In this article, we propose and study a stochastic and relaxed preconditioned Douglas–Rachford splitting method to solve saddle-point problems that have separable dual variables. We prove the almost sure convergence of the iteration sequences in Hilbert spaces for a class of convex-concave and nonsmooth saddle-point problems. We also provide the sublinear convergence rate for the ergodic sequence concerning the expectation of the restricted primal-dual gap functions. Numerical experiments show the high efficiency of the proposed stochastic and relaxed preconditioned Douglas–Rachford splitting methods.
成果 2
数学学院教师向田与其博士生陈元琳及硕士生刘硕合作的论文《Global dynamics in a Keller-Segel system with rotation and indirect signal production》在期刊《Nonlinearity》发表。
In this work, we study global dynamics in a Neumann-initial boundary value problem for a fully parabolic minimal Keller-Segel model with matrix-valued sensitivity and indirect signal production. This work provides quantitative and qualitative effects of the sensitivity matrix S on global boundedness, preservation of radial symmetry, blow-up and exponential convergence of bounded solutions in the minimal indirect Keller–Segel model. The exponential convergence is new and fills up a gap even for the (scalar) indirect Keller–Segel model. Therefore, this work significantly generalizes the previous knowledge on the (scalar) indirect Keller–Segel model and the (direct) Keller–Segel model with rotational sensitivity to the indirect Keller–Segel model with rotational effect.
成果 3
数学学院教师刘双与清华大学林勇老师合作的论文《Gradient estimates for unbounded Laplacians with ellipticity condition on graphs》在期刊《Journal of Mathematical Analysis and Applications》发表。
In this article, we prove various gradient estimates for unbounded graph Laplacians which satisfy the ellipticity condition. Unlike common assumptions for unbounded Laplacians, i.e. completeness and non-degenerate measure, the ellipticity condition is purely local that is easy to verify on a graph. First, we establish an equivalent semigroup property, namely the gradient estimate of exponential curvature-dimension inequality, which is a modification of the curvature-dimension inequality and can be viewed as a notion of curvature on graphs. Additionally, we use the semigroup method to prove the Li-Yau inequalities and the Hamilton inequality for unbounded Laplacians on graphs with the ellipticity condition.
