Convergence of differentiable non-monotone schemes for fully nonlinear parabolic equations
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arXiv
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| Format: | Preprint |
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2018
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| _version_ | 1866909020599615488 |
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| author | Nakano, Yumiharu |
| author_facet | Nakano, Yumiharu |
| contents | We develop a convergence theory for non-monotone approximation schemes for fully nonlinear parabolic partial differential equations. Modern computational methods such as kernel-based collocation, spectral methods, physics-informed neural networks, and deep Galerkin methods are typically non-monotone, since they produce smooth approximate solutions and compute spatial derivatives directly from gradients of the chosen ansatz. Such schemes therefore lie outside the scope of the classical Barles and Souganidis convergence theory. We introduce an abstract framework that replaces strict monotonicity by two pointwise consistency conditions, on the PDE residual and on the terminal mismatch, both directly verifiable for a smooth approximating sequence. The technical key is a max-min representation of the nonlinearity that converts a vanishing classical residual into the viscosity subsolution and supersolution inequalities, and so dispenses with monotonicity in the abstract argument. The framework yields qualitative convergence under standard hypotheses, together with a quantitative error bound for Hamilton-Jacobi-Bellman equations on an unbounded spatial domain, in which the residual on an expanding truncation cylinder is balanced against an exponentially decaying tail term coming from the controlled stochastic differential equation underlying the value function. As a concrete realization, we analyze kernel-based collocation with Wendland radial basis functions, and present numerical experiments on a benchmark Hamilton-Jacobi-Bellman problem in one and two spatial dimensions that confirm the predicted convergence behaviour. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1803_09446 |
| institution | arXiv |
| publishDate | 2018 |
| record_format | arxiv |
| spellingShingle | Convergence of differentiable non-monotone schemes for fully nonlinear parabolic equations Nakano, Yumiharu Numerical Analysis Optimization and Control We develop a convergence theory for non-monotone approximation schemes for fully nonlinear parabolic partial differential equations. Modern computational methods such as kernel-based collocation, spectral methods, physics-informed neural networks, and deep Galerkin methods are typically non-monotone, since they produce smooth approximate solutions and compute spatial derivatives directly from gradients of the chosen ansatz. Such schemes therefore lie outside the scope of the classical Barles and Souganidis convergence theory. We introduce an abstract framework that replaces strict monotonicity by two pointwise consistency conditions, on the PDE residual and on the terminal mismatch, both directly verifiable for a smooth approximating sequence. The technical key is a max-min representation of the nonlinearity that converts a vanishing classical residual into the viscosity subsolution and supersolution inequalities, and so dispenses with monotonicity in the abstract argument. The framework yields qualitative convergence under standard hypotheses, together with a quantitative error bound for Hamilton-Jacobi-Bellman equations on an unbounded spatial domain, in which the residual on an expanding truncation cylinder is balanced against an exponentially decaying tail term coming from the controlled stochastic differential equation underlying the value function. As a concrete realization, we analyze kernel-based collocation with Wendland radial basis functions, and present numerical experiments on a benchmark Hamilton-Jacobi-Bellman problem in one and two spatial dimensions that confirm the predicted convergence behaviour. |
| title | Convergence of differentiable non-monotone schemes for fully nonlinear parabolic equations |
| topic | Numerical Analysis Optimization and Control |
| url | https://arxiv.org/abs/1803.09446 |