Arthur Bousquet

Jinchao Xu, Yukun Li, Shuonan Wu et al.

We study in this paper the accuracy and stability of partially and fully implicit schemes for phase field modeling. Through theoretical and numerical analysis of Allen-Cahn and Cahn-Hillard models, we investigate the potential problems of using partially implicit schemes, demonstrate the importance of using fully implicit schemes and discuss the limitation of energy stability that are often used to evaluate the quality of a numerical scheme for phase-field modeling. In particular, we make the following observations: 1. a convex splitting scheme (CSS in short) can be equivalent to some fully implicit scheme (FIS in short) with a much different time scaling and thus it may lack numerical accuracy; 2. most implicit schemes (in discussions) are energy-stable if the time-step size is sufficiently small; 3. a traditionally known conditionally energy-stable scheme still possess an unconditionally energy-stable physical solution; 4. an unconditionally energy-stable scheme is not necessarily better than a conditionally energy-stable scheme when the time step size is not small enough; 5. a first-order FIS for the Allen-Cahn model can be devised so that the maximum principle will be valid on the discrete level and hence the discrete phase variable satisfies $|u_h(x)|\le 1$ for all $x$ and, furthermore, the linearized discretized system can be effectively preconditioned by discrete Poisson operators.

Arthur Bousquet, Mickaël D. Chekroun, Youngjoon Hong et al.

We aim to study a finite volume scheme to solve the two dimensional inviscid primitive equations of the atmosphere with humidity and saturation, in presence of topography and subject to physically plausible boundary conditions to the system of equations. In that respect, a version of a projection method is introduced to enforce the compatibility condition on the horizontal velocity field, which comes from the boundary conditions. The resulting scheme allows for a significant reduction of the errors near the topography when compared to more standard finite volume schemes. In the numerical simulations, we first present the associated good convergence results that are satisfied by the solutions simulated by our scheme when compared to particular analytic solutions. We then report on numerical experiments using realistic parameters. Finally, the effects of a random small-scale forcing on the velocity equation is numerically investigated. The numerical results show that such a forcing is responsible for recurrent large-scale patterns to emerge in the temperature and velocity fields.

Shahrzad Jamshidi, Arthur Bousquet, Sugata Banerji et al.

Age prediction using brain imaging, such as MRIs, has achieved promising results, with several studies identifying the model's residual as a potential biomarker for chronic disease states. In this study, we developed a brain age predictive model using a dataset of 1,220 U.S. veterans (18--80 years) and convolutional neural networks (CNNs) trained on two-dimensional slices of axial T2-weighted fast spin-echo and T2-weighted fluid attenuated inversion recovery MRI images. The model, incorporating a degree-3 polynomial ensemble, achieved an $R^{2}$ of 0.816 on the testing set. Images were acquired at the level of the anterior commissure and the frontal horns of the lateral ventricles. Residual analysis was performed to assess its potential as a biomarker for five ICD-coded conditions: hypertension (HTN), diabetes mellitus (DM), mild traumatic brain injury (mTBI), illicit substance abuse/dependence (SAD), and alcohol abuse/dependence (AAD). Residuals grouped by the number of ICD-coded conditions demonstrated different trends that were statistically significant ($p = 0.002$), suggesting a relationship between disease states and predicted brain age. This association was particularly pronounced in patients over 49 years, where negative residuals (indicating advanced brain aging) correlated with the presence of multiple ICD codes. These findings support the potential of residuals as biomarkers for detecting latent health conditions.