Gradient methods

We introduce this method to determine the minimum in the optimal control problem of Section 6.1.1 (p. 153) in which both final time and final state are 

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Fig.3 Corrosion calibration standard for eddy curent instrument using the gradient method.

When using the gradient method described above, conventional flat-bottom hole standards can not be used. Tbe probe will see only the edges of the holes and they all have the same gradient, i.e. vertical steepness, and will give the same signal responses. 

The gradient of the PES (force) can in principle be calculated by finite difference methods. This is, however, extremely inefficient, requiring many evaluations of the wave function. Gradient methods in quantum chemistiy are fortunately now very advanced, and analytic gradients are available for a wide variety of ab initio methods [123-127]. Note that if the wave function depends on a set of parameters X], for example, the expansion coefficients of the basis functions used to build the orbitals in molecular orbital (MO) theory. 

Davis, M. E., McCammon, J. A. Solving the finite difference linearized Poisson-Boltzmann equation A comparison of relaxation and conjugate gradients methods.. J. Comp. Chem. 10 (1989) 386-394. 

A con jugate gradicri I method differs from the steepest descent technique by using both the current gradient and the previous search direction to drive the rn in im i/ation. , A conjugate gradient method is a first order in in im i/er. 

HyperChem provides two versions of the conjugate gradient method, Fletcher-Reeves and Bolak-Rihiere. Polak-Ribiere is more refined and is the default eh oiee in HyperChem,... 

Tbis tecbnic ne is available only for the MM-i- force field. As is true for the conjugate gradient methods, yon should noi use this algorithm when the initial interatomic forces are very large (meaning, the molecular structure is far from a inmiimim). 

Several variants of the conjugate gradients method have been proposed. The formulatior given in Equation (5.7) is the original Fletcher-Reeves algorithm. Polak and Ribiere proposed an alternative form for the scalar constant 7) ... 

Tafe/e 5.1 A comparison of the steepest descents and conjugate gradients methods for an initial refinement and a stringent minimisation. 

Note Because of its neglect of off-diagonal blocks, this optimizer can sometimes oscillate and fail to converge. In this case, use a conjugate gradient method. 

Conjugate Gradient methods compute the conjugate directions hj by iterative computation involving the gradient gj without... 

The procedure uses second derivative information and can be quite efficient compared to conjugate gradient methods. However, th e neglect of couplin g in th e Hessian m atrix can lead to situation s where oscillation is possible. Conjugate gradient methods. 

Hestenes, M. R. Conjugate Gradient Methods in Optimization, Springer-Verlag (1980). 

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