Stochastic Methods in Scientific Computing: From Foundations to Advanced Techniques (Chapman & Hall/CRC Numerical Analysis and Scientific Computing Series)

✍ Scribed by Massimo D'Elia, Kurt Langfeld, Biagio Lucini

• Publisher: Chapman and Hall/CRC
• Year: 2024
• Tongue: English
• Leaves: 401
• Edition: 1
• Category: Library

✦ Synopsis

Stochastic Methods in Scientific Computing: From Foundations to Advanced Techniques introduces the reader to advanced concepts in stochastic modelling, rooted in an intuitive yet rigorous presentation of the underlying mathematical concepts. A particular emphasis is placed on illuminating the underpinning Mathematics, and yet have the practical applications in mind. The reader will find valuable insights into topics ranging from Social Sciences and Particle Physics to modern-day Computer Science with Machine Learning and AI in focus. The book also covers recent specialised techniques for notorious issues in the field of stochastic simulations, providing a valuable reference for advanced readers with an active interest in the field.

Features

• Self-contained, starting from the theoretical foundations and advancing to the most recent developments in the field
• Suitable as a reference for post-graduates and researchers or as supplementary reading for courses in numerical methods, scientific computing, and beyond
• Interdisciplinary, laying a solid ground for field-specific applications in finance, physics and biosciences on common theoretical foundations
• Replete with practical examples of applications to classic and current research problems in various fields.

