\section{Application to inverse problems}
\subsection{Inverse problem formulation}\label{inverse problem formulation}

\begin{frame}[fragile]
\frametitle{Inverse problem}

\begin{itemize}
\setlength\itemsep{2em}
	\item[$\rightarrow$] Detection of cracks in existing structures
	\item[$\rightarrow$] Measurements are available
	\item[$\rightarrow$] A computational model is employed
	\item[$\rightarrow$] The difference between the two is minimized
	\item[$\rightarrow$] Information regarding the cracks is obtained
\end{itemize}

\end{frame}

\begin{frame}[fragile]
\frametitle{Inverse problem}

Mathematical formulation:

\vspace{\baselineskip}

\begin{align*}\label{eq:inverse_problem}
	\textit{Find } \beta_i \textit{ such that} \\ \nonumber
	\mathcal{F} \left( r \left( \beta_i \right) \right) \rightarrow \text{min}
\end{align*}

where

\vspace{\baselineskip}

\begin{description}%[$\mathcal{N}^s$]
\setlength\itemsep{1em}
\item[$\beta_i$] Parameters describing the crack geometry
\item[$r\left(\cdot\right)$] Norm of the difference between measurements and computed values
\item[$\mathcal{F}$] Some function of the residual
\end{description}

\vspace{\baselineskip}

The CMA-ES algorithm is employed to solve the problem.

\end{frame}

\begin{frame}[fragile]
\frametitle{Inverse problem}

Solution process:

\vspace{\baselineskip}

\begin{itemize}
\setlength\itemsep{2em}
\setlength\itemsep{2em}
	\item[$\rightarrow$] Generation of initial population ($\beta_i$) with CMA-ES
	\item[$\rightarrow$] Fitness function ($\mathcal{F} \left( r \left( \beta_i \right) \right)$) evaluation using XFEM and measurements
	\item[$\rightarrow$] Population is updated with CMA-ES
	\item[$\rightarrow$] The procedure is repeated until convergence
\end{itemize}

\end{frame}

\begin{frame}[fragile]
\frametitle{Inverse problem}

During the optimization proccess: 

\vspace{\baselineskip}

\begin{itemize}
\setlength\itemsep{2em}
	\item A large number of crack geometries is tested
	\item The computational model is solved several times
	\item An efficient and robust method is required
\end{itemize}

\end{frame}