Gradient norm

The advantage of the NR method is that the convergence is second-order near a stationary point. If the function only contains tenns up to second-order, the NR step will go to the stationary point in only one iteration. In general the function contains higher-order terms, but the second-order approximation becomes better and better as the stationary point is approached. Sufficiently close to tire stationary point, the gradient is reduced quadratically. This means tlrat if the gradient norm is reduced by a factor of 10 between two iterations, it will go down by a factor of 100 in the next iteration, and a factor of 10 000 in the next ... 

Expectation Values of the Hamiltonian, (//), Virial Coefficients, r, and Squared Gradient Norms, grad p. For the Ground State of (R1 = R2 = R3 = 1.65) for the 75-Term Wave Function... 

Starting Geometries (in parentheses), Energies, H), Virial Coefficients, r. Squared Gradient Norms, II grad Ip, and Optimized Geometries for the Hydrogen Clusters"... 

All methods need good initial guesses for the parameters, otherwise they may not converge or end in a local minimum. Here, the linearization technique is useful to provide these. The parameter iteration continues until a certain criterion is satisfied or the maximum number of function evaluations is exceeded. These criteria may be that the relative change in the SSR value, or in the parameter values is below a preset value or the norm of the gradient is less than a certain value (in the minimum this gradient norm vanishes). 

In addition to conditions [21a,b], another test involving f may be imposed on the gradient norm to evaluate the optimality of the converging iterate ... 

Tables 4 and 5 summarize performance results for two different starting points. The first (xl) is closer to a minimum than the second (x2) and has lower function value and gradient norm by about four orders of magnitude (see table footnotes for details). From both starting points, we first note how well preconditioning works in TN. The residual truncation criterion of [54] and [55] was used here with cr = 0.5. With preconditioning, the number of inner (PCG) iterations is reduced by two to three orders of magnitude. Even the number of Newton iterations is reduced, and the time is accelerated by a factor of 2 to 3. Not only is precision of the resulting gradient norm not sacrificed it improves. This is a typical observation with good preconditioning.

Gradient Norm Minimizations 14.5.9 Netvton-Raphson Methods 14.5.10 Gradient Extremal Methods Constrained Optimization Problems Locating the Global Minimum and Conformational Sampling 333 333 338 338 339 Appendix C First and Second Quantization Reference 411 411 412... 

TIil/Il- iii iy liC iiioiC uiiC Tkj two miiiuiid. mciiiy example of a one-dimensional function and its associated gradient norm. It is clear that a... 

Figure 14.11 An example of a function and the associated gradient norm known of these is perhaps the GDIIS (Geometry Direct Inversion in the Iterative Subspace) which is directly analogous to the DIIS for electronic wave functions... 

Gradient Extremal (GE), 338 Gradient norm minimization, 333 Gradient of a function, 238 Greens function, 257 GROMOS force field, 40 Gross atomic charge, 218... 

Since transition structures are points where the gradient is zero, they may in principle be located by minimizing the gradient norm. This is in general not a good approach for two reasons ... 

There are typically many points where the gradient norm has a minimum without being zero. 

Any stationary point has a gradient norm of zero, thus all types of saddle points and minima/maxima may be found, not just TS s. 

Figure 12.10 shows an example of a one-dimensional function and its associated gradient norm. It is clear that a gradient norm minimization will only locate one of the two stationary points if started near x = 1 or x = 9. Most other starting points will converge on the shallow part of the function near x = 5. The often very small convergence radius makes gradient norm minimizations impractical for routine use. 

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