Non-Stationary Schemes

All of the above has dealt with stationary schemes where the coefficients in the affine combinations giving refined points remain the same at all steps of the refinement. However, some of the properties we have considered depend primarily on the early steps (such as the support and the artifacts), some on the later ones (such as derivative continuity), and so it seems sensible to consider varying the coefficients between the steps. 

We look at two examples which have been suggested in the literature, introduce a new criterion, that of step-independence, and open a Pandora s box of schemes which satisfy that criterion. 

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Stationary means that the same stencils are used at every step of refinement. A non-stationary scheme could have different stencils used for each step. 

With a non-stationary scheme, however, for a given initial spatial frequency of polygon, (i.e. a fixed known number of vertices per complete cycle, or vertices forming a regular polygon), the coefficients can vary at each step so that the halving relative frequency of the signal is tracked by a zero of the kernel. 

UP is definitely an exception. Most of the interesting non-stationary schemes can be looked at as letting the coefficients of a scheme take some trajectory in the design space of fixed finite dimension (and fixed arity) considered above. Except when the coefficient of the widest box-spline happens to drop to zero, the support will remain constant, at that given by the widest scheme included in the linear combinations. 

The design of a non-stationary scheme can be regarded then as the design of a trajectory in design space. 

Non-stationary schemes where the trajectory is pre-defined have a big inelegance that if one step of refinement is made, perhaps to permit the addition of short-wavelength features, then either the implementation has to remember that the first step has already been made, and start instead at the second, or... 

A non-stationary scheme does not have the necessary eigenvectors to apply the above directly to the original polygon. However, in cases where the scheme converges adequately fast towards its own limit, the eigenvectors of the limit scheme can be used with good accuracy after a relatively small number of refinements. How many such refinements are needed has to be determined for each scheme individually. 

The Fourier domain arguments do appear to apply well to non-stationary schemes, and they deserve more attention now. 

Because of this, there is a real need for designing the general method, by means of which economical schemes can be created for equations with variable and even discontinuous coefhcients as well as for quasilinear non-stationary equations in complex domains of arbitrary shape and dimension. As a matter of experience, the universal tool in such obstacles is the method of summarized approximation, the framework of which will be explained a little later on the basis of the heat conduction equation in an arbitrary domain G of the dimension p with the boundary F... 

In order to estimate kinetic constants for elementary processes in template polymerization two general approaches can be applied. The first is based on the generalized kinetic model for radical-initiated template polymerizations published by Tan and Alberda van Ekenstein. The second is based on the direct measurement of the polymerization rate in a non-stationary state by rotating sector procedure or by post-effect in photopolymerization. The first approach involves partial absorption of the monomer on the template. Polymerization proceeds according to zip mechanism (with propagation rate constant kp i) in the sequences filled with the monomer, and according to pick up mechanism (with rate constant kp n) at the sites in which monomer is outside the template and can be connected by the macroradical placed onto template. This mechanism can be illustrated by the following scheme ... 

The advantage of such an optimization scheme is that the SCF iterations do not converge to a non-stationary energy on the unconstrained potential energy surface that may only represent an energy minimum on the constrained potential energy surface, but to a true energy minimum, where the final local spin values—within a certain threshold—may differ from the ideal ones. 

Schemes to introduce instabilities and irregularities into purely periodic waveforms have tended to be less successful than simply using a loop with enough periods of a natural waveform to seem non-stationary. For many instruments, a loop of a less than...

About thirty years ago, all cases of polymerization kinetics used to be solved as statinary reactions. Hayes and Pepper [27] were the first to call attention to the non-stationary character of ionic polymerizations. They noticed the premature decay of styrene polymerization initiated by H2S04 (see Fig. 8). This was a simple case of non-stationarity caused by the slow decay of rapidly generated active centres [27, 28]. They assumed that the polymerization proceeds according to a rather conventional scheme represented in simplified form (without transfer) by the reactions... 

The solution of this scheme in a way analc us to the solutton of Eq. (152) leads to Eq.(158) which can be applied to the early non-stationary stage and which gives access to both kp and ktt-... 

It is not surprising to find therefore that acids such as perchloric lnd sulphuric are more efficient initiators than, say, simple hydrogen halides. As already pointed out, the polymerization of styrene by these acids has been widely studied [10—22, 68], In the case of sulphuric acid in ethylene dichloride, there is an initial fast reaction which stops abruptly before all the monomer is consumed [11]. Pepper et al. [11] have analysed this non-stationary state polymerization employing the scheme... 

Finally we look at what can be achieved by changing the rules fundamentally, by considering non-stationary subdivision, in which the mask changes from step to step. In particular we consider geometry-sensitive schemes where the mask is itself determined locally and at every step from the geometry of the polygon. 

Such schemes will typically look non-uniform and non-stationary when re-expressed in terms of points, but all the theory is still in fact regular when standing in the dual world. 

N. Dyn and D.Levin Stationary and Non-stationary subdivision schemes. pp209-216 in Mathematical methods in Computer Aided Geometric Design (eds Lyche and Schumaker), 1992... 

N. Dyn, D.Levin and A.Luzzato Non-stationary interpolatory subdivision schemes reproducing spaces of exponential polynomials. Found Comput Math, ppl97-206, 2003... 

C.Beccari, G.Gasciola and L.Romani An interpolating 4-point C2 ternary non-stationary subdivision scheme with tension control. CAGD 23(4), pp210-219, 2007... 

S.Daniel and P.Shunmugaraj An approximating C2 non-stationary subdivision scheme. CAGD 26(7), pp810-821, 2009 N.Dyn, M.S.Floater and K.Hormann Four-point curve subdivision based on iterated chordal and centripetal parameterizations. CAGD 26(3), pp279-286, 2009... 

When evaluating the results of ternary copolymerizations by means of the Alfrey-Gold-finger scheme, the stationary character of copropagation should be critically established. With non-stationary processes, the uncertainty of interpretation becomes more serious, even when the experimental results agree with theory. 

To construct special schemes, it is possible to use well-developed methods either fitted methods or methods with special condensed grids. Fitted methods are attractive because they allow one to use grids with an arbitrary distribution of nodes, in particular, uniform grids (see, e.g., [13, 14, 30-34]). However, even for the simplest singularly perturbed non-stationary diffusion equation, fitted methods are found to be inapplicable for the construction of e-uniformly convergent schemes. Fitted methods are inapplicable for more general elliptic and parabolic equations in the case when parabolic boundary layers, that is, layers described by parabolic equations, appear [4, 23, 29]. Therefore, the use of methods with special condensed grids is necessary for the construction of special schemes. 

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