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switchingcontrol_ell.m
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switchingcontrol_ell.m
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function switchingcontrol_ell
% This function solves the elliptic switching control problem
% min 1/2 \|y-yd\|^2 + alpha/2 \int |u|^2 + beta |u_1u_2|_0 dx_1
% s.t. -\Delta y = \chi_{\omega_1} u_1 + \chi_{\omega_2} u_2
% using the approach described in the paper
% "A convex analysis approach to optimal controls with
% switching structure for partial differential equations"
% by Christian Clason, Kazufumi Ito and Karl Kunisch, see
% http://math.uni-graz.at/mobis/publications/SFB-Report-2014-001.pdf.
%
% August 7, 2014 Christian Clason <[email protected]>
% http://udue.de/clason
%% setup
N = 128; % number of spatial grid points
alpha = 1e-3; % quadratic penalty
beta = 1e-3; % switching penalty
maxit = 99; % max iterations in semismooth Newton method
tmin = 1e-12; % min step size for line search
tol = 1e-6; % terminate Newton if residual norm smaller than tol
% grid, differential operators
[A,M,xx,yy] = assembleFEM(N);
tplot = @(n,f,s) tplot_(n,f,s,N,xx,yy);
% control domains
om1 = xx(1,:) < 1/4;
om2 = xx(1,:) > 3/4;
dplot = @(n,f,s) dplot_(n,f,s,N,xx,yy,om1,om2);
% control operator
BT = [kron(speye(N),om1')';kron(speye(N),om2')']*M'; BB = BT';
% target
yd = xx.*sin(2*pi*xx).*sin(2*pi*yy);
yd = yd(:);
dplot(1,yd,'target and control domains');
%% initialization
y = zeros(N*N,1);
p = zeros(N*N,1);
As_old = zeros(N,7);
sqab = sqrt(2*alpha*beta);
% constant blocks of Newton matrix
Lyy = M;
Lpy = A; Lyp = A';
% continuation strategy
for gamma = 10.^(-(0:16))
fprintf('\nSolving for gamma = %1.0e\n', gamma);
fprintf('It\tupdate\tresidual\tstep size\tQ1\tQ2\tQ0\tQ10\tQ20\tQ00\tQ12\n');
% semismooth Newton iteration
it = 1; nold = 1e99; tau = 1;
while true
% compute active sets
q = BT*p; q1 = q(1:N); q2 = q(1+N:end);
As(:,1) = (abs(q1) > (1+gamma/alpha)*abs(q2)) & (abs(q2) < sqab);
As(:,2) = (abs(q2) > (1+gamma/alpha)*abs(q1)) & (abs(q1) < sqab);
As(:,3) = (abs(q1) > (1+gamma/alpha)*sqab) & (abs(q2) > (1+gamma/alpha)*sqab);
As(:,4) = (sqab <= abs(q1)) & (abs(q1) <= (1+gamma/alpha)*sqab) & ...
(abs(q2) > (1+gamma/alpha)*sqab);
As(:,5) = (sqab <= abs(q2)) & (abs(q2) <= (1+gamma/alpha)*sqab) & ...
(abs(q1) > (1+gamma/alpha)*sqab);
As(:,6) = (sqab <= abs(q1)) & (abs(q1) <= (1+gamma/alpha)*sqab) & ...
(sqab <= abs(q2)) & (abs(q2) <= (1+gamma/alpha)*sqab) & ...
((abs(q1)+abs(q2)) > (2+gamma/alpha)*sqab);
As(:,7) = (alpha/(alpha+gamma)*abs(q2) <= abs(q1)) & ...
(abs(q1) <= (1+gamma/alpha)*abs(q2)) & ...
((abs(q1)+abs(q2)) <= (2+gamma/alpha)*sqab);
Hg = [1/(alpha+gamma)*(As(:,1)+As(:,3)+As(:,4)).*q1 + ...
1/gamma*(As(:,5)+As(:,6)).*(q1-sign(q1)*sqab) + ...
1/(gamma*(2*alpha+gamma))*As(:,7).*((alpha+gamma)*q1-alpha*sign(q1).*abs(q2)); ...
1/(alpha+gamma)*(As(:,2)+As(:,3)+As(:,5)).*q2 + ...
1/gamma*(As(:,4)+As(:,6)).*(q2-sign(q2)*sqab) + ...
