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\author{Davide Baroli,\\ St\'ephane Bordas,\\ Lars Beex\\ Jack Hale  }
\title{Advanced Reduced Order Methods: From POD to coupled Nitsche}
\setlength{\algomargin}{0.3in}
%\titlegraphic{\includegraphics[height=1cm,width=2cm]{logo1}}
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\begin{document}
\begin{frame}
\maketitle

\end{frame}


 \frame{
           \frametitle{Outline}
           \centering
             {\scriptsize
                     \tableofcontents
             }
  
 }

\section{Introduction to Model Reduction} 
\begin{frame}
\frametitle{Reliable and fast method: model reduction method}
Given a solution set (a discrete manifold) $\mathcal{M}_h=\left\{u_h(\mu) \in V_h : \mu \in \mathcal{P} \right\}$ of high-fidelity problem.
\begin{equation}
 a(u_h(\mu),v_h;\mu) = f (v_h; \mu ) \forall v in V_h
\end{equation}
\begin{figure}
  \includegraphics[width=0.3\paperwidth]{RB.png} 
  \caption{From ``Reduced Basis Method for PDEs''(A.Quarteroni et al.)}
\end{figure}
\end{frame}
\begin{frame}
The goal of reduced order methods  is to find a few functions $V_N=\left\{\xi_i\right\}_{i=1}^N \subset V_h$ with 
$N << dim(V_h)$ such that for any parameter $\mu$ the solution of parametrized problem is approximated by:
\begin{equation*}
  u_h(\mu) = \sum_{i=1}^N u_N(\mu)^i \xi_i 
\end{equation*}
where the coefficients $u_N(\mu)$ could be obtained via Galerkin reduced basis approximation:
\begin{equation*}
 [\xi_1|\ldots|\xi_N]^T \mathbb{A}_h(\mu) [\xi_1|\ldots|\xi_N]  u_N =[\xi_1|\ldots|\xi_N]^T \mathbf{f}_h(\mu) 
\end{equation*}
with $\mathbb{A}_h$ and $\mathbf{f}_h$ are the discrete counterpart of the differential operator.
\end{frame}

\begin{frame}
\frametitle{Accuracy of reduced order method}
Ansantz:
		\begin{equation*}
		\| u(\mu) - u_{ROM}(\mu)\| \leq \| u(\mu)- u_h(\mu)\| + \|u_h(\mu) -u_{ROM}(\mu)\|
	   \end{equation*}
How efficient is the reduced order method representation? \\
Let 
		\begin{equation*}
				E(M_h,V_N)= \sup_{u_h \in \mathcal{M}_h} \inf_{u_{ROM} \in V_N} \| u_h - u_{ROM} \|
		\end{equation*}
The Kolmogorov N-width of $M_h$ in $V_N$ is defined as:
		\begin{equation*}
				d_N(M_h)= \inf_{V_{N}\subset V_h} E(M_h,V_{N})
	\end{equation*}
If N-width decays as N increases then the discrete manifold is well approximated by 
		a $V_{ROM}$ with a small dimension.
%		Alternative upper bound (using Cea's lemma):
%		\begin{equation*}
%   \|u(\mu) -u_{ROM}(\mu) \| \leq \left(1+ \frac{\gamma(\mu)}{\alpha(\mu)}\right)
%				\inf_{u_{ROM}\in V_{N}} \| u(\mu) -u_{ROM}\|
%		\end{equation*}
%where $\alpha$ and $\gamma$ are the coercivity and continuity constant.

\end{frame}

\begin{frame}
\frametitle{Outline of workflow}
Summary of workflow:
\begin{itemize}
\item Offline: Compute snapshots of solution of PDE across parameter space.
\item Offline: Construct reduced order basis (POD, RB).
\item Online: Run reduced-order model.
\end{itemize}
\end{frame}
\subsection{Proper Orthogonal Decomposition}
\begin{frame}
\frametitle{Proper Orthogonal Decomposition}
The goal of POD is to investigate the sampling of $M$ $\mu$ in the parameter space ($\mathbb{P}$)
and compress the discrete manifold of high-fidelity solution corresponding to sampled parameter, 
retaining the most energetic information.\\
Regarding the accurancy, the $V_N$ generated by POD
minimize the following quantity
		\begin{equation*}
				\sqrt{\frac{1}{M}\sum_{\mu \in \mathbb{P}} \inf_{u_{ROM} \in V_N}
				\| u_h(\mu) - u_{ROM}\|^2_{V} }
		\end{equation*}
    
