arXiv subjects

Tsiry Avisoa Randrianasolo

Publications and source records attributed to Tsiry Avisoa Randrianasolo.

4 recordsLinked to original sources

arXiv2026

Reduced dynamics for models of pattern formation

The goal of this work is to analyze the long-term behavior of reaction-diffusion systems arising in two-species chemical models and to identify finite sets of modes that determine their asymptotic dynamics. The models considered include, as particular cases the Gray--Scott and the Glycolysis models. These systems are described by coupled reaction-diffusion equations and admit a finite-dimensional representation based on a limited number of spatial Fourier modes that capture their essential reduced dynamics. The concept of determining modes, introduced in this context, is closely related to other approaches that seek finite-dimensional representations of infinite-dimensional dynamics, such as the Proper Orthogonal Decomposition and the construction of Approximate Inertial Manifolds. We prove that the dynamics of the system can be completely characterized by a finite number of low modes, since all higher modes are asymptotically determined by them, thus providing an analytical foundation for reduced dynamics in models of pattern formation.

math.AP↗

arXiv2025

A discrete data assimilation algorithm for the reconstruction of Gray--Scott dynamics

The Gray--Scott model governs the interaction of two chemical species via a system of reaction-diffusion equations. Despite its simple form, it produces extremely rich patterns such as spots, stripes, waves, and labyrinths. That makes it ideal for studying emergent behavior, self-organization, and instability-driven pattern formation. It is also known for its sensitivity to poorly observed initial conditions. Using such initial conditions alone quickly leads simulations to deviate from the true dynamics. The present paper addresses this challenge with a nudging-based data assimilation algorithm: coarse, cell-averaged measurements are injected into the model through a feedback (nudging) term, implemented as a finite-volume interpolant. We prove two main results. (i) For the continuous problem, the nudged solution synchronizes with the true dynamics, and the $L^2$-error decays exponentially under conditions that tie observation resolution, nudging gains, and diffusion. (ii) For the fully discrete semi-implicit finite-volume scheme, the same synchronization holds, up to a mild time-step restriction. Numerical tests on labyrinthine patterns support the theory. They show recovery of fine structure from sparse data and clarify how the observation resolution, the nudging gain, and the frequency of updates affect the decay rate.

math.NA↗

arXiv2025

Convergence and error analysis of a semi-implicit finite volume scheme for the Gray--Scott system

We analyze a semi-implicit finite volume scheme for the Gray--Scott system, a model for pattern formation in chemical and biological media. We prove unconditional well-posedness of the fully discrete problem and establish qualitative properties, including positivity and boundedness of the numerical solution. A convergence result is obtained by compactness arguments, showing that the discrete approximations converge strongly to a weak solution of the continuous system. Under additional regularity assumptions, we further derive a priori error estimates in the $L^2$ norm. Numerical experiments validate the theoretical analysis, confirm a rate of convergence of order 1, and illustrate the ability of the scheme to capture classical Gray--Scott patterns.

math.NA↗

arXiv2024

Numerical approximation of Dynkin games with asymmetric information

We propose an implementable, neural network-based structure preserving probabilistic numerical approximation for a generalized obstacle problem describing the value of a zero-sum differential game of optimal stopping with asymmetric information. The target solution depends on three variables: the time, the spatial (or state) variable, and a variable from a standard $(I-1)$-simplex which represents the probabilities with which the $I$ possible configurations of the game are played. The proposed numerical approximation preserves the convexity of the continuous solution as well as the lower and upper obstacle bounds. We show convergence of the fully-discrete scheme to the unique viscosity solution of the continuous problem and present a range of numerical studies to demonstrate its applicability.

math.NA↗