Optimal interpolation

Its waveform retains the random quality of the original signal, and likewise the excitation signal in the gap matches the surrounding excitation. Hence the sub-optimal interpolant is likely to sound more convincing to the listener than the LSAR reconstruction. 

Bujgess, T. M., and R. Webster. 1980a."Optimal interpolation and isarilhmic mapping of soil properties, I, The semi-variogram and punctual kriging, Journal of Soil Science. 31 315-331. 

To obtain an algebraic approximation to Eq. (7), the for fluxes expressions on each face must be discretized. The optimal interpolation formula used to evaluate the variables and their derivatives depends on the local Peclet number. Nevertheless, the formulas for the east and west faces will have the following forms ... 

The fast FT allows optimal interpolation of data. The original N points are folded 0 to make 2N points and are then shifted by - - N (Figure 3). The fast FT then creates 2N points in the frequency domain, which are padded by the desired number of extra zeros in the appropriate location in the middle (e.g., 6N zeros in total for fourfold interpolation) and transformed back. (The extra zeros are added in the middle because of the aliasing of points from - Vmax to 0 into v ,ax to 2v ,ax 3S shown in Figure 3.) This procedure creates interpolated points between the original data points. 

Summary. In this chapter, we are concerned with the problem of multivariate data interpolation. The main focus hes on the concept of minimizing a quadratic form which, in practice, emerges from a physical model, subject to the interpolation constraints. The approach is a natural extension of the one-dimensional polynomial spline interpolation. Besides giving a basic outline of the mathematical framework, we design a fast numerical scheme and analyze the performance quality. We finally show that optimal interpolation is closely related to standard hnear stochastic estimation methods. 

We now generalize the setting of the previous section in order to cover a wide range of optimal interpolation which is more flexible for application purposes. 

For optimal interpolation, we need to specify a semi-norm. Let us consider piecewise continuous weight functions Wa, a = m, that are positive and bounded. The bilinear form... 

T. Werther (2003) Optimal Interpolation in Semi-Hilbert Spaces. PhD thesis. University of Vienna. 

In simple relaxation (the fixed approximate Hessian method), the step does not depend on the iteration history. More sophisticated optimization teclmiques use infonnation gathered during previous steps to improve the estimate of the minunizer, usually by invoking a quadratic model of the energy surface. These methods can be divided into two classes variable metric methods and interpolation methods. 

All numerical computations inevitably involve round-off errors. This error increases as the number of calculations in the solution procedure is increased. Therefore, in practice, successive mesh refinements that increase the number of finite element calculations do not necessarily lead to more accurate solutions. However, one may assume a theoretical situation where the rounding error is eliminated. In this case successive reduction in size of elements in the mesh should improve the accuracy of the finite element solution. Therefore, using a P C" element with sufficient orders of interpolation and continuity, at the limit (i.e. when element dimensions tend to zero), an exact solution should be obtaiiied. This has been shown to be true for linear elliptic problems (Strang and Fix, 1973) where an optimal convergence is achieved if the following conditions are satisfied ... 

A related idea is used in the Line Then Plane (LTP) algorithm where the constrained optimization is done in the hyperplane perpencheular to the interpolation line between the two end-points, rather than on a hypersphere. 

Because physicochemical cause-and-effect models are the basis of all measurements, statistics are used to optimize, validate, and calibrate the analytical method, and then interpolate the obtained measurements the models tend to be very simple (i.e., linear) in the concentration interval used. 

Initially, the coordinates x for the extra images added to the path are approximated by a linear interpolation between the converged points for the core set. In the case of the environment set, the initial coordinates are approximated by the environment set of the immediate neighboring converged point. That is, if only one image is added between each pair of optimized points, the initial coordinates of the core set for the image added between the optimized points xo and xi are given by a linear interpolation between xq and xi. The environment coordinates are set to correspond... 

Another class of methods of unidimensional minimization locates a point x near x, the value of the independent variable corresponding to the minimum of /(x), by extrapolation and interpolation using polynomial approximations as models of/(x). Both quadratic and cubic approximation have been proposed using function values only and using both function and derivative values. In functions where/ (x) is continuous, these methods are much more efficient than other methods and are now widely used to do line searches within multivariable optimizers. 

Each wafer has 100 chip sites with 0.25 cm2 active area. The daily production level is to be 2500 finished wafers. Find the resist thickness to be used to maximize the number of good chips per hour. Assume 0.5 < f < 2.5 as the expected range. First use cubic interpolation to find the optimal value of t, t. How many parallel production lines are required for t, assuming 20 h/day operation each How many iterations are needed to reach the optimum if you use quadratic interpolation ... 

We can reach the minimum of fix) in two stages using first s° and then s1. Can we use the search directions in reverse order From x° = [1 l]T we can carry out a numerical search in the direction s° = [—4 —2]T to reach the point x1. Quadratic interpolation can obtain the exact optimal step length because /is quadratic, yielding a = 0.27778. Then... 

For the next stage, the search direction is s1 = [1 —4]r, and the optimal step length calculated by quadratic interpolation is a1 = 0.1111. Hence... 

Table 3.2 lists the optimal values of the interpolation coefficients estimated by Berman and Brown (1987) for the most common oxide constituents of rock-forming minerals. These coefficients, through equations 3.78.1, 3.78.2, and 3.78.3, allow the formulation of polynomials of the same type as equation 3.54, whose precision is within 2% of experimental Cp values in the T range of applicability. However, the tabulated coefficients cannot be applied to phases with lambda transitions (see section 2.8). 

---