SteadyStateApproximation/pbe_existence_region.py at master · MathBioCU/SteadyStateApproximation · GitHub

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# -*- coding: utf-8 -*-
#Created on Oct 14, 2015
#@author: Inom Mirzaev

"""
    This program computes existence and stability regions of the population
    balance equations. Computed regions are plotted in 3D and saved in 'images'
    folder. All the model rates should be changed in 'pbe_model_rates.py' file.
    The program has been written in parallel. Therefore, for the faster
    computation, the parameter 'ncpus' in 'pbe_model_rates' should
    be set to the maximum number of cores available.

"""


from pbe_model_rates import *
from scipy.optimize import fsolve


import multiprocessing as mp
import time , os


start = time.time()


x = np.ravel( grid_x )
y = np.ravel( grid_y )
z = np.ravel( grid_z )

myarray = np.array([x, y, z]).T


def region_plots(nn, myarray = myarray):

    pos_sol = 0
    eigs2=0

    #Initialize the approximate operators
    An, Ain, Aout, nu, N, dx = initialization( 50 , myarray[nn , 0] , myarray[nn , 1] , myarray[nn, 2] )

    root_finding  = partial( approximate_IG , An=An, Aout=Aout, Ain=Ain )
    exact_jacobian = partial( jacobian_IG , An=An, Aout=Aout, Ain=Ain)

    #Search the root of the system for 10 different initial seeds
    for mm in range( 10 ):

        seed = 2**mm * np.ones(N)
        sol = fsolve( root_finding ,  seed , fprime = exact_jacobian , xtol = 1e-8 , full_output=1 )

        #Break the loop if a positive solution is found
        if sol[2]==1 and np.linalg.norm( sol[0] ) > 1 and np.all( sol[0] > 0 ):
            pos_sol = 1
            eigs2 = np.max(  np.real ( np.linalg.eig(  exact_jacobian( sol[0] ) )[0] ) )
            break

    return ( myarray[nn , 0] , myarray[nn , 1] , myarray[nn, 2] , pos_sol ,  eigs2 )


if __name__ == '__main__':

    #Number of CPUs used for computations
    pool = mp.Pool( processes = ncpus )
    ey_nana = range( len( myarray) )
    result = pool.map( region_plots , ey_nana )

    #Save the output in the 'data_files' folder
    output = np.asarray(result)
    fname = 'pbe_data'
    np.save( os.path.join( 'data_files' , fname ) , output )


from scipy.spatial import  ConvexHull
from mpl_toolkits.mplot3d import Axes3D
from scipy.interpolate import griddata

import matplotlib.pyplot as plt
import matplotlib


fname = 'pbe_data.npy'

output=np.load( os.path.join( 'data_files' , fname ) )
output = output[ np.nonzero( output[: , 3 ] ) ]


points = output[ :, 0:3]
values = output[ : , -1 ]


eigs = griddata( points , values , ( grid_x , grid_y , grid_z ) )

out = np.array([np.ravel(grid_x) , np.ravel(grid_y)  , np.ravel(grid_z) , np.ravel(eigs)] ).T

mypts = out[ np.nonzero( np.isnan(out[:, 3] )==False )[0] ]


"""
    Plots the existence region
"""
plt.close('all')

fig = plt.figure(0)
ax = fig.add_subplot(111, projection='3d')


hull = ConvexHull( mypts[ : , 0:3] )
simp = hull.points[ hull.vertices ]

ax.plot_trisurf(mypts[:, 0] , mypts[:, 1] , mypts[:, 2] , triangles=hull.simplices,
                linewidth=0, color='#8A2BE2', shade=False)


ax.view_init(  azim=115 , elev=25 )

ax.set_xlabel( '$a$'    , fontsize=20 )
ax.set_ylabel( '$b$'    , fontsize=20 )
ax.set_zlabel( '$c$'    , fontsize=20 )
ax.set_xlim( amin , amax )
ax.set_ylim( bmin , bmax )
ax.set_zlim( cmin , cmax )


plt.savefig( os.path.join( 'images' , 'existence_region.png' ) , dpi=400 ,bbox_inches='tight')


"""
    Plots the stability region
"""
fig = plt.figure(1)

ax = fig.add_subplot(111, projection='3d')

neg_jac = np.nonzero( mypts[ : , -1] < 0 )[0]
pos_jac = np.nonzero( mypts[ : , -1] >= 0 )[0]

ax.scatter( mypts[ neg_jac , 0] , mypts[ neg_jac , 1 ] , mypts[ neg_jac , 2 ] , color='red' , label = 'stable' )
ax.scatter( mypts[ pos_jac , 0] , mypts[ pos_jac , 1 ] , mypts[ pos_jac , 2 ] , color='blue' , label = 'unstable' )

scatter1_proxy = matplotlib.lines.Line2D( [0],[0] , linestyle="none" , c='blue', marker = 'o' )
scatter2_proxy = matplotlib.lines.Line2D( [0],[0] , linestyle="none" , c='red', marker = 'o' )
ax.legend( [ scatter1_proxy , scatter2_proxy ] , [ 'stable' , 'unstable' ] , numpoints = 1)

ax.view_init(  azim=115 , elev=25 )
ax.set_xlabel( '$a$'    , fontsize=20 )
ax.set_ylabel( '$b$'    , fontsize=20 )
ax.set_zlabel( '$c$'    , fontsize=20 )
ax.set_xlim( amin , amax )
ax.set_ylim( bmin , bmax )
ax.set_zlim( cmin , cmax )

plt.savefig( os.path.join( 'images' , 'stability_region.png' ) , dpi=400 , bbox_inches='tight' )

end = time.time()


print "Time elapsed ", round( end - start , 2) , " seconds"