科研成果

The first nontrivial lower bound of the worst-case approximation ratio for the maxcut problem was achieved via the dual Cheeger problem, whose optimal value is referred to the dual Cheeger constant $h^+$, and later improved through its modification $\widehat{h}^+$. However, the dual Cheeger problem and its modification themselves are relatively unexplored, especially lack of effective approximate algorithms. To this end, we first derive equivalent spectral formulations of $h^+$ and  $\widehat{h}^+$ within the framework of the nonlinear spectral theory of signless 1-Laplacian, present their interactions with the Laplacian matrix and 1-Laplacians, and then use them to develop an inverse power algorithm that leverages the local linearity of the objective functions involved. We prove that the inverse power algorithm monotonically converges to a ternary-valued eigenvector, and provide the approximate values of $h^+$ and  $\widehat{h}^+$ on G-set for the first time. The recursive spectral cut algorithm for the maxcut problem can be enhanced by integrating into the inverse power algorithms, leading to significantly improved approximate values on G-set. Finally, we show that the lower bound of the worst-case approximation ratio for the maxcut problem within the recursive spectral cut framework can not be improved beyond $0.769$.

Persulfate-based advanced oxidation processes (PS-AOPs) catalyzed by carbon-based catalysts are promising for removing organic pollutants via radical/non-radical pathways. However, the activation efficiency of peroxymonosulfate (PMS) or peroxydisulfate (PDS) usage and the reaction mechanism remain insufficiently understood. In this study, the effects of PMS/PDS dosage on the degradation of bisphenol A (BPA, 10 mg/L) were evaluated using N-doped biochar (N-BC, 0.2 g/L) assisted PS-AOPs. The reaction pathways were comprehensively investigated through a combination of characterization techniques and molecular simulations. With low PS dosages (0.05 and 0.1 mM), the degradation rate constants () were higher in N-BC/PDS (0.04 and 0.07 min−1) compared to N-BC/PMS (0.02 and 0.04 min−1), likely due to higher PDS utilization, which enhanced the contribution of the non-radical pathway. Interestingly, with higher PS dosages (0.5 and 1.5 mM), the  values were 0.16 min−1 and 0.18 min−1 in N-BC/PMS, respectively, significantly exceeding those determined in N-BC/PDS (0.11 and 0.11 min−1). This result stemmed from the greater adsorption capacity of N-BC for PMS compared to PDS, leading to increased formation of 1O2. The contribution of non-radical pathways for both PMS and PDS increased with higher PS dosage. The results highlighted that BPA degradation improved significantly with the increase in PMS dosage; meanwhile, BPA degradation was insensitive to PDS dosage. The optimal PMS dosage for BPA degradation was found to be 1.5 mM and 0.1 mM for PDS. This study offered valuable insights for optimizing PS-AOPs in environmental remediation, helping to guide the selection of appropriate oxidants and dosages for maximizing pollutant removal.

Numerical resolution of moderately high-dimensional nonlinear PDEs remains a huge challenge due to the curse of dimensionality for the classical numerical methods including finite difference, finite element and spectral methods. Starting from the weak formulation of the Lawson-Euler scheme, this paper proposes a stochastic particle method (SPM) by tracking the deterministic motion, random jump, resampling and reweighting of particles. Real-valued weighted particles are adopted by SPM to approximate the high-dimensional solution, which automatically adjusts the point distribution to intimate the relevant feature of the solution. A piecewise constant reconstruction with virtual uniform grid is employed to evaluate the nonlinear terms, which fully exploits the intrinsic adaptive characteristic of SPM. Combining both, SPM can achieve the goal of adaptive sampling in time. Numerical experiments on the 6-D Allen-Cahn equation and the 7- D Hamiltonian-Jacobi-Bellman equation demonstrate the potential of SPM in solving moderately high-dimensional nonlinear PDEs efficiently while maintaining an acceptable accuracy