Numerical Analysis and Computation Commons^™

Greedy Signal Recovery Review, Deanna Needell, Joel A. Tropp, Roman Vershynin

CMC Faculty Publications and Research

The two major approaches to sparse recovery are L1-minimization and greedy methods. Recently, Needell and Vershynin developed Regularized Orthogonal Matching Pursuit (ROMP) that has bridged the gap between these two approaches. ROMP is the first stable greedy algorithm providing uniform guarantees.

Even more recently, Needell and Tropp developed the stable greedy algorithm Compressive Sampling Matching Pursuit (CoSaMP). CoSaMP provides uniform guarantees and improves upon the stability bounds and RIC requirements of ROMP. CoSaMP offers rigorous bounds on computational cost and storage. In many cases, the running time is just O(NlogN), where N is the ambient dimension of the signal. This …

Electrothermal Imaging In One And Two Dimensions, Michael Janus, David Kibbling

Mathematical Sciences Technical Reports (MSTR)

Developing methods for the nondestructive testing of materials is an important area of research for industry. Situations often arise in which the integrity of an object is questioned, but testing it is very difficult. For example, a support bar may be embedded in a larger structure so that testing the bar’s integrity directly would require the impractical task of breaking down the larger structure. Instead, the ends of the bar might be accessible without dismantling the enclosing structure. The goal of nondestructive testing is to use methods that require taking measurements at the ends of the bar alone to give …

Determining The Shape Of A Resistor Grid, Esther Chiew, Vincent Selhorst-Jones

Mathematical Sciences Technical Reports (MSTR)

Impedance imaging has received a lot of attention in the past two decades, as a means for non-destructively imaging the interior of a conductive object. One injects a known electrical current pattern into an object at the exterior boundary, then measures the induced potential (voltage) on some portion of the boundary. The goal is to recover information about the interior conductivity of the object, which (we hope) influences the voltages we measure. Of course one can also use multiple input currents and measured voltages. A variation on this problem is that of "boundary identification," in which some portion of the …

Transient Analysis Of Heat And Mass Transfer By Natural Convection In Power-Law Fluid Past A Vertical Plate Immersed In A Porous Medium (Numerical Study), Nasser S. Elgazery

Applications and Applied Mathematics: An International Journal (AAM)

This paper attempted a numerical examination of the problem of unsteady free convection with heat and mass transfer from an isothermal vertical flat plate to a non-Newtonian fluid saturated porous medium. The flow in the porous medium was described via the Darcy-Brinkman-Forchheimer model. The simultaneous development of the problem of boundary layers was obtained numerically by finite difference method. Boundary layer and Boussinesq approximations had been incorporated. Numerical calculations were carried out for the various parameters entering into the problem. Velocity, temperature and concentration profiles were shown graphically and the physical quantities of the problem were given in tables. It …

Homotopy Perturbation Method And Padé Approximants For Solving Flierl-Petviashivili Equation, Syed T. Mohynd-Din, Muhammad A. Noor

Applications and Applied Mathematics: An International Journal (AAM)

In this paper, we present a reliable combination of homotopy perturbation method and Padé approximants to investigate the Flierl-Petviashivili (FP) equation. The approach introduces a new transformation necessary for the conversion of the Flierl-Petviashivili equation to a first order initial value problem and a reliable framework designed to overcome the difficulty of the singular point at x = 0. The proposed homotopy perturbation method is applied to the reformulated first order initial value problem which leads the solution in terms of transformed variable. The desired series solution is obtained by making use of the inverse transformation. The suggested algorithm may …

On A-Ary Subdivision For Curve Design Ii. 3-Point And 5-Point Interpolatory Schemes, Jian-Ao Lian

Applications and Applied Mathematics: An International Journal (AAM)

The a-ary 3-point and 5-point interpolatery subdivision schemes for curve design are introduced for arbitrary odd integer a greater than or equal to 3. These new schemes further extend the family of the classical 4- and 6-point interpolatory schemes.

A Reliable Approach For Higher-Order Integro-Differential Equations, Muhammad A. Noor, Syed T. Mohyud-Din

Applications and Applied Mathematics: An International Journal (AAM)

In this paper, we apply the variational iteration method (VIM) for solving higher-order integro differential equations by converting the problems into system of integral equations. The proposed technique is applied to the re-formulated system of integro-differential equations. Numerical results show the accuracy and efficiency of the suggested algorithm. The fact that the VIM solves nonlinear problems without calculating Adomian’s polynomials is a clear advantage of this technique over the decomposition method.

