Gradient-based techniques

As noted in the introduction, energy-only methods are generally much less efficient than gradient-based techniques. The simplex method [9] (not identical with the similarly named method used in linear programming) was used quite widely before the introduction of analytical energy gradients. The intuitively most obvious method is a sequential optimization of the variables (sequential univariate search). As the optimization of one variable affects the minimum of the others, the whole cycle has to be repeated after all variables have been optimized. A one-dimensional minimization is usually carried out by finding the... 

For example, In the following situation, two wells have penetrated the same reservoir sand. The updip well finds the sand gas bearing, with gas down to (GOT) at the base of the sands, while the downdip well finds the same sand to be fully oil bearing, with an oil up to (OUT) at the top of the sand. Pressures taken at intervals in each well may be used to predict where the possible gas-oil contact (PGOC) lies. This method is known as the gradient intercept technique. 

An important extension to rigid-body fitting is the so-called directed tweak technique [105]. Directed tweak allows for an RMS fit, simultaneously considering the molecular flexibility. By the use of local coordinates for the handling of rotatable bonds, it is possible to formulate analytical derivatives of the objective function. With a gradient-based local optimizer flexible RMS fits are obtained extremely fast. However, no torsional preferences may be introduced. Therefore, directed tweak may result in energetically unfavorable conformations. 

Conventional gradient base optimisation techniques are not effective to deal with objective functions with multiple local minima and can be trapped in local minima. Particle swam optimisation (PSO) is a recently developed optimisation technique that can cope with multiple local minima. This paper proposes using PSO and stacked neural networks to find the optimal control policy for batch processes. A standard PSO algorithm and three new PSO algorithms with local search were developed. In order to enhance the reliability of the obtained optimal control policy, an additional term is added to the optimisation objective function to penalise wide model prediction confidence bormds. 

A novel gradient-based optimisation framework for large-scale steady-state input/output simulators is presented. The method uses only low-dimensional Jacobian and reduced Hessian matrices calculated through on-line model-reduction techniques. The typically low-dimensional dominant system subspaces are adaptively computed using efficient subspace iterations. The corresponding low-dimensional Jacobians are constructed through a few numerical perturbations. Reduced Hessian matrices are computed numerically from a 2-step projection, firstly onto the dominant system subspace and secondly onto the subspace of the (few) degrees of freedom. The tubular reactor which is known to exhibit a rich parametric behaviour is used as an illustrative example. 

Using anion exchange chromatography with pulsed amperometric detection, polymers up to DP70 may be analyzed. The necessary gradient elution technique is based on the combination of sodium hydroxide and sodium acetate eluents described above. 

Several NMR pulse sequences have been reported for the accurate measurement of 1,b/c,h values of carbohydrates, including frequency/line-separation-based techniques or quantitative / spectroscopy33 using either carbon or proton detection. These techniques have developed rapidly over the years with the introduction of gradient spectroscopy and with the advent of improved selective excitation schemes. Such methodological advancements have been comprehensively reviewed recently.34 In this section we will concentrate mainly on those methods that are typically applied to carbohydrates. 

A. Herbelin, J. Ruzicka, Pulse modulation. A novel approach to gradient-based flow injection techniques, Collect. Czech. Chem. Commun. 66 (2001) 1219. 

A strategy suggested by Chang (1982) is based on the assumption that the perturbed trajectory is in the vicinity of the nominal trajectory and uses one-step iteration in a gradient search technique to improve the perturbed performance index. This approach assumes that the moments of the CSD (the states of the lumped model resulting from the method of moments) can be measured or estimated. 

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