Convergence rates of Gaussian ODE filters

Area of Science:

• Numerical Analysis
• Applied Mathematics
• Computational Science

Background:

• Probabilistic solvers incorporating uncertainty awareness are emerging for solving ordinary differential equations (ODEs).
• Gaussian (Kalman) filtering is a key technique applied to initial value problems within these solvers.
• These methods model the solution and its derivatives as a Gauss-Markov process, updated iteratively with new information.

Purpose of the Study:

• To establish worst-case local and global convergence rates for Gaussian ODE filters.
• To analyze the impact of approximate function evaluations on convergence rates.
• To assess the calibration of posterior credible intervals in globally convergent cases.

Main Methods:

• Theoretical analysis of convergence rates for Gaussian ODE filters.
• Modeling the ODE solution and its derivatives as a Gauss-Markov process.
• Investigating the effect of inaccurate function evaluations on convergence.
• Analyzing the contraction rate of posterior credible intervals.

Main Results:

• Established worst-case local convergence rates of order $$ for various Gaussian ODE filter versions.
• Proved global convergence rates of order q for specific priors (e.g., integrated Brownian motion).
• Demonstrated that posterior credible intervals contract at the same rate as the truncation error in globally convergent cases.
• Analyzed the influence of inaccurate function evaluations on convergence rates.

Conclusions:

• Gaussian ODE filters exhibit robust convergence properties, both locally and globally.
• The posterior credible intervals are well-calibrated, providing reliable uncertainty quantification.
• Numerical experiments suggest potential generalizability of these findings.