Shuyang Xiang

Philippe G. LeFloch, Shuyang Xiang

We study the dynamical behavior of compressible fluids evolving on the outer domain of communication of a Schwarzschild background. To this end, we design several numerical methods which take the Schwarzschild geometry into account and we treat, both, the relativistic Burgers equation and the relativistic Euler system under the assumption that the flow is spherically symmetric. All the schemes we construct are proven to be well-balanced and therefore to preserve the family of steady state solutions for both models. They enable us to study the nonlinear stability of fluid equilibria, and in particular to investigate the behavior of the fluid near the blackhole horizon. We state and numerically demonstrate several conjectures about the late-time behavior of perturbations of steady solutions.

Eric Cancès, Virginie Ehrlacher, Frederic Legoll et al.

This contribution is the numerically oriented companion article of the work [E. Cancès, V. Ehrlacher, F. Legoll, B. Stamm and S. Xiang, arxiv preprint 1807.05131]. We focus here on the numerical resolution of the embedded corrector problem introduced in [E. Cancès, V. Ehrlacher, F. Legoll and B. Stamm, CRAS 2015; E. Cancès, V. Ehrlacher, F. Legoll, B. Stamm and S. Xiang, arxiv preprint 1807.05131] in the context of homogenization of diffusion equations. Our approach consists in considering a corrector-type problem, posed on the whole space, but with a diffusion matrix which is constant outside some bounded domain. In [E. Cancès, V. Ehrlacher, F. Legoll, B. Stamm and S. Xiang, arxiv preprint 1807.05131], we have shown how to define three approximate homogenized diffusion coefficients on the basis of the embedded corrector problems. We have also proved that these approximations all converge to the exact homogenized coefficients when the size of the bounded domain increases. We show here that, under the assumption that the diffusion matrix is piecewise constant, the corrector problem to solve can be recast as an integral equation. In case of spherical inclusions with isotropic materials, we explain how to efficiently discretize this integral equation using spherical harmonics, and how to use the fast multipole method (FMM) to compute the resulting matrix-vector products at a cost which scales only linearly with respect to the number of inclusions. Numerical tests illustrate the performance of our approach in various settings.

Eric Cancès, Virginie Ehrlacher, Frederic Legoll et al.

This article is the first part of a two-fold study, the objective of which is the theoretical analysis and numerical investigation of new approximate corrector problems in the context of stochastic homogenization. We present here three new alternatives for the approximation of the homogenized matrix for diffusion problems with highly-oscillatory coefficients. These different approximations all rely on the use of an embedded corrector problem (that we previously introduced in [Cancès, Ehrlacher, Legoll and Stamm, C. R. Acad. Sci. Paris, 2015]), where a finite-size domain made of the highly oscillatory material is embedded in a homogeneous infinite medium whose diffusion coefficients have to be appropriately determined. The motivation for considering such embedded corrector problems is made clear in the companion article [Cancès, Ehrlacher, Legoll, Stamm and Xiang, in preparation], where a very efficient algorithm is presented for the resolution of such problems for particular heterogeneous materials. In the present article, we prove that the three different approximations we introduce converge to the homogenized matrix of the medium when the size of the embedded domain goes to infinity.

Shuyang Xiang, Hao Guan

Large language models typically represent Chinese characters as discrete index-based tokens, largely ignoring their visual form. For logographic scripts, visual structure carries semantic and phonetic information, which may aid prediction. We investigate whether low-resolution visual inputs can serve as an alternative for character-level modeling. Instead of token IDs, our decoder receives grayscale images of individual characters, with resolutions as low as $8 \times 8$ pixels. Remarkably, these inputs achieve 39.2\% accuracy, comparable to the index-based baseline of 39.1\%. Such low-resource settings also exhibit a pronounced \emph{hot-start} effect: by 0.4\% of total training, accuracy reaches above 12\%, while index-based models lag at below 6\%. Overall, our results demonstrate that minimal visual structure can provide a robust and efficient signal for Chinese language modeling, offering an alternative perspective on character representation that complements traditional index-based approaches.