Monte Carlo simulation is a numerical method that estimates the means of random variables
and multidimensional integrals. This method returns the arithmetic average of function …

Function approximation, integration, and optimization are three fundamental mathematical
problems. They are especially challenging when the functions involved fluctuate wildly in …

This thesis investigates how to solve univariate integration problems using numerical
methods, including the trapezoidal rule and the Simpson's rule. Most existing guaranteed …

Automatic and adaptive approximation, optimization, or integration of functions in a cone
with guarantee of accuracy is a relatively new paradigm. Our purpose is to create an open …

We discuss adaptive mesh point selection for the solution of scalar initial value problems.
We consider a method that is optimal in the sense of the speed of convergence, and we aim …

We study potential advantages of adaptive mesh point selection for the solution of systems
of initial value problems. For an optimal order discretization method, we propose an …

A Brief Introduction to GAIL (Guaranteed Automatic Integration Library) Version 2.3 Page 1 A
Brief Introduction to GAIL (Guaranteed Automatic Integration Library) Version 2.3 Sou-Cheng …

It is known that, for systems of initial-value problems, algorithms using adaptive information
perform much better in the worst case setting than the algorithms using nonadaptive …