Gradient Decay in the Boltzmann theory of Non-isothermal boundary

We consider the Boltzmann equation in convex domain with non-isothermal boundary of diffuse reflection. For both unsteady/steady problems, we construct solutions belong to $W^{1,p}_x$ for any $p<3$. We prove that the unsteady solution converges to the steady solution in the same Sobolev space exponentially fast as $t \rightarrow \infty$

Macroscopic estimate of the linear Boltzmann and Landau equations with Specular reflection boundary

In this short note, we prove an $L^6$-control of the macroscopic part of the linear Boltzmann and Landau equations. This result is an extension of the test function method of Esposito-Guo-Kim-Marra~\cite{EGKM}\cite{EGKM2} to the specular reflection boundary condition, in which we crucially used the Korn's inequality and the system of symmetric Poisson equations

A Zeroth-Order Variance-Reduced Method for Decentralized Stochastic Non-convex Optimization

In this paper, we consider a distributed stochastic non-convex optimization problem, which is about minimizing a sum of $n$ local cost functions over a network with only zeroth-order information. A novel single-loop Decentralized Zeroth-Order Variance Reduction algorithm, called DZOVR, is proposed, which combines two-point gradient estimation, momentum-based variance reduction technique, and gradient tracking. Under mild assumptions, we show that the algorithm is able to achieve $\mathcal{O}(dn^{-1}\epsilon^{-3})$ sampling complexity at each node to reach an $\epsilon$-accurate stationary point and also exhibits network-independent and linear speedup properties. To the best of our knowledge, this is the first stochastic decentralized zeroth-order algorithm that achieves this sampling complexity. Numerical experiments demonstrate that DZOVR outperforms the other state-of-the-art algorithms and has network-independent and linear speedup properties

The Sequence Optimization of the Railway Tree-Shaped Special Line\u27s Shunting for Taking-out and Placing-in of Wagons

Shunting for taking-out and placing-in of wagons (STPW) is an important work of freight stations, technical stations with more cargo operations, and large intermediate stations.The sequence optimization of STPW can effectively reduce the total shunting time and reduce operating costs, which is of great significance. When the optimization goal is to minimize the total shunting time, there will be many optimal solutions. Therefore, it is necessary to introduce a second optimization objective in order to seek a more reasonable solution from numerous optimal solutions. However, there is no relevant research on this in existing literature. In response to this situation, this paper proposes two optimization objectives: the first is to minimize the total time of STPW, and the second is to minimize the wagon-hour based on minimizing the total time of STPW. Based on the above two optimization objectives, a mathematical model is established, and a three-stage optimization solution strategy is proposed. The first stage is to solve the first optimization objective by using the improved ant colony algorithm; the second stage is to introduce and improve the crossover operation of the genetic algorithm to improve the diversity of the optimal path; the third stage is to solve the second optimization objective. Finally, an example is given to verify the feasibility of the model and the solution strategy