Convergence Rates of Gaussian ODE Filters
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A recently-introduced class of probabilistic (uncertainty-aware) solvers for ordinary differential equations (ODEs) applies Gaussian (Kalman) filtering to initial value problems. These methods model the true solution x and its first q derivatives a priori as a Gauss--Markov process X, which is then iteratively conditioned on information about ẋ. We prove worst-case local convergence rates of order h^q+1 for a wide range of versions of this Gaussian ODE filter, as well as global convergence rates of order h^q in the case of q=1 and an integrated Brownian motion prior, and analyze how inaccurate information on ẋ coming from approximate evaluations of f affects these rates. Moreover, we present explicit formulas for the steady states and show that the posterior confidence intervals are well calibrated in all considered cases that exhibit global convergence---in the sense that they globally contract at the same rate as the truncation error.
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