def run_example(with_plots=True):
r"""
This example shows how to use Assimulo and CVode for simulating sensitivities
for initial conditions.
.. math::
\dot y_1 &= -(k_{01}+k_{21}+k_{31}) y_1 + k_{12} y_2 + k_{13} y_3 + b_1\\
\dot y_2 &= k_{21} y_1 - (k_{02}+k_{12}) y_2 \\
\dot y_3 &= k_{31} y_1 - k_{13} y_3
with the parameter dependent inital conditions
:math:`y_1(0) = 0, y_2(0) = 0, y_3(0) = 0` . The initial values are taken as parameters :math:`p_1,p_2,p_3`
for the computation of the sensitivity matrix,
see http://sundials.2283335.n4.nabble.com/Forward-sensitivities-for-initial-conditions-td3239724.html
on return:
- :dfn:`exp_mod` problem instance
- :dfn:`exp_sim` solver instance
"""
def f(t, y, p):
y1, y2, y3 = y
k01 = 0.0211
k02 = 0.0162
k21 = 0.0111
k12 = 0.0124
k31 = 0.0039
k13 = 0.000035
b1 = 49.3
yd_0 = -(k01 + k21 + k31) * y1 + k12 * y2 + k13 * y3 + b1
yd_1 = k21 * y1 - (k02 + k12) * y2
yd_2 = k31 * y1 - k13 * y3
return N.array([yd_0, yd_1, yd_2])
#The initial conditions
y0 = [0.0, 0.0, 0.0] #Initial conditions for y
p0 = [0.0, 0.0, 0.0] #Initial conditions for parameters
yS0 = N.array([[1, 0, 0], [0, 1, 0], [0, 0, 1.]])
#Create an Assimulo explicit problem
exp_mod = Explicit_Problem(f,
y0,
p0=p0,
name='Example: Computing Sensitivities')
#Sets the options to the problem
exp_mod.yS0 = yS0
#Create an Assimulo explicit solver (CVode)
exp_sim = CVode(exp_mod)
#Sets the paramters
exp_sim.iter = 'Newton'
exp_sim.discr = 'BDF'
exp_sim.rtol = 1e-7
exp_sim.atol = 1e-6
exp_sim.pbar = [
1, 1, 1
] #pbar is used to estimate the tolerances for the parameters
exp_sim.report_continuously = True #Need to be able to store the result using the interpolate methods
exp_sim.sensmethod = 'SIMULTANEOUS' #Defines the sensitvity method used
exp_sim.suppress_sens = False #Dont suppress the sensitivity variables in the error test.
#Simulate
t, y = exp_sim.simulate(400) #Simulate 400 seconds
#Basic test
nose.tools.assert_almost_equal(y[-1][0], 1577.6552477, 5)
nose.tools.assert_almost_equal(y[-1][1], 611.9574565, 5)
nose.tools.assert_almost_equal(y[-1][2], 2215.88563217, 5)
nose.tools.assert_almost_equal(exp_sim.p_sol[0][1][0], 1.0)
#Plot
if with_plots:
title_text = r"Sensitivity w.r.t. ${}$"
legend_text = r"$\mathrm{{d}}{}/\mathrm{{d}}{}$"
P.figure(1)
P.subplot(221)
P.plot(t,
N.array(exp_sim.p_sol[0])[:, 0], t,
N.array(exp_sim.p_sol[0])[:, 1], t,
N.array(exp_sim.p_sol[0])[:, 2])
P.title(title_text.format('p_1'))
P.legend((legend_text.format('y_1',
'p_1'), legend_text.format('y_1', 'p_2'),
legend_text.format('y_1', 'p_3')))
P.subplot(222)
P.plot(t,
N.array(exp_sim.p_sol[1])[:, 0], t,
N.array(exp_sim.p_sol[1])[:, 1], t,
N.array(exp_sim.p_sol[1])[:, 2])
P.title(title_text.format('p_2'))
P.legend((legend_text.format('y_2',
'p_1'), legend_text.format('y_2', 'p_2'),
legend_text.format('y_2', 'p_3')))
P.subplot(223)
P.plot(t,
N.array(exp_sim.p_sol[2])[:, 0], t,
N.array(exp_sim.p_sol[2])[:, 1], t,
N.array(exp_sim.p_sol[2])[:, 2])
P.title(title_text.format('p_3'))
P.legend((legend_text.format('y_3',
'p_1'), legend_text.format('y_3', 'p_2'),
legend_text.format('y_3', 'p_3')))
P.subplot(224)
P.title('ODE Solution')
P.plot(t, y)
P.suptitle(exp_mod.name)
P.show()
return exp_mod, exp_sim
