🤖 AI Summary
This work addresses the estimation of the mean of discretization errors in numerical solutions of ordinary differential equations by proposing a Bayesian inference framework. The approach models the error as a random variable and constructs a linear Gaussian state-space model by incorporating a Markovian prior informed by classical error analysis and dependent on the solver’s step size. The ensemble Kalman filter is employed to efficiently infer the temporal evolution of the error mean. Theoretical analysis establishes that this prior exhibits a well-defined probabilistic convergence rate as the step size tends to zero. Experimental results on the simple pendulum system and the FitzHugh–Nagumo model demonstrate the method’s effectiveness and practicality in accurately estimating the mean of discretization errors.
📝 Abstract
We propose a Bayesian framework to quantify discretization errors in numerical solutions of ODE models based on observational data. The discretization error is modeled as a random variable, and its mean-referred to as the discretization error mean-is inferred from the observations. By introducing a Markov prior on the temporal evolution of the discretization error mean, we formulate the problem as a state-space model with a linear Gaussian observation process, which enables efficient inference via the Ensemble Kalman Filter. We also propose a specific form of a Markov prior motivated by classical discretization error analysis, in which global errors accumulate from local errors. The proposed prior depends on the step size of a numerical solver, and we establish its convergence rate in probability as the step size tends to zero. Numerical experiments on the pendulum system and the FitzHugh-Nagumo model demonstrate the effectiveness of the proposed approach.