Searching Relaxation Methods

Park, I., C. Johnson, and D. Gabriel, Probe diffusion in polyacrylamide gels as observed by means of holographic relaxation methods search for a universal equation. Macromolecules, 1990, 23, 1548-1553. 

In constrained optimization, the task of finding the optimum point is divided into two parts. The constrained optimization problem is converted to an equivalent unconstrained problem followed by a constrained search. The methods of converting constrained problems to unconstrained ones are beyond the scope of this writing, but a description of them can be found in Pierre and Lxjwe (1975). The objective of an unconstrained search is to find the parameter values that minimize (or maximize) the objective function. In the case of the relaxation modulus, the objective is to find the relaxation times and relaxation strengths that minimize the error between the approximating function and the experimental data. 

Let us calculate the relaxation time of particles in this potential (escape time over a barrier) which agrees with inverse of the lowest nonvanishing eigenvalue Yj. Using the method of eigenfunction analysis as presented in detail in Refs. 2, 15, 17, and 18 we search for the solution of the Fokker-Planck equation in the... 

The modern branch-and-bound algorithms for MILPs use branch-and-bound with integer relaxation, i.e., the branch-and-bound algorithm performs a search on the integer components while lower bounds are computed from the integer relaxation of the MILP by linear programming methods. The upper bound is taken from the best integer solution found prior to the actual node. 

A method based on that of Go and Scheraga " was used to search conformation space to solve the first part of the problem. The relaxation itself entails the propagation of a conformational defect from one unit cell to the next... 

The search for local minima in the neighborhood of a given local minimum is usually performed by the excitation of the system from this state followed by the relaxation of the system. If the relaxation of the excited system results in a state different from the initial state (and explored earlier), then a new local minimum is found, otherwise the evolution of the excited system is continued. The ways of moving out of the initial state can be different in temperature accelerated dynamics (TAD) by Sorensen and Voter [78], MD is used at high temperatures in the activation-relaxation technique (ART) by Mousseau and Barkema [79] and the local activated Monte Carlo method (LAMC) [80], the system evolves along the direction opposite to the direction of the force in the long-scale kinetic Monte Carlo... 

The angular velocity and angular momentum acfs themselves are important to any dynamical theory of molecular liquids but are very difficult to extract directly from spectral data. The only reliable method available seems to be spin-rotation nuclear magnetic relaxation. (An approximate method is via Fourier transformation of far-infrared spectra.) The simulated torque-on acfs in this case become considerably more oscillatory, and, which is important, the envelope of its decay becomes longer-lived as the field strength increases. This is dealt with analytically in Section III. In this case, computer simulation is particularly useful because it may be used to complement the analytical theory in its search for the forest among the trees. Results such as these for autocorrelation functions therefore supplement our... 

The BB method (Gupta and Ravindran, 1985 Nabar and Schrage, 1991 Borchers and Mitchell, 1992) starts by solving first the continuous NLP relaxation. If all discrete variables take discrete values, the search is stopped. Otherwise, it performs a tree search in the space of the integer variables y/, i G /,... 

A graphical approach was also used by Millet and Pons to analyse anisotropy of rotational diffusion in proteins. The values of Z)j and DJD compatible with R IRi ratios are presented as a contour plot. The intersection of the contour plots for different residues provides the values of anisotropy parameters compatible with experimental data. The obtained parameters can be used as starting values for further optimisation. The method is apphcable to axially symmetric rotation. A combination of approximate and exact methods was used by Ghose et al. to reduce the computational time of the determination of rotational diffusion tensor from NMR relaxation data. The initial values of the tensor components and its orientation are evaluated from the approximate solution, which substantially reduces the search space during the exact calculations. The method was applied for the estimation of relative domain orientation of a dual domain protein. 

There are many modulation methods used in ENDOR. Since the details of relaxation processes are seldom understood, the usual experimental approach is unfortunately to search rather blindly for experimental conditions that yield good signals. 

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