APPROXIMATING THE STIELTJES INTEGRAL VIA A WEIGHTED TRAPEZOIDAL RULE WITH APPLICATIONS

Abstract. In this paper we provide sharp error bounds in approximating the weighted Riemann-Stieltjes integral Applications for continuous functions of selfadjoint operators in complex Hilbert spaces are given as well

Mercer–Trapezoid Rule for the Riemann–Stieltjes Integral with Applications

In this paper several new error bounds for the Mercer - Trapezoid quadrature rule for the Riemann-Stieltjes integral under various assumptions are proved. Applications for functions of selfadjoint operators on complex Hilbert spaces are provided as well

Two Ostrowski type inequalities for the stieltjes integral of monotonic functions

Two integral inequalities of Ostrowski type for the Stieltjes integral are given. The first is for monotonie integrators and Holder continuous integrands while the second considers the dual case, that is, for monotonie integrands and Holder continuous integrators. Applications for the mid-point inequality that are useful in the numerical analysis of Stieltjes integrals are exhibited. Some connections with the generalised trapezoidal rule are also presented. Copyright Clearance Centre, Inc.postprin

Inequalities for Functions of Selfadjoint Operators on Hilbert Spaces

The main aim of this book is to present recent results concerning inequalities for continuous functions of selfadjoint operators on complex Hilbert spaces. It is intended for use by both researchers in various fields of Linear Operator Theory and Mathematical Inequalities, domains which have grown exponentially in the last decade, as well as by postgraduate students and scientists applying inequalities in their specific areas

Parameter Dimension Reduction for Scientific Computing

Advances in computational power have enabled the simulation of increasingly complex physical systems. Mathematically, we represent these simulations as a mapping from inputs to outputs. Studying this map—e.g., performing optimization, quantifying uncertainties, etc.—is a critical component of computational science research. Such studies, however, can suffer from the curse of dimensionality—i.e., an exponential increase in computational cost resulting from increases in the input dimension. Dimension reduction combats this curse by determining relatively important (or unimportant) directions in the input space. The problem is then reformulated to emphasize the important directions while the unimportant directions are ignored. Functions that exhibit this sort of low-dimensional structure through linear transformations of the input space are known as ridge functions. Ridge functions appear as the basic components in various approximation and regression techniques such as neural networks, projection pursuit regression, and multivariate Fourier series expansion. This work focuses on how to discover, interpret, and exploit ridge functions to improve scientific computing.In this thesis, we examine relationships between the ridge recovery technique active subspaces and the physically-motivated Buckingham Pi Theorem in magnetohydrodynamics (MHD) models. We show that active subspaces can recover known unitless quantities from MHD such as the Reynolds and Hartmann numbers through a log transformation of the inputs. We then study the relationship between ridge functions and statistical dimension reduction for regression problem—i.e., sufficient dimension reduction (SDR). We show that a class of SDR methods called inverse regression methods provide a gradient-free approach to ridge recovery when applied to deterministic functions. We examine the numerical properties of these methods as well as their failure cases. We also introduce novel algorithms for computing the underlying population matrices of these inverse regression methods using classical iterative methods for generating orthogonal polynomials. Finally, we introduce a new method for cheaply constructing accurate polynomial surrogates on one-dimensional ridge functions by obtaining generalized Gauss-Christoffel quadrature rules with respect to the marginal density on the one-dimensional ridge subspace

Error bounds for Lanczos-based matrix function approximation

We analyze the Lanczos method for matrix function approximation (Lanczos-FA), an iterative algorithm for computing $f(\mathbf{A}) \mathbf{b}$ when $\mathbf{A}$ is a Hermitian matrix and $\mathbf{b}$ is a given mathbftor. Assuming that $f : \mathbb{C} \rightarrow \mathbb{C}$ is piecewise analytic, we give a framework, based on the Cauchy integral formula, which can be used to derive {\em a priori} and \emph{a posteriori} error bounds for Lanczos-FA in terms of the error of Lanczos used to solve linear systems. Unlike many error bounds for Lanczos-FA, these bounds account for fine-grained properties of the spectrum of $\mathbf{A}$, such as clustered or isolated eigenvalues. Our results are derived assuming exact arithmetic, but we show that they are easily extended to finite precision computations using existing theory about the Lanczos algorithm in finite precision. We also provide generalized bounds for the Lanczos method used to approximate quadratic forms $\mathbf{b}^\textsf{H} f(\mathbf{A}) \mathbf{b}$, and demonstrate the effectiveness of our bounds with numerical experiments