Solving Higher Dimensional Initial Boundary Value Problems By Variational Iteration Decomposition Method, Muhammad A. Noor, Syed T. Mohyud-Din

Applications and Applied Mathematics: An International Journal (AAM)

In this paper, we apply a relatively new technique which is called the variational iteration decomposition method (VIDM) by combining the traditional variational iteration and the decomposition methods for solving higher dimensional initial boundary value problems. The proposed method is an elegant combination of variational iteration and the decomposition methods. The analytical results of the problems have been obtained in terms of convergent series with easily computable components. The method is quite efficient and is practically well suited for use in these problems. Several examples are given to verify the accuracy and efficiency of the proposed technique.

Chebyshev Collocation Method For The Effect Of Variable Thermal Conductivity On Micropolar Fluid Flow Over Vertical Cylinder With Variable Surface Temperature, Nasser S. Elgazery, Nader Y. Abd Elazem

Applications and Applied Mathematics: An International Journal (AAM)

An analysis is performed to study the role of a variable thermal conductivity on unsteady free convection in a micro-polar fluid past a semi-infinite vertical cylinder with variable surface temperature in the presence of magnetic filed and radiation. The surface temperature is measured to vary as a power of the axial coordinate measured from the leading edge of the cylinder. The governing non-linear partial differential equations are transformed into a linear algebraic system utilizing Chebyshev collocation method in spatial and Crank-Nicolson method in time. Numerical results for the velocity, angular velocity and temperature profiles as well as for the local …

Model-Based Clustering Of Methylation Array Data: A Recursive-Partitioning Algorithm For High-Dimensional Data Arising As A Mixture Of Beta Distributions, E. Andres Houseman, Brock C. Christensen, Ru-Fang Yeh, Carmen J. Marsit, Margaret R. Karagas, Margaret Wrensch, Heather H. Nelson, Joseph Wiemels, Shichun Zheng, John K. Wiencke, Karl T. Kelsey

Harvard University Biostatistics Working Paper Series

No abstract provided.

Uniform Uncertainty Principle And Signal Recovery Via Regularized Orthogonal Matching Pursuit, Deanna Needell, Roman Vershynin

CMC Faculty Publications and Research

This paper seeks to bridge the two major algorithmic approaches to sparse signal recovery from an incomplete set of linear measurements—L1-minimization methods and iterative methods (Matching Pursuits). We find a simple regularized version of Orthogonal Matching Pursuit (ROMP) which has advantages of both approaches: the speed and transparency of OMP and the strong uniform guarantees of L1-minimization. Our algorithm, ROMP, reconstructs a sparse signal in a number of iterations linear in the sparsity, and the reconstruction is exact provided the linear measurements satisfy the uniform uncertainty principle.

On A-Ary Subdivision For Curve Design: I. 4-Point And 6-Point Interpolatory Schemes, Jian-Ao Lian

Applications and Applied Mathematics: An International Journal (AAM)

The classical binary 4-point and 6-point interpolatery subdivision schemes are generalized to a-ary setting for any integer a greater than or equal to 3. These new a-ary subdivision schemes for curve design are derived easily from their corresponding two-scale scaling functions, a notion from the context of wavelets.

Variational Iteration Method For Solving Initial And Boundary Value Problems Of Bratu-Type, Muhammad A. Noor, Syed T. Mohyud-Din

Applications and Applied Mathematics: An International Journal (AAM)

In this paper, we present a reliable framework to solve the initial and boundary value problems of Bratu-type which are widely applicable in fuel ignition of the combustion theory and heat transfer. The algorithm rests mainly on a relatively new technique, the variational iteration method. Several examples are given to confirm the efficiency and the accuracy of the proposed algorithm.

Signed Decomposition Of Fully Fuzzy Linear Systems, Tofigh Allahviranloo, Nasser Mikaeilvand, Narsis A. Kiani, Rasol M. Shabestari

Applications and Applied Mathematics: An International Journal (AAM)

System of linear equations is applied for solving many problems in various areas of applied sciences. Fuzzy methods constitute an important mathematical and computational tool for modeling real-world systems with uncertainties of parameters. In this paper, we discuss about fully fuzzy linear systems in the form AX = b (FFLS). A novel method for finding the non-zero fuzzy solutions of these systems is proposed. We suppose that all elements of coefficient matrix A are positive and we employ parametric form linear system. Finally, Numerical examples are presented to illustrate this approach and its results are compared with other methods.

