stochrare.fokkerplanck

Numerical solvers for the Fokker-Planck equations

This module contains numerical solvers for the Fokker-Planck equations associated to diffusion processes.

For now, it only contains a basic finite difference solver for the 1D case.

class stochrare.fokkerplanck.FokkerPlanck1D(drift, diffusion)

Bases: object

Solver for the 1D Fokker-Planck equation.

\(\partial_t P(x,t) = - \partial_x a(x,t)P(x,t) + D \partial^2_{xx} P(x,t)\)

Parameters:
  • drift (function with two variables) – The drift coefficient \(a(x, t)\).
  • diffusion (float) – The constant diffusion coefficient \(D\).

Notes

This is just the legacy code which was migrated from the stochrare.dynamics.DiffusionProcess1D class. It should be rewritten with a better structure. In particular, it only works with a constant diffusion for now.

classmethod gaussian1d(mean, std, X)

Return a 1D Gaussian pdf.

Parameters:
  • mean (float) –
  • std (float) –
  • X (ndarray) – The sample points.
Returns:

pdf – The Gaussian pdf at the sample points.
Return type:

ndarray
fpintegrate(t0, T, **kwargs)

Numerical integration of the associated Fokker-Planck equation, or its adjoint.

Parameters:
  • t0 (float) – Initial time.
  • T (float) – Integration time.
Keyword Arguments:
 
  • bounds (float 2-tuple) – Domain where we should solve the equation (default (-10.0,10.0))
  • npts (ints) – Number of discretization points in the domain (i.e. spatial resolution). Default: 100.
  • dt (float) – Timestep (default choice suitable for the heat equation with forward scheme)
  • bc (stochrare.edpy.BoundaryCondition object or tuple) – Boundary conditions (either a BoundaryCondition object or a tuple sent to _fpbc)
  • method (str) – Numerical scheme: explicit (‘euler’, default), implicit, or crank-nicolson
  • adjoint (bool) – Integrate the adjoint FP rather than the forward FP (default False).
  • P0 (str) – Initial condition: ‘gauss’ (default), ‘dirac’ or ‘uniform’.
Returns:

t, X, P – Final time, sample points and solution of the Fokker-Planck equation at the sample points.
Return type:

float, ndarray, ndarray
fpintegrate_generator(*args, **kwargs)

Numerical integration of the associated Fokker-Planck equation, generator version.

Parameters:*args (variable length argument list) – Times at which to yield the pdf.
Yields:t, X, P (float, ndarray, ndarray) – Time, sample points and solution of the Fokker-Planck equation at the sample points.

Classes

FokkerPlanck1D(drift, diffusion) Solver for the 1D Fokker-Planck equation.