Estimating the parameters of a PDEs system using Fipy

Question

I am solving a system of PDEs using Fipy which involve two parameters or constants, so I would like to know whether is also possible to estimate those parameters in Fipy, or what others libraries would be more adequate for that.

Note: I know that scipy has some functions for that (optimize.minimize for MLE), but I am not sure if is adequate apply them to a code of Fipy.

UPDATE: For the below PDE system I want to estimate the two unknown parameters: "Beta" and "m"

The function for solving this PDE in Fipy would be something like that:

import scipy as sci
import fipy as fipy
import numpy as np
from fipy import *

# Grid
nx = 100
ny = 100

dx = 1.
dy = dx

mesh = Grid2D(nx=nx, ny=ny, dx=dx, dy=dy)

x = mesh.cellCenters[0]
y = mesh.cellCenters[1]

# Setting variable of results and adding inicial conditions

u = CellVariable(name="Individual 1", mesh=mesh, value=0.)
u.setValue(1., where=(50. < x) & (70. > x) & (50. < y) & (70. > y))

v = CellVariable(name="Individual 2", mesh=mesh, value=0.)
v.setValue(1., where=(40. < x) & (60. > x) & (40. < y) & (60. > y))

p = CellVariable(name= "Marks Individual 1", mesh=mesh, value=0.)
p.setValue(1., where=(50. < x) & (70. > x) & (50. < y) & (70. > y))

q = CellVariable(name= "Marks Individual 2", mesh=mesh, value=0.)
q.setValue(1., where=(40. < x) & (60. > x) & (40. < y) & (60. > y))

# Plotting inicial conditions
if __name__ == '__main__':
    viewer = Viewer(u, v, datamin=0., datamax=1.)
    viewer.plot()

# Setting PDE 
def HRMLE(params):
    m = params[0]
    beta = params[1]
    D = 1.

    CU = CellVariable(mesh=mesh, rank=1)
    CU[:]= 1.
    CU.setValue(-1., where = (x > 60.) * [[[1], [0]]])
    CU.setValue(-1., where = (y > 60.) * [[[0], [1]]])

    CV = CellVariable(mesh=mesh, rank=1)
    CV[:]=1.
    CV.setValue(-1., where = (x > 50.) * [[[1], [0]]])
    CV.setValue(-1., where = (y > 50.) * [[[0], [1]]])

    # Transient formulation
    eqU = TransientTerm() == DiffusionTerm(coeff=D) -\
                     ConvectionTerm(coeff=CU*q.value*beta) 

    eqV = TransientTerm() == DiffusionTerm(coeff=D) -\
                     ConvectionTerm(coeff=CV*p.value*beta) 

    eqP = TransientTerm() == u*(1 + m*q) - p

    eqQ = TransientTerm() == v*(1 + m*p) - q

    # Solving Transient term
    timeStepDuration = 1.
    steps = 50
    t = timeStepDuration * steps

    for step in range(steps):
        eqU.solve(var=u, dt=timeStepDuration)
        eqV.solve(var=v, dt=timeStepDuration)
        eqP.solve(var=p, dt=timeStepDuration)
        eqQ.solve(var=q, dt=timeStepDuration)

    # Plotting results
    #if __name__ == '__main__':
    #    vieweru = Viewer(u, datamin=0., datamax=1.)
    #    viewerv = Viewer(v, datamin=0., datamax=1.)
    #    vieweru.plot()
    #    viewerv.plot()

    loglink = np.sum(np.log(u.value)) + np.sum(np.log(v.value))
    return(loglink)

Finally, for Maximun likelihood criteria I would like to maximise:

Setting an initial value, and using scipy

mb = [0., .5]
mb

results = sci.optimize.minimize(HRMLE, mb, method='Nelder-Mead')
results

The results show values that always are close to my initial values for the parameters, that's the reason I am not sure the my procedure is right. Any suggestions will be quite appreciated.

Answer 1

Well, I have realised that for maximise the function is necessary multiply by -1 the output of the function and then minimise it. So the output of my function should be:

loglink = np.sum(np.log(u.value)) + np.sum(np.log(v.value))*-1
return(loglink)

On the other hand, actually maximisation is just for a list of points in the grid, and not for all values of u1 and u2.