Lijie Ji

Lijie Ji, Yanlai Chen, Zhenli Xu

In numerical simulations of many charged systems at the micro/nano scale, a common theme is the repeated solution of the Poisson-Boltzmann equation. This task proves challenging, if not entirely infeasible, largely due to the nonlinearity of the equation and the high dimensionality of the physical and parametric domains with the latter emulating the system configuration. In this paper, we for the first time adapt a mathematically rigorous and computationally efficient model order reduction paradigm, the so-called reduced basis method (RBM), to mitigate this challenge. We adopt a finite difference method as the mandatory underlying scheme to produce the {\em truth approximations} of the RBM upon which the fast algorithm is built and its performance is measured against. Numerical tests presented in this paper demonstrate the high efficiency and accuracy of the fast algorithm, the reliability of its error estimation, as well as its capability in effectively capturing the boundary layer.

Lijie Ji, Sabrina Rashid, Yanlai Chen et al.

Traditional projection-based reduced-order modeling approximates the full-order model by projecting it onto a linear subspace. With a fast-decaying Kolmogorov $n$-width of the solution manifold, the resulting reduced-order model (ROM) can be an efficient and accurate emulator. However, for parametric partial differential equations with slowly decaying Kolmogorov $n$-width, the dimension of the linear subspace required for a reasonable accuracy becomes very large, which undermines computational efficiency. To address this limitation, quadratic manifold methods have recently been proposed. These data-driven methods first identify a quadratic mapping by minimizing the linear projection error over a large set of snapshots, often with the aid of regularization techniques to solve the associated minimization problem, and then use this mapping to construct ROMs. In this paper, we propose and test a novel enhancement to this quadratic manifold approach by introducing a first-of-its-kind double-greedy algorithm on the regularization parameters coupled with a standard greedy algorithm on the physical parameter. Our approach balances the trade-off between the accuracy of the quadratic mapping and the stability of the resulting nonlinear ROM, leading to a highly efficient and data-sparse algorithm. Numerical experiments conducted on equations such as linear transport, acoustic wave, advection-diffusion, and Burgers' demonstrate the accuracy, efficiency, and stability of the proposed algorithm.

Tansheng Zhu, Hongyu Zhou, Ke Jin et al.

Bayesian optimization is highly effective for optimizing expensive-to-evaluate black-box functions, but it faces significant computational challenges due to the high computational complexity of Gaussian processes, which results in a total time complexity that is quartic with respect to the number of iterations. To address this limitation, we propose the Bayesian Optimization by Kernel regression and density-based Exploration (BOKE) algorithm. BOKE uses kernel regression for efficient function approximation, kernel density for exploration, and integrates them into the confidence bound criteria to guide the optimization process, thus reducing computational costs to quadratic. Our theoretical analysis rigorously establishes the global convergence of BOKE and ensures its robustness in noisy settings. Through extensive numerical experiments on both synthetic and real-world optimization tasks, we demonstrate that BOKE not only performs competitively compared to Gaussian process-based methods and several other baseline methods but also exhibits superior computational efficiency. These results highlight BOKE's effectiveness in resource-constrained environments, providing a practical approach for optimization problems in engineering applications.