Joubine Aghili

Joubine Aghili, Daniele A. Di Pietro

In this work, we study advection-robust Hybrid High-Order discretizations of the Oseen equations. For a given integer $k\ge 0$, the discrete velocity unknowns are vector-valued polynomials of total degree $\le k$ on mesh elements and faces, while the pressure unknowns are discontinuous polynomials of total degree $\le k$ on the mesh. From the discrete unknowns, three relevant quantities are reconstructed inside each element: a velocity of total degree $\le(k+1)$, a discrete advective derivative, and a discrete divergence. These reconstructions are used to formulate the discretizations of the viscous, advective, and velocity-pressure coupling terms, respectively. Well-posedness is ensured through appropriate high-order stabilization terms. We prove energy error estimates that are advection-robust for the velocity, and show that each mesh element $T$ of diameter $h_T$ contributes to the discretization error with an $\mathcal{O}(h_T^{k+1})$-term in the diffusion-dominated regime, an $\mathcal{O}(h_T^{k+\frac12})$-term in the advection-dominated regime, and scales with intermediate powers of $h_T$ in between. Numerical results complete the exposition.

Joubine Aghili, Rémi Imbach, Anne Pallarès et al.

Time-Resolved Spectroscopy (TRS) is a powerful modality for non-invasive characterization of turbid media. However, extracting optical properties, absorption $μ_a$ and reduced scattering $μ_s'$, from 3D stochastic measurements remains computationally expensive for real-time applications. In this paper, we propose a data-efficient, physics-informed transfer learning strategy using a Bidirectional Long Short-Term Memory (Bi-LSTM) network. By leveraging a fast deterministic solver to establish a physical prior before fine-tuning on a restricted set of 3D Monte Carlo simulations, our model successfully bridges the analytical-to-stochastic domain gap. The proposed method eliminates the systematic bias of analytical models while maintaining a competitive error with near-instantaneous inference time.

Joubine Aghili

We introduce a Hybrid High-Order (HHO) method for the Schrödinger equation in the presence of a magnetic vector potential. In quantum mechanics, physical observables are invariant under continuous gauge transformations, which must be kept at the discrete level to avoid unphysical artifacts. To address this, we construct a discrete covariant gradient operator on arbitrary polyhedral meshes. We prove that the resulting discrete bilinear form guarantees gauge covariance asymptotically at the discrete level. The resulting scheme achieves optimal convergence rates and preserves a discrete Garding inequality, guaranteeing a stable ground state. The theoretical properties of the scheme are corroborated by numerical experiments, including the computation of the Fock-Darwin fundamental energy and replicating the Aharonov-Bohm effect.