Alternating projection methods

Interest primarily in alternating projection methods has been sufficient to schedule a special Topical Meeting on Signal Recovery and Synthesis with Incomplete Information and Partial Constraints (Lake Tahoe, Nevada, January 12-14, 1983). 

When we adjust the momenta by scaling in order to satisfy the energy constraint, we are, in effect, projecting the preliminary numerical approximation onto an energy surface, using one particular type of projection. There are many alternative projection methods which we could use for this purpose, which might alter both the positions and momenta. Modifying the positions means that we will somehow need to solve the equation... 

Alternate projection methods aim to reduce the data dimensionality by optimizing the representation in the lower-dimension space so that the distances between points in the projected space are as similar as possible to the distances between the corresponding points in the original space. We will describe here a class of methods known as multidimensional scaling (MDS). The aim of these methods is to project data from a pseudo-metric space (i.e., one characterized by a dissimilarity measure) onto a metric space. Such methods are especially useful for preprocessing non-metric data in order to use algorithms valid only for metric input. 

J. Tropp, I. Dhillon, R. Heath, Jr, and T. Strohmer, Designing stmc-tural tight frames via an alternating projection method, IEEE Trans.Inf. Theory, vol. 51, no. 1, pp. 188-209, 2005. 

The method of allocating overheads can seriously affect the assigned costs of a project and hence the apparent cash flows for that project. Since these cash flows are used to assess profitability by the net-present-value (NPV) and discounted-cash-flow-rate-of-retum (DCFRR) methods, unfair allocation of overhead costs can result in a wrong choice between alternative projects. 

An alternative method would be to use complete spectra and hope that discrepancies between the linear calibration model and the real world data are leveraged out by the large pool of spectral information. For example, the PLS method is said to be suitable for such a type of P-matrix analysis. However, deterioration of measured spectra is more likely to be attributed to systematic physical effects than to uncorrelated random noise. Despite excellent results obtained using full spectra PLS calibrations and predictions compared to other linear projection methods, prediction errors can stiQ be significantly higher than for... 

It is sometimes necessary to determine the Miller indices of a given pole on a crystal projection, for example the pole A in Fig. 2-39(a), which applies to a cubic crystal. If a detailed standard projection is available, the projection with the unknown ])ole can be superimposed on it and its indices will be disclosed by its coincidence with one of the known poles on the standard. Alternatively, the method illustrated in Fig. 2-39 may be used. The pole A defines a direction in space, normal to the plane Qikl) whose indices are required, and this direction makes angles p, <7, t with the coordinate axes a, b, c. These angles are measured on the projection as shown in (a). Let the perpendicular distance between the origin and the hkl) plane nearest the origin be d [Fig. 2-39(b)], and let the... 

Unfortunately, if spatial symmetrization (in contrast to spin symmetrization considered above) is used, discontinuities will arise when spatial symmetrybreaking nuclear motion is considered. Thus broken spatial symmetry is best treated via the methods discussed in the next section that unfortunately do not yield states of pure spatial symmetry. (The alternative energy-projection method of Noodleman when self-consistently applied also becomes discontinuous during nuclear symmetry breaking.)... 

Previous hydroponic and soil pot research investigations have identified a number of Festuca rubra and Festuca arundinacea cultivars suitable for binding and holding various heavy metals on or within their root tissue (Kucharski and Sas-Nowosielska, 2004a). Festuca"s ability to tolerate heavy metal-contaminated soils, combined with proven cultivation methods, renders it an excellent candidate for phytostabilization projects at metal-contaminated sites where alternative remediation methods may not be economically feasible. 

Alternatively, the constraints can be treated using projection methods [123]. Instead of minimizing the absolute value of the energy difference, l 2 — il, it is advantageous to use the square of the energy difference, ( 2 — ] ), since this quantity is better suited for quasi-Newton optimization methods. For the remaining — 7 or — 8... 

ABSTRACT Within the EU project PAMINA different Sensitivity Analysis (SA) techniques have been applied to models that simulate the behaviour of different radioactive waste repositories. This paper summarises the main results and highlights specific aspects of regression/correlation techniques, the use of alternative imputation methods for datasets with missing values, the use of derived input parameters, the use of Monte Carlo filtering techniques in combination with cobweb plots and of variance based techniques. 

Nowadays, most programmes in sensory science include rapid sensory methods as well. However, it is noted that in many cases, professors naturally tend to teach techniques they have developed, or that they are used to applying in their research projects. As a matter of fact, it is difficult to have a comprehensive view of all existing sensory profiling techniques and related methods. In an online survey that we conducted in 2011 with attendees at the Pangbom Sensory Science Symposium, we observed a lack of knowledge and trust in alternative DA methods (Rogeaux et ai, 2011). 

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