Solving Hamilton-Jacobi equations by minimizing residuals of monotone discretizations
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| author | Bokanowski, Olivier Esteve-Yagüe, Carlos Tsai, Richard |
| author_facet | Bokanowski, Olivier Esteve-Yagüe, Carlos Tsai, Richard |
| contents | We establish a well-posedness and error-estimation framework that solves Hamilton-Jacobi equations by minimizing the least-squares residual of monotone finite-difference discretizations. This approach also applies naturally to second-order elliptic and parabolic problems. We prove that, under suitable monotonicity conditions, every critical point of the residual loss functional is the unique global minimizer and coincides with the solution of the discrete scheme. We derive \emph{a~posteriori} error estimates that bound the approximation error by the magnitude of the residual with explicit, computable constants, and extend the full analysis to time-dependent problems with implicit discretization of the time derivatives. A spectral analysis of the linearized system shows that the condition number scales as $O(Δx^{-1})$ for proper schemes, and as $O(\exp(Δx^{-1}))$ under a uniform ellipticity condition. These results quantify the increasing difficulty of solving the optimization problem on finer meshes, and motivates a progressive multi-level warm-start strategy using Artificial Neural Networks. Combined with the convergence theorem of Barles and Souganidis for monotone and consistent schemes, our results guarantee that the solutions obtained converge to the unique viscosity solution as the mesh is refined. Numerical experiments demonstrate the scalability of the approach to high-dimensional Eikonal equations, level-set problems, and Hamilton--Jacobi--Isaacs equations with genuine second-order diffusion arising from stochastic differential games. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_21764 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Solving Hamilton-Jacobi equations by minimizing residuals of monotone discretizations Bokanowski, Olivier Esteve-Yagüe, Carlos Tsai, Richard Numerical Analysis Optimization and Control 49L25, 35A15, 68T07, 49K20, 65N06 We establish a well-posedness and error-estimation framework that solves Hamilton-Jacobi equations by minimizing the least-squares residual of monotone finite-difference discretizations. This approach also applies naturally to second-order elliptic and parabolic problems. We prove that, under suitable monotonicity conditions, every critical point of the residual loss functional is the unique global minimizer and coincides with the solution of the discrete scheme. We derive \emph{a~posteriori} error estimates that bound the approximation error by the magnitude of the residual with explicit, computable constants, and extend the full analysis to time-dependent problems with implicit discretization of the time derivatives. A spectral analysis of the linearized system shows that the condition number scales as $O(Δx^{-1})$ for proper schemes, and as $O(\exp(Δx^{-1}))$ under a uniform ellipticity condition. These results quantify the increasing difficulty of solving the optimization problem on finer meshes, and motivates a progressive multi-level warm-start strategy using Artificial Neural Networks. Combined with the convergence theorem of Barles and Souganidis for monotone and consistent schemes, our results guarantee that the solutions obtained converge to the unique viscosity solution as the mesh is refined. Numerical experiments demonstrate the scalability of the approach to high-dimensional Eikonal equations, level-set problems, and Hamilton--Jacobi--Isaacs equations with genuine second-order diffusion arising from stochastic differential games. |
| title | Solving Hamilton-Jacobi equations by minimizing residuals of monotone discretizations |
| topic | Numerical Analysis Optimization and Control 49L25, 35A15, 68T07, 49K20, 65N06 |
| url | https://arxiv.org/abs/2601.21764 |