L}_{u_i}(u_i, m, p_i), \tilde{u}_{i}\rangle = 0, \quad \forall i \\ & \langle\mathcal{L}_{m}(u_i, m, p_i), \tilde{m}\rangle = 0. \end{align}$
Through a tedious calculation, we finally obtain formula (21).

Notes for "A-optimal encoding we

Formula (21)

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