Methods Of Assessing And Ranking Probable Sources Of Error, Nataniel Greene

Publications and Research

A classical method for ranking n potential events as sources of error is Bayes' theorem. However, a ranking based on Bayes' theorem lacks a fundamental symmetry: the ranking in terms of blame for error will not be the reverse of the ranking in terms of credit for lack of error. While this is not a flaw in Bayes' theorem, it does lead one to inquire whether there are related methods which have such symmetry. Related methods explored here include the logical version of Bayes' theorem based on probabilities of conditionals, probabilities of biconditionals, and ratios or differences of credit to …

Caching And Distributing Statistical Analyses In R, Roger D. Peng

Johns Hopkins University, Dept. of Biostatistics Working Papers

We present the cacher package for \proglang{R}, which provides tools for caching statistical analyses and for distributing these analyses to others in an efficient manner. The cacher package takes objects created by evaluating R expressions and stores them in key-value databases. These databases of cached objects can subsequently be assembled into packages for distribution over the web. The cacher package also provides tools to help readers examine the data and code in a statistical analysis and reproduce, modify, or improve upon the results. In addition, readers can easily conduct alternate analyses of the data. We describe the design and implementation …

Study Of Multibody Dynamics Using Matlab, Kalyani Soman

Mechanical & Aerospace Engineering Theses & Dissertations

Research in computational dynamics has gained importance in the past few years because of the demand for analysis and simulation of various multi-body systems in the growing mechanical and aerospace industry. These multi-body systems are made of individual bodies that are interconnected via mechanical joints. Mathematically, these joints that connect the bodies can be described as constraint equations imposed upon the motions of the involved free bodies. This process will result in an equation of motion expressed in the form of a differential-algebraic equation. It is the main objective of this thesis to develop a solution algorithm that can take …

A Wavelet-Based Method For Overcoming The Gibbs Phenomenon, Nataniel Greene

Publications and Research

The Gibbs phenomenon refers to the lack of uniform convergence which occurs in many orthogonal basis approximations to piecewise smooth functions. This lack of uniform convergence manifests itself in spurious oscillations near the points of discontinuity and a low order of convergence away from the discontinuities. Here we describe a numerical procedure for overcoming the Gibbs phenomenon called the inverse wavelet reconstruction method. The method takes the Fourier coefficients of an oscillatory partial sum and uses them to construct the wavelet coefficients of a non-oscillatory wavelet series.

Hardware Algorithm Implementation For Mission Specific Processing, Jason W. Shirley

Theses and Dissertations

There is a need to expedite the process of designing military hardware to stay ahead of the adversary. The core of this project was to build reusable, synthesizeable libraries to make this a possibility. In order to build these libraries, Matlab® commands and functions, such as Conv2, Round, Floor, Pinv, etc., had to be converted into reusable VHDL modules. These modules make up reusable libraries for the Mission Specific Process (MSP) which will support AFRL/RY. The MSP allows the VLSI design process to be completed in a mere matter of days or months using an FPGA or ASIC design, as …

An Overview Of Conditionals And Biconditionals In Probability, Nataniel Greene

Publications and Research

Conditional and biconditional statements are a standard part of symbolic logic but they have only recently begun to be explored in probability for applications in artificial intelligence. Here we give a brief overview of the major theorems involved and illustrate them using two standard model problems from conditional probability.

A Method For Visualizing Multivariate Time Series Data, Roger D. Peng

Johns Hopkins University, Dept. of Biostatistics Working Papers

Visualization and exploratory analysis is an important part of any data analysis and is made more challenging when the data are voluminous and high-dimensional. One such example is environmental monitoring data, which are often collected over time and at multiple locations, resulting in a geographically indexed multivariate time series. Financial data, although not necessarily containing a geographic component, present another source of high-volume multivariate time series data. We present the mvtsplot function which provides a method for visualizing multivariate time series data. We outline the basic design concepts and provide some examples of its usage by applying it to a …

Fourier Series Of Orthogonal Polynomials, Nataniel Greene

Publications and Research

Explicit formulas for the Fourier coefficients of the Legendre polynomials can be found in the Bateman Manuscript Project. However, similar formulas for more general classes of orthogonal polynomials do not appear to have been worked out. Here we derive explicit formulas for the Fourier series of Gegenbauer, Jacobi, Laguerre and Hermite polynomials.

