Regularization Function

The aim of this section is to present all the regularization functions that can be used as part of the regularization term in the inverse problem.

Denote \(\rho_k\) the control vector associated to the \(k^{\text{th}}\) parameter \(\boldsymbol{\theta}\) or initial state \(\boldsymbol{h}\), \(\rho_{0_k}\) the prior value associated to \(\rho_k\) and \(\phi\) the mapping operator (see section Mapping).

prior

A reminder to prior value

\[j_{prior} = \sum_{k=1}^{N_{\theta} + N_h} \sum_{i=1}^N \left(\rho_k(i) - \rho_{0_k}(i)\right)^2\]

smoothing

A spatial smoothing taking into account the prior value \(\rho_{0_k}\)

\begin{eqnarray} &m_k& &=& &\phi(\rho_k) - \phi(\rho_{0_k})\\ &j_{smoothing}& &=& &\sum_{k=1}^{N_{\theta} + N_h} \sum_{x=1}^{N_x} \sum_{y=1}^{N_y} &\left(m_k(x - 1, y) - 2m_k(x, y) + m_k(x + 1, y)\right)^2 \\ &&&&&&+ \left(m_k(x, y - 1) - 2m_k(x, y) + m_k(x, y + 1)\right)^2 \end{eqnarray}

with the following boundary conditions

\begin{eqnarray} &&m_k(N_x + 1, y)& =& &m_k(N_x, y)\\ &&m_k(0, y)& =& &m_k(1, y)\\ &&m_k(x, N_y + 1)& =& &m_k(x, N_y)\\ &&m_k(x, 0)& =& &m_k(x, 1)\\ &&m_k(N_x + 1, N_y + 1)& =& &m_k(N_x, N_y)\\ &&m_k(0, 0)& =& &m_k(1, 1)\\ &&m_k(N_x + 1, 0)& =& &m_k(N_x, 1)\\ &&m_k(0, N_y + 1)& =& &m_k(1, N_y) \end{eqnarray}

hard-smoothing

A spatial smoothing without taking into account the prior value \(\rho_{0_k}\)

\begin{eqnarray} &m_k& &=& &\phi(\rho_k)\\ &j_{hard-smoothing}& &=& &j_{smoothing} \end{eqnarray}