Algebraic Core and Convex Calculus without Topology

Algebraic Lipschitz and Subdifferential Calculus in General Vector Spaces

The main contribution of this paper is that every convex function with non-empty relative algebraic interior of its domain is Lipschitz and subdifferentiable in some algebraic sense without any additional topological constraints. The proposed approach uses slightly modified Clarke's subdifferential for functions defined on a convex symmetric set and Lipschitz with respect to a Minkowski functional. Following this, Clarke's subdifferential calculus is generalized to vector spaces and, where continuity properties are claimed, to topological vector spaces.

基于有限元外微积分的希尔伯特复空间最优控制

Optimal Control in Hilbert Complex Spaces with Finite Element Exterior Calculus

该研究开发了基于有限元外微积分的希尔伯特复空间PDE约束最优控制框架，建立了相关理论结果，通过数值实验验证了其在不同区域的控制效果，并扩展到时变麦克斯韦控制场景。

Algebraic Approach for Performance Bound Calculus on Transportation Networks (Road Network Calculus)

We propose in this article an adaptation of the basic techniques of the deterministic network calculus theory to the road traffic flow theory. Network calculus is a theory based on min-plus algebra. It uses algebraic techniques to compute performance bounds in communication networks, such as maximum end-to-end delays and backlogs. The objective of this article is to investigate the application of such techniques for determining performance bounds on road networks, such as maximum bounds on travel times. The main difficulty to apply the network calculus theory on road networks is the modeling o…

形状微积分与自动微分在非拟合有限元多相水平集拓扑优化中的应用

Shape calculus and automatic differentiation for multi-phase level-set topology optimisation with unfitted finite elements

本文提出形状微积分与可扩展自动形状微分框架，用于非拟合多相拓扑优化，经验证高精度，支持大规模并行，并应用于各向异性扩散、线弹性和流固耦合问题。

Extremal Systems of Convex Sets with Applications to Convex Calculus in Vector Spaces

In this paper we introduce and study the concept of set extremality for systems of convex sets in vector spaces without topological structures. Characterizations of the extremal systems of sets are obtained in the form of the convex extremal principle, which is shown to be equivalent to convex separation under certain qualification conditions expressed via algebraic cores. The obtained results are applied via a variational geometric approach to deriving enhanced calculus rules for normals to convex sets, coderivatives of convex set-valued mappings, and subgradients of extended-real-valued conv…