\end{frame}
\begin{frame}
\frametitle{Proper orthogonal decomposition algorithm}
Given $\mu_1,\ldots, \mu_M \in \mathbb{P}$ and the high-fidelity approximation 
		$u(\mu_1),\ldots,u(\mu_M)\in \mathcal{M}_h$.\\
\begin{itemize}
		\item Construct the symmetric correlation matrix $C\in \mathbb{R}^{M\times M}$  defined by
				\begin{equation*}
				C_{i,j}= \langle u(\mu_i), u(\mu_j) \rangle_V \,  1\leq i,j\leq M
	\end{equation*}
\item Compute the eigenvalue-eigenvector pairs $(\lambda_n,\zeta_n)$ of the matric $C$ such that
		$\|\zeta_n\|_{\ell^2(\mathbb{R}^M)}=1$ and 
				\begin{equation*}
C \zeta_n = \lambda_n \zeta_n 
				\end{equation*}
				with descending eigenvalue order.
\item Reconstruct the POD basis
\begin{equation*}
		\xi_n= \frac{1}{\sqrt{\lambda_n}} [ u(\mu_1) | \ldots | u(\mu_M)] \zeta_n 	
\end{equation*}

\end{itemize}

\end{frame}


\begin{frame}
\frametitle{Application: Non-linear parametrized fkpp equation}
						Find  $u(\mathbf{x}, t) \in H^1(\Omega,(0,T])$  such that  \\
			\begin{equation}\label{eq:fkpp}
		\begin{array}{rcll}
					\partial_t u - \Delta u &=& c u (1-u) &\text{in } \Omega=[0,3]^2 \times (0, T],\\
						u &=& 0 &\text{on } \partial \Omega \times (0, T], \\
						u(\mathbf{x}, 0) &=& u_0(\mathbf{x}) &\forall \mathbf{x} \in \Omega,
				\end{array}
		\end{equation}
		

\end{frame}

\begin{frame}
Following the standard Galerkin procedure of forming the $L^2$ inner product
		with test functions $v \in \hat{V}$ and performing integration by parts,
		the semi-linear weak residual formulation of \eqref{eq:fkpp} can be written as:
		\begin{equation}\label{eq:fkpp_semi_linear_continuous}
				\begin{aligned}
						&\text{Find } u(x, t) \in V \text{ such that:} \\
						&F(u, c; v) := (\partial_t u; v) + (\nabla u; \nabla v) - (c u; v) + (c u^2; v) = 0 \quad \forall v \in \hat{V}.
				\end{aligned}
		\end{equation}

