This code implements a discrete data assimilation algorithm for the Gray--Scott reaction-diffusion model, using a semi-implicit (IMEX) finite-volume scheme built with FiPy. The continuous form of the data assimilation algorithm reads $$\partial_t \tilde{u} = d_u \Delta \tilde{u} - \tilde{u} \tilde{v}^2 + F(1-\tilde{u}) + \mu_u (\mathcal{I}_H\tilde{u} - \mathcal{I}_H u ), \qquad\partial_t \tilde{v} = d_v \Delta \tilde{v} + \tilde{u} \tilde{v}^2 - (F+k)\tilde{v} + \mu_v (\mathcal{I}_H\tilde{v} - \mathcal{I}_H v ),$$ with Neumann boundary conditions and a finite volume interpolant $\mathcal{I}_H$ that characterizes the coarse observation we have on the Truth $(u,v)$. The framework supports a multigrid (multiresolution) approach, where the observations are defined on a coarse grid (low resolution) and the model state is reconstructed on a fine grid (high resolution). This enables the recovery of fine-scale Gray--Scott patterns from sparse or low-resolution observations. The underlying IMEX solver can also be used on its own to simulate the Gray--Scott system by setting $\mu_u = \mu_v = 0$. The data assimilation module updates the model state at discrete, possibly sparse, times using coarse or noisy observations, while the IMEX solver advances the forecast between updates. It can therefore both generate synthetic data and reconstruct the full state from partial measurements.

@misc{randrianasolo2025a, title={A discrete data assimilation algorithm for the reconstruction of Gray--Scott dynamics}, author={Tsiry Avisoa Randrianasolo}, year={2025}, eprint={2510.03972}, archivePrefix={arXiv}, primaryClass={math.NA}, url={https://arxiv.org/abs/2510.03972}, }