With a stationary multiscale algorithm, the condition number of the discrete matrix can be relatively small, and computation can be performed in O ( N ) operations. The third method is stationary multilevel interpolation. Beyond that, there is no case known where the error and the sensitivity are both reasonably small. Madych indicated that the interpolation error goes to zero as ε → 0, this does not seem to be true in practice. Some discussion of the existing parametrization schemes is provided in books. A number of strategies for choosing a “good” value of ε have been suggested, such as the power function as indicator, the Cross Validation algorithm and the Contour–Padé algorithm. The general observation was that a large ε leads to a very well-conditioned linear system but also a poor approximation rate, whereas a smaller ε yields excellent approximation at the price of a badly conditioned system. The choice of ε has a profound influence on both the approximation accuracy and numerical stability of the solution to interpolation problem. Obviously, a smaller value of ε causes the function to become flatter, whereas increasing ε leads to a more peaked radial function, and therefore localizes its influence. ∥ by a shape parameter ε in the practical approximation. ![]() ![]() ![]() We have become accustomed to scaling one of radial functions through multiplying the independent variable ∥ The first method is finding optimal shape parameter ε (which is related to the scattered centers distribution, and usually is inversely proportional to mesh norm).
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