\end{frame}

\begin{frame}
\begin{center}
\includegraphics[scale=1.0]{femsolution.pdf}

Three snapshots in time for the same initial condition.
\end{center}
\end{frame}

\begin{frame}
\begin{center}
\includegraphics[scale=0.5]{spectrum.pdf}

Eigenvalues of the matrix $C$.
\end{center}
\end{frame}

\begin{frame}
\begin{center}
\includegraphics[scale=0.5]{rapod.pdf}

First three eigenvectors of the matrix $C$.
\end{center}
\end{frame}

\begin{frame} 
\frametitle{Where is the greatest computational effort in the "on-fly" model reduction?}
\begin{itemize}
\item Online Galerkin projection when the parametrized projection require to "re-assemble" the stiffness matrix $K$ or the right-hand side $f$.
%\item Acquisition of full-state data for generating basis 
\end{itemize}
\end{frame}
 \section{Reduced Assembly}
 \begin{frame}
 \frametitle{Sequential reduced assembly}
\only<1->{ Given a subspace generated by a set of orthonormal basis $V_{RB}=\left\{\xi_1, \ldots, \xi_N\right\}$, 
 quadrature rule order $m$ and a coarse mesh $\Omega_H:=\mathcal{T}_H(\Omega)$:\\
 Let start with the first basis function $\xi_k$ $(k=0)$ and a given tolerance $tol$.\\}
 \begin{enumerate}
  \item<1> \only<1>{ Compute the error :
  \begin{equation*}
  \eta(\xi_k)=\frac{|\int_{\Omega_H} \xi_k - \int_{\Omega_h} \xi_i|}{\int_{\Omega_h} \xi_k}
  \end{equation*}}
\item<2> \only<2>{ If $err > tol$ then \\
  Evaluate cell-wise the estimator:
  \begin{equation}
  \eta_{rel}(\xi_i,\Omega_H) = {\int_{K} |I_{m}(\xi_k)| -  \int_{K} |I_{m+1}(\xi_k)|} \quad \forall\, K \in \Omega_H
  \end{equation}}
\item<3> \only<3>{ Set  $\alpha:=\sum_{K\in \Omega_H}\eta_K(\xi_i,\Omega_H)$.\\ 
     If $\alpha >tol$ then\\
       For $i=1,\ldots,N_{cells}$\\
       if $\sum_{j=1}^i \eta_{K_j}(\xi_i,\Omega_H) >diam(K_i)\cdot tol$:\\
       then refine($K_i$) untill the $\alpha<tol$ \\
	}
 \item<4> Go to 1 and $k:=k+1$ and use as coarser mesh the last obtained.
\end{enumerate}
\end{frame}

\begin{frame}
\begin{center}
\includegraphics[scale=0.5]{rapod.pdf}

Left: First three eigenvectors of the matrix $C$ and right: first three RAPOD meshes.
\end{center}
\end{frame}

\begin{frame}
\frametitle{Reduced Assembly POD}
\begin{center}
\includegraphics[height=0.7\paperheight]{finalmode.pdf}
\end{center}
5128 cells vs ~200000 in original mesh for 1\% error in all modes.
\end{frame}
\section{Validation of RAPOD for FKPP in 2D and 3D}

\begin{frame}
		\frametitle{Computational demands}
\begin{figure}
		\includegraphics[width=0.8\paperwidth]{rapod_linear.pdf} 
\end{figure}
\end{frame}
\begin{frame}
\begin{figure}
		\frametitle{Wall time}
		\includegraphics[width=0.8\paperwidth]{rapod_log.pdf} 
\end{figure}		
\end{frame}
\begin{frame}
				\frametitle{Reliability of POD vs RAPOD}
\begin{figure}
		\includegraphics[width=0.8\paperwidth]{plot.pdf} 
\end{figure}	
		\end{frame}
	

\section{Implementation with SLEPc}
\begin{frame}
\frametitle{Reduced Basis technique: Greedy algorithm using SLEPc}
\only<1->{Given tol and set of parameters $\left\{\mu_1,\ldots,\mu_n\right\}$.\\}
\only<1>{Initialize the Reduced Basis space:\\
 \pythoninline{reducedbasis = SLEPc.BV().create(Comm)}}
\begin{enumerate}
\item<2-> \only<2->{Compute high-fidelity solution $u_h(\mu_1)$ for $\mu_1$ }
\item<3-> \only<3->{ Set $\mathbf{V}_{rb}=span\left\{u_h(\mu_1)\right\}$.}
\only<3>{\pythoninline{reducedbasis.insertVec(index, function)}} 
\item<4-> \only<4->{ For each $\mu\in \mathcal{P}$ \\
   Compute $u_{RB}=\sum_{i=1}^N u_N^i \xi_i(\mu)$ where $\xi_i$ are the basis of $\mathbf{V}_{rb}$. \\
  Evaluate the error estimator $\eta(\mu)=\frac{(res_{RB},M^{-1} res_{RB})}{\alpha_{LB}}$} 
 \item<5-> \only<5->{ Choose $\tilde{\mu}= arg\,max_{\mu \in \mathcal{P}} \eta(\mu)$}
 \item<6-> \only<6->{ If $\eta(\tilde{\mu})>tol$  and $V_{RB}=V_{RB}\cup \left\{u(\tilde{\mu})\right\}$}
\only<6>{ \pythoninline{reducedbasis.orthogonalizeVec(function)}}
\only<6>{ \pythoninline{reducedbasis.scaleColumn(norm)}}
\end{enumerate}
\end{frame}
\begin{frame}
\frametitle{Online Galerkin Project with SLEPc}
\begin{equation*}
 [\xi_1|\ldots|\xi_N]^T \mathbb{A}_h(\mu) [\xi_1|\ldots|\xi_N]  u_N =[\xi_1|\ldots|\xi_N]^T \mathbf{f}_h(\mu) 
\end{equation*}
with $\mathbb{A}_h$ and $\mathbf{f}_h$ are the discrete counterpart of the differential operator.\\

