Call number:
9783319673714 (e-book)
Description / Table of Contents:
This book showcases powerful new hybrid methods that combine numerical and symbolic algorithms. Hybrid algorithm research is currently one of the most promising directions in the context of geosciences mathematics and computer mathematics in general. One important topic addressed here with a broad range of applications is the solution of multivariate polynomial systems by means of resultants and Groebner bases. But that’s barely the beginning, as the authors proceed to discuss genetic algorithms, integer programming, symbolic regression, parallel computing, and many other topics. The book is strictly goal-oriented, focusing on the solution of fundamental problems in the geosciences, such as positioning and point cloud problems. As such, at no point does it discuss purely theoretical mathematics. "The book delivers hybrid symbolic-numeric solutions, which are a large and growing area at the boundary of mathematics and computer science." Dr. Daniel Li chtbau
Description / Table of Contents:
Solution of algebraic polynomial systems -- Homotopy solution of nonlinear systems -- Over and underdeterminated systems -- Simulated annealing -- Genetic algorithm -- Particle swarm optimization -- Integer programming -- Multiobjective optimization -- Approximation with radial bases functions -- Support vector machines (SVM) -- Symbolic regression -- Quantile regression -- Robust regression -- Stochastic modeling -- Parallel computations
Type of Medium:
12
Pages:
1 Online-Ressource (xxvii, 596 Seiten)
,
Illustrationen, Diagramme
ISBN:
9783319673714
,
978-3-319-67371-4
URL:
Ebook (access only within the AWI network)
DOI:
10.1007/978-3-319-67371-4
Language:
English
Note:
Contents
Part I Solution of Nonlinear Systems
1 Solution of Algebraic Polynomial Systems
1.1 Zeros of Polynomial Systems
1.2 Resultant Methods
1.2.1 Sylvester Resultant
1.2.2 Dixon Resultant
1.3 Gröbner Basis
1.3.1 Greatest Common Divisor of Polynomials
1.3.2 Reduced Gröbner Basis
1.3.3 Polynomials with Inexact Coefficients
1.4 Using Dixon-EDF for Symbolic Solution of Polynomial Systems
1.4.1 Explanation of Dixon-EDF
1.4.2 Distance from a Point to a Standard Ellipsoid
1.4.3 Distance from a Point to Any 3D Conic
1.4.4 Pose Estimation
1.4.5 How to Run Dixon-EDF
1.5 Applications
1.5.1 Common Points of Geometrical Objects
1.5.2 Nonlinear Heat Transfer
1.5.3 Helmert Transformation
1.6 Exercises
1.6.1 Solving a System with Different Techniques
1.6.2 Planar Ranging
1.6.3 3D Resection
1.6.4 Pose Estimation
References
2 Homotopy Solution of Nonlinear Systems
2.1 The Concept of Homotopy
2.2 Solving Nonlinear Equation via Homotopy
2.3 Tracing Homotopy Path as Initial Value Problem
2.4 Types of Linear Homotopy
2.4.1 General Linear Homotopy
2.4.2 Fixed-Point Homotopy
2.4.3 Newton Homotopy
2.4.4 Affine Homotopy
2.4.5 Mixed Homotopy
2.5 Regularization of the Homotopy Function
2.6 Start System in Case of Algebraic Polynomial Systems
2.7 Homotopy Methods in Mathematica
2.8 Parallel Computation
2.9 General Nonlinear System
2.10 Nonlinear Homotopy
2.10.1 Quadratic Bezier Homotopy Function
2.10.2 Implementation in Mathematica
2.10.3 Comparing Linear and Quadratic Homotopy
2.11 Applications
2.11.1 Nonlinear Heat Conduction
