Error Bounds for Boundary-Stopped Positive Cubature Schemes for Hamilton--Jacobi--Bellman Equations
Abstract
We derive error bounds for a boundary-stopped positive cubature scheme for degenerate parabolic Hamilton--Jacobi--Bellman equations with Dirichlet data on bounded domains. The scheme balances antipodal branches at their first boundary contacts and uses a control-dependent effective time. Its stopped-barrier remainder is O(τΔtγ/2), and the total mass of interpolated branches is at most 2τ/Δt, where τ is the effective time. After division by the effective time, these estimates give a consistency bound that is uniform as τ→0. Under a common strict boundary barrier and the stated mesh conditions, interior consistency, coefficient shaking, switching, and localized comparison yield |(u−uh)+|∞≤ C(Δt1/4+hΔt−1/2), |(uh−u)+|∞≤ C(Δt1/10+h1/2Δt−1/4). Thus Δtasymp h10/7 gives an O(h1/7) maximum-norm bound. The analysis applies to any fixed centrally symmetric positive degree-two cubature and permits rank-deficient diffusion. Numerical examples examine parallel heat loss to a cold wall and control-dependent, rank-deficient diffusion.
Related
- Stochastic differential games with random coefficients and state constraints: associated stochastic Hamilton-Jacobi-Bellman-Isaacs equations with Neumann boundary conditions
- Parametric Families of Minimal Positive Cubature Formulas of Degrees 4 and 6 on the 2-Sphere
- Positive cubature on S2:low-degree rigidity and uniform bounds
- Positive Cubature Compression and Optimal Hyperinterpolation Stability on a Conic Surface
- Scaled boundary cubature scheme in higher dimensions: integration over polytopes and curved regions
BibTeX
@misc{xu2026error,
title = {{Error Bounds for Boundary-Stopped Positive Cubature Schemes for Hamilton--Jacobi--Bellman Equations}},
author = {Haoran Xu and Xingye Yue},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.17186},
eprint = {2609.17186},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.17186},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Error Bounds for Boundary-Stopped Positive Cubature Schemes for Hamilton--Jacobi--Bellman Equations
AU - Haoran Xu
AU - Xingye Yue
PY - 2026
DA - 2026/09/15
PB - arXiv
DO - 10.48550/arxiv.2609.17186
UR - https://arxiv.org/abs/2609.17186
AB - We derive error bounds for a boundary-stopped positive cubature scheme for degenerate parabolic Hamilton--Jacobi--Bellman equations with Dirichlet data on bounded domains. The scheme balances antipodal branches at their first boundary contacts and uses a control-dependent effective time. Its stopped-barrier remainder is O(τΔt^γ/2), and the total mass of interpolated branches is at most 2τ/Δt, where τ is the effective time. After division by the effective time, these estimates give a consistency bound that is uniform as τ→0. Under a common strict boundary barrier and the stated mesh conditions, interior consistency, coefficient shaking, switching, and localized comparison yield |(u−u_h)^+|_∞≤ C(Δt^1/4+hΔt^−1/2), |(u_h−u)^+|_∞≤ C(Δt^1/10+h^1/2Δt^−1/4). Thus Δtasymp h^10/7 gives an O(h^1/7) maximum-norm bound. The analysis applies to any fixed centrally symmetric positive degree-two cubature and permits rank-deficient diffusion. Numerical examples examine parallel heat loss to a cold wall and control-dependent, rank-deficient diffusion.
ER -