 \pythoninline{reducedbasis.matProject(A, reducedbasis)}\\ 
 \pythoninline{reducedbasis.dotVec(f)}

 \end{frame}
 

\begin{frame}
 \frametitle{Parametrized compressible linear elasticity}
 The linear compressible elasticity problem reads: find the displacement $u$ such that 
\begin{align*}
 - div(\sigma(\mathbf{u},\mu,\lambda))= f(\beta,\rho,c_i)  &\quad \textrm{in}\, \Omega \\
  \mathbf{u} =0 &\quad \textrm{on}\, \partial \Gamma_{D}\\
 \sigma(\mathbf{u},\mu,\lambda)\cdot \mathbf{n} = \mathbf{g}  &\quad \textrm{on}\,\partial \Gamma_{N}
\end{align*}
 Here the stress tensor $\sigma$  is related to the displacement by Hooke's law:
 \begin{equation*}
  \sigma(\mathbf{u},\mu,\lambda) = 2 \mu \epsilon(\mathbf{u}) + \lambda tr(\epsilon(\mathbf{u})) \mathbf{I}  \quad\textrm{in}\, \Omega
 \end{equation*}
And the source is defined as :
$f(\beta,\rho,c_i)= (0.0,\, -\rho\cdot 9.8 \cdot e^{- \sum_{i=1}^3 c_i \cdot(x_i-\beta_i)^2})$
 \end{frame}
 % \epsilon = strain tensor
 % mu and \lambda Lame parameters


\section{Coupling Nitsche}
\begin{frame}
\frametitle{Locally reduced assembly}
Due to local behavior of modes/basis which is sensitive to the zone of application of the external force/moments, we combine the domain-decomposition paradigm "divide-et-impera'' with reduced assembly technique.\\
The algorithm consist in 
\begin{itemize}
 \item  Decompose the physical domain in part (regular - ad hoc)
 \item  Restrict each reduced basis on each subdomain, constructing the space $V_{RB,i}$ for $i=1,\ldots,\textrm{NbSubDomains}$
 \item  Perform in each subdomain the reduced-integration 
 \item  Re-orthonormalization of each local space.
\end{itemize}
\end{frame}

\begin{frame}
\frametitle{Non-conforming impera algorithm}
\begin{figure}
  \includegraphics[width=0.5\paperwidth]{GluingMesh.png} 
\end{figure}
Some domain decomposition technique: FETI-DP, BDDC, Mortar, Discontinuous Galerkin, InterNODES, Nitsche, \ldots  \\
ROM + domain decomposition: Reduced Basis Element [RBE],  DG-RBE, static condensation reduced basis element, Latin Multiscale

\end{frame}
\begin{frame}
 \frametitle{Why we choose Nitsche technique?}
Pros.\\
 \begin{itemize}
 \item do not require any additional unknown
 \item has the same accurancy property of the underling discretization
 \item avoid of mesh burden near interface
 \item could treat also overlapping subdomains
 \end{itemize}
Cons. \\
The stability of Nitsche method depends on the choose of a penalty scalar value. However, it is possible to estimate this constant using trace inequality ( S. Claus et al.).
\end{frame}