✦ Table of Contents

Cover
Half Title
Series Page
Title Page
Copyright Page
Dedication
Contents
Preface
Author
1. Random Numbers
1.1. Random numbers and probability distribution
1.1.1. Quantifying randomness
1.1.2. Pseudo randomness
1.1.3. Designing probability distributions
1.1.4. Applications: Poisson and exponential distributions
1.2. Central limit theorem
1.2.1. Conditions and theorem
1.2.2. The normal distribution
1.2.3. Independent measurements and error propagation
1.3. Beyond the normal distribution
1.3.1. Cauchy-Lorentz distribution and the failure of error reduction
1.3.2. Pareto distribution and applications
1.4. Exercises
2. Random walks
2.1. Random walk as a Markov process
2.2. Random walks in 1 and 2 dimensions
2.2.1. Random walk on Z
2.2.2. Stability
2.2.3. Limits: Wiener process and diffusion equation
2.2.4. Gaussian random walk and correlations
2.3. Levy flight
2.3.1. Definition and simulations
2.3.2. Generalised Central Limit Theorem and stability
2.3.3. Computer experiments
2.4. Random walks with potentials
2.4.1. The Langevin diffusion
2.4.2. Diffusion in external potentials
2.4.3. Langevin simulations
2.5. Exercises
3. Monte Carlo methods
3.1. Objectives and concepts
3.1.1. Simulation and error bar
3.1.2. Estimating expectation values
3.2. Monte Carlo integration
3.2.1. The accept-reject method
3.2.2. The importance of a proper sampling
3.3. Markov Chain Monte Carlo
3.3.1. Ensemble of Markov chains and some theorems on the spectrum of W
3.3.2. Equilibrium condition and the detailed balance principle
3.3.3. The Metropolis algorithm
3.3.4. Heat-bath Algorithms
3.3.5. Composition of Markov chains
3.4. Advanced Error Analysis Techniques
3.5. Error estimate in the presence of autocorrelations
3.5.1. Data blocking techniques
3.5.2. Markov chain optimization: looking for the highest possible efficiency
3.6. Error estimate for non-trivial estimators: The Jackknife, and the Bootstrap
3.6.1. The bootstrap
3.6.2. The jackknife
3.6.3. Extension to the case of correlated data sets
3.7. Biased Estimators
3.8. Exercises
4. Statistical models
4.1. An introduction to thermodynamics
4.1.1. Thermodynamic systems and associated states
4.1.2. First look at differential forms
4.1.3. First law of thermodynamics
4.1.4. Second law of thermodynamics
4.1.5. Legendre transform and thermodynamic potentials
4.1.6. Maxwell’s relations
4.1.7. Heat capacities
4.2. From thermodynamics to statistical mechanics
4.2.1. The micro-canonical ensemble
4.2.2. The canonical ensemble
4.2.3. The grand canonical ensemble
4.2.4. From classical to quantum system
4.3. Phase transitions
4.3.1. Classification of phase transitions
4.3.2. The role of symmetries in phase transitions
4.3.3. An application: the Van der Waals gas
4.4. The Ising model
4.4.1. Ferromagnetism and the Ising model
4.4.2. Exact solution of the 1D Ising model
4.4.3. Mean-field solution of the 2D Ising model
4.4.4. Dualisation and critical temperature
4.4.5. Comparing mean-field with the exact solution
4.4.6. Monte Carlo simulation and phase transition
4.4.7. Upper and lower critical dimension
4.5. An overview of other models
4.6. Exercises
5. Advanced Monte Carlo simulation techniques
5.1. Hamiltonian (Hybrid) Monte Carlo (HMC) simulations
5.1.1. Molecular dynamics simulations and HMC
5.1.2. Practical implementation of the HMC algorithm
5.1.3. Effectiveness of the HMC algorithm and further improvements
5.1.4. An example of the HMC implementation
5.2. Non-local Monte Carlo update
5.2.1. Critical slowing down
5.2.2. Cluster update algorithms
5.2.3. Improved operators and correlation length
5.2.4. Flux algorithms
5.3. Micro-canonical simulations
5.3.1. What are micro-canonical ensembles?
5.3.2. The demon method
5.3.3. Performance and auto-correlations
5.4. Flat histogram methods
5.4.1. Reweighting
5.4.2. Multicanonical sampling
5.4.3. Wang-Landau sampling
5.4.4. Do flat histogram methods work?
5.5. The Linear Logarithmic Relaxation (LLR) method
5.5.1. The Linear Logarithmic Relaxation formulation
5.5.2. The Robbins-Monro solution
5.5.3. LLR and observables
5.5.4. LLR showcase: the Ising model
5.6. Exercises
6. From Statistical Systems to Quantum Field Theory*
6.1. Invitation: The O(2) model
6.1.1. Model and Symmetry
6.1.2. Wolff cluster algorithm
6.1.3. Phase transition and symmetry breaking
6.1.4. Quantum Field Theory rising
6.2. The Bridge to QFT: the Feynman path-integral
6.2.1. A working example: the harmonic oscillator and Brownian motion
6.2.2. Extension to Quantum Field Theory
6.2.3. Energy gaps, continuum limit and critical points
6.3. Gauge Theories
6.3.1. Continuum Yang-Mills theories and their lattice counterpart
6.3.2. Numerical algorithms: the SU(2) case
6.3.3. Extension to SU(N) gauge theories
6.3.4. Scaling, asymptotic freedom and dimensional transmutation
6.4. Adding fermion fields
6.4.1. Positivity of the fermion determinant
6.4.2. Non-locality of the fermion determinant
6.4.3. Monte Carlo sampling and the HMC algorithm
6.5. Exercises
7. Current challenges in Monte Carlo Simulations
7.1. Sign and overlap problems
7.1.1. What is the issue?
7.1.2. Theories with sign problems
7.2. Introduction to overlap problems
7.2.1. Prototype issue
7.2.2. Exponential Error Suppression with LLR
7.2.3. Solution to overlap problems using LLR
7.2.4. The Ising model showcase
7.3. Estimating probability density functions
7.3.1. The worm algorithm
8. Data Analytics and Statistical Systems
8.1. Model regression – L2 norm
8.2. Gaussian Process
8.2.1. Stochastic generation of smooth curves
8.2.2. Choosing hyperparamters
8.2.3. Conditional Gaussian distributions
8.2.4. Regression and prediction
8.2.5. Guided learning
8.3. Machine learning with graphs
8.3.1. Graph theory basics for machine learning
8.3.2. Random fields on graphs and the Hammersley-Clifford theorem
8.3.3. The ising model as a Markov random field
8.4. Emulation of statistical systems with Machine Learning
8.4.1. Restricted Boltzmann machines
8.4.2. Training a Boltzmann machine
8.4.3. The 2d Ising model showcase
8.5. Categorisation in statistical physics: Naive Bayes
8.6. Machine learning classification of phase transitions
8.6.1. Principal Component Analysis – PCA
8.6.2. The Ising phase transition from unsupervised learning
Bibliography
Index