Local adaption for approximation and minimization of univariate functions

Abstract

Most commonly used adaptive algorithms for univariate real-valued function approximation and global minimization lack theoretical guarantees. Our new locally adaptive algorithms are guaranteed to provide answers that satisfy a user-specified absolute error tolerance for a cone, C, of non-spiky input functions in the Sobolev space W 2,∞ [a, b]. Our algorithms automatically determine where to sample the function-sampling more densely where the second derivative is larger. The computational cost of our algorithm for approximating a univariate function f on a bounded interval with L ∞ -error no greater than ε is O f 1 2 /ε as ε → 0. This is the same order as that of the best function approximation algorithm for functions in C. The computational cost of our global minimization algorithm is of the same order and the cost can be substantially less if f significantly exceeds its minimum over much of the domain. Our Guaranteed Automatic Integration Library (GAIL) contains these new algorithms. We provide numerical experiments to illustrate their superior performance.

Asymptotically tight worst case complexity bounds for initial-value problems with nonadaptive information

“…We briefly recall what is known about the potential of nonadaption for (1). It has been shown in [3] and [4] that the following holds.…”

“…In what follows, for a positive function γ = γ(ε), the asymptotic expressions O(γ(ε)), Ω(γ(ε)) and Θ(γ(ε)) will always be meant as ε → 0. It is known for many years that for problem (1) adaptive information is much more efficient in the worst case setting than nonadaptive one. It was shown for adaptive information that the ε-complexity of (1) is, (see [3]):…”

Adaptive mesh point selection for the efficient solution of scalar IVPs

“…In this paper we present results explaining potential gain of adaptive mesh point selection for a regular problem (1). In particular, we rigorously discuss the accuracy and cost of an adaptive process for a well precised class of problems, not only for a number of computational examples.…”

Adaptive mesh selection asymptotically guarantees a prescribed local error for systems of initial value problems

“…It is shown in [9] that adaption does not improve the order of convergence, but it can reduce the asymptotic constant of the method. In the similar spirit, adaption has been considered for univariate approximation and minimization in [1]. For scalar autonomous problems (1), adaptive mesh selection has been recently studied in [4].…”

“…In the similar spirit, adaption has been considered for univariate approximation and minimization in [1]. For scalar autonomous problems (1), adaptive mesh selection has been recently studied in [4]. An adaptive strategy has been proposed and the cost analyzed, based on specific properties of scalar autonomous equations.…”