Error estimates for finite difference schemes associated with Hamilton-Jacobi equations on a junction

This paper is concerned with monotone (time-explicit) finite difference schemes associated with first order Hamilton-Jacobi equations posed on a junction. They extend the schemes recently introduced by Costeseque, Lebacque and Monneau (2013) to general junction conditions. On the one hand, we prove the convergence of the numerical solution towards the viscosity solution of the Hamilton-Jacobi equation as the mesh size tends to zero for general junction conditions. On the other hand, we derive optimal error estimates of order $($\Delta$x)^{\frac{1}{2}}$ in $L\_{loc}^{\infty}$ for junction conditions of optimal-control type at least if the flux is "strictly limited".Comment: 39 pages. In the initial version, the proof of the error estimate only works for Hamiltonians with the same minimum with no flux limiter. In the revised version, we can handle general quasi-convex Hamiltonians and flux limiters. We also provide numerical simulation

Quantitative regularity for parabolic De Giorgi classes

We deal with the De Giorgi Hölder regularity theory for parabolic equations with rough coefficients and parabolic De Giorgi classes which extend the notion of solution. We give a quantitative proof of the interior Hölder regularity estimate for both using De Giorgi method. Recently, the De Giorgi method initially introduced for elliptic equation has been extended to parabolic equation in a non quantitative way. Here we extend the method to the parabolic De Giorgi classes in a quantitative way. To this aim, we get a quantitative version of the non quantitative step of the method, the parabolic intermediate value lemma, one of the two main tools of the De Giorgi method sometimes called ``second lemma of De Giorgi''

Quantitative de Giorgi Methods in Kinetic Theory

We consider hypoelliptic equations of kinetic Fokker-Planck type, also known as Kolmogorov or ultraparabolic equations, with rough coefficients in the drift-diffusion operator. We give novel short quantitative proofs of the De Giorgi intermediate-value Lemma as well as weak Harnack and Harnack inequalities. This implies H{\"o}lder continuity with quantitative estimates. The paper is self-contained

A New Interpretable Neural Network-Based Rule Model for Healthcare Decision Making

In healthcare applications, understanding how machine/deep learning models make decisions is crucial. In this study, we introduce a neural network framework, $\textit{Truth Table rules}$ (TT-rules), that combines the global and exact interpretability properties of rule-based models with the high performance of deep neural networks. TT-rules is built upon $\textit{Truth Table nets}$ (TTnet), a family of deep neural networks initially developed for formal verification. By extracting the necessary and sufficient rules $\mathcal{R}$ from the trained TTnet model (global interpretability) to yield the same output as the TTnet (exact interpretability), TT-rules effectively transforms the neural network into a rule-based model. This rule-based model supports binary classification, multi-label classification, and regression tasks for small to large tabular datasets. After outlining the framework, we evaluate TT-rules' performance on healthcare applications and compare it to state-of-the-art rule-based methods. Our results demonstrate that TT-rules achieves equal or higher performance compared to other interpretable methods. Notably, TT-rules presents the first accurate rule-based model capable of fitting large tabular datasets, including two real-life DNA datasets with over 20K features.Comment: This work was presented at IAIM23 in Singapore https://iaim2023.sg/. arXiv admin note: substantial text overlap with arXiv:2309.0963

Regularity and trend to equilibrium for a non-local advection-diffusion model of active particles

We establish regularity and, under suitable assumptions, convergence to stationary states for weak solutions of a parabolic equation with a non-linear non-local drift term; this equation was derived from a model of active Brownian particles with repulsive interactions in a previous work, which incorporates advection-diffusion processes both in particle position and orientation. We apply De Giorgi's method and differentiate the equation with respect to the time variable iteratively to show that weak solutions become smooth away from the initial time. This strategy requires that we obtain improved integrability estimates in order to cater for the presence of the non-local drift. The instantaneous smoothing effect observed for weak solutions is shown to also hold for very weak solutions arising from distributional initial data; the proof of this result relies on a uniqueness theorem in the style of M.~Pierre for low-regularity solutions. The convergence to stationary states is proved under a smallness assumption on the drift term.Comment: 37 page

Error estimates for a finite difference scheme associated with Hamilton–Jacobi equations on a junction

This paper is concerned with monotone (time-explicit) finite difference scheme associated with first order Hamilton-Jacobi equations posed on a junction. It extends the scheme introduced by Costeseque, Lebacque and Monneau (2015) to general junction conditions. On the one hand, we prove the convergence of the numerical solution towards the viscosity solution of the Hamilton-Jacobi equation as the mesh size tends to zero for general junction conditions. On the other hand, we derive some optimal error estimates of in $L^\infty_{\text{loc}}$ for junction conditions of optimal-control type.The authors acknowledge the support of Agence Nationale de la Recherche through the funding of the project HJnet ANR-12-BS01-0008-01. The second author’s PhD thesis is supported by UL-CNRS Lebanon

A case of non-Hodgkin's lymphoma associated with hypercalcemia.

A patient with a diffuse, small cleaved cell, non-Hodgkin's lymphoma associated with marked hypecalcemia was described. Antibody to the adult T-cell leukemia-lymphoma virus was absent. Although bone marrow was infiltrated by lymphoma cells, destructive or lytic bone lesions could not be detected. The serum level of immunoreactive parathyroid hormone C-terminal (PTH-C) was normal. The serum level of 1, 25-dihydroxyvitamin D was lower than normal. This case suggests that other humoral substances produced by lymphoma cells may be responsible for hypercalcemia.&#60;/P&#62;</p