2.11.2 Local Coordinates via GNSS
2.12 Exercises
2.12.1 GNSS Positioning N-Point Problem
References
3 Overdetermined and Underdetermined Systems
3.1 Concept of the Over and Underdetermined Systems
3.1.1 Overdetermined Systems
3.1.2 Underdetermined Systems
3.2 Gauss–Jacobi Combinatorial Solution
3.3 Gauss–Jacobi Solution in Case of Nonlinear Systems
3.4 Transforming Overdetermined System into a Determined System
3.5 Extended Newton–Raphson Method
3.6 Solution of Underdetermined Systems
3.6.1 Direct Minimization
3.6.2 Method of Lagrange Multipliers
3.6.3 Method of Penalty Function
3.6.4 Extended Newton–Raphson
3.7 Applications
3.7.1 Geodetic Application—The Minimum Distance Problem
3.7.2 Global Navigation Satellite System (GNSS) Application
3.7.3 Geometric Application
3.8 Exercises
3.8.1 Solution of Overdetermined System
3.8.2 Solution of Underdetermined System
Part II Optimization of Systems
4 Simulated Annealing
4.1 Metropolis Algorithm
4.2 Realization of the Metropolis Algorithm
4.2.1 Representation of a State
4.2.2 The Free Energy of a State
4.2.3 Perturbation of a State
4.2.4 Accepting a New State
4.2.5 Implementation of the Algorithm
4.3 Algorithm of the Simulated Annealing
4.4 Implementation of the Algorithm
4.5 Application to Computing Minimum of a Real Function
4.6 Generalization of the Algorithm
4.7 Applications
4.7.1 A Packing Problem
4.7.2 The Traveling Salesman Problem
4.8 Exercise
5 Genetic Algorithms
5.1 The Genetic Evolution Concept
5.2 Mutation of the Best Individual
5.3 Solving a Puzzle
5.4 Application to a Real Function
5.5 Employing Sexual Reproduction
5.5.1 Selection of Parents
5.5.2 Sexual Reproduction: Crossover and Mutation
5.6 The Basic Genetic Algorithm (BGA)
5.7 Applications
5.7.1 Nonlinear Parameter Estimation
5.7.2 Packing Spheres with Different Sizes
5.7.3 Finding All the Real Solutions of a Non-algebraic System
5.8 Exercises
5.8.1 Foxhole Problem
References
6 Particle Swarm Optimization
6.1 The Concept of Social Behavior of Groups of Animals
6.2 Basic Algorithm
6.3 The Pseudo Code of the Algorithm
6.4 Applications
6.4.1 1D Example
6.4.2 2D Example
6.4.3 Solution of Nonlinear Non-algebraic System
6.5 Exercise
Reference
7 Integer Programming
7.1 Integer Problem
7.2 Discrete Value Problems
7.3 Simple Logical Conditions
7.4 Some Typical Problems of Binary Programming
7.4.1 Knapsack Problem
7.4.2 Nonlinear Knapsack Problem
7.4.3 Set-Covering Problem
7.5 Solution Methods
7.5.1 Binary Countdown Method
7.5.2 Branch and Bound Method
7.6 Mixed–Integer Programming
7.7 Applications
7.7.1 Integer Least Squares
7.7.2 Optimal Number of Oil Wells
7.8 Exercises
7.8.1 Study of Mixed Integer Programming
7.8.2 Mixed Integer Least Square
References
8 Multiobjective Optimization
8.1 Concept of Multiobjective Problem
8.1.1 Problem Definition
8.1.2 Interpretation of the Solution
8.2 Pareto Optimum
8.2.1 Nonlinear Problems
8.2.2 Pareto-Front and Pareto-Set
8.3 Computation of Pareto Optimum
8.3.1 Pareto Filter
8.3.2 Reducing the Problem to the Case of a Single Objective
8.3.3 Weighted Objective Functions
8.3.4 Ideal Point in the Function Space
8.3.5 Pareto Balanced Optimum
8.3.6 Non-convex Pareto-Front
8.4 Employing Genetic Algorithms
8.5 Application
8.5.1 Nonlinear Gauss-Helmert Model
8.6 Exercise
