Dexuan Xie

Efficient Implementation of the Truncated-Newton Algorithm for Large-Scale Chemistry Applications

To efficiently implement the truncated-Newton (TN) optimization method for large-scale, highly nonlinear functions in chemistry, an unconventional modified Cholesky (UMC) factorization is proposed to avoid large modifications to a problem-derived preconditioner, used in the inner loop in approximating the TN search vector at each step. The main motivation is to reduce the computational time of the overall method: large changes in standard modified Cholesky factorizations are found to increase the number of total iterations, as well as computational time, significantly. Since the UMC may generate an indefinite, rather than a positive definite, effective preconditioner, we prove that directions of descent still result. Hence, convergence to a local minimum can be shown, as in classic TN methods, for our UMC-based algorithm. Our incorporation of the UMC also requires changes in the TN inner loop regarding the negative-curvature test (which we replace by a descent direction test) and the choice of exit directions. Numerical experiments demonstrate that the unconventional use of an indefinite preconditioner works much better than the minimizer without preconditioning or other minimizers available in the molecular mechanics package CHARMM. Good performance of the resulting TN method for large potential energy problems is also shown with respect to the limited-memory BFGS method, tested both with and without preconditioning.

New Parallel SOR Method by Domain Partitioning

In this paper we propose and analyze a new parallel SOR method, the PSOR method, formulated by using domain partitioning and interprocessor data communication techniques. We prove that the PSOR method has the same asymptotic rate of convergence as the Red/Black (R/B) SOR method for the five-point stencil on both strip and block partitions, and as the four-color (R/B/G/O) SOR method for the nine-point stencil on strip partitions. We also demonstrate the parallel performance of the PSOR method on four different MIMD multiprocessors (a KSR1, an Intel Delta, a Paragon, and an IBM SP2). Finally, we compare the parallel performance of PSOR, R/B SOR, and R/B/G/O SOR. Numerical results on the Paragon indicate that PSOR is more efficient than R/B SOR and R/B/G/O SOR in both computation and interprocessor data communication.

An Efficient Projection Protocol for Chemical Databases: Singular Value Decomposition Combined with Truncated-Newton Minimization

A rapid algorithm for visualizing large chemical databases in a low-dimensional space (2D or 3D) is presented as a first step in database analysis and design applications. The projection mapping of the compound database (described as vectors in the high-dimensional space of chemical descriptors) is based on the singular value decomposition (SVD) combined with a minimization procedure implemented with the efficient truncated-Newton program package (TNPACK). Numerical experiments on four chemical datasets with real-valued descriptors (ranging from 58 to 27 255 compounds) show that the SVD/TNPACK projection duo achieves a reasonable accuracy in 2D, varying from 30% to about 100% of pairwise distance segments that lie within 10% of the original distances. The lowest percentages, corresponding to scaled datasets, can be made close to 100% with projections onto a 10-dimensional space. We also show that the SVD/TNPACK duo is efficient for minimizing the distance error objective function (especially for scaled datasets), and that TNPACK is much more efficient than a current popular approach of steepest descent minimization in this application context. Applications of our projection technique to similarity and diversity sampling in drug design can be envisioned.

New solution decomposition and minimization schemes for Poisson-Boltzmann equation in calculation of biomolecular electrostatics

The Poisson-Boltzmann equation (PBE) is one widely-used implicit solvent continuum model in the calculation of electrostatic potential energy for biomolecules in ionic solvent, but its numerical solution remains a challenge due to its strong singularity and nonlinearity caused by its singular distribution source terms and exponential nonlinear terms. To effectively deal with such a challenge, in this paper, new solution decomposition and minimization schemes are proposed, together with a new PBE analysis on solution existence and uniqueness. Moreover, a PBE finite element program package is developed in Python based on the FEniCS program library and GAMer, a molecular surface and volumetric mesh generation program package. Numerical tests on proteins and a nonlinear Born ball model with an analytical solution validate the new solution decomposition and minimization schemes, and demonstrate the effectiveness and efficiency of the new PBE finite element program package.

Remark on Algorithm 702—the updated truncated Newton minimization package

A truncated Newton minimization package, TNPACK, was described in ACM Transactions on Mathematical Software 14, 1 (Mar. 1992), pp.46–111. Modifications to enhance performance, especially for large-scale minimization of molecular potential functions, are described here. They involve three program segments of TNPACK: negative curvature test, modified Cholesky factorization, and line-search stopping rule.

A Fast Solver for a Nonlocal Dielectric Continuum Model

The nonlocal continuum dielectric model is an important extension of the classical Poisson dielectric model, but it is very expensive to be solved in general. In this paper, we prove that the solution of one commonly used nonlocal continuum dielectric model of water can be split as a sum of two functions, and these two functions are simply the solutions of one Poisson equation and one Poisson-like equation. With this new solution splitting formula, we develop a fast finite element algorithm and a program package in Python based on the DOLFIN program library such that a nonlocal dielectric model can be solved numerically in an amount of computation that merely doubles that of solving a classic Poisson dielectric model. Using the new solution splitting formula, we also derive the analytical solutions of two nonlocal model problems. We then solve these two nonlocal model problems numerically by our program package and validate the numerical solutions through a comparison with the analytical solutions. Finally, our study of free energy calculation by a nonlocal Born ion model demonstrates that the nonlocal dielectric model is a much better predictor of the solvation free energy of ions than the local Poisson dielectric model.

