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Papers by Kyle Riley

Research paper thumbnail of Using Spreadsheets to Estimate the Volatility of Stock Prices

Mathematics and Computer Education, 2002

INTRODUCTION The concepts of mean and variance arise naturally in modeling the erratic behavior o... more INTRODUCTION The concepts of mean and variance arise naturally in modeling the erratic behavior of stock prices. We present a simple method for estimating the volatility of stock prices and use a spreadsheet (Microsoft Excel is used here) to apply our method to actual stock prices. This method is a nice application of parameter estimation and can be used in an introductory statistics course. STOCK PRICES, FINANCIAL OPTIONS, AND VOLATILITY Even the most naive investor knows that stock prices vary greatly over any period of time and this fluctuation introduces an element of risk into stock market investing. The plot of one year's closing prices in Figure 1 illustrates the erratic motion of stock prices for a given stock over a year-long period. Note that there are 249 active trading days for the US stock market in a 365 day year. The volatility of a stock is a measure of how great the stock price can fluctuate. A stock with a high volatility will be more susceptible to greater flu...

Research paper thumbnail of Two-Level Preconditioners for Ill-Conditioned Linear Systems with Semidefinite Regularization

A family preconditioners for the solution of discrete linear systems arising in regularized ill-p... more A family preconditioners for the solution of discrete linear systems arising in regularized ill-posed problems is presented. These preconditioners are based on a two-level splitting of the solution space, and were previously developed by Hanke and Vogel for positive definite regularization operators. The work presented here extends previous results to the case where the regularization operator has a nontrivial null space. Key words: Conjugate Gradient, Preconditioners, Iterative Methods, Image Deblurring 1 Introduction Our goal is to efficiently solve very large symmetric positive definite linear systems of the form Au = b (1a) where A = K K + ffL: (1b) The matrix K is assumed to be highly ill-conditioned and full, the matrix L is sparse and symmetric positive semidefinite, and ff is a small positive Preprint submitted to Elsevier Preprint 24 March parameter. Systems with this structure arise, for example, in image deblurring. Blurred, noisy image data is modeled by d ij = Z Z...

Research paper thumbnail of <title>Preconditioners for linear systems arising in image reconstruction</title>

Advanced Signal Processing Algorithms, Architectures, and Implementations VIII, 1998

For the numerical solution of large linear systems, the preconditioned conjugate gradient algorit... more For the numerical solution of large linear systems, the preconditioned conjugate gradient algorithm can be very effective if one has a good preconditioner. Two distinctly different approaches to preconditioning are discussed for solving systems derived from continuous linear operators of the form K + (alpha) L, where K is a convolution operator, L is a regularization operator, and (alpha) is a small positive parameter. The first approach is circulant preconditioning. The second, less standard, approach is based on a two-level decomposition of the solution space. A comparison of the two approaches is given for a model problem arising in atmospheric image deblurring.

Research paper thumbnail of Two-level preconditioners for regularized ill-posed problems

The goal of this thesis is the solution of large symmetric positive definite linear systems which... more The goal of this thesis is the solution of large symmetric positive definite linear systems which arise when Tikhonov regularization is applied to solve an ill-posed problem. The coefficient matrix for these systems is the sum of two terms. The first term comes from the discretization of a compact operator and is a dense matrix, i.e., not sparse. The second term, which is called the regularization matrix, is a sparse matrix that is either the identity or the discretization of a diffusion operator. In addition, the regularization matrix is scaled by a small positive parameter, which is called the regularization parameter. In practice, these systems are moderately ill-conditioned. To solve such systems, we apply the preconditioned conjugate gradient algorithm with two-level preconditioners. These preconditioners were previously developed by Hanke and Vogel for positive definite regularization matrices. The contribution of this thesis is extension to the case where the regularization matrix is positive semidefinite. Also there is a compilation of the two-level preconditioning algorithms, and an examination of computational cost issues. To evaluate performance the preconditioners are applied to a problem from image processing known as image deblurring.

Research paper thumbnail of Conducting an External Review of a Mathematics Program and Department

PRIMUS, 2003

The Mathematical Association of America Guidelines for Programs and Departments in Undergraduate ... more The Mathematical Association of America Guidelines for Programs and Departments in Undergraduate Mathematical Sciences recommends the use of a periodic external review as an assessment tool to evaluate the mathematics program and the department. However, there is surprisingly little information on how to conduct an external review for a mathematics program. In this paper, we will present an argument as to why a mathematics department should go through a periodic external review. In addition, we will present some helpful hints on how to conduct an external review.

Research paper thumbnail of Pattern formation in the Gray–Scott model

Nonlinear Analysis: Real World Applications, 2004

We investigate an elliptic system that arises in cubic autocatalytic reactions known as the Gray-... more We investigate an elliptic system that arises in cubic autocatalytic reactions known as the Gray-Scott model. Complicated patterns were reported by Pearson in a numerical study of this system. We produce the bifurcation analysis to support the existing numerical evidence for patterns. Speciÿcally bifurcation results and C 2 bounds for nonuniform steady states are derived.

Research paper thumbnail of A priori bounds for reaction–diffusion systems arising in chemical and biological dynamics

Applied Mathematics and Computation, 2005

The authors investigate reaction-diffusion equations which arise in chemical and biological dynam... more The authors investigate reaction-diffusion equations which arise in chemical and biological dynamics. It is shown that several common systems share a useful property, a structure on the non-linearity which arises from conservation of mass or population. This conservation property is used to demonstrate a priori bounds for the parabolic problems and the associated elliptic problem. The types of systems included in the analysis are the Gray-Scott system, SIR model, and the Selkov model of glycolysis.

