Newton algorithm

A novel optimization approach based on the Newton-Kantorovich iterative scheme applied to the Riccati equation describing the reflection from the inhomogeneous half-space was proposed recently [7]. The method works well with complicated highly contrasted dielectric profiles and retains stability with respect to the noise in the input data. However, this algorithm like others needs the measurement data to be given in a broad frequency band. In this work, the method is improved to be valid for the input data obtained in an essentially restricted frequency band, i.e. when both low and high frequency data are not available. This... 

Although it was originally developed for locating transition states, the EF algoritlnn is also efficient for minimization and usually perfonns as well as or better than the standard quasi-Newton algorithm. In this case, a single shift parameter is used, and the method is essentially identical to the augmented Hessian method. 

But the methods have not really changed. The Verlet algorithm to solve Newton s equations, introduced by Verlet in 1967 [7], and it s variants are still the most popular algorithms today, possibly because they are time-reversible and symplectic, but surely because they are simple. The force field description was then, and still is, a combination of Lennard-Jones and Coulombic terms, with (mostly) harmonic bonds and periodic dihedrals. Modern extensions have added many more parameters but only modestly more reliability. The now almost universal use of constraints for bonds (and sometimes bond angles) was already introduced in 1977 [8]. That polarisability would be necessary was realized then [9], but it is still not routinely implemented today. Long-range interactions are still troublesome, but the methods that now become popular date back to Ewald in 1921 [10] and Hockney and Eastwood in 1981 [11]. 

We may conclude that the matter of optimal algorithms for integrating Newton s equations of motion is now nearly settled however, their optimal and prudent use [28] has not been fully exploited yet by most programs and may still give us an improvement by a factor 3 to 5. 

T. Schlick and A. Fogelson. TNPACK — A truncated Newton minimization package for large-scale problems I. Algorithm and usage. ACM Trans. Math. Softw., 14 46-70, 1992. 

Extending time scales of Molecular Dynamics simulations is therefore one of the prime challenges of computational biophysics and attracted considerable attention [2-5]. Most efforts focus on improving algorithms for solving the initial value differential equations, which are in many cases, the Newton s equations of motion. 

A molecular dynamics simulation samples the phase space of a molecule (defined by the position of the atoms and their velocities) by integrating Newton s equations of motion. Because MD accounts for thermal motion, the molecules simulated may possess enough thermal energy to overcome potential barriers, which makes the technique suitable in principle for conformational analysis of especially large molecules. In the case of small molecules, other techniques such as systematic, random. Genetic Algorithm-based, or Monte Carlo searches may be better suited for effectively sampling conformational space. 

IlyperChem supplies three types of optimi/ers or algorithms steepest descent, conjugate gradient (Fletcher-Reeves and Polak-Ribiere), and block diagonal (Newton-Raph son). 

I he eigenvector-following (or Hessian mode) method implemented in HyperChem is based on an effieien t quasi-Newton like algorithm for loca tin g tran sitiori states, wh ieh can locate tran si-tion states for alternative rearran gern eri t/dissoeiation reactions, even when startin g from th e wron g regio n on th e poten tial en ergy surface. 

This formula is exact for a quadratic function, but for real problems a line search may be desirable. This line search is performed along the vector — x. . It may not be necessary to locate the minimum in the direction of the line search very accurately, at the expense of a few more steps of the quasi-Newton algorithm. For quantum mechanics calculations the additional energy evaluations required by the line search may prove more expensive than using the more approximate approach. An effective compromise is to fit a function to the energy and gradient at the current point x/t and at the point X/ +i and determine the minimum in the fitted function. 

The HE, GVB, local MP2, and DFT methods are available, as well as local, gradient-corrected, and hybrid density functionals. The GVB-RCI (restricted configuration interaction) method is available to give correlation and correct bond dissociation with a minimum amount of CPU time. There is also a GVB-DFT calculation available, which is a GVB-SCF calculation with a post-SCF DFT calculation. In addition, GVB-MP2 calculations are possible. Geometry optimizations can be performed with constraints. Both quasi-Newton and QST transition structure finding algorithms are available, as well as the SCRF solvation method. 

The Newton-Raphson block diagonal method is a second order optimizer. It calculates both the first and second derivatives of potential energy with respect to Cartesian coordinates. These derivatives provide information about both the slope and curvature of the potential energy surface. Unlike a full Newton-Raph son method, the block diagonal algorithm calculates the second derivative matrix for one atom at a time, avoiding the second derivatives with respect to two atoms. 

In HyperChem, two different methods for the location of transition structures are available. Both arethecombinationsofseparate algorithms for the maximum energy search and quasi-Newton methods. The first method is the eigenvector-following method, and the second is the synchronous transit method. 

There are several reasons that Newton-Raphson minimization is rarely used in mac-romolecular studies. First, the highly nonquadratic macromolecular energy surface, which is characterized by a multitude of local minima, is unsuitable for the Newton-Raphson method. In such cases it is inefficient, at times even pathological, in behavior. It is, however, sometimes used to complete the minimization of a structure that was already minimized by another method. In such cases it is assumed that the starting point is close enough to the real minimum to justify the quadratic approximation. Second, the need to recalculate the Hessian matrix at every iteration makes this algorithm computationally expensive. Third, it is necessary to invert the second derivative matrix at every step, a difficult task for large systems. 

Equation 13-39 is a cubic equation in terms of the larger aspect ratio R2. It can be solved by a numerical method, using the Newton-Raphson method (Appendix D) with a suitable guess value for R2. Alternatively, a trigonometric solution may be used. The algorithm for computing R2 with the trigonometric solution is as follows ... 

Molecular dynamics, in contrast to MC simulations, is a typical model in which hydrodynamic effects are incorporated in the behavior of polymer solutions and may be properly accounted for. In the so-called nonequilibrium molecular dynamics method [54], Newton s equations of a (classical) many-particle problem are iteratively solved whereby quantities of both macroscopic and microscopic interest are expressed in terms of the configurational quantities such as the space coordinates or velocities of all particles. In addition, shear flow may be imposed by the homogeneous shear flow algorithm of Evans [56]. 

It uses a linear or quadratic synchronous transit approach to get closer to the quadratic region of the transition state and then uses a quasi-Newton or eigenvalue-following algorithm to complete the optimization. 

In a recent version, the LST or QST algorithm is used to find an estimate of the maximum, and a Newton method is then used to complete the optimization (Peng and Schlegel, 1993). 

Pseudo-Newton-Raphson methods have traditionally been the preferred algorithms with ab initio wave function. The interpolation methods tend to have a somewhat poor convergence characteristic, requiring many function and gradient evaluations, and have consequently primarily been used in connection with semi-empirical and force field methods. 

---