Jianbo Cui

Jianbo Cui, Raffaele D'Ambrosio, Stefano Di Giovacchino et al.

We study strong temporal approximation of periodic stochastic Korteweg--de Vries equation driven by small additive \(Q\)-Wiener noise of amplitude \(\mathcal O(\varepsilon)\), \(0<\varepsilon\ll1\). Strong error analysis for temporal approximations of stochastic KdV is a challenging problem, due to the additional derivative term in the nonlinearity and thanks to the lack of suitable exponential moment bounds for the exact solutions. Exploiting the small-noise regime, we first decompose the solution into a deterministic KdV flow and a stochastic component; then we linearize the obtained stochastic equation and approximate the resulting equation by means of Fourier analytic techniques. Combining the small-noise linearization error, the discretization error of the linearized equation, and the deterministic temporal approximation error, we prove strong convergence rates of order \(\mathcal O(\max(\varepsilon^2,τ,\varepsilonτ^{1/2}))\) under \(H^1\)-regularity and \(\mathcal O(\max(\varepsilon^2,τ))\) under \(H^2\)-regularity, for the obtained approximation of the original stochastic KdV. To the best of our knowledge, these are the first explicit strong convergence rates shown for numerical time approximations of the stochastic KdV.

Jianbo Cui, Jialin Hong

In this paper, we investigate the damped stochastic nonlinear Schrödinger(NLS) equation with multiplicative noise and its splitting-based approximation. When the damped effect is large enough, we prove that the solutions of the damped stochastic NLS equation and the splitting scheme are exponential stable and possess some exponential integrability. These properties lead that the strong order of the scheme is $\frac 12$ and independent of time. Meanwhile, we analyze the regularity of the Kolmogorov equation with respect to the equation. As a consequence, the weak order of the scheme is shown to be twice the strong order and independent of time.

Jianbo Cui, Jialin Hong, Zhihui Liu et al.

We indicate that the nonlinear Schrödinger equation with white noise dispersion possesses stochastic symplectic and multi-symplectic structures. Based on these structures, we propose the stochastic symplectic and multi-symplectic methods, which preserve the continuous and discrete charge conservation laws, respectively. Moreover, we show that the proposed methods are convergent with temporal order one in probability. Numerical experiments are presented to verify our theoretical results.

Jianbo Cui, Jialin Hong, Liying Sun

Approximating the invariant measure and the expectation of the functionals for parabolic stochastic partial differential equations (SPDEs) with non-globally Lipschitz coefficients is an active research area and is far from being well understood. In this article, we study such problem in terms of a full discretization based on the spectral Galerkin method and the temporal implicit Euler scheme. By deriving the a priori estimates and regularity estimates of the numerical solution via a variational approach and Malliavin calculus, we establish the sharp weak convergence rate of the full discretization. When the SPDE admits a unique $V$-uniformly ergodic invariant measure, we prove that the invariant measure can be approximated by the full discretization. The key ingredients lie on the time-independent weak convergence analysis and time-independent regularity estimates of the corresponding Kolmogorov equation. Finally, numerical experiments confirm the theoretical findings.

Qujiangxue Chen, Jianbo Cui, Luca Dieci et al.

In this paper, we study dynamical optimal transport on a connected graph from the perspective of the Benamou-Brenier formulation, where densities are assigned to vertices and velocities to edges. However, directly using Newton's method on the resulting nonlinear systems encounters two potential difficulties: (i) if the graph contains cycles, edge variables are not unique, and (ii) there is no guarantee that the density variables remain positive. To address these challenges, we introduce a finite-difference-type Newton method that eliminates cycle-induced redundancies through a spanning-tree gauge, resulting in a reduced set of independent variables and a well-posed, sparse linear system. For the lattice graph arising from the continuous optimal transport problem, density positivity can also be guaranteed by using an upwind discretization subject to a CFL-type condition. We further demonstrate the versatility of the proposed scheme by applying it to a range of problems, including optimal transport on lattices and random graphs, inverse optimal transport problems, and social network analysis.

Jianbo Cui, Tonghe Dang

We prove first-order convergence of semi-discrete monotone finite difference schemes for Hamilton--Jacobi equations on the Wasserstein space over a finite graph. A central challenge is the boundary degeneracy of the Wasserstein simplex, which prevents the direct use of the standard $L^1$ adjoint method and limits doubling-of-variables arguments to the suboptimal rate $\mathcal O(h^{\frac 12})$ \cite{CDM25}. We address this issue by introducing a weighted $L^1$ framework with a boundary-vanishing weight and by analyzing the corresponding weighted adjoint equation for the linearized operator of the scheme, featuring a new geometric drift term. Our proof relies on uniform bounds for the weighted adjoint variable and the mesh-parameter derivative of the numerical solution. These estimates are derived from discrete gradient and semi-concavity bounds, obtained through a bootstrap argument for two classes of monotone Hamiltonians.

Jianbo Cui, Mihály Kovács, Derui Sheng

We study the numerical approximation of a class of degenerate parabolic stochastic partial differential equations on non-compact metric graphs, which naturally arise in the asymptotic analysis of Hamiltonian flows under small noise perturbations. The numerical discretization of these equations faces several challenges, including the non-compactness of the graph, the degeneracy of the differential operator near vertices, and the non-symmetry of the associated bilinear form. To address these issues, we propose a multi-step numerical strategy combining graph truncation, localized coefficient regularization, and finite element spatial discretization. By incorporating localization techniques, tightness arguments, and resolvent estimates, we establish the strong convergence of the proposed scheme in a weighted $L^2$-space. Our results provide a systematic methodology that is potentially extensible to more general non-compact graphs and degenerate operators.