Search results

(1 - 2 of 2)
FUNCTION APPROXIMATION WITH KERNEL METHODS
Title
FUNCTION APPROXIMATION WITH KERNEL METHODS
Creator
Zhou, Xuan
Date
2015, 2015-12
Description
This dissertation studies the problem of approximating functions of d variables in a separable Banach space Fd. In particular we are...
Show moreThis dissertation studies the problem of approximating functions of d variables in a separable Banach space Fd. In particular we are interested in convergence and tractability results in the worst case setting and in the average case setting. The symmetric positive definite kernel in both settings is of a product form Kd(x, t) := d =1 1 − α2 + α2 Kγ (x , t ) for all x, t ∈ Rd. The kernel Kd generalizes the anisotropic Gaussian kernel, whose tractability properties have been established in the literature. For a fixed d, we study rates of convergence, which indicate how quickly approximation errors decay. Since rates of convergence can deteriorate quickly as d increases, it is desirable to have dimension-independent convergence rates, which corresponds to the concept of strong polynomial tractability. We present sufficient conditions on {α }∞ =1 and {γ }∞ =1 under which strong polynomial tractability holds for function approximation problems in Fd. Numerical examples are presented to support the theory and guaranteed automatic algorithms are provided to solve the function approximation problem in a straightforward and efficient way. viii
Ph.D. in Applied Mathematics, December 2015
Show less
GUARANTEED ADAPTIVE UNIVARIATE FUNCTION APPROXIMATION
Title
GUARANTEED ADAPTIVE UNIVARIATE FUNCTION APPROXIMATION
Creator
Ding, Yuhan
Date
2015, 2015-12
Description
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