TheoremStatement: AI-adaptedProof: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-6-luna)

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Smooth consistency of the diagonal-diffusion Bellman stencil

Statement

Assume the coefficients are bounded and periodic and a periodic test function $ϕ$ is smooth with bounded derivatives $ϕ_{t t}$ , $ϕ_{x_{j} x_{j}}$ , and $ϕ_{x_{j} x_{j} x_{j} x_{j}}$ , and the coefficients are bounded. For the scheme of Explicit monotone diagonal-diffusion Bellman grid update, define the normalized local residual $R_{h, τ} (x_{i}, t_{n}) = \frac{ϕ (x_{i}, t_{n} + τ) - S_{h, τ} ϕ (\cdot, t_{n})_{i}}{τ} .$ Then $R_{h, τ} = ϕ_{t} - \max_{a \in A} {\sum_{j} (α_{j}^{a} ϕ_{x_{j} x_{j}} + b_{j}^{a} ϕ_{x_{j}}) + f^{a}} + O (τ + \max_{j} [α_{j, \max} h_{j}^{2} ∥ ϕ_{x_{j}^{4}} ∥_{\infty} + ∣ b_{j} ∣_{\max} h_{j} ∥ ϕ_{x_{j} x_{j}} ∥_{\infty}]),$ where $α_{j, \max} : = \sup_{a, x, t} α_{j}^{a} (x, t)$ and $∣ b_{j} ∣_{\max} : = \sup_{a, x, t} ∣ b_{j}^{a} (x, t) ∣$ . The bound is uniform over nodes and times before the terminal step. The $O (\cdot)$ is a smooth-test residual estimate, not a solution-error rate.

Facts & Assumptions

Given: The finite-control diagonal diffusion scheme, the coefficient bounds, and a periodic smooth test function with the stated derivative bounds.

[F1]

The time update is $U^{n + 1} = U^{n} + τ \max_{a} {\sum_{j} (α_{j}^{a} D_{j j}^{h} U + b_{j}^{a} D_{j, a}^{u p} U) + f^{a}}$ with the derivative conventions of the algorithm (Explicit monotone diagonal-diffusion Bellman grid update).

[F2]

A Taylor remainder of order $m$ is bounded by the $(m + 1)$ st derivative supremum times the segment length to that power divided by $(m + 1)!$ (A uniform derivative bound gives a uniform Taylor remainder bound).

[F3]

The continuous operator is $u_{t} - \max_{a} {\sum_{j} (α_{j}^{a} u_{x_{j} x_{j}} + b_{j}^{a} u_{x_{j}}) + f^{a}} = 0$ (Periodic diagonal controlled-diffusion HJB Cauchy problem).

Proof

1.1F1F2givenalgebra

Applying [F2] at $x_{i} \pm h_{j} e_{j}$ and cancelling the odd terms gives $D_{j j}^{h} ϕ (x_{i}, t_{n}) = ϕ_{x_{j} x_{j}} (x_{i}, t_{n}) + O (h_{j}^{2} ∥ ϕ_{x_{j}^{4}} ∥_{\infty}) .$ Applying [F2] to the one-sided difference in [F1] gives $D_{j, a}^{u p} ϕ (x_{i}, t_{n}) = ϕ_{x_{j}} (x_{i}, t_{n}) + O (h_{j} ∥ ϕ_{x_{j} x_{j}} ∥_{\infty})$ for either sign of $b_{j}^{a}$ . The coefficient bounds make these estimates uniform over the finite action and coordinate sets.

2.1F1F2step 1.1F3algebra

Taylor expansion in time gives $\frac{ϕ (x_{i}, t_{n} + τ) - ϕ (x_{i}, t_{n})}{τ} = ϕ_{t} (x_{i}, t_{n}) + O (τ ∥ ϕ_{t t} ∥_{\infty}) .$ Substitute the spatial estimates of step 1.1 into the update in [F1]. Since $∣ \max_{a} X_{a} - \max_{a} Y_{a} ∣ \leq \max_{a} ∣ X_{a} - Y_{a} ∣$ , the uniform spatial residual is bounded by the sum displayed in the statement. Combining with the time estimate gives its formula and its explicit smoothness dependence.

3.1step 2.1∎

The mesh consistency residual tends to zero when $τ \to 0$ and $h_{\max} \to 0$ with bounded coefficients. It supplies the local consistency input for the half-relaxed-limit result and makes no assertion of a universal convergence rate.

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Sources

Kristian Debrabant and Espen R. Jakobsen, Semi-Lagrangian schemes for linear and fully non-linear Hamilton-Jacobi-Bellman equations (standard reference, not scraped)
Guy Barles and Panagiotis E. Souganidis, Convergence of approximation schemes for fully nonlinear second order equations (standard reference, not scraped)