Extrap: updates after July meeting

Research/Extrapolation methods/Code/ALGO_extrap_AltS_v2_20210713.m

-function [u1,u2] = ALGO_extrap_AltS_v2_20210713(A,b,f,B1,B2,ind1,ind2)
+function [u1,u2,G] = ALGO_extrap_AltS_v2_20210713(A,b,f,u2,B1,B2,ind1,ind2,n,m,x)
 % ALTS performs alternating Schwarz, in a compact implementation
 %   [u1,u2] = AltS(?)
 
-m=1; N=10;
-
-I=speye(size(A));
-R1=I(ind1,:); P1=I(:,ind1); A1=R1*A*P1; b1=R1*A*b;
-R2=I(ind2,:); P2=I(:,ind2); A2=R2*A*P2; b2=R2*A*b;
-B1=R1*A*B1*P2; B2=R2*A*B2*P1;
-for i=1:N
+N=length(A); I=speye(N);
+R1=I(ind1,:); P1=I(:,ind1);     % restriction and prolongation operators
+R2=I(ind2,:); P2=I(:,ind2); 
+A1=R1*A*P1; b1=R1*b;            % restrict and prolong A and b
+A2=R2*A*P2; b2=R2*b;
+C1=R1*A*B1*P2; C2=R2*A*B2*P1;   % info passed between subdomains
+u1=zeros(N,1); u1=u1(ind1);     % initial guess for u1
+for i=1:n
+  for j=1:m
+    u1 = A1 \ (f(u1) - C1*u2 - b1);
+  end
   for j=1:m
-    u1 = A1 \ (f(u1) - B1*u2 - b1);
-    u2 = A2 \ (f(u2) - B2*u1 - b2);
+    u2 = A2 \ (f(u2) - C2*u1 - b2);
+  end
+    
+  if nargin==11
+      plot(x(ind1),u1,'r',x(ind2),u2,'b')
+      axis([0,1,-1,0])
+      pause(0.1)
+  end
+  if nargout==3
+    G=B1*P2*u2;
   end
 end
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Research/Extrapolation methods/Code/EX_extrap_1DLaplace_v2_20210714.m

+% EX_extrap_1DLaplace_v2_20210714
+%   1DLAPLACE solves the 1D Laplace problem on [0,1] using alternating Schwarz
+
+N=64; x=1:N-1; x=x'/N; M=N-1;
+L=ones(M,1); L=spdiags([L,-2*L,L],0:2,M,M+2)*N^2;
+B=[L(:,1),sparse(M,M),L(:,end)]; L=L(:,2:end-1);
+u=zeros(N+1,1); % BCs
+
+% Subdomains: [0,a] and [b,1]
+a=0.6; b=0.4;
+ind1=x<a; ind2=x>b; l1=sum(ind1); l2=M-sum(ind2);
+B1=sparse(l1+1,l1+1,1,M,M);
+B2=sparse(l2,l2,1,M,M);
+
+gamma=-2:0.01:2; G=zeros(size(gamma)); alpha=3.6;
+for i=1:length(gamma)
+  u2=gamma(i)*ones(M,1); u2=u2(ind2);
+  [u1,u2,bc] = ALGO_extrap_AltS_v2_20210713(L,B*u,@(v) sin(alpha*v),...
+                                       u2,B1,B2,ind1,ind2,1,100);
+  bc=bc(~ind1); G(i)=bc(1);
+end
+
+plot(gamma,G,'b-')
+hold on
+for k=-1:0.25:1
+  plot(gamma,k*sqrt(abs(gamma))+gamma,'k--')
+end
+hold off
+axis([-2,2,-2,2])
+axis square
+xlabel('\gamma')
+ylabel('G(\gamma)')
\ No newline at end of file

Research/Extrapolation methods/notes.tex

@@ -39,6 +39,10 @@
 
 \begin{document}
 
+\section{Meeting notes}
+
+\subsection{}
+
 Nonlinear preconditioning: Krylov as extrapolation methods.
 There exists proof of quadratic convergence (Sidi, \textit{Extrapolation vs. projection methods for linear systems of equations}, copy here)
 
@@ -99,6 +103,20 @@ Questions:
 \item do the sequences of Schwarz methods have distinct forms? Can they be predicted/exploited?
 \end{itemize}
 
+\subsection{Meeting 14.07.2021}
+
+Goal is to take Newton's method and connect it to an extrapolation method and then a Krylov method, in an attempt to unify these types of nonlinear solves.
+Also to apply extrapolation methods to the sinusoid altS example to see how they deal with these problems.
+(Extrapolation methods are unlikely to find fault with this example as they will only accelerate a monotonically converging sequence; only a method seeking to quadratically converge would dramatically change the convergence behaviour.)
+
+To read:
+\begin{itemize}
+\item Brezinski, Numerical stability of a quadratic method for solving systems of nonlinear equations
+\item Shafer, On quadratic approximation
+\item Brezinski, Méthodes d'acceleartion de la convergence en analyse numerique
+\item Gekeler, On the solution of systems of equations by the epsilon algorithm of Wynn
+\end{itemize}
+
 \section{Framework of acceleration methods}
 
 Take a sequence $(S_n) \to S$.