Morphologies And Kinetics Of A Dewetting Ultrathin Solid Film, Mikhail Khenner

Mathematics Faculty Publications

The surface evolution model based on geometric partial differential equation is used to numerically study the kinetics of dewetting and dynamic morphologies for the localized pinhole defect in the surface of the ultrathin solid film with the strongly anisotropic surface energy. Depending on parameters such as the initial depth and width of the pinole, the strength of the attractive substrate potential and the strength of the surface energy anisotropy, the pinhole may either extend to the substrate and thus rupture the film, or evolve to the quasiequilibrium shape while the rest of the film surface undergoes phase separation into a …

Enhanced Stability Of A Dewetting Thin Liquid Film In A Single-Frequency Vibration Field, Mikhail Khenner

Mathematics Faculty Publications

Dynamics of a thin dewetting liquid film on a vertically oscillating substrate is considered. We assume moderate vibration frequency and large (compared to the mean film thickness) vibration amplitude. Using the lubrication approximation and the averaging method, we formulate the coupled sets of equations governing the pulsatile and the averaged fluid flows in the film, and then derive the nonlinear amplitude equation for the averaged film thickness. We show that there exists a window in the frequency-amplitude domain where the parametric and shear-flow instabilities of the pulsatile flow do not emerge. As a consequence, in this window the averaged description …

Enhanced Stability Of A Dewetting Thin Liquid Film In A Single-Frequency Vibration Field, Sergey Shklyaev, Mikhail Khenner, Alexei Alabuzhev

Mathematics Faculty Publications

Dynamics of a thin dewetting liquid film on a vertically oscillating substrate is considered. We assume moderate vibration frequency and large (compared to the mean film thickness) vibration amplitude. Using the lubrication approximation and the averaging method, we formulate the coupled sets of equations governing the pulsatile and the averaged fluid flows in the film, and then derive the nonlinear amplitude equation for the averaged film thickness. We show that there exists a window in the frequency-amplitude domain where the parametric and shear-flow instabilities of the pulsatile flow do not emerge. As a consequence, in this window the averaged description …

Enhanced Stability Of A Dewetting Thin Liquid Film In A Single-Frequency Vibration Field, Sergey Shklyaev, Mikhail Khenner, Alexei Alabuzhev

Mathematics Faculty Publications

Dynamics of a thin dewetting liquid film on a vertically oscillating substrate is considered. We assume moderate vibration frequency and large (compared to the mean film thickness) vibration amplitude. Using the lubrication approximation and the averaging method, we formulate the coupled sets of equations governing the pulsatile and the averaged fluid flows in the film, and then derive the nonlinear amplitude equation for the averaged film thickness. We show that there exists a window in the frequency-amplitude domain where the parametric and shear-flow instabilities of the pulsatile flow do not emerge. As a consequence, in this window the averaged description …

Morphologies And Kinetics Of A Dewetting Ultrathin Solid Film, Mikhail Khenner

Mathematics Faculty Publications

The surface evolution model based on geometric partial differential equation is used to numerically study the kinetics of dewetting and dynamic morphologies for the localized pinhole defect in the surface of the ultrathin solid film with the strongly anisotropic surface energy. Depending on parameters such as the initial depth and width of the pinole, the strength of the attractive substrate potential and the strength of the surface energy anisotropy, the pinhole may either extend to the substrate and thus rupture the film, or evolve to the quasiequilibrium shape while the rest of the film surface undergoes phase separation into a …

Formulas For The Fourier Series Of Orthogonal Polynomials In Terms Of Special Functions, Nataniel Greene

Publications and Research

An explicit formula for the Fourier coefficient of the Legendre polynomials can be found in the Bateman Manuscript Project. However, formulas for more general classes of orthogonal polynomials do not appear to have been worked out. Here we derive explicit formulas for the Fourier series of Gegenbauer, Jacobi, Laguerre and Hermite polynomials. The methods described here apply in principle to a class of polynomials, including non-orthogonal polynomials.

Inverse Wavelet Reconstruction For Resolving The Gibbs Phenomenon, Nataniel Greene

Publications and Research

The Gibbs phenomenon refers to the lack of uniform convergence which occurs in many orthogonal basis approximations to piecewise smooth functions. This lack of uniform convergence manifests itself in spurious oscillations near the points of discontinuity and a low order of convergence away from the discontinuities. Here we describe a numerical procedure for overcoming the Gibbs phenomenon called the inverse wavelet reconstruction method. The method takes the Fourier coefficients of an oscillatory partial sum and uses them to construct the wavelet coefficients of a non-oscillatory wavelet series.

Risk Classification And Ratemaking In Insurance, Erin Conrad

Honors Program Theses

Actuaries in insurance companies strive to put a price tag on future risk. They use a variety of analyses and models to accurately price insurance products. Models discussed in this paper include the pure premium method, the loss ratio method, the minimum bias procedure, and generalized linear models. Each of these pricing mechanisms are used to determine accurate and actuarially sound rates for insurance products. Competition in the industry forces companies to find more refined ways to predict the future costs of a risk. Classification systems allow companies to include a variety of rating variables so that each risk is …