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Preface; Contents; Part I Modern Challenges and Interdisciplinary Interactions via Mathematical, Statistical, and Computational Models; Modern Challenges and Interdisciplinary Interactionsvia Mathematical, Statistical, and Computational Models; 1 The Role of Mathematical and Statistical Models; 2 Application Areas and State-of-the-Art Developments; 2.1 Large Deviation Theory and Random Perturbations of Dynamical Systems with Applications; 2.2 Nonlinear waves, Hyperbolic Problems, and their Applications; 2.3 Group-Theoretical Approaches to Conservation Lawsand Their Applications
2.4 Materials Science, Engineering, and New Technologies2.5 Finance and Systemic Risk; 2.6 Life and Environmental Sciences; 2.7 Number Theory and Algebraic Geometry in Cryptography and Other Applications; 2.8 Sustainability and Cooperation; 3 Conclusions; References; Part II Large Deviation Theory and Random Perturbations of Dynamical Systems with Applications; Long Term Effects of Small Random Perturbations on Dynamical Systems: Theoretical and Computational Tools; 1 Introduction; 2 Freidlin-Wentzell Large Deviation Theory (LDT); 2.1 Some Key Concepts in LDT; 2.2 Geometric Action Functional
3 Numerical Minimization of the Geometric Action3.1 Geometric Minimum Action Method; 3.2 A Simplified gMAM; 3.3 Connection to gMAM; 3.4 Simplifications for SDEs with Additive Noise; 3.5 Simplifications for General SDEs (Multiplicative Noise); 3.6 Comments on Improving the Numerical Efficiency; 3.7 SPDEs with Additive Noise; 4 Illustrative Applications; 4.1 Maier-Stein Model; 4.2 Allen-Cahn/Cahn-Hilliard System; 4.2.1 Reduced Allen-Cahn/Cahn-Hilliard Model; 4.2.2 Full Allen-Cahn/Cahn-Hilliard Model; 4.3 Burgers-Huxley Model; 4.4 Noise-Induced Transitions Between Climate Regimes
4.4.1 Metastable Climate Regimes in Egger's Model4.4.2 Metastable Climate Regimes in the Charney-DeVore Model; 4.5 Generalized Voter/Ising Model; 4.6 Bi-Stable Reaction-Diffusion Model; 4.7 Slow-Fast Systems; 5 Concluding Remarks; References; Part III Nonlinear Waves, Hyperbolic Problems, and their Applications; Long Time Dynamics and Coherent States in NonlinearWave Equations; 1 Introduction; 2 General Hamiltonian Formulation; 3 Coherent States; 3.1 Variational Methods: Existence and Stability of Ground States; 3.2 Bifurcation Methods; 3.3 Orbital Stability
4 Asymptotic Stability of Coherent States5 Conclusions; References; About Non Linear Stabilization for Scalar Hyperbolic Problems; 1 Introduction; 2 Formulation of Residual Distribution Schemes; 2.1 Definition, Connection to Finite Element Methods; 2.2 Structure Conditions; 2.2.1 Conservation; 2.2.2 Accuracy; 3 Construction of Monotonicity Preserving Arbitrary Accurate Schemes; 3.1 A Preliminary Remark; 3.2 Explicit Construction; 3.3 Filtering; 3.3.1 Streamline Filtering; 3.3.2 Jump Filtering; 4 Numerical Examples; 5 Conclusions; References
2.4 Materials Science, Engineering, and New Technologies2.5 