### Extrap: update to notes, initial commit on code folder and AltS algorithm (4 lines)

parent 492662b6
 function [u1,u2] = ALGO_extrap_AltS_v1_20210708(F1,F2,u2,N) % ALTS Performs alternating Schwarz % [u1,u2] = AltS(F1,F2,u2,N) returns u1 and u2, the solution on the first % and second subdomains, respectively, using N iterations of the % alternating Schwarz method. The functionals F1 and F2 compute the % solutions on the respective subdomains by taking the solution on the % previous subdomain for its boundary data. for i=1:N u1=F1(u2); u2=F2(u1); end \ No newline at end of file
 ... ... @@ -73,6 +73,7 @@ d^n = & G d^{n-1} = G^n d^0 \\ \sum \gamma_j d^j = & \sum \gamma_j G^j d^0 = p_n(G) d^0. \end{align*} Minimizing this last equation is called Modified Polynomial Extrapolation. (nb: no it's not, it's called Minimized Polynomial Extrapolation) Overdetermine the system: \begin{equation*} ... ... @@ -191,7 +192,7 @@ which may be readily obtained by rearranging the relation $R$. \section{Summary of Sidi proof of MPE=Arnoldi} Let $(x_n)$ be a sequence to be accelerated, then we produce approximations Let $(x_n)$ be a vector sequence to be accelerated, then we produce approximations \begin{equation} T_n^{(k)} = \sum_{j=0}^k \gamma_j^{(n,k)} x_{n+j} \end{equation} ... ... @@ -200,7 +201,7 @@ where \sum_{j=0}^k \gamma_j^{(n,k)} = 1 . \end{equation*} The $\gamma_j^{(n,k)}$ also satisfy \begin{equation} \begin{equation} \label{eq:gamma 2} \sum_{j=0}^k \langle \Delta x_{n+i}, \Delta x_{n+j} \rangle \gamma_j^{(n,k)} = 0, \quad 0 \leq i \leq k-1 \end{equation} where $\langle \cdot, \cdot \rangle$ is the L2 inner product. ... ... @@ -235,7 +236,11 @@ Moreover, \end{equation} Thus, \begin{equation} r(T_n^{(k)}) = \sum \gamma_i^{(n,k)} \Delta x_{n+i} \in \Span \set{\Delta x_n, A \Delta x_n, \dots , A^k \Delta x_n}. r(T_n^{(k)}) = \sum \gamma_i^{(n,k)} \Delta x_{n+i} \in \Span \set{\Delta x_n, A \Delta x_n, \dots , A^k \Delta x_n} = K_{k-1}(A,\Delta x_n) \end{equation} where $K_{k+1}(A,\Delta x_n)$ is the Krylov subspace of dimension $k+1$ generated by $A$ and $\Delta x_n$. By Equation (\ref{eq:gamma 2}) we have that $\langle v, r(T_n^{(k)}) \rangle = 0$ for all $v \in K_{k}(A,\Delta x_n)$. Therefore, MPE is an orthogonal projection method. In fact, it is identical in methodology to the Arnoldi process. \end{document} \ No newline at end of file
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