1/(gamma*(2*alpha+gamma))*As(:,7).*((alpha+gamma)*q2-alpha*sign(q2).*abs(q1))];
% gradient
Ly = A'*p + M*(y-yd);
Lp = A*y - BB*Hg;
F = -[Ly; Lp];
ng = norm(F);
% line search
if ng >= nold % if no decrease: backtrack (never on first iteration)
tau = tau/2;
y = y - tau*dx(1:N*N);
p = p - tau*dx(1+N*N:end);
if tau < tmin % terminate Newton iteration
disp('step size too small')
break;
else % bypass rest of while loop; compute new gradient
continue;
end
end
% decrease (or on first iteration): accept step
update = nnz(As-As_old);
fprintf('%i\t%d\t\t%1.3e\t%1.3e\t%d\t%d\t%d\t%d\t%d\t%d\t%d\n',...
it,update,ng,tau,sum(As,1));
% terminate Newton?
if update == 0 && ng < tol % success, solution found
break;
elseif it == maxit % failure, too many iterations
break;
end
% otherwise update information, continue
it = it+1; nold = ng; tau = 1; As_old = As;
% Newton matrix: iteration dependent block
H11 = spdiags(1/(alpha+gamma)*(As(:,1)+As(:,3)+As(:,4)) + ...
1/gamma*(As(:,5)+As(:,6)) + ...
(alpha+gamma)/(gamma*(2*alpha+gamma))*As(:,7),0,N,N);
H12 = spdiags(-alpha/(gamma*(2*alpha+gamma))*As(:,7).*sign(q1).*sign(q2),0,N,N);
H22 = spdiags(1/(alpha+gamma)*(As(:,2)+As(:,3)+As(:,5)) + ...
1/gamma*(As(:,4)+As(:,6)) + ...
(alpha+gamma)/(gamma*(2*alpha+gamma))*As(:,7),0,N,N);
Lpp = -BB*[H11 H12; H12 H22]*BT;
H = [Lyy Lyp; Lpy Lpp];
% solve for Newton direction
dx = H\F;
y = y + dx(1:N*N);
p = p + dx(1+N*N:2*N*N);
end % Newton
% compute control from optimality condition, plot
u = [Hg(1:N),Hg(1+N:end)];
figure(2);plot(xx(1,:),u); title('control')
tplot(3,y,'state');
% terminate continuation if Newton iteration converged in one step
if it==1 && gamma < alpha
break;
end
end % continuation
end
%%
function [K,M,xx,yy] = assembleFEM(n)
a = 0; b = 1; % computational domain [a,b]^2
nel = 2*(n-1)^2; % number of nodes
h2 = ((b-a)/(n-1))^2; % Jacobi determinant of transformation (2*area(T))
% nodes
[xx,yy] = meshgrid(linspace(0,1,n));
% triangulation
tri = zeros(nel,3);
ind = 1;
for i = 1:n-1
for j = 1:n-1
node = (i-1)*n+j+1; % two triangles at node
tri(ind,:) = [node node-1 node+n]; % triangle 1 (lower left)
tri(ind+1,:) = [node+n-1 node+n node-1]; % triangle 2 (upper right)
ind = ind+2;
end
end
% Mass and stiffness matrices
Ke = 1/2 * [2 -1 -1 -1 1 0 -1 0 1]'; % elemental stiffness matrix
Me = h2/24 * [2 1 1 1 2 1 1 1 2]'; % elemental mass matrix
ent = 9*nel;
row = zeros(ent,1);
col = zeros(ent,1);
valk = zeros(ent,1);
valm = zeros(ent,1);
ind = 1;
for el=1:nel
ll = ind:(ind+8); % local node indices
gl = tri(el,:); % global node indices
row(ll) = gl([1;1;1],:); rg = gl';
col(ll) = rg(:,[1 1 1]);
valk(ll) = Ke;
valm(ll) = Me;
ind = ind+9;
end
M = sparse(row,col,valm);
K = sparse(row,col,valk);
% modify matrices for homogenenous Dirichlet conditions
bdnod = [find(abs(xx-a) < eps); find(abs(yy-a) < eps); ...
find(abs(xx-b) < eps); find(abs(yy-b) < eps)];
M(bdnod,:) = 0;
K(bdnod,:) = 0; K(:,bdnod) = 0;
for j = bdnod'
K(j,j) = 1; %#ok<SPRIX>
end
end
%%
function tplot_(n,f,s,N,x,y)
figure(n);
surf(x,y,reshape(f,N,N));
colormap summer
shading interp; lighting phong; camlight headlight; alpha(0.8);
title(s); xlabel('x_1'); ylabel('x_2');
drawnow;
end % tplot_ function
function dplot_(n,f,s,N,x,y,om1,om2)
figure(n);
surf(x,y,reshape(f,N,N));
colormap summer
shading interp; lighting phong; camlight headlight; alpha(0.8);
hold on
contour(x,y,reshape(1.0*(om1+om2)'*ones(1,N),N,N),1,'LineWidth',2);
hold off
title(s); xlabel('x_1'); ylabel('x_2');
drawnow;
end % dplot_ function