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Copy pathFE_Example.m
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151 lines (112 loc) · 4.43 KB
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function FE_Example
hodlroption('threshold', 1e-8);
show_plot = false;
Ns = 2.^( 9 : 16 );
for k = 1 : 4
switch mod(k, 2)
case 1
beta1 = 1.3;
beta2 = 1.7;
case 0
beta1 = 1.7;
beta2 = 1.9;
end
times = zeros(1, length(Ns));
ranks = zeros(1, length(Ns));
qsranks = zeros(1, length(Ns));
for i = 1 : length(Ns)
n = Ns(i);
h = 1 / (n+2);
[am1, ap1] = symbol_FE2(beta1, n);
[am2, ap2] = symbol_FE2(beta2, n);
L1 = (h^(-beta1) / gamma(4-beta1)) * hodlr('toeplitz', am1, ap1, n);
L2 = (h^(-beta2) / gamma(4-beta2)) * hodlr('toeplitz', am2, ap2, n);
L1 = L1 + L1';
L2 = L2 + L2';
% Mass matrix
sM = spdiags(ones(n, 1) * [1/6 2/3 1/6], -1:1, n, n);
M = hodlr('banded', sM, 1, 1);
% Choose a reasonable time step for this problem
dt = .1;
L1 = dt * (M\L1) - eye(n, 'like', L1) / 2;
L2 = dt * (M\L2) - eye(n, 'like', L2) / 2;
tic;
ranks(i) = 0;
if k <= 2
L1s = ek_struct(L1, false);
L2s = ek_struct(L2, false);
qsranks(i) = max(hodlrrank(L1), hodlrrank(L2));
else
pp1 = ones(n, 1); % pp(t', beta1);
pm1 = ones(n, 1); % pm(t', beta1);
tau1 = dt / h^(beta1) / gamma(4-beta1);
tau2 = dt / h^(beta2) / gamma(4-beta2);
nrm = 1;
D1 = .5 * tau1 * spdiags(pp1, 0, n, n);
D2 = .5 * tau1 * spdiags(pm1, 0, n, n);
if beta1 < .5
B = spdiags(ones(n,1) * [ 1 -1 ], 0 : 1, n, n);
else
B = spdiags(ones(n,1) * [ -1 2 -1 ], -1 : 1, n, n);
end
[LL1, UU1] = lu(D1 * B + D2 * B' + .5 * speye(n));
% Construct symbol of the Toeplitz part of the linear coefficients
tam1 = am1 * tau1;
tam1(1) = tam1(1) - .25 * sM(1,1); tam1(2) = tam1(2) - .25 * sM(2,1);
tap1 = ap1 * tau1; tap1(1) = tam1(1); tap1(2) = tap1(2) - sM(1,2) * .25;
L1s = ek_gmres_struct(@(x) sM \ (nrm \ mat_mul1D(tam1, tap1, pp1, pm1, 0, x)), ...
@(x) UU1 \ (LL1 \ x), norm(L1));
qp1 = ones(n, 1); % qp(t', beta2);
qm1 = ones(n, 1); % qm(t', beta2);
D1 = .5 * tau2 * spdiags(qp1, 0, n, n);
D2 = .5 * tau2 * spdiags(qm1, 0, n, n);
if beta2 < .5
B = spdiags(ones(n,1) * [ 1 -1 ], 0 : 1, n, n);
else
B = spdiags(ones(n,1) * [ -1 2 -1 ], -1 : 1, n, n);
end
[LL2, UU2] = lu(D1 * B + D2 * B' + .5 * speye(n));
tam2 = am2 * tau2;
tam2(1) = tam2(1) - sM(1,1) * .25; tam2(2) = tam2(2) - sM(2,1) * .25;
tap2 = ap2 * tau2; tap2(1) = tam2(1); tap2(2) = tap2(2) - sM(1,2) * .25;
L2s = ek_gmres_struct(@(x) nrm \ mat_mul1D(tam2, tap2, qp1, qm1, 0, x), ...
@(x) UU2 \ (LL2 \ x), norm(L2));
end
Xu = [ zeros(3*n/8, 1) ; .5 ; ones(n/4-2,1) ; .5 ; zeros(3*n/8, 1) ];
Xv = Xu;
f1 = Xu; f2 = Xv;
timesteps = 8;
if show_plot
t = linspace(0, 1, n);
if mod(timesteps, 2) == 0
subplot(2, timesteps / 2, 1);
else
subplot(1, timesteps, 1);
end
[XX, YY] = meshgrid(t, t);
mesh(XX, YY, Xu * Xv');
pause
end
for j = 1 : timesteps - 1
% Make sure the RHS is in compressed form
[UU, VV] = compress_low_rank([dt * f1, -Xu], [f2, Xv], 1e-6);
[Xu, Xv] = ek_sylv(L1s, L2s, -UU, VV, inf, ...
@(r, nrm) r < 1e-6 * nrm * n, false, 'fro');
ranks(i) = max(ranks(i), size(Xu, 2));
if show_plot
if mod(timesteps, 2) == 0
subplot(2, timesteps / 2, j+1);
else
subplot(1, timesteps, j+1);
end
mesh(XX, YY, Xu * Xv')
end
end
times(i) = toc;
fprintf('N = %d, time = %e, rank = %d, qsrank = %d\n', n, times(i), ranks(i), qsranks(i));
end
dlmwrite(sprintf('fe-times_%d.dat', k), [ Ns ; times ; ranks ; qsranks ]', '\t');
end
%V = [ dlmread('fe-times_1.dat'), dlmread('fe-times_3.dat') ];
%dlmwrite('fe-times_13.dat', V, '\t');
end