Guowei Wei

Lin Mu, Junping Wang, Guowei Wei et al.

Weak Galerkin methods refer to general finite element methods for PDEs in which differential operators are approximated by their weak forms as distributions. Such weak forms give rise to desirable flexibilities in enforcing boundary and interface conditions. A weak Galerkin finite element method (WG-FEM) is developed in this paper for solving elliptic partial differential equations (PDEs) with discontinuous coefficients and interfaces. The paper also presents many numerical tests for validating the WG-FEM for solving second order elliptic interface problems. For such interface problems, the solution possesses a certain singularity due to the nonsmoothness of the interface. A challenge in research is to design high order numerical methods that work well for problems with low regularity in the solution. The best known numerical scheme in the literature is of order one for the solution itself in $L_\infty$ norm. It is demonstrated that the WG-FEM of lowest order is capable of delivering numerical approximations that are of order 1.75 in the usual $L_\infty$ norm for $C^1$ or Lipschitz continuous interfaces associated with a $C^1$ or $H^2$ continuous solutions. Theoretically, it is proved that high order of numerical schemes can be designed by using the WG-FEM with polynomials of high order on each element.

Bao Wang, Chengzhang Wang, Guowei Wei

In this work, a systematic protocol is proposed to automatically parametrize implicit solvent models with polar and nonpolar components. The proposed protocol utilizes the classical Poisson model or the Kohn-Sham density functional theory (KSDFT) based polarizable Poisson model for modeling polar solvation free energies. For the nonpolar component, either the standard model of surface area, molecular volume, and van der Waals interactions, or a model with atomic surface areas and molecular volume is employed. Based on the assumption that similar molecules have similar parametrizations, we develop scoring and ranking algorithms to classify solute molecules. Four sets of radius parameters are combined with four sets of charge force fields to arrive at a total of 16 different parametrizations for the Poisson model. A large database with 668 experimental data is utilized to validate the proposed protocol. The lowest leave-one-out root mean square (RMS) error for the database is 1.33k cal/mol. Additionally, five subsets of the database, i.e., SAMPL0-SAMPL4, are employed to further demonstrate that the proposed protocol offers some of the best solvation predictions. The optimal RMS errors are 0.93, 2.82, 1.90, 0.78, and 1.03 kcal/mol, respectively for SAMPL0, SAMPL1, SAMPL2, SAMPL3, and SAMPL4 test sets. These results are some of the best, to our best knowledge.

Sean Cottrell, Rui Wang, Guowei Wei

Over the years, Principal Component Analysis (PCA) has served as the baseline approach for dimensionality reduction in gene expression data analysis. It primary objective is to identify a subset of disease-causing genes from a vast pool of thousands of genes. However, PCA possesses inherent limitations that hinder its interpretability, introduce classification ambiguity, and fail to capture complex geometric structures in the data. Although these limitations have been partially addressed in the literature by incorporating various regularizers such as graph Laplacian regularization, existing improved PCA methods still face challenges related to multiscale analysis and capturing higher-order interactions in the data. To address these challenges, we propose a novel approach called Persistent Laplacian-enhanced Principal Component Analysis (PLPCA). PLPCA amalgamates the advantages of earlier regularized PCA methods with persistent spectral graph theory, specifically persistent Laplacians derived from algebraic topology. In contrast to graph Laplacians, persistent Laplacians enable multiscale analysis through filtration and incorporate higher-order simplicial complexes to capture higher-order interactions in the data. We evaluate and validate the performance of PLPCA using benchmark microarray datasets that involve normal tissue samples and four different cancer tissues. Our extensive studies demonstrate that PLPCA outperforms all other state-of-the-art models for classification tasks after dimensionality reduction.

Bao Wang, Guowei Wei

Developing accurate solvers for the Poisson Boltzmann (PB) model is the first step to make the PB model suitable for implicit solvent simulation. Reducing the grid size influence on the performance of the solver benefits to increasing the speed of solver and providing accurate electrostatics analysis for solvated molecules. In this work, we explore the accurate coarse grid PB solver based on the Green's function treatment of the singular charges, matched interface and boundary (MIB) method for treating the geometric singularities, and posterior electrostatic potential field extension for calculating the reaction field energy. We made our previous PB software, MIBPB, robust and provides almost grid size independent reaction field energy calculation. Large amount of the numerical tests verify the grid size independence merit of the MIBPB software. The advantage of MIBPB software directly make the acceleration of the PB solver from the numerical algorithm instead of utilization of advanced computer architectures. Furthermore, the presented MIBPB software is provided as a free online sever.

Bao Wang, Guowei Wei

Differential geometry (DG) based solvation models are a new class of variational implicit solvent approaches that are able to avoid unphysical solvent-solute boundary definitions and associated geometric singularities, and dynamically couple polar and nonpolar interactions in a self-consistent framework. Our earlier study indicates that DG based nonpolar solvation model outperforms other methods in nonpolar solvation energy predictions. However, the DG based full solvation model has not shown its superiority in solvation analysis, due to its difficulty in parametrization, which must ensure the stability of the solution of strongly coupled nonlinear Laplace-Beltrami and Poisson-Boltzmann equations. In this work, we introduce new parameter learning algorithms based on perturbation and convex optimization theories to stabilize the numerical solution and thus achieve an optimal parametrization of the DG based solvation models. An interesting feature of the present DG based solvation model is that it provides accurate solvation free energy predictions for both polar and nonploar molecules in a unified formulation. Extensive numerical experiment demonstrates that the present DG based solvation model delivers some of the most accurate predictions of the solvation free energies for a large number of molecules.

Bao Wang, Kelin Xia, Guowei Wei

Elastic materials are ubiquitous in nature and indispensable components in man-made devices and equipments. When a device or equipment involves composite or multiple elastic materials, elasticity interface problems come into play. The solution of three dimensional (3D) elasticity interface problems is significantly more difficult than that of elliptic counterparts due to the coupled vector components and cross derivatives in the governing elasticity equation. This work introduces the matched interface and boundary (MIB) method for solving 3D elasticity interface problems. The proposed MIB method utilizes fictitious values on irregular grid points near the material interface to replace function values in the discretization so that the elasticity equation can be discretized using the standard finite difference schemes as if there were no material interface. The interface jump conditions are rigorously enforced on the intersecting points between the interface and the mesh lines. Such an enforcement determines the fictitious values. A number of new technique are developed to construct efficient MIB schemes for dealing with cross derivative in coupled governing equations. The proposed method is extensively validated over both weak and strong discontinuity of the solution, both piecewise constant and position-dependent material parameters, both smooth and nonsmooth interface geometries, and both small and large contrasts in the Poisson's ratio and shear modulus across the interface. Numerical experiments indicate that the present MIB method is of second order convergence in both $L_\infty$ and $L_2$ error norms.