Restricted step method

Bofill J M 1994 Updated Hessian matrix and the restricted step method for locating transition structures J. Comput. Chem. 15 1... 

In the trust region or restricted step method we determine in each iteration the global minimizer of the RSO model Eq. (3.10). In the global region a step is taken to the boundary of the trust region... 

In molecular quantum mechanics, the analytical calculation of G is very time consuming. Furthermore, as discussed later, the Hessian should be positive definite to ensure a step in the direction of the local minimum. One solution to this later problem is to precondition the Hessian matrix and this is discussed for the restricted step methods. The Quasi-Newton methods, presented next, provides alternative solution to both of these problems. 

The restricted step method of the type discussed below were originally proposed by Levenberg and Marquardt [37,38] and extended to minimization algorithms by Goldfeld, Quandt and Trotter [39]. Recently Simons [13] discussed the restricted step method with respect to molecular energy hypersurfaces. The basic idea again is that the energy hypersurface E(x) can reasonably be approximated, at least locally, by the quadratic function... 

The restricted step method requires the two parameters A and X to be defined. Various numerical tests have been suggested for varying the trust region size A depending upon the quality of the quadratic surfaces [9,37,38]. 

In our experience the Newton and Quasi-Newton methods with the line search described is faster to reach a minimum than the restricted step method with confidence region. This latter method is a more "conservative" method as discussed in the results section. The restricted step methods, however, are very effective in searching for transition states [40,41]. 

How is it possible to overcome the discussed shortcomings of line search methods and to embed more information about the function into the search for the local minimum One answer are trust-region methods (or restricted step methods). They do a search in a restricted neighborhood of the current iterate and try to minimize a quadratic model of /. For example in the double-dogleg implementation, it is a restricted step search in a two-dimensional subspace, spanned by the actual gradient and the Newton step (and further reduced to a non-smooth curve search). For information on trust region methods see e.g. Dennis and Schnabel [2], pp. 129ff. 

In the trust-region or restricted-step method [19], the Newton step (10.8.2) is taken only if it is smaller than or equal to the trust radius h ... 

Another way to overcome the step-size restriction fc < is to use multiple-time-stepping methods [4] or implicit methods [17, 18, 12, 3). In this paper, we examine the latter possibility. But for large molecular systems, fully implieit methods are very expensive. For that reason, we foeus on the general class of scmi-implicit methods depicted in Fig. 1 [12]. In this scheme. Step 3 of the nth time step ean be combined with Step 1 of the (n - - l)st time step. This then is a staggered two-step splitting method. We refer to [12] for further justification. 

In addition to examining Newton steps, we have also examined methods of the restricted step or augmented Hessian type. 

The first method is that of Car and Parinello, in which one performs molecular dynamics simulations for a large number (hundreds) of molecules, calculating the forces and energy from quantum mechanics for the whole assembly of molecules at each step. However, in order to be tractable, the calculations are carried out with DFT, but even then only for small molecules and requiring greater computational resources than are usually available. Also, the use of DFT restricts this method to systems in which electrostatic forces dominate ab initio methods could be used instead, but only at very large supercomputer centers. 

The conformationally restricted cyclic, disulfide containing, enkephalin analogue (D-Pen, D-Pen) enkephalin (DPDPE) was synthesized by solid-phase methods. Its purification was accomplished previously by partition on Sephadex G-25 block polymerizate using the solvent system (1-butanol acetic acid water 4 1 5), followed by gel filtration on Sephadex G-15 with 30% acetic acid as the eluent. DuCCC demonstrated a highly efficient and one step method for the purification of DPDPE. The crude DPDPE (500 mg), which contained impurities and salts, was purified by DuCCC with a two-phase solvent system... 

All critical steps, method parameters or restrictions for any aspect of the stepwise procedure that may have been established during method robustness testing (Section 9.8.4) should also be described in the appropriate section of the procedure. If any special test injections or system suitability procedures have been established (Section 9.8.3) a detailed description of these procedures, along with instructions and examples for the preparation of any reqnired test solutions, should be described. 

In this section we will present one-step methods for DAEs. Though there are successful implementations of extrapolation methods for differential algebraic equations [DHZ87] and mainly for constrained mechanical systems [Lub91] we restrict our discussion of one-step methods to Runge-Kutta methods. 

We will restrict us here to the presentation of half-explicit one-step methods. They have been first introduced in [HLR89]. 

Above, we have studied in detail the stability properties of simple single-step methods of the class (4.150). More generally, explicit methods always have time restrictions of the form (4.169), and the more a method makes use of past data to predict the fiiture, the more strict the time step restrictions become. We consider here the stabiUty properties of implicit multi-step BDF methods... 

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