Ellipsoid algorithm

Related Approaches Target Function Minimization, the Diffusion Equation Method, and the Ellipsoid Algorithm... 

M. Billeter, T. F. Havel, and K. Wiithrich,/. Comput. Chem., 8(2), 132(1987). The Ellipsoid Algorithm as a Method for the Determination of Polypeptide Conformations from Experimental Distance Constraints and Energy Minimization. 

M. Billeter, A, E. Howard, 1. D. Kuntz, and P. A. Kollman,/. Am. Chem. Soc., 110, 8383 (1988), A New Technique to Calculate Low-Energy Conformations of Cyclic Molecules Utilizing the Ellipsoid Algorithm and Molecular Dynamics Application to 18-Crown-6. 

Fig. 4.4 The match search algorithm creates a matrix with one cell for each pair ofdirected tree edges. The cell stores the overall similarity of the two subtrees. The similarity value is calculated with a dynamic programming scheme shown on the right. First, an extension match (blue ellipsoid) is searched. Then the subtrees are cut and matched in all possible combinations. For each combination, a similarity value can be extracted from the matrix (exemplarily shown by the blue arrows). A maximum-weight bipartite matching solves the assignment of the subtrees.

The hidden layer parameters to be determined are the parameters of hyperellipsoids that partition the input data into discrete clusters or regions. The parameters locate the centers (i.e., the means) of each ellipsoid region s basis function and describe the extent or spread of the region (i.e., the variance or standard deviations). There are many ways of doing this. One is to use random samples of the input data as the cluster centers and add or subtract clusters as needed to best represent the data. Perhaps the most common method is called the K-means algorithm (Kohonen, 1997 Linde et al 1980) ... 

The simplest model potentials that form liquid crystals are the hard ellipsoid fluid and the hard cylinder fluid [4]. Linear and angular momenta are constant between collisions so that very efficient molecular dynamics algorithms can be devised. Unfortunately, when transport coefficients are calculated external fields and thermostats are often applied. That means that the particles accelerate between collisions. The advantages of using hard body fluids is conse-... 

This algorithm has been applied to calculate the thermal conductivity of a variant of the Gay-Beme fluid where the Lennard-Jones core has been replaced by a purely repulsive 1/r core [20]. Two systems were studied, one consisting of prolate ellipsoids with a length to width ratio of 3 1 and another one consisting of oblate ellipsoids with a length to width ratio of 1 3. The potential parameters are given in Appendix II. They both form nematic phases at high densities. 

The director constraint algorithm makes it very easy to calculate the Mies-owicz viscosities. One simply fixes the director in the desired direction and calculates the shear stress. In a liquid crystal consisting of prolate ellipsoids one has % > It is easy to realise that T7j must be the smallest viscosity... 

We may also adopt a different procedure in which the adaptive distances are computed at each iteration in the GFNM algorithm. At the first iteration the diameters are all equal 5, = 1, / = 1,2,..., n. The diameter of the fuzzy class A, obtained at iteration k induces an adaptive distance. The adaptive distances are used to compute the fuzzy classes at iteration k + For a relatively large data set the computation of diameters may involve the storage of a large distance matrix. We may avoid the computational difficulties in such cases by using a simpler adaptive distance. We have supposed the classes are approximately spherical (or ellipsoidal). The mean of a fuzzy class may thus be considered as an approximation of the geometric center of the class. On this basis we may define the radius r, of the fuzzy class A, as... 

The stepwidth vector s is defined prior to the optimization and kept constant during the optimization run. Due to the normalization of d (step 2), all generated configurations are placed on a rotation ellipsoid of dimensionality corresponding to s. This makes it difficult for the algorithm to locate the exact extreme of an optimization function. 

The optimization algorithm of the Domains in accordance with MPDs approach has been described in references [26,38]. An atomic or ELF basin, defined oti a grid, or another domain chosen by the user (a sphere, an ellipsoid, a cube, or a previously obtained MPD) can be a first guess for the MPDs. 

Using the same theoretical example as mentioned above. Fig. 6.16 illustrates the effect of a systematically erroneous arrival time on the source localization. For an array of 40 by 40 AE-sources the theoretical arrival times at the four sensors were calculated. To introduce an error, 5 ps were added to the arrival times of Sensor 1. The iterative localization algorithm then yields AE-source locations that minimize the travel time residuals and thus distribute the error in arrival times over all sensors. Fig. 6.16 top depicts the difference between the actual and the calculated AE-source location, on the left side as error vectors and on the right side as a density function of the error value. Fig. 6.16 bottom left shows a density function of the minimized travel time residuals (mean value over all sensors) and bottom right the major axis of the error ellipsoid. In most cases the size and orientation of calculated location uncertainties (bottom right) corresponds well to the actual error vector (top left). 

The aggregation of polymers and smfactants has been studied with DPD (212). This algorithm provides an alternative to Langevin or Brownian d5mam-ics, and can even be used with Widom insertion (212). The ad hoc potential that is used in DPD is not unlike a soft ellipsoidal model for polymer molecules (213,214), which suggests that DPD could be extended to polymer melts. Along these lines a... 

There are a number of computational models used to investigate granular media. Event-driven or hard-sphere algorithms are based on the calculation of changes from distinct collisions between single grains that are often approximated by spheres, ellipsoids, or polyhedral.MD of soft-particle models are another common way to simulate granular materials. In this approach, the repulsive contact force in normal direction is typically proportional to the particle... 

The GK algorithm (Gustafson and Kessel, 1979), searches for ellipsoidal clusters. It can be used for linear or planar clusters because this type of cluster can be viewed as a special case of ellipsoids for which one or more radii are zero. 

A larger number of rules provide only a marginally better result. It should be noted that the clustering algorithm used here is slightly different from the previous two, in this case spherical clusters are used since it is assumed that the points are bounded by a hypercube (Chiu, 1994), whereas in the first two cases ellipsoidal clusters were used. 

The main disadvantage of the presented method is the fact that the results of the reconstruction depend, for certain shapes, from the orientation of the chosen coordinate system. Fig. 8 depicts this fact. A thin circular plate is registered in two different coordinate systems. System A is the one used in the presented results. From all three directions the projections of the analyzed object are of elliptic shape. Therefore the reconstruction algorithm produces an ellipsoid as the three dimensional reconstruction. In example B, the same object is viewed in a special coordinate system adapted to the shape. In consequence, the reconstructed structure corresponds much better to the original. 

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