\begin{frame}
\only<1>{
The Nitsche-type weak formulation associated to this problem is: Find a $u_h \in V^k_h$ such that 
\begin{equation*}
 a_h(u_h,v_h)- b_h(u_h,v_h) - b_h(v_h,u_h)  + j_h(u_h,v_h)= L_h(v_h) \quad \forall \, v_h \in V^k_h
\label{eq:NFull}
 \end{equation*}
}
\only<2>{The bilinear form $a_h$ and $b_h$ are defined as
\begin{equation*}
a(u_h,v_h) = \sum_{i=1}^{2} \int_{\Omega_i} \sigma(u^i_h,mu_i,\lambda_i): \epsilon(v^i_h) dX 
\end{equation*}
\begin{equation*}
b(u_h,v_h) = \int_\Gamma \avgw{(\sigma(u_h,mu,\lambda))}{\kappa},\jump{(v\cdot n)} d\Gamma
\end{equation*}
and the penalty-ghost form is
\begin{equation*}
j(u_h,v_h)=   \int_\Gamma \frac{\alpha}{\avg{h}} \jump{u^h}\jump{v^h}d\Gamma
\end{equation*}}
\only<3>{
Create decomposed mesh:\\
\pythoninline{multimesh=dolfin.MultiMesh()}\\
\pythoninline{multimesh.add(mesh)}\\
\pythoninline{multimesh.build()}\\
The MultiMeshForm for the first subdomain:\\ 
 \pythoninline{a = inner(sigma(u,mu0,lmbda0), grad(v))*dX }\\
 \pythoninline{b_u= - inner(w_avg(sigma(u,mu0,lmbda0),kappa0,kappa1),\ }\\
\hspace{2cm}  \pythoninline{tensor_jump(v,n))*dI}\\
\pythoninline{b_v=- inner(w_avg(sigma(v,mu0,lmbda0),kappa0,kappa1), \ }\\
\hspace{2cm}  \pythoninline{tensor_jump(u,n))*dI}\\
\pythoninline{j= alpha/avg(h)*inner(jump(u), jump(v))*dI}
}
\only<4->{The bilinear form $a_h$ and $b_h$ are defined as
\begin{equation*}
a(u_h,v_h) = \sum_{i=1}^{2} \int_{\Omega_i} \sigma(u^i_h,mu_i,\lambda_i): \epsilon(v^i_h) dX 
\end{equation*}
\begin{equation*}
b(u_h,v_h) = \int_\Gamma \avgw{(\sigma(u_h,mu,\lambda))}{\kappa},\jump{(v\cdot n)} d\Gamma
\end{equation*}
and the penalty-ghost form is
\begin{equation*}
j(u_h,v_h)=   \int_\Gamma \frac{\alpha}{\avg{h}} \jump{u^h}\jump{v^h}d\Gamma
\end{equation*}}
\only<4>{
where the penalty parameter is
\begin{equation*}
\alpha= \frac{C_I(d,p)\cdot (E_1\cdot E_2)}{E_1+E_2} 
\end{equation*}
with $d$ is physical dimension and $p$ the order of the polynomial Lagrange discretization space (Tomar remark). 
}
\only<5>{
\begin{equation*}
 \avgw{\mu  u}{\kappa}=(\kappa_1 \mu_1  u^1+  \kappa_2 \mu_2  u^2)_{|\Gamma}
\end{equation*}
where the weights are 
\begin{equation*}
 \kappa_1= \frac{E_1}{E_1+E_2}, \quad \kappa_2= \frac{E_2}{E_1+E_2}
\end{equation*}
}

\end{frame}

\begin{frame}
\frametitle{Numerical test with single material}
\begin{figure}
  \includegraphics[width=0.5\paperwidth]{MatchingMesh.png} 
  \includegraphics[width=0.5\paperwidth]{NonMatchingMesh2.png} 
\end{figure}
\end{frame}

 \begin{frame}
 \frametitle{Numerical test with two materials}
 \begin{figure}
 \includegraphics[width=0.5\paperwidth]{DifferentMaterial.png} 
 \end{figure}
 \end{frame}

 \section{Validation test of iRB-Nitsche: 2D and 3D cantiliver bar}
\begin{frame}
 \frametitle{Numerical Validation Test:2D}
\begin{figure}
  \includegraphics[width=0.45\paperwidth]{Deformation.png}
  \includegraphics[width=0.45\paperwidth]{DeformationYWrap.png}
\end{figure}
Number of  dof of high-fidelity discretization:7442\\
Number of  dof of reduced assembly discretization: 2796\\
Speed-up: x40 time with iRB+Nitsche\\
Speed-up: x10 time with reduced assembly.\\
Error: $L_2$-norm:=3.92e-07, $H_1$-norm=2.03e-06 (1st SubDomain) \\
$L_2$-norm:=2.23e-08, $H_1$-norm=5.47e-08 (2nd SubDomain)\\
\end{frame}

\begin{frame}
\frametitle{Numerical Validation Test:3D}
\begin{figure}
  \includegraphics[width=0.45\paperwidth]{MeshReduced3D.png}
  \includegraphics[width=0.45\paperwidth]{ReducedSol3D.png}
\end{figure}
Number of  dof of high-fidelity discretization:9600\\
Number of  dof of reduced assembly discretization: 6953\\
\end{frame}






 

 
 
\end{document}
