An efficient algorithm for the numerical evaluation of boundary integral equations

CHAPTER 1. AN INTRODUCTION TO DATA AND FUNCTIONS. 1.1 Describing Single-Variable Data. 1.2 Describing Relationships between Two Variables. 1.3 An Introduction to Functions. 1.4 The Language of Functions. 1.5 Visualizing Functions. CHAPTER 2. RATES OF CHANGE AND LINEAR FUNCTIONS. 2.1 Average Rates of Change. 2.2 Change in the Average Rate of Change. 2.3 The Average Rate of Change Is a Slope. 2.4 Putting a Slant on Data. 2.5 Linear Functions: When Rates of Change Are Constant. 2.6 Visualizing Linear Functions. 2.7 Constructing Graphs and Equations of Linear Functions. 2.8 Special Cases. 2.9 Breaking the Line: Piecewise Linear Functions. 2.10 Constructing Linear Models of Data. 2.11 Looking for Links between Education and Earnings: A Case Study on Using Regression Lines. CHAPTER 3. WHEN LINES MEET: LINEAR SYSTEMS. 3.1 Interpreting Intersection Points: Linear and Nonlinear Systems. 3.2 Visualizing and Solving Linear Systems. 3.3 Reading between the Lines: Linear Inequalities. 3.4 Systems with Piecewise Linear Functions: Tax Plans. CHAPTER 4. THE LAWS OF EXPONENTS AND LOGARITHMS: MEASURING THE UNIVERSE. 4.1 The Numbers of Science: Measuring Time and Space. 4.2 Positive Integer Exponents. 4.3 Zero, Negative, and Fractional Exponents. 4.4 Converting Units. 4.5 Orders of Magnitude. 4.6 Logarithms as Numbers. CHAPTER 5. GROWTH AND DECAY: AN INTRODUCTION TO EXPONENTIAL FUNCTIONS. 5.1 Exponential Growth. 5.2 Exponential Decay. 5.3 Comparing Linear and Exponential Functions. 5.4 Visualizing Exponential Functions. 5.5 Exponential Functions: A Constant Percent Change. 5.6 More Examples of Exponential Growth and Decay. 5.7 Compound Interest and the Number e. 5.8 Semi-Log Plots of Exponential Functions. CHAPTER 6. LOGARITHMIC LINKS: LOGARITHMIC AND EXPONENTIAL FUNCTIONS. 6.1 Using Logarithms to Solve Exponential Equations. 6.2 Using Natural Logarithms to Solve Exponential Equations Base e. 6.3 Visualizing and Applying Logarithmic Functions. 6.4 Using Semi-Log Plots to Construct Exponential Models for Data. C H A P T E R 7. POWER FUNCTIONS. 7.1 The Tension between Surface Area and Volume. 7.2 Direct Proportionality: Power Functions with Positive Powers. 7.3 Visualizing Positive Integer Power Functions. 7.4 Comparing Power and Exponential Functions. 7.5 Inverse Proportionality: Power Functions with Negative Powers. 7.6 Visualizing Negative Integer Power Functions. 7.7 Using Logarithmic Scales to Find the Best Functional Model. CHAPTER 8. QUADRATICS AND THE MATHEMATICS OF MOTION. 8.1 An Introduction to Quadratic Functions: The Standard Form. 8.2 Visualizing Quadratics: The Vertex Form. 8.3 The Standard Form vs. the Vertex Form. 8.4 Finding the Horizontal Intercepts: The Factored Form. 8.5 The Average Rate of Change of a Quadratic Function. 8.6 The Mathematics of Motion. CHAPTER 9. NEW FUNCTIONS FROM OLD. 9.1 Transformations. 9.2 The Algebra of Functions. 9.3 Polynomials: The Sum of Power Functions. 9.4 Rational Functions: The Quotient of Polynomials. 9.5 Composition and Inverse Functions. 9.6 Exploring, Extending & Expanding. APPENDIX Student Data Tables for Exploration 2.1. Data Dictionary for FAM1000 Data. SOLUTIONS For all Algebra Aerobics and Check Your Understanding problems for odd-numbered problems in the Exercises and Chapter Reviews. All solutions are grouped by chapter ANS-1. INDEX.

Conjugate gradient methods for the solution of boundary integral equations on a piecewise smooth boundary

We study the numerical solution of the boundary integral equation formulation for the scattering of time-harmonic electromagnetic waves from infinite cylinders. For smooth boundaries of the cylinder cross section we describe a Nystro¨m, a collocation and a Galerkin method based on an approximation by trigonometric polynomials on an equidistant mesh. For smooth data in each of the three methods the convergence is exponential. From the three approaches the Nystro¨m method is the most efficient since it requires the least computational effort. For cross sections with corners we develop a Nystro¨m method on a graded mesh based on the idea of transforming the nonsmooth case to a smooth periodic case via an appropriate substitution. We conclude the paper with some considerations on the corresponding three dimensional problem.

