Minimization problems

Solver includes 3 different minimization problems.

Classical form

\[\min_{\Theta} \sum_{\alpha=1}^{q} \left[\| \bar{S} \bar{u}^\alpha - f^\alpha \|_{i} + \lambda \|\bar{b} \bar{u}^\alpha - g^\alpha\|_{j} \right]_{X}\]

Weak form

\[\min_{\Theta} \sum_{\alpha=1}^{q} \int_{X} \left[\bar{S}(\bar{u}^\alpha) - f^\alpha\right] \cdot \phi^\alpha \, \mathrm{d}X + \left[\lambda \|\bar{b} \bar{u}^\alpha - g^\alpha\|_{j} \right]_{X}\]

Causal form

\[\min_{\Theta} \sum_{\alpha=1}^{q} \left[\omega * \| \bar{S} \bar{u}^\alpha - f^\alpha \|_{i} + \lambda \|\bar{b} \bar{u}^\alpha - g^\alpha\|_{j} \right]_{X}\]

where

\[\omega = {\omega_0 ... \omega_i}\]

\[\omega_i = \exp(-\epsilon \sum_{k=1}^{i-1}\bar{S} \bar{u}^\alpha - f^\alpha),\]

for i = 2,3,…N_t, N_t is number of grid points, epsilon is constant.