Thien P. B. Nguyen

Alain Bensoussan, Thien P. B. Nguyen, Minh-Binh Tran et al.

We propose a splitting approach to solve the second-order Hamilton--Jacobi equation, reducing it to a heat step and a purely first-order step. The latter is implemented using a gradient value policy iteration algorithm, enabling efficient characteristic-based machine learning methods. We establish convergence rates for the splitting method. In particular, the $L^\infty$ error is bounded below by $\mathcal{O}(h)$ and above by $\mathcal{O}(h^{1/7})$ for Lipschitz initial data; this improves to $\mathcal{O}(h^{1/5})$ for semiconcave data and to $\mathcal{O}(h^{1/3})$ for $C^2$ data. We also prove an upper $L^1$ error estimate of order $\mathcal{O}(h^{1/2})$ in the periodic setting, where $h$ is the splitting step. For the first-order step, we provide a weighted $L^2$ error analysis that shows exponential convergence. Each iteration solves linear characteristic equations and learns the value function by minimizing a weighted value gradient loss. The approach yields stable and accurate numerical results.

Cong B. Van, Thien P. B. Nguyen, Minh-Binh Tran et al.

We study an inverse initial data problem for a convection-diffusion equation with memory, where the goal is to recover the unknown initial condition from final-time data. The model includes convection, an instantaneous Laplacian term, and a nonlocal-in-time memory term involving the Laplacian of the past states, which leads to a severely ill-posed backward problem. We prove uniqueness in a spatially independent coefficient setting by applying the Fourier transform and using an analyticity argument for a scalar Volterra equation. For the variable-coefficient case, we develop a computational method based on Legendre spatial dimensional reduction and Tikhonov regularization. The solution is approximated by a finite tensor-product Legendre expansion, thereby reducing the inverse problem to a finite-dimensional terminal-value system for the time-dependent coefficients. We solve the reduced problem by a Tikhonov-regularized least-squares method with an $H^2$ penalty. For a fixed truncation order, we prove that the regularized minimizers converge to the finite-dimensional minimum-norm solution as the noise level and the regularization parameter vanish, under a suitable choice of the regularization parameter. Some two-dimensional numerical examples are presented to illustrate the performance of the proposed method.