Boundary least-squares method

The spline surface S(x,y) consists of a set of bicubic polynomials, one in each panel, joined together with continuity up to the second derivative across the panel boundaries. Because each B-spline only extends over four adjacent knot intervals, the functions B.(x)C.(y) are each non-zero only over a rectangle composed of 16 adjacent panels in a 4 x 4 arrangement. The amount of calculation required to evaluate the coefficients y may be reduced by making use of this property. As before, least-squares methods may be used if the number of data exceeds (h+4)(jJ+4), which is usually the case. 

Crystalline phases of glass-ceramic specimens were identified by X-ray diffraction (XRD) method. The lattice parameters of the N5-type hexagonal unit cell were calculated by a least-squares method using the XRD peaks of (054), (044), (134), (440) and (024). Glass-ceramics of Y3+-contained NaRPSi were subjected to scanning (SEM) and transmission electron microscope (TEM) for microstructural analysis. Electron diffraction and compositional analyses were also performed to characterize the structure of the grain boundary. 

Considering the weak formulation of the least-squares method. Based on the boundary conditions presented in this section, the boundary condition matrix B, defined by (12.481), is given as ... 

The choice of test function distinguishes between the most commonly used spectral schemes, the Galerkin, tan, collocation, and least squares versions [22, 51, 84, 89] (see also [60, 132, 54, 17]). In the Galerkin approach, the test functions are the same as the trail functions, whereas in the collocation approach the test functions are translated Dirac delta functions centered at special, so-called collocation points. The collocation approach thus requires that the differential equation is satisfied exactly at the collocation points. Spectral tau methods are close to Galerkin methods, but they differ in the treatment of boundary conditions. 

We will state in this chapter the mathematical task of parameter identification and discuss the corresponding numerical methods. Techniques from various branches of numerical mathematics are required, e.g. numerical solution of differential equations, numerically solving nonlinear problems especially large-scale constrained nonlinear least squares problem. Thus, some of the methods discussed in the previous chapters will reappear here. We will see how parameter identification problems can be treated efficiently by boundary value problem (BVP) methods and extend the discussion of solution techniques for initial value problems (IVPs) to those for BVPs. 

The results of the phase boundary experiments are summarized in Figure 1.2 for adamantane. The equation representing adamantane has been presented by the least squares linear regression method. The result of this regression can be expressed in the following form. 

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