Adaptive methods

Hence, as the second class of techniques, we discuss adaptive methods for accurate short-term integration (Sec. 4). For this class, it is the major requirement that the discretization allows for the stepsize to adapt to the classical motion and the coupling between the classical and the quantum mechanical subsystem. This means, that we are interested in discretization schemes which avoid stepsize restrictions due to the fast oscillations in the quantum part. We can meet this requirement by applying techniques recently developed for evaluating matrix exponentials iteratively [12]. This approach yields an adaptive Verlet-based exponential integrator for QCMD. 

A more convincing approach leads to an adaptive method based on the symmetric second order scheme (9). As a first step, we have to introduce a first order scheme substituting p of the previous section. In what follows, we use the following pair of schemes ... 

Selenski investigated the use of chiral enol ether auxiliaries in order to adapt method F-H for enantioselective syntheses. After surveying a variety of substituted and unsubstituted enol ethers derived from a vast assortment of readily available chiral alcohols, she chose to employ enol ethers derived from trans-1,2-phenylcyclohexanol such as 73 and 74 (Fig. 4.37). These derivatives were found to undergo highly diastereoselective cycloadditions resulting in the formation of 75 and 76 in respective... 

The problems associated with sampling in many areas of chemical testing have been addressed and methods have been validated and published (see ref. 3 for more details). Where specific methods are not available, the analytical chemist should rely upon experience or adapt methods from similar applications. When in doubt, the material of interest and any samples taken from it... 

Chow, S-C., Chang, M., and Pong, A., 2005, Statistical consideration of adaptive methods in clinical development, Journal of Biopharmaceutical Statistics, 15 575-591. 

Finally, the use of ultrasound to entrain Reformatsky reactions deserves mentioning.37 Since a conventional laboratory cleaning bath is usually sufficiently effective, this easily adaptable method should find wider use in particular for large scale preparations. The same may hold true for the use of sacrificial anodes and/or for Reformatsky-type reactions triggered by electrochemical means.38... 

The above models are based on physical insights gained from analysis of typically two site problems under simplified assumptions. To provide a quantitative basis for these models, it becomes necessary to deal with larger and more realistic systems. In this context, VB method becomes an invaluable tool, being a spin adapted method. Exact solution of large model systems afforded by the VB method allows introducing parameters that make the models more realistic. 

The most influential method of analysis of orthogonal saturated designs yet proposed is the robust adaptive method of Lenth (1989). The quick and easy method that he proposed is based on the following estimator of the standard deviation, op, of the effect estimators Pi. This estimator is robust and adaptive , concepts which are explained in detail after the following description of the method. 

Sene, C., Santambien, P., and Pouradier Duteil, X. (1990). Purification of IgG monoclonal antibodies. A simple and adaptable method. Chim. Oggi 8, 15-21. 

Riedmiller, M. Braun, H. (1993). A (Greet adaptive method for faster backpropagation learning the Rprop algorithm. In Proceedings of the IEEE International Conference on Neural Networks (ICNN 93). (ed. Ruspini H.), pp. 586-91. 

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