References
Part III Approximation of Functions and Data
9 Approximation with Radial Bases Functions
9.1 Basic Idea of RBF Interpolation
9.2 Positive Definite RBF Function
9.3 Compactly Supported Functions
9.4 Some Positive Definite RBF Function
9.4.1 Laguerre-Gauss Function
9.4.2 Generalized Multi-quadratic RBF
9.4.3 Wendland Function
9.4.4 Buchmann-Type RBF
9.5 Generic Derivatives of RBF Functions
9.6 Least Squares Approximation with RBF
9.7 Applications
9.7.1 Image Compression
9.7.2 RBF Collocation Solution of Partial Differential Equation
9.8 Exercise
9.8.1 Nonlinear Heat Transfer
References
10 Support Vector Machines (SVM)
10.1 Concept of Machine Learning
10.2 Optimal Hyperplane Classifier
10.2.1 Linear Separability
10.2.2 Computation of the Optimal Parameters
10.2.3 Dual Optimization Problem
10.3 Nonlinear Separability
10.4 Feature Spaces and Kernels
10.5 Application of the Algorithm
10.5.1 Computation Step by Step
10.5.2 Implementation of the Algorithm
10.6 Two Nonlinear Test Problems
10.6.1 Learning a Chess Board
10.6.2 Two Intertwined Spirals
10.7 Concept of SVM Regression
10.7.1 e-Insensitive Loss Function
10.7.2 Concept of the Support Vector Machine Regression (SVMR)
10.7.3 The Algorithm of the SVMR
10.8 Employing Different Kernels
10.8.1 Gaussian Kernel
10.8.2 Polynomial Kernel
10.8.3 Wavelet Kernel
10.8.4 Universal Fourier Kernel
10.9 Applications
10.9.1 Image Classification
10.9.2 Maximum Flooding Level
10.10 Exercise
10.10.1 Noise Filtration
References
11 Symbolic Regression
11.1 Concept of Symbolic Regression
11.2 Problem of Kepler
11.2.1 Polynomial Regression
11.2.2 Neural Network
11.2.3 Support Vector Machine Regression
11.2.4 RBF Interpolation
11.2.5 Random Models
11.2.6 Symbolic Regression
11.3 Applications
11.3.1 Correcting Gravimetric Geoid Using GPS Ellipsoidal Heights
11.3.2 Geometric Transformation
11.4 Exercise
11.4.1 Bremerton Data
References
12 Quantile Regression
12.1 Problems with the Ordinary Least Squares
12.1.1 Correlation Height and Age
12.1.2 Engel’s Problem
12.2 Concept of Quantile
12.2.1 Quantile as a Generalization of Median
12.2.2 Quantile for Probability Distributions
12.3 Linear Quantile Regression
12.3.1 Ordinary Least Square (OLS)
12.3.2 Median Regression (MR)
12.3.3 Quantile Regression (QR)
12.4 Computing Quantile Regression
12.4.1 Quantile Regression via Linear Programming
12.4.2 Boscovich’s Problem
12.4.3 Extension to Linear Combination of Nonlinear Functions
12.4.4 B-Spline Application
12.5 Applications
12.5.1 Separate Outliers in Cloud Points
12.5.2 Modelling Time-Series
12.6 Exercise
12.6.1 Regression of Implicit-Functions
References
13 Robust Regression
13.1 Basic Methods in Robust Regression
13.1.1 Concept of Robust Regression
13.1.2 Maximum Likelihood Method
13.1.3 Danish Algorithm
13.1.4 Danish Algorithm with PCA
13.1.5 RANSAC Algorithm
13.2 Application Examples
13.2.1 Fitting a Sphere to Point Cloud Data
13.2.2 Fitting a Cylinder
13.3 Problem
13.3.1 Fitting a Plane to a Slope
References
14 Stochastic Modeling
14.1 Basic Stochastic Processes
14.1.1 Concept of Stochastic Processes
14.1.2 Examples for Stochastic Processes
14.1.3 Features of Stochastic Processes
14.2 Time Series
14.2.1 Concept of Time
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