Parameter Sweep of ODEs
This is a parameter sweep study of a 2nd order ODE system.
We solve the ODE for a time span of 0 to 25 seconds, with initial conditions 
and 
. We sweep the parameters 
and 
and record the peak values of 
for each condition. At the end, we plot a surface of the results.
Contents
Initialize Problem
m = 5; % mass bVals = linspace(0.1, 5, 50); % damping values kVals = linspace(1.5, 5, 70); % stiffness values [kGrid, bGrid] = meshgrid(bVals, kVals); peakVals = nan(size(kGrid));
Parameter Sweep (Serial)
We'll first do the parameter sweep computation in a serial for loop.
disp('Computing in serial...');drawnow; tic; for idx = 1:numel(kGrid) % Solve ODE [T,Y] = ode45(@(t,y) odesystem(t, y, m, bGrid(idx), kGrid(idx)), ... [0, 25], ... % simulate for 25 seconds [0, 1]) ; % initial conditions % Determine peak value peakVals(idx) = max(Y(:,1)); end t1 = toc; fprintf('Elapsed time is %0.2f seconds.\n', t1);
Computing in serial... Elapsed time is 16.98 seconds.
Parameter Sweep (Parallel)
Next, we convert the for loop to a parfor loop and start a pool or MATLAB workers.
disp('Computing in parallel...');drawnow; matlabpool open tic; parfor idx = 1:numel(kGrid) % Solve ODE [T,Y] = ode45(@(t,y) odesystem(t, y, m, bGrid(idx), kGrid(idx)), ... [0, 25], ... % simulate for 25 seconds [0, 1]) ; % initial conditions % Determine peak value peakVals(idx) = max(Y(:,1)); end t2 = toc; fprintf('Elapsed time is %0.2f seconds.\n', t2); % Close MATLAB Pool matlabpool close
Computing in parallel... Starting matlabpool using the 'local' configuration ... connected to 2 labs. Elapsed time is 8.45 seconds. Sending a stop signal to all the labs ... stopped.
Visualize
figure; surf(bVals, kVals, peakVals); xlabel('Damping (b)'); ylabel('Stiffness (k)'); zlabel('Peak Response'); view(50, 30)
Speed Up
We can see that we get a very good speed up. Since this loop has very minimal overhead in terms of data transfer, we may see a super-linear speed up, which sometimes happens due to the Just In Time acceleration.
fprintf('\n\nSpeed up (time serial / time parallel): %0.2f\n', t1/t2);
Speed up (time serial / time parallel): 2.01