arXiv subjects

Stefano Di Giovacchino

Publications and source records attributed to Stefano Di Giovacchino.

7 recordsLinked to original sources

arXiv2026

Backward error analysis of stochastic Poisson integrators

We address our attention to the numerical time discretization of stochastic Poisson systems via Poisson integrators. The aim of the investigation regards the backward error analysis of such integrators to reveal their ability of being structure-preserving, for long times of integration. In particular, we first provide stochastic modified equations suitable for such integrators and then we rigorously study them to prove accurate estimates on the long-term numerical error along the dynamics generated by stochastic Poisson integrators, with reference to the preservation of the random Hamiltonian conserved along the exact flow of the approximating Wong-Zakai Poisson system. Finally, selected numerical experiments confirm the effectiveness of the theoretical analysis.

math.NA↗

arXiv2026

Strong error analysis of a temporal approximation for stochastic Korteweg-de Vries equation with small additive noise

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.

math.NA↗

arXiv2026

Resonance-based integrators for stochastic Schrödinger equations. Convergence and long-time error bounds

We develop resonance-based low-regularity numerical integrators for stochastic Schr"odinger equations with additive $Q$-Wiener noise, covering both the linear equation with rough potential and the cubic nonlinear case. For the linear problem, we prove strong and almost sure convergence, achieving first-order accuracy in $H^σ$ for solutions in $H^{σ+1}$, improving the classical $H^{σ+2}$ requirement. In a regime of $O(\varepsilon^2)$ potentials and $O(\varepsilon)$ noise, we establish uniform moment bounds up to times $O(\varepsilon^{-2})$ and construct a non-resonant scheme with long-time error $O(\varepsilon^2τ)$. For the cubic case, we derive analogous pathwise convergence results at low regularity. In the weakly nonlinear stochastic regime, we obtain long-time pathwise errors of size $O(\varepsilon^2τ^δ)$, for any $δ<1$, up to times $O(\varepsilon^{-2})$. The analysis relies on a novel extension of the regularity-compensation oscillation (RCO) technique to the stochastic setting, overcoming the loss of temporal regularity induced by stochastic convolutions and yielding an $O(\varepsilon^2)$ improvement in long-time error bounds. To the best of our knowledge, this is the first work establishing long-time error bounds for low-regularity integrators for stochastic dispersive equations. Numerical experiments support the theory.

math.NA↗

arXiv2026

Fully discrete approximation of the semilinear stochastic wave equation on the sphere

The semilinear stochastic wave equation on the sphere driven by multiplicative Gaussian noise is discretized by a stochastic trigonometric integrator in time and a spectral Galerkin approximation in space based on the spherical harmonic functions. Strong and almost sure convergence of the explicit fully discrete numerical scheme are shown. Furthermore, these rates are confirmed by numerical experiments.

math.NA↗

arXiv2023

Backward error analysis and the qualitative behaviour of stochastic optimization algorithms: Application to stochastic coordinate descent

Stochastic optimization methods have been hugely successful in making large-scale optimization problems feasible when computing the full gradient is computationally prohibitive. Using the theory of modified equations for numerical integrators, we propose a class of stochastic differential equations that approximate the dynamics of general stochastic optimization methods more closely than the original gradient flow. Analyzing a modified stochastic differential equation can reveal qualitative insights about the associated optimization method. Here, we study mean-square stability of the modified equation in the case of stochastic coordinate descent.

math.OC↗

arXiv2021

A Hierarchy of Network Models Giving Bistability Under Triadic Closure

Triadic closure describes the tendency for new friendships to form between individuals who already have friends in common. It has been argued heuristically that the triadic closure effect can lead to bistability in the formation of large-scale social interaction networks. Here, depending on the initial state and the transient dynamics, the system may evolve towards either of two long-time states. In this work, we propose and study a hierarchy of network evolution models that incorporate triadic closure, building on the work of Grindrod, Higham, and Parsons [Internet Mathematics, 8, 2012, 402--423]. We use a chemical kinetics framework, paying careful attention to the reaction rate scaling with respect to the system size. In a macroscale regime, we show rigorously that a bimodal steady-state distribution is admitted. This behavior corresponds to the existence of two distinct stable fixed points in a deterministic mean-field ODE. The macroscale model is also seen to capture an apparent metastability property of the microscale system. Computational simulations are used to support the analysis.

cs.SI↗

arXiv2020

Mean-square contractivity of stochastic $θ$-methods

The paper is focused on the nonlinear stability analysis of stochastic $θ$-methods. In particular, we consider nonlinear stochastic differential equations such that the mean-square deviation between two solutions exponentially decays, i.e., a mean-square contractive behaviour is visible along the stochastic dynamics. We aim to make the same property visible also along the numerical dynamics generated by stochastic $θ$-methods: this issue is translated into sharp stepsize restrictions depending on parameters of the problem, here accurately estimated. A selection of numerical tests confirming the effectiveness of the analysis and its sharpness is also provided.

math.NA↗