Converged properties

Finally, it is shown in terms of the presented example that the proposed adaptive reconstruction algorithm is valuable for image reconstruction from projections without any prior information even in the case of noisy data. The number of required projections can be determined by investigating the convergence properties of the reconstruction algorithm. 

Ahlrichs R 1976 Convergence properties of the intermolecular force series (l/r expansion) Theor. Chim. Acta 41 7... 

CASSCF is a version of MCSCF theory in which all possible configurations involving the active orbitals are included. This leads to a number of simplifications, and good convergence properties in the optimization steps. It does, however, lead to an explosion in the number of configurations being included, and calculations are usually limited to 14 elections in 14 active orbitals. 

IlyperChem has a set of optimi/.ers available to es.plore potential surfaces. These differ in their generality, convergence properties and conipntaiioiial requirements. One must be somewhat pragmatic about optirn i/ation andswitch optirn izers or restart an optimizer when It encounters specific problems. 

All of these methods use just the new and current points to update the inverse Hessian. The default algorithm used in the Gaussian series of molecular orbital programs [Schlegel 1982] makes use of more of the previous points to construct the Hessian (and thence the inverse Hessian), giving better convergence properties. Another feature of this method is its use... 

Swarbrick, S.J. and Nassehi, V., 1992b. A new decoupled finite element algorithm for viscoelastic flow. Part 2 convergence properties of the algorithm. Int. J. Numer. Methods Fluids 14, 1377-1382. 

As stated above, MC simulations are popular in many diverse fields. Their popularity is due mainly to their ease of use and their good convergence properties. Nonetheless, straightforward and application of MC methods to biomolecules is often problematic due... 

When natural orbitals are determined from a wave function which only includes a limited amount of electron correlation (i.e. not full Cl), the convergence property is not rigorously guaranteed, but since most practical methods recover 80-90% of the total electron correlation, the occupation numbers provide a good guideline for how important a given orbital is. This is the reason why natural orbitals are often used for evaluating which orbitals should be included in an MCSCF wave function (Section 4.6). 

The transformation from a set of Cartesian coordinates to a set of internal coordinates, wluch may for example be distances, angles and torsional angles, is an example of a non-linear transformation. The internal coordinates are connected with the Cartesian coordinates by means of square root and trigonometric functions, not simple linear combinations. A non-linear transformation will affect the convergence properties. This may be illustrate by considering a minimization of a Morse type function (eq. (2.5)) with D = a = ] and x = AR. 

The natural expansion has here also another important optimum convergency property. If this expansion is interrupted after r terms, the renormalized truncated function Wr has the smallest total deviation from the exact solution ... 

There are different variants of the conjugate gradient method each of which corresponds to a different choice of the update parameter C - Some of these different methods and their convergence properties are discussed in Appendix D. The time has been discretized into N time steps (f, = / x 8f where i = 0,1, , N — 1) and the parameter space that is being searched in order to maximize the value of the objective functional is composed of the values of the electric field strength in each of the time intervals. 

Scales (1986) recommends the Polak Ribiere version because it has slightly better convergence properties. Scales also gives an algorithm which is used for both methods that differ only in the formula for the updating of the search vector. 

The actual properties of this transformation combined with the convergence properties of molecular electron densities implies analyticity almost everywhere on the compact manifold. Consequently, this four-dimensional representation of the molecular electron density satisfies the conditions of a theorem of analytic continuation, that establishes the holographic properties of molecular electron densities represented on the compact manifold S3. 

If the electron density partitioning results in subsystems without boundaries and with convergence properties which closely resemble the convergence properties of the complete system, then it is possible to avoid one of the conditions of the Holographic Electron Density Fragment Theorem , by generating fuzzy electron density fragments which do not have boundaries themselves, but then the actual subsystems considered cannot be confined to any finite domain D of the ordinary three-dimensional space E3. 

It fix) and g(x) are nonconvex, additional difficulties can occur. In this case, nonunique, local solutions can be obtained at intermediate nodes, and consequently lower bounding properties would be lost. In addition, the nonconvexity in g(x) can lead to locally infeasible problems at intermediate nodes, even if feasible solutions can be found in the corresponding leaf node. To overcome problems with nonconvexities, global solutions to relaxed NLPs can be solved at the intermediate nodes. This preserves the lower bounding information and allows nonlinear branch and bound to inherit the convergence properties from the linear case. However, as noted above, this leads to much more expensive solution strategies. 

Although expressions (2.8) and (2.9) are formally equivalent, their convergence properties may be quite different. As will be discussed in detail in Chap. 6, this means that there is a preferred direction to carry out the required transformation between the two states. 

Convergence properties of free energy calculations Alpha-cyclodextrin complexes as a case study. J. Am. Chem. Soc. 116 6293 (1994). 

Ursin, B. (1980). Asymptotic convergence properties of the extended Kalman filter using filtered state estimates. IEEE Trans. Autom. Control AC-25, 1207-1211. 

In order to get a good starting value, we could solve the standard unweighted least-square set of 11 equations. Convergence properties are better shown, however, by taking nearly arbitrary starting values, e.g., a=0.1 and / = 15. Some other choices may not converge, which shows that saddle points of the function c2 distant from the actual minimum do exist. 

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