This book is a rigorous but practical presentation of the techniques of uncertainty quantification, with applications in R and Python. This volume includes mathematical arguments at the level necessary to make the presentation rigorous and the assumptions clearly established, while maintaining a focus on practical applications of uncertainty quantification methods. Practical aspects of applied probability are also discussed, making the content accessible to students. The introduction of R and Python allows the reader to solve more complex problems involving a more significant number of variables. Users will be able to use examples laid out in the text to solve medium-sized problems.
The list of topics covered in this volume includes linear and nonlinear programming, Lagrange multipliers (for sensitivity), multi-objective optimization, game theory, as well as linear algebraic equations, and probability and statistics. Blending theoretical rigor and practical applications, this volume will be of interest to professionals, researchers, graduate and undergraduate students interested in the use of uncertainty quantification techniques within the framework of operations research and mathematical programming, for applications in management and planning.
Author(s): Eduardo Souza de Cursi
Series: International Series in Operations Research & Management Science, 335
Publisher: Springer
Year: 2023
Language: English
Pages: 767
City: Cham
Introduction
Contents
Chapter 1: Some Tips to Use R and RStudio
1.1 How to Install R and RStudio
1.2 How to Include a Third-Part Add-In
1.3 How to Create a Document with RStudio
1.4 How to Create a Script with RStudio
1.5 How to Manipulate Numeric Variables, Vectors, and Factors
1.6 How to Manipulate Matrices and Arrays
1.7 How to Use Lists
1.8 Using data.frames
1.9 Plotting with RStudio
1.10 Programming with R
1.11 Classes in R
1.12 How to Solve Differential Equations with R
1.12.1 Initial Value Problems for Ordinary Differential Equations
1.12.2 Boundary Value Problems for Ordinary Differential Equations
1.13 Optimization with R
1.13.1 Linear Programming
1.13.2 Nonlinear Programming
1.13.3 Duality Methods
1.13.4 Multiobjective Optimization
1.14 Solving Equations with R
1.14.1 Systems of Linear Equations
1.14.2 Systems of Nonlinear Equations
1.14.3 Optimization and Systems of Equations
1.15 Interpolation and Approximation with R
1.15.1 Variational Approximation
1.15.2 Smoothed Particle Approximation
1.16 Integrals and Derivatives with R
1.16.1 Variational Approximation of the Derivatives
1.16.2 Smoothed Particle Approximation of the Derivative
Chapter 2: Probabilities and Random Variables
2.1 Notation
2.2 Probability
2.2.1 Mass Functions and Mass Densities
2.2.2 Combinatorial Probabilities
2.3 Independent Events
2.4 Numerical Variables on Finite Populations
2.4.1 Couples of Numerical Variables
2.4.2 Independent Numerical Variables
2.5 Numerical Variables as Elements of Hilbert Spaces
2.5.1 Conditional Probabilities as Orthogonal Projections
2.5.2 Means as Orthogonal Projections
2.5.3 Affine Approximations and Correlations
2.5.4 Conditional Mean
2.6 Random Variables
2.6.1 Numerical Evaluation of Statistics
2.7 Random Vectors
2.8 Discrete and Continuous Random Variables
2.8.1 Discrete Variables
2.8.2 Continuous Variables Having a PDF
2.9 Sequences of Random Variables
2.10 Samples
2.10.1 Maximum-Likelihood Estimators
2.10.2 Samples from Random Vectors
2.10.3 Empirical CDF and Empirical PDF
2.10.4 Testing Adequacy of a Sample to a Distribution
2.10.5 Testing the Independence a Couple of Variables
2.11 Generating Triangular Random Numbers
2.12 Generating Random Numbers by Inversion
2.13 Generating Random Vectors with a Given Covariance Matrix
2.14 Generating Regular Random Functions
2.15 Generating Regular Random Curves
Chapter 3: Representation of Random Variables
3.1 The UQ Approach for the Representation of Random Variables
3.2 Collocation
3.3 Variational Approximation
3.4 Moments Matching Method
3.4.1 The Standard Formulation of M3
3.4.2 Alternative Formulations of M3
3.5 Multidimensional Expansions
3.5.1 Case Where U Is Multidimensional
3.5.2 Case Where X Is Multidimensional
3.6 Random Functions
3.7 Random Curves
3.8 Mean, Variance, and Confidence Intervals for Random Functions or Random Curves
Chapter 4: Stochastic Processes
4.1 Ergodicity
4.2 Determination of the Distribution of a Stationary Process
4.3 White Noise
4.4 Moving Average Processes
4.5 Autoregressive Processes
4.6 ARMA Processes
4.7 Markov Processes
4.8 Diffusion Processes
4.8.1 Time Integral and Derivative of a Process
4.8.2 Simulation of the Time Integral of a White Noise
4.8.3 Brownian Motion
4.8.4 Random Walks
4.8.5 Itô´s Integrals
4.8.6 Itô´s Calculus
4.8.7 Numerical Simulation of Stochastic Differential equations
Chapter 5: Uncertain Algebraic Equations
5.1 Uncertain Linear Systems
5.1.1 Very Small Linear Systems
5.2 Nonlinear Equations and Adaptation of an Iterative Code
5.3 Iterative Evaluation of Eigenvalues
5.3.1 Very Small Matrices
5.4 The Variational Approach for Uncertain Algebraic Equations
Chapter 6: Random Differential Equations
6.1 Linear Differential Equations
6.2 Nonlinear Differential Equations
6.3 Adaptation of ODE Solvers
6.4 Uncertainties on Curves Connected to Differential Equations
Chapter 7: UQ in Game Theory
7.1 The Language from Game Theory
7.2 A Simplified Odds and Evens Game
7.2.1 GT Strategies When p = (p1,p2) Is Known
7.2.2 Strategies When p Is Unknown
7.2.3 Strategies for the Stochastic Game
7.2.4 Replicator Dynamics
7.3 The Prisoner´s Dilemma
7.3.1 Replicator Dynamics
7.4 The Goalie´s Anxiety at the Penalty Kick
7.5 Hawks and Doves
Chapter 8: Optimization Under Uncertainty
8.1 Using the Methods of Representation
8.2 Using the Adaptation of a Descent Method
8.3 Combining Statistics of the Objective, the Constraints, and Expansions
Chapter 9: Reliability
9.1 Limit State Curves
9.2 Design Point
9.3 Multiple Failure Conditions
9.4 Reliability Analysis
9.5 Hasofer-Lind Reliability Index
9.5.1 The General Situation
9.5.2 The Case of Affine Limit State Equations
9.5.3 Convex Failure Regions
9.6 Using the Reliability Index to Estimate the Probability of Failure
9.6.1 The Case of Affine Limit State Equations
9.6.2 The Case of a Convex Failure Region
9.6.3 General Failure Regions
9.7 The Transformations of Rosenblatt and Nataf
9.8 FORM and SORM
9.8.1 First-Order Reliability Method (FORM)
9.8.2 Second Order Reliability Method (SORM)
9.9 Reliability Based Design Optimization
9.9.1 The Bilevel or Double Loop Approach for a Desired β
9.9.2 The Bilevel or Double Loop Approach for a Desired Objective
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Bibliography
Index
