Numerical Methods for Fokker–Planck Equations II

A grid that keeps mass, positivity and its equilibrium is only the beginning. Escape from a well can take longer than any grid will run; a drift can depend on the density itself; in six dimensions a full grid no longer fits; and a diffusion model generates its samples by running a Fokker–Planck equation backwards.

This second volume takes the structure-keeping schemes of Volume I beyond the grid: spectral methods and the spectral gap that sets an escape rate, every scheme read as a Markov chain whose forward and backward equations agree to round-off, nonlinear and nonlocal equations from phase transitions to chemotaxis and coupled neurons, particle methods and the samplers of diffusion models, and tensor trains and score networks measured against the grid on problems with exact answers. A closing guide matches the features of a problem to a method, with a test suite of exact solutions.

Every main result is proved. Every figure and every number comes from the Python printed whole in the appendix, checked against an exact solution and computed two ways.

Sanne Kooistra is a numerical analyst from Delft who has simulated electrons in semiconductors and fusion plasmas and the firing of neuron populations, and has counted the mass at every step.