We consider iterated function systems on the real line that consist of continuous, piecewise linear functions. Under a mild separation condition, we show that the Hausdorff and box dimensions of the attractor are equal to the minimum of 1 and the exponent which comes from the most natural system of covers of the attractor.

Nonlinear effects of a power amplifier (PA) in communications systems include spectral spreading, which causes adjacent channel interference. For predistorter design, it is crucial to employ a model that can sufficiently model the intermodulation distortion (IMD) caused by the PA. The order of IMD generated by a polynomial is determined by its highest order basis. For a piecewise linear function, the order of nonlinearity cannot be directly deduced from its basis function. This paper investigates how IMD is introduced by the simplicial canonical piecewise linear (SCPWL) function. We examine the operation imposed by the SCPWL basis function on sinusoid and multi-tone signal in the time domain and frequency domain. Our analysis results in expressions that can be used to predict the output spectrum of the SCPWL basis. The SCPWL basis operation involves spreading of the input signal spectrum. The output spectrum of each SCPWL basis includes infinite IMD components. The IMD order of the SCPWL model is not determined by the highest order basis. Instead, the number of SCPWL parameters, i.e., the number of linear affine segments that defines the function, determines how accurately it can model the IMD components.

Topology design of remote patient monitoring system concerning qualitative and quantitative issues

Piecewise linear function (PLF) is not only a generalization of univariate segmented linear function in multivariate case, but also an important bridge to study the approximation of continuous function by Mamdani and Takagi-Sugeno fuzzy systems. In this paper, the definitions of the PLF and subdivision are introduced in the hyperplane, the analytic expression of PLF is given by using matrix determinant, and the concept of approximation factor is first proposed by using m-mesh subdivision. Secondly, the vertex coordinates and their changing rules of the n-dimensional small polyhedron are found by dividing a three-dimensional cube, and the algebraic cofactor and matrix norm of corresponding determinants of piecewise linear functions are given. Finally, according to the method of solving algebraic cofactors and matrix norms, it is proved that the approximation factor has nothing to do with the number of subdivisions, but the approximation accuracy has something to do with the number of subdivisions. Furthermore, the process of a specific binary piecewise linear function approaching a continuous function according to infinite norm in two dimensions space is realized by a practical example, and the validity of PLFs to approximate a continuous function is verified by t-hypothesis test in Statistics.

Fuzzy aspects of the boundary element method

This book provides an extensive introduction to the numerical solution of a large class of integral equations. The initial chapters provide a general framework for the numerical analysis of Fredholm integral equations of the second kind, covering degenerate kernel, projection and Nystrom methods. Additional discussions of multivariable integral equations and iteration methods update the reader on the present state of the art in this area. The final chapters focus on the numerical solution of boundary integral equation (BIE) reformulations of Laplace's equation, in both two and three dimensions. Two chapters are devoted to planar BIE problems, which include both existing methods and remaining questions. Practical problems for BIE such as the set up and solution of the discretised BIE are also discussed. Each chapter concludes with a discussion of the literature and a large bibliography serves as an extended resource for students and researchers needing more information on solving particular integral equations.

This paper describes iterative procedures for determining linear and piecewise linear discriminant functions for multicategory pattern classifiers. While classifiers with the same structure have often been proposed, it is less well known that their parameters can be efficiently determined by simple adjustment procedures. For linear discriminant functions, convergence proofs are given for procedures that are guaranteed to yield error-free solutions on design samples, provided only that such solutions exist. While no similar results are known for piecewise linear discriminant functions, simple procedures are given that have been effective in various experiments. The results of experiments with artificially generated multimodal data and with hand-printed alphanumeric characters are given to show that this approach compares favorably with other classification methods.

An approximate method of solving an integral equation of the potential theory for a Dirichlet problem for the Laplace operator is proposed in the case when domains are curvilinear polygons with piecewise analytic boundaries. The proposed method is exponentially convergent with respect to the number of quadrature nodes in use.

The major problem encountered by the solution of boundary integral equations is related to the nonuniqueness of its solution. The problem is also known as internal resonance problem. As a remedy the combined-field integral equation (CFIE) technique is widely used which contains a linear combination of the magnetic and electric field integral equation to provide a unique stable solution. A second effective technique is the constrained conjugate gradient method (CCG) that minimizes a cost functional consisting of two terms. The first term is the error norm with respect to boundary integral equation, while the second term is the error norm with respect to the interior equation over a closed interior surface. This paper presents the efficiency of both approaches. The results are compared for the solution of a sphere.

We give a convergence and error analysis for a Nystrom method on a graded mesh for the numerical solution of boundary integral equations for the harmonic Dirichlet problem in plane domains with corners.

This paper deals with approximate solutions to integral equations arising in boundary value problems for the biharmonic equation in simply connected piecewise smooth domains. The approximation method considered demonstrates excellent convergence even in the case of boundary conditions discontinuous at corner points. In an application we obtain very accurate approximations for some characteristics of two‐dimensional Stokes flow in non‐smooth domains. (© 2013 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

A fourth-order kernel-free boundary integral method for implicitly defined surfaces in three space dimensions