Research paper thumbnail of Using Spreadsheets to Estimate the Volatility of Stock Prices

Mathematics and Computer Education, 2002

INTRODUCTION The concepts of mean and variance arise naturally in modeling the erratic behavior o... more INTRODUCTION The concepts of mean and variance arise naturally in modeling the erratic behavior of stock prices. We present a simple method for estimating the volatility of stock prices and use a spreadsheet (Microsoft Excel is used here) to apply our method to actual stock prices. This method is a nice application of parameter estimation and can be used in an introductory statistics course. STOCK PRICES, FINANCIAL OPTIONS, AND VOLATILITY Even the most naive investor knows that stock prices vary greatly over any period of time and this fluctuation introduces an element of risk into stock market investing. The plot of one year's closing prices in Figure 1 illustrates the erratic motion of stock prices for a given stock over a year-long period. Note that there are 249 active trading days for the US stock market in a 365 day year. The volatility of a stock is a measure of how great the stock price can fluctuate. A stock with a high volatility will be more susceptible to greater flu...

Research paper thumbnail of Two-Level Preconditioners for Ill-Conditioned Linear Systems with Semidefinite Regularization

A family preconditioners for the solution of discrete linear systems arising in regularized ill-p... more A family preconditioners for the solution of discrete linear systems arising in regularized ill-posed problems is presented. These preconditioners are based on a two-level splitting of the solution space, and were previously developed by Hanke and Vogel for positive definite regularization operators. The work presented here extends previous results to the case where the regularization operator has a nontrivial null space. Key words: Conjugate Gradient, Preconditioners, Iterative Methods, Image Deblurring 1 Introduction Our goal is to efficiently solve very large symmetric positive definite linear systems of the form Au = b (1a) where A = K K + ffL: (1b) The matrix K is assumed to be highly ill-conditioned and full, the matrix L is sparse and symmetric positive semidefinite, and ff is a small positive Preprint submitted to Elsevier Preprint 24 March parameter. Systems with this structure arise, for example, in image deblurring. Blurred, noisy image data is modeled by d ij = Z Z...

Research paper thumbnail of <title>Preconditioners for linear systems arising in image reconstruction</title>

Advanced Signal Processing Algorithms, Architectures, and Implementations VIII, 1998

For the numerical solution of large linear systems, the preconditioned conjugate gradient algorit... more For the numerical solution of large linear systems, the preconditioned conjugate gradient algorithm can be very effective if one has a good preconditioner. Two distinctly different approaches to preconditioning are discussed for solving systems derived from continuous linear operators of the form K + (alpha) L, where K is a convolution operator, L is a regularization operator, and (alpha) is a small positive parameter. The first approach is circulant preconditioning. The second, less standard, approach is based on a two-level decomposition of the solution space. A comparison of the two approaches is given for a model problem arising in atmospheric image deblurring.

Research paper thumbnail of Two-level preconditioners for regularized ill-posed problems

The goal of this thesis is the solution of large symmetric positive definite linear systems which... more The goal of this thesis is the solution of large symmetric positive definite linear systems which arise when Tikhonov regularization is applied to solve an ill-posed problem. The coefficient matrix for these systems is the sum of two terms. The first term comes from the discretization of a compact operator and is a dense matrix, i.e., not sparse. The second term, which is called the regularization matrix, is a sparse matrix that is either the identity or the discretization of a diffusion operator. In addition, the regularization matrix is scaled by a small positive parameter, which is called the regularization parameter. In practice, these systems are moderately ill-conditioned. To solve such systems, we apply the preconditioned conjugate gradient algorithm with two-level preconditioners. These preconditioners were previously developed by Hanke and Vogel for positive definite regularization matrices. The contribution of this thesis is extension to the case where the regularization matrix is positive semidefinite. Also there is a compilation of the two-level preconditioning algorithms, and an examination of computational cost issues. To evaluate performance the preconditioners are applied to a problem from image processing known as image deblurring.

Research paper thumbnail of Conducting an External Review of a Mathematics Program and Department

PRIMUS, 2003

The Mathematical Association of America Guidelines for Programs and Departments in Undergraduate ... more The Mathematical Association of America Guidelines for Programs and Departments in Undergraduate Mathematical Sciences recommends the use of a periodic external review as an assessment tool to evaluate the mathematics program and the department. However, there is surprisingly little information on how to conduct an external review for a mathematics program. In this paper, we will present an argument as to why a mathematics department should go through a periodic external review. In addition, we will present some helpful hints on how to conduct an external review.

Research paper thumbnail of Pattern formation in the Gray–Scott model

Nonlinear Analysis: Real World Applications, 2004

We investigate an elliptic system that arises in cubic autocatalytic reactions known as the Gray-... more We investigate an elliptic system that arises in cubic autocatalytic reactions known as the Gray-Scott model. Complicated patterns were reported by Pearson in a numerical study of this system. We produce the bifurcation analysis to support the existing numerical evidence for patterns. Speciÿcally bifurcation results and C 2 bounds for nonuniform steady states are derived.

Research paper thumbnail of A priori bounds for reaction–diffusion systems arising in chemical and biological dynamics

Applied Mathematics and Computation, 2005

The authors investigate reaction-diffusion equations which arise in chemical and biological dynam... more The authors investigate reaction-diffusion equations which arise in chemical and biological dynamics. It is shown that several common systems share a useful property, a structure on the non-linearity which arises from conservation of mass or population. This conservation property is used to demonstrate a priori bounds for the parabolic problems and the associated elliptic problem. The types of systems included in the analysis are the Gray-Scott system, SIR model, and the Selkov model of glycolysis.