Functional estimation problem space

Here the B-spline Bim(zf, Xj) is the ith B-spline basis function on the extended partition Xj (which contains locations of the knots in the Zj direction), and is a coefficient. We use cubic splines and sufficient numbers of uniformly spaced knots so that the estimation problem is not affected by the partition. The estimation problem now involves determining the set of B-spline coefficients that minimizes Eq. (4.1.26), subject to the state equations [Eqs. (4.1.24 and 4.1.25)], for a suitable value of the regularization parameter. At this point, the minimization problem corresponds to a nonlinear programming problem. 

In studying problem (12) we refer to the space H of all grid functions defined on the grid and vanishing for i = 0 and the operator Ay - y, for which the estimate... 

Let, for example, u be a solution of problem (5) and t/ = be a solution provided by this or that additive scheme, Hh, where Hh is the set of grid functions. Following established practice, the difference — — wjj, where = Vh u and Vh is a linear operator from the space Hq, into the space Hh u Hq, Uh Hh), needs investigation. To be more specific, we are interested in the possible estimates of the quantity t/l -... 

Assessment of the state of a complicated system like the NSS requires several indicators to characterize its state by whatever criteria are chosen (e.g., at U.N. level). Essentially, it is a question of introducing some rule or norm to estimate any deviation of the NSS from its prescribed state. Formalization of this process is reduced to the choice of some functional R(xi,..., xn) where xt is the vector of the state of the NSS. The functional R determines the estimation rule of any deviation of the NSS from its optimal state. The problem consists in choosing the kind of R. The problem consists in determining this space and choosing the kind of R. 

A key factor in modeling is parameter estimation. One usually needs to fit the established model to experimental data in order to estimate the parameters of the model both for simulation and control. However, a task so common in a classical system is quite difficult in a chaotic one. The sensitivity of the system s behavior to the initial conditions and the control parameters makes it very hard to assess the parameters using tools such as least squares fitting. However, efforts have been made to deal with this problem [38]. For nonlinear data analysis, a combination of statistical and mathematical tests on the data to discern inner relationships among the data points (determinism vs. randomness), periodicity, quasiperiodicity, and chaos are used. These tests are in fact nonparametric indices. They do not reveal functional relationships, but rather directly calculate process features from time-series records. For example, the calculation of the dimensionality of a time series, which results from the phase space reconstruction procedure, as well as the Lyapunov exponent are such nonparametric indices. Some others are also commonly used ... 

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