Steepest descent direction

The reaction path is defined by Fukui [83] as the line q(.s) leading down from a transition state along the steepest descent direction... 

A starting poin t is defined an d th e in itial conjugate direction is chosen to be the steepest descent direction, h[, = g... 

The conjugate direction is reset to the steepest descent direction every 3N search direction s or cycles, or if the en ergy rises between cycles. 

For quadratic functions this is identical to the Fletcher-Reeves formula but there is some evidence that the Polak-Ribiere may be somewhat superior to the Fletcher-Reeves procedure for non-quadratic functions. It is not reset to the steepest descent direction unless the energy has risen between cycles. 

The gradient methods, like those of Newton, Gauss-Newton, Fletcher, and Levenberg-Marquardt, use the derivative vector of the SSR with respect to the parameter directions to determine the direction where this gradient changes most, the steepest-descent direction. 

The steepest descent direction, that is, the negative gradient of the Born-Oppenheimer potential energy hypersurfacecan be determined from the forces -dE/dR acting on nuclei a of coordinates X, Y, and Z. In some applications, the equivalent representation as the Hellmann-Feynman force (F > is more advantageous. 

Figure 5-3 The top part of the figure shows the isolines of the misfit functional map and the steepest descent path of the iterative solutions in the space of model parameters. The bottom part presents a magnified element of this map with just one iteration step shown, from iteration (n. — 1) to iteration number ti. According to the line search principle, the direction of the steepest ascent at iteration number n must be perpendicular to the misfit isoline at the minimum point along the previous direction of the steepest descent. Therefore, many steps may be required to reach the global minimum, because every subsequent steepest descent direction is perpendicular to the previous one, similar to the path of experienced slalom skiers.

Equation (5.69) follows from (5.62), and equation (5.70) holds because in the previous step we moved along the search line in the direction I i to the minimum, so the steepest descent direction 1 at the minimum point will be perpendicular to l i. Also, it can be shown that... 

This method uses the same ideas as the conventional conjugate gradient method. However, the iteration process is based on the calculation of the regularized steepest descent directions ... 

Figure 9.3 Photodissociation Spectra of ABC. A laser pulse transfers the ground state vibrational wavefunction to the repulsive electronically excited potential energy surface. The wavepacket moves and spreads (to, ti, <2, 3 snapshots) on the excited surface, being accelerated in the steepest descent direction (force is negative gradient of the potential) toward A + BCt products (where and f refer respectively to electronic and vibrational excitation). Information about the photodissociation mechanism may be obtained from the (structureless)...

According to Appendix 7.A (p. 229), the reduction in J per unit size of p is maximum along the direction of — VJ, i.e., opposite to that of the gradient VJ. This direction is known as the direction of the steepest descent. It can be easily seen that Equations (7.2) and (7.3) already utilize the steepest descent direction to improve u as a continuous function of t, and t(. 

Now a change of magnitude eo in p along the steepest descent direction is given by... 

One widely adopted technique is to move in the steepest descent direction with a step dt, which progressively increases (e.g., with dti+i = 2 dti), until an interval of... 

Of course, the surface is not quadratic and the Hessian is not constant from step to step. However, near a critical point, the Newton-Raphson method (Eq. (2)) will converge rapidly. The main difficulty is that the convergence of the Newton-Raphson method is local. Thus the method will converge to the nearest critical point to the starting point. Consequently, one must start the optimization with a Hessian that contains no or one negative eigenvalue according to whether a minimum or a saddle point structure is required. For a minimum, the steepest descent direction... 

At s 0, this path follows the steepest-descent direction at large negative s, the limit is Newton s method. 

The GE passing this CCI belt, and the contact line dividing the flanks of ridge r and valley v, do not coincide, see Fig. 15. They do, however, cross at the point (0.685, 0.920), where the Hessian H has a zero eigenvector parallel to a tangent t in a steepest descent direction. At this point the GE Eg. (16) becomes... 

As with the case for the impenetrability constraint, the tangential (stick-slip) constraints are also gradually enforced. The frictional slip is determined from a line search along the steepest descent direction M (Gr) (Pxr ), and the rate constraints are active during the Newton iterations while assuming sticking conditions, that is. 

Move through the interior in directions that show promise of moving quickly to the optimal solution (i.e., the steepest descent direction). 

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