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Stable and Scalable Probabilistic Numerical Solvers for Stiff and High-Dimensional ODEs
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Proposes MatfreeEK1, a matrix-free, stochastically block-diagonal ODE filter with exact Jacobian and linear-in-dimension cost, plus iterated EKF re-linearization, achieving stability on stiff problems while scaling linearly.
Problem
Probabilistic ODE filters are either stable (A-/L-stable) but cost cubic in ODE dimension, or linear-cost but unstable due to Jacobian approximations, which hinders stiff high-dimensional ODEs/PDEs.
Approach
Perform the EKF update with exact Jacobian via JVPs/VJPs and iterative solvers (CG, block-GMRES) without instantiating the Jacobian, estimate the block-diagonal posterior covariance stochastically from samples, and add iterated EKF re-linearization to make the solver fully implicit.
Key results
- MatfreeEK1 scales linearly in ODE dimension like EK0/DiagonalEK1 while EK1 and ExpEK scale cubically (Lorenz96 per-step runtime).
- On the stiffness sweep of the 1D Brusselator, MatfreeEK1 keeps step count essentially constant as stiffness alpha grows, whereas EK0 and DiagonalEK1 require more steps.
- On 2D Fisher-KPP (d=4096), MatfreeEK1 solves across the full tolerance range with a runtime of 27 min at the tightest tolerance, while EK1 would take ~1h45m even at the loosest tolerance.
Surprising findings
- IEKF re-linearization did not improve MatfreeEK1 in the adaptive setting, though it did help at fixed step sizes and substantially helped DiagonalEK1.
Limitations
- No theoretical A-stability guarantee for MatfreeEK1 due to covariance approximation.
- Effect of the number of covariance estimator samples not studied systematically.
- Posterior uncertainty calibration not directly evaluated; only point-estimate metrics reported.
Experimental setup
Datasets: Lorenz96 1D Brusselator 1D Burgers 1D Fisher-KPP 2D Fisher-KPP 2D FitzHugh-Nagumo
Benchmarks: Lorenz96 1D Brusselator 1D Burgers 2D Fisher-KPP 2D FitzHugh-Nagumo
Metrics: runtime per step number of steps final error work-precision
Baselines: EK0 EK1 DiagonalEK1 ExpEK DiagonalEK1+IEKF
Builds on: EK1 (Schober et al. 2019) DiagonalEK1 (Krämer et al. 2021) ExpEK (Bosch et al. 2023) iterated extended Kalman filter Krylov methods
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