Vincenzo Citro

Johannes Lang, Vincenzo Citro, Luca Leuzzi et al.

Understanding the dynamics of systems evolving in complex and rugged energy landscapes is central across physics, economics, biology, and computer science. Disordered mean-field models provide a powerful framework, as exact Dynamical Mean-Field Equations (DMFE) can be derived. However, solving the DMFE -- a set of coupled integral-differential equations for two-time functions -- remains a major numerical challenge. So far, large-time solutions of DMFE rely either on analytical approaches, such as the Cugliandolo--Kurchan ansatz based on assumptions like weak ergodicity breaking (which is known to fail in some cases), or on numerical integrations that reliably reach times $O(10^3)$ and extend further only via poorly controlled approximations. Consequently, no general method currently exists to solve DMFE at very long times, limiting the study of slow dynamics in complex landscapes. We present \textsc{Dynamite} (DYNAmical Mean-fIeld Time Evolution solver), a high-performance framework for solving DMFE up to unprecedented times $t=O(10^7)$. It combines non-uniform interpolation, adaptive time stepping, and numerical `renormalization' of memory, enabling accurate evaluation of history integrals. Its asymptotic runtime is linear, with sublinear memory scaling. Stability and precision are ensured via an adaptive Runge--Kutta scheme and periodic sparsification of the past. \textsc{Dynamite} achieves orders-of-magnitude speedups over uniform-grid methods while maintaining accuracy and reproducibility on CPU and GPU architectures. Benchmarks on glassy mean-field models, including the mixed spherical $p$-spin system, demonstrate access to aging and relaxation regimes previously out of reach. The framework provides a reproducible and extensible foundation for studying long-memory dynamical systems.

Vincenzo Citro, Davide Palitta

BoostConv has been introduced in earlier works as an effective acceleration technique for nonlinear iterative processes and has been successfully employed in a variety of applications to enhance convergence rates or to compute unstable fixed points that are otherwise inaccessible through standard approaches. Despite its demonstrated practical effectiveness, the theoretical properties of the method have not yet been fully characterized. In this work, we present a more robust formulation of the BoostConv algorithm and, for the first time, provide a rigorous proof of its convergence. The proposed analysis places BoostConv within a precise mathematical framework, clarifying its interpretation as a nonlinear convergence accelerator and establishing sufficient conditions under which convergence to a fixed point is guaranteed. The theoretical findings are illustrated through several numerical examples, spanning from a linear problem to a low-dimensional benchmark and a large-scale incompressible Navier-Stokes simulation. These results demonstrate the robustness and practical relevance of the proposed method and bridge the gap between empirical performance and rigorous analysis, paving the way for further developments and applications to complex nonlinear problems.