Numerical algorithms for univariate function approximation attempt to provide approximate solutions that differ from the original function by... Show moreNumerical algorithms for univariate function approximation attempt to provide approximate solutions that differ from the original function by no more than a user-specified error tolerance. The computational cost is often determined adaptively by the algorithm based on the function values sampled. While adaptive algorithms are widely used in practice, most lack guarantees, i.e., conditions on input functions that ensure the error tolerance is met. In this dissertation we establish guaranteed adaptive numerical algorithms for univariate function approximation using piecewise linear splines. We introduce a guaranteed globally adaptive algorithm, funappxglobal g, in Chapter 2, along with sufficient conditions for the success of funappxglobal g. Two-sided bounds on the computational cost are given in Theorem 1. These bounds are of the same order as the computational cost for an algorithm that knows the infinity norm of the second derivative of the input function as a priori. Lower bound on the complexity of the problem is also provided in Theorem 3. To illustrate the advantages of funappxglobal g, corresponding numerical experiments are presented in Section 2.7. The cost of a globally adaptive algorithm is determined by the most peaky part of the input function. In contrast, locally adaptive algorithms sample more points where the function is peaky and fewer points elsewhere. In Chapter 3, we establish a locally adaptive algorithm, funappx g, with sufficient conditions for its success. An upper bound on the computational cost is also given in Theorem 4. One GUI example is presented to show how funappx g works. Some interesting function approximation problems in computational graphics are also presented. The key to analyzing these adaptive algorithms is looking at the error for cones of input functions rather than balls of input functions. Non-convex cones provide a setting where adaption may be beneficial. Ph.D. in Applied Mathematics, December 2015 Show less
