Haoran Sun

Haoran Sun, Wancheng Wu, Kun Wang

This paper develops a family of fast, structure-preserving numerical algorithms for the nonlinear Maxwell-Ampere Nernst-Planck equations. For the first-order scheme, the Slotboom transformation rewrites the Nernst-Planck equation to enable positivity preservation. The backward Euler method and centered finite differences discretize the transformed system. Two correction strategies are introduced: one enforces Gauss's law via a displacement correction, and the other preserves Faraday's law through potential reconstruction. The fully discrete scheme exactly satisfies mass conservation, concentration positivity, energy dissipation, Gauss's law, and Faraday's law, with established error estimates. The second-order scheme adopts BDF2 time discretization while retaining the same structure-preserving strategies, exactly conserving mass, Gauss's law, and Faraday's law. Numerical experiments validate both schemes using analytical solutions, confirming convergence orders and positivity preservation. Simulations of ion transport with fixed charges demonstrate exact preservation of Gauss's and Faraday's laws over long-time evolution, reproducing electrostatic attraction, ion accumulation, and electric field screening. The results fully support the theoretical analysis and the schemes' stability and superior performance.

Zhijian Duan, Haoran Sun, Yichong Xia et al.

Identifying high-revenue mechanisms that are both dominant strategy incentive compatible (DSIC) and individually rational (IR) is a fundamental challenge in auction design. While theoretical approaches have encountered bottlenecks in multi-item auctions, there has been much empirical progress in automated designing such mechanisms using machine learning. However, existing research primarily focuses on randomized auctions, with less attention given to the more practical deterministic auctions. Therefore, this paper investigates the automated design of deterministic auctions and introduces OD-VVCA, an objective decomposition approach for automated designing Virtual Valuations Combinatorial Auctions (VVCAs). Firstly, we restrict our mechanism to deterministic VVCAs, which are inherently DSIC and IR. Afterward, we utilize a parallelizable dynamic programming algorithm to compute the allocation and revenue outcomes of a VVCA efficiently. We then decompose the revenue objective function into continuous and piecewise constant discontinuous components, optimizing each using distinct methods. Extensive experiments show that OD-VVCA achieves high revenue in multi-item auctions, especially in large-scale settings where it outperforms both randomized and deterministic baselines, indicating its efficacy and scalability.