Abstract. To e ciently implement the truncated-Newton TN optimization method for largescale, highly nonlinear functions in chemistry, an unconventional modi ed Cholesky UMC factorization is proposed to avoid large modi cations 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 modi ed Cholesky factorizations are found to increase the number of total iterations, as well as computational time, signi cantly. Since the UMC may generate an inde nite, rather than a positive de nite, e ective 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 inde nite preconditioner works much better than the minimizer without preconditioning or other minimizers available in the molecular mechanics and dynamics 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.
The nonlinear Poisson-Boltzmann equation (PBE) is a widely-used implicit solvent model in biomolecular simulations. This paper formulates a new PBE nonlinear algebraic system from a mortar finite element approximation, and proposes a new minimization protocol to solve it efficiently. In particular, the PBE mortar nonlinear algebraic system is proved to have a unique solution, and is equivalent to a unconstrained minimization problem. It is then solved as the unconstrained minimization problem by the subspace trust region Newton method. Numerical results show that the new minimization protocol is more efficient than the traditional merit least squares approach in solving the nonlinear system. At least 80 percent of the total CPU time was saved for a PBE model problem.
We present a nonlocal electrostatic formulation of nonuniform ions and water molecules with interstitial voids that uses a Fermi-like distribution to account for steric and correlation effects in electrolyte solutions. The formulation is based on the volume exclusion of hard spheres leading to a steric potential and Maxwell's displacement field with Yukawa-type interactions resulting in a nonlocal electric potential. The classical Poisson-Boltzmann model fails to describe steric and correlation effects important in a variety of chemical and biological systems, especially in high field or large concentration conditions found in and near binding sites, ion channels, and electrodes. Steric effects and correlations are apparent when we compare nonlocal Poisson-Fermi results to Poisson-Boltzmann calculations in electric double layer and to experimental measurements on the selectivity of potassium channels for K + over Na + . The present theory links atomic scale descriptions of the crystallized KcsA channel with macroscopic bulk conditions. Atomic structures and macroscopic conditions determine complex functions of great importance in biology, nanotechnology, and electrochemistry.Continuum electrostatic theory is a fundamental tool for studying physical and chem-
The nonlocal dielectric approach has been studied for more than forty years but only limited to water solvent until the recent work (SISC, 35(6):B1267-1284, 2013). 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.
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