Matrix, Numerical, and Optimization Methods in Science and Engineering

✍ Scribed by Kevin W. Cassel

• Publisher: Cambridge University Press
• Year: 2021
• Tongue: English
• Leaves: 600
• Edition: 1
• Category: Library

✦ Synopsis

Address vector and matrix methods necessary in numerical methods and optimization of linear systems in engineering with this unified text. Treats the mathematical models that describe and predict the evolution of our processes and systems, and the numerical methods required to obtain approximate solutions. Explores the dynamical systems theory used to describe and characterize system behaviour, alongside the techniques used to optimize their performance. Integrates and unifies matrix and eigenfunction methods with their applications in numerical and optimization methods. Consolidating, generalizing, and unifying these topics into a single coherent subject, this practical resource is suitable for advanced undergraduate students and graduate students in engineering, physical sciences, and applied mathematics.

✦ Table of Contents

Copyright
Dedications
Contents
Preface
Part I Matrix Methods
1 Vector and Matrix Algebra
1.1 Introduction
1.2 Definitions
1.3 Algebraic Operations
1.4 Systems of Linear Algebraic Equations – Preliminaries
1.5 Systems of Linear Algebraic Equations – Solution Methods
1.6 Vector Operations
1.7 Vector Spaces, Bases, and Orthogonalization
1.8 Linear Transformations
1.9 Note on Norms
1.10 Briefly on Bases
Exercises
2 Algebraic Eigenproblems and Their Applications
2.1 Applications of Eigenproblems
2.2 Eigenvalues and Eigenvectors
2.3 Real Symmetric Matrices
2.4 Normal and Orthogonal Matrices
2.5 Diagonalization
2.6 Systems of Ordinary Differential Equations
2.7 Schur Decomposition
2.8 Singular-Value Decomposition
2.9 Polar Decomposition
2.10 QR Decomposition
2.11 Briefly on Bases
2.12 Reader’s Choice
Exercises
3 Differential Eigenproblems and Their Applications
3.1 Function Spaces, Bases, and Orthogonalization
3.2 Eigenfunctions of Differential Operators
3.3 Adjoint and Self-Adjoint Differential Operators
3.4 Partial Differential Equations – Separation of Variables
3.5 Briefly on Bases
Exercises
4 Vector and Matrix Calculus
4.1 Vector Calculus
4.2 Tensors
4.3 Extrema of Functions and Optimization Preview
4.4 Summary of Vector and Matrix Derivatives
4.5 Briefly on Bases
Exercises
5 Analysis of Discrete Dynamical Systems
5.1 Introduction
5.2 Phase-Plane Analysis – Linear Systems
5.3 Bifurcation and Stability Theory – Linear Systems
5.4 Phase-Plane and Stability Analysis – Nonlinear Systems
5.5 Poincaré and Bifurcation Diagrams – Duffing Equation
5.6 Attractors and Periodic Orbits – Saltzman–Lorenz Model
Part II Numerical Methods
6 Computational Linear Algebra
6.1 Introduction to Numerical Methods
6.2 Approximation and Its Effects
6.3 Systems of Linear Algebraic Equations – Direct Methods
6.4 Systems of Linear Algebraic Equations – Iterative Methods
6.5 Numerical Solution of the Algebraic Eigenproblem
6.6 Epilogue
Exercises
7 Numerical Methods for Differential Equations
7.1 General Considerations
7.2 Formal Basis for Finite-Difference Methods
7.3 Formal Basis for Spectral Numerical Methods
7.4 Formal Basis for Finite-Element Methods
7.5 Classification of Second-Order Partial Differential Equations
Exercises
8 Finite-Difference Methods for Boundary-Value Problems
8.1 Illustrative Example from Heat Transfer
8.2 General Second-Order Ordinary Differential Equation
8.3 Partial Differential Equations
8.4 Direct Methods for Linear Systems
8.5 Iterative (Relaxation) Methods
8.6 Boundary Conditions
8.7 Alternating-Direction-Implicit (ADI) Method
8.8 Multigrid Methods
8.9 Compact Higher-Order Methods
8.10 Treatment of Nonlinear Terms
Exercises
9 Finite-Difference Methods for Initial-Value Problems
9.1 Introduction
9.2 Single-Step Methods for Ordinary Differential Equations
9.3 Additional Methods for Ordinary Differential Equations
9.4 Partial Differential Equations
9.5 Explicit Methods
9.6 Numerical Stability Analysis
9.7 Implicit Methods
9.8 Boundary Conditions – Special Cases
9.9 Treatment of Nonlinear Convection Terms
9.10 Multidimensional Problems
9.11 Hyperbolic Partial Differential Equations
9.12 Coupled Systems of Partial Differential Equations
9.13 Parallel Computing
9.14 Epilogue
Exercises
Part III Least Squares and Optimization
10 Least-Squares Methods
10.1 Introduction to Optimization
10.2 Least-Squares Solutions of Algebraic Systems of Equations
10.3 Least-Squares with Constraints
10.4 Least-Squares with Penalty Functions
10.5 Nonlinear Objective Functions
10.6 Conjugate-Gradient Method
10.7 Generalized Minimum Residual (GMRES) Method
10.8 Summary of Krylov-Based Methods
Exercises
11 Data Analysis: Curve Fitting and Interpolation
11.1 Linear Regression
11.2 Polynomial Regression
11.3 Least-Squares Regression as an Overdetermined System
11.4 Least Squares with Orthogonal Basis Functions – Fourier Series
11.5 Polynomial Interpolation
11.6 Spline Interpolation
11.7 Curve Fitting and Interpolation of Multidimensional Data
11.8 Linear Regression Using Singular-Value Decomposition
11.9 Least-Squares Regression as State Estimation
11.10 Definitions of the Residual
Exercises
12 Optimization and Root Finding of Algebraic Systems
12.1 Introduction
12.2 Nonlinear Algebraic Equations – Root Finding
12.3 Optimization
12.4 Nonlinear Unconstrained Optimization
12.5 Numerical Methods
12.6 Nonlinear Constrained Optimization
12.7 Linear Programming – Formulation
12.8 Linear Programming – Simplex Method
12.9 Optimal Control
Exercises
13 Data-Driven Methods and Reduced-Order Modeling
13.1 Introduction
13.2 Projection Methods for Continuous Systems
13.3 Galerkin Projection and Reduced-Order Modeling for Continuous Systems
13.4 Projection Methods for Discrete Systems
13.5 Galerkin Projection and Reduced-Order Modeling for Discrete Systems
13.6 Proper-Orthogonal Decomposition (POD) for Continuous Data
13.7 Proper-Orthogonal Decomposition (POD) for Discrete Data
13.8 Extensions and Alternatives to POD
13.9 System Identification
13.10 Epilogue
References
Index