A More Lenient Stopping Rule for Line Search Algorithms

An iterative univariate minimizer (line search) is often used to generate a steplength in each step of a descent method for minimizing a multivariate function. The line search performance strongly depends on the choice of the stopping rule enforced. This termination criterion and other algorithmic details also affect the overall efficiency of the multivariate minimization procedure. Here we propose a more lenient stopping rule for the line search that is suitable for objective univariate functions that are not necessarily convex in the bracketed search interval. We also describe a remedy to special cases where the minimum point of the cubic interpolant constructed in each line search iteration is very close to zero. Results in the context of the truncated Newton package TNPACK for 18 standard test functions, as well as molecular potential functions, show that these strategies can lead to modest performance improvements in general, and significant improvements in special cases.

A new finite element and finite difference hybrid method for computing electrostatics of ionic solvated biomolecule

The Poisson-Boltzmann equation (PBE) is one widely-used implicit solvent continuum model for calculating electrostatics of ionic solvated biomolecule. In this paper, a new finite element and finite difference hybrid method is presented to solve PBE efficiently based on a special seven-overlapped box partition with one central box containing the solute region and surrounded by six neighboring boxes. In particular, an efficient finite element solver is applied to the central box while a fast preconditioned conjugate gradient method using a multigrid V-cycle preconditioning is constructed for solving a system of finite difference equations defined on a uniform mesh of each neighboring box. Moreover, the PBE domain, the box partition, and an interface fitted tetrahedral mesh of the central box can be generated adaptively for a given PQR file of a biomolecule. This new hybrid PBE solver is programmed in C, Fortran, and Python as a software tool for predicting electrostatics of a biomolecule in a symmetric 1:1 ionic solvent. Numerical results on two test models with analytical solutions and 12 proteins validate this new software tool, and demonstrate its high performance in terms of CPU time and memory usage. Propose special 7-box partition with one solute box surrounded by six solvent boxes.Define a finite element and finite difference hybrid PBE solver on the 7-box partition.Program PCG with multigrid V-cycle preconditioning for linear system on solvent box.Generate the 7-box partition and an interface-fitted mesh of solute box adaptively.Reduce the total CPU time of a finite element PBE solver up to 70% in protein tests.Verify the new hybrid PBE solver to be numerically stable in computing solvation free energy.

Principal component analysis combined with truncated-Newton minimization for dimensionality reduction of chemical databases

Abstract. The similarity and diversity sampling problems are two challenging optimization tasks that arise in the analysis of chemical databases. As a first step to their solution, we propose an efficient projection/ refinement protocol based on the principal component analysis (PCA) and the truncated-Newton minimization method implemented by our package TNPACK (PCA/TNPACK). We show that PCA can provide the same initial guess as the singular value decomposition (SVD) for the optimization task of solving the distance-geometry optimization problem if each column of a database matrix has a mean of zero. Hence, our PCA/TNPACK approach is analogous to the SVD/TNPACK projection/refinement protocol that we developed recently for visualizing large chemical databases. Using PCA/TNPACK and the Merck MDDR database (MDL Drug Data Report), we further investigate the projection/refinement procedure with regards to the preservation of the original clusters of chemical compounds, the accuracy of similarity and diversity sampling of chemical compounds, and the potential application in the study of structure activity relationships. We also explore by simple experiments accuracy and efficiency aspects of the PCA/TNPACK procedure compared to those of a global optimization algorithm (simulated annealing, as implemented by the program package SIMANN) in terms of producing the projection mapping of a database. Numerical results show that the 2D PCA/TNPACK mapping can preserve the distance relationships of the original database and is thus valuable as a first step in similarity and diversity applications. Of course, the generation of a global rather than local minimizer and its interpretation in terms of pharameceutical applications remains a challenge. Since all numerical tests are performed on the Merck MDDR database, results are representative of realistic cases encountered in the field of drug design, and may help analyze properties of medicinal compounds.

A nonlocal modified Poisson–Boltzmann equation and finite element solver for computing electrostatics of biomolecules

Abstract The nonlocal dielectric approach has been studied for more than forty years but only limited to water solvent until the recent work of Xie et al. (2013) [20] . As the development of this recent work, in this paper, a nonlocal modified Poisson–Boltzmann equation (NMPBE) is proposed to incorporate nonlocal dielectric effects into the classic Poisson–Boltzmann equation (PBE) for protein in ionic solvent. The focus of this paper is to present an efficient finite element algorithm and a related software package for solving NMPBE. Numerical results are reported to validate this new software package and demonstrate its high performance for protein molecules. They also show the potential of NMPBE as a better predictor of electrostatic solvation and binding free energies than PBE.