Ordinary Differential Equations
This site accompanies our recent publication, Probability Measures for Numerical Solutions of Differential Equations. It proposes a new strategy for randomising existing solvers for ordinary and partial differential equations, which rigorously models the uncertainty introduced by the numerical methods. This correct handling of uncertainty is crucial when numerical solvers are used for statistical analysis, for example, in Bayesian inference. This work is part of the emerging field of Probabilistic Numerics, which frames numerical methods as statistical inference tasks, allowing the tools of statistics to be combined with classical numerical analysis.
For an example of our construction, consider integrating the chaotic Lorenz oscillator:
A classical 4th order Runge-Kutta does not provide any measure of uncertainty. In contrast, the figure below shows draws from our randomised solver in blue. Our construction reveals that while early times are integrated accurately with this step-size, later times are increasingly inaccurate.
In fact, the distribution is quite complex, exhibiting strong correlations in time and highly non-Gaussian structures. Uncertainty increases as time evolves.