Provides a better understanding of random dynamical systems and stochastic differential equations
Offers fundamental ideas and tools for the modeling, analysis, and prediction of complex phenomena
Presents elementary tool for turbulent flow simulations
Mathematical analyses and computational predictions of the behavior of complex systems are needed to effectively deal with weather and climate predictions, for example, and the optimal design of technical processes. Given the random nature of such systems and the recognized relevance of randomness, the equations used to describe such systems usually need to involve stochastics.
The basic goal of this book is to introduce the mathematics and application of stochastic equations used for the modeling of complex systems. A first focus is on the introduction to different topics in mathematical analysis. A second focus is on the application of mathematical tools to the analysis of stochastic equations. A third focus is on the development and application of stochastic methods to simulate turbulent flows as seen in reality.
This book is primarily oriented towards mathematics and engineering PhD students, young and experienced researchers, and professionals working in the area of stochastic differential equations and their applications. It contributes to a growing understanding of concepts and terminology used by mathematicians, engineers, and physicists in this relatively young and quickly expanding field.
Related Subjects: Probability Theory and Stochastic Processes, Engineering Fluid Dynamics,
Computational Science and Engineering, Nonlinear Dynamics, Complexity
Author(s): Stefan Heinz, Hakima Bessaih
Series: Mathematical Engineering
Edition: 2015
Publisher: Springer
Year: 2015
Language: English
Pages: C, VII, 192
An Introduction to the Malliavin Calculus and Its Applications
1 Introduction
1.1 Finite Dimensional Case
1.2 Malliavin Calculus on the Wiener Space
1.3 Sobolev Spaces
1.4 The Divergence as a Stochastic Integral
2 Wiener Chaos and the Ornstein-Uhlenbeck Semigroup
2.1 Multiple Stochastic Integrals
2.2 The Ornstein-Uhlenbeck Semigroup
3 Clark Ocone's Formula
4 Application of Malliavin Calculus to Regularity and Estimations of Densities
4.1 First Density Formula
4.2 Second Density Formula
4.3 Existence and Smoothness of Densities
5 Malliavin Calculus and Normal Approximation
5.1 Stein's Method for Normal Approximation
5.2 Normal Approximation on a Fixed Wiener Chaos
5.3 Convergence of Densities
References
Fractional Brownian Motion and an Application to Fluids
1 Introduction
1.1 Fractional Brownian Motion
1.2 Properties of Fractional Brownian Motion
1.3 An Application of fBm to Fluids
2 Some Preliminaries
2.1 The Integrodifferential Equation
2.2 Assumptions
2.3 Functional Setting
3 Some a Priori Estimates
3.1 Fractional Integrals and Derivatives
3.2 A Priori Estimates
4 Stochastic Integrals with Respect to a fBm
5 Main Results with Proofs
5.1 Main Result
5.2 Fixed Point Argument
5.3 Proof of Theorem 1
References
An Introduction to Large Deviations and Equilibrium Statistical Mechanics for Turbulent Flows
1 Introduction
2 Models of Turbulent Flows
2.1 3D and 2D Hydrodynamics
2.2 Global Invariants
2.3 Geophysical Flows
2.4 Discretized Form for 2D Euler Flows and Analogies with Toy Models of Magnetic Systems
3 Mean-Field Theory for 2D Flows
3.1 Microcanonical Measure and Large Deviations for the Energy and Vorticity Distribution
3.2 Large Deviations for the Macrostates
3.3 Thermodynamic Limit and Mean-Field Equation
3.4 Non-equivalence of Ensembles
3.5 Large Deviations for the Coarse-Grained Vorticity Field
3.6 The Energy-Enstrophy Measure
4 Conclusion
References
Recent Developments on the Micropolar and Magneto-Micropolar Fluid Systems: Deterministic and Stochastic Perspectives
1 Introduction
2 Deterministic Case
2.1 Global Regularity of the 2D MMPF System with Zero Angular Viscosity
2.2 Regularity Criterion of the 3D MMPF System in Terms of Two Velocity Components
3 Stochastic Case
3.1 Existence of Weak Martingale Solution for the 3D MPF and the MMPF Systems with Non-Lipschitz Multiplicative Noise
3.2 Unique Strong Solution for the 2D MPF and the MMPF Systems with Lipschitz Multiplicative Noise
4 Conclusion
References
Pathwise Sensitivity Analysis in Transient Regimes
1 Introduction
2 Time-Dependent Sensitivity Analysis
2.1 Decomposition of the Pathwise Relative Entropy
2.2 Sensitivity Analysis and Fisher Information Matrix
3 Discrete-Time Markov Chains
4 Continuous-Time Markov Chains
5 Stochastic Differential Equations---Markov Processes
6 Demonstration Example
6.1 Continuous Time Markov Chains: An EGFR Model
7 Conclusions
References
The Langevin Approach: A Simple Stochastic Method for Complex Phenomena
1 Introduction
2 The Langevin Approach
2.1 Processes in Scale
2.2 Necessary Conditions: Stationarity and the Markov Property
2.3 The Fokker-Planck Equation for Increments
2.4 Langevin Processes in Scale
3 Applying the Langevin Approach to Turbulence
3.1 The Langevin Approach in Laboratory Turbulence
3.2 The Langevin Approach in Simulated Turbulence
3.3 Comparative Analysis
4 Discussion and Conclusions
References
Monte Carlo Simulations of Turbulent Non-premixed Combustion using a Velocity Conditioned Mixing Model
1 Introduction
2 Stochastic Modeling of Turbulent Non-premixed Combustion
2.1 Deterministic Conservation Equations
2.2 Simplified Chemical Description
2.3 Statistical Description
2.4 Monte Carlo Method
2.5 Hybrid Solution Algorithm
3 Micro Mixing Models
3.1 Description of IEM and IECM Models
3.2 The Correct Local Scalar Isotropy
3.3 Implementation of the IECM Micro-Mixing Model
4 Test Case and Numerical Details
4.1 Delft III Flame
4.2 Numerical Setup
5 Results and Discussion
5.1 Velocity Statistics
5.2 Results for the Mixture Fraction Fields
5.3 Results for the Temperature
5.4 Discussion on Parameters of the IECM Algorithm
6 Conclusions
References
Massively Parallel FDF Simulation of Turbulent Reacting Flows
1 Introduction
2 Filtered Density Function
3 Irregularly Portioned FDF Simulator
3.1 Static Block Decomposition
3.2 Irregularly Portioned Lagrangian Monte Carlo (IPLMC)
3.3 Irregularly Portioned Lagrangian Monte Carlo Finite Difference (IPLMCFD)
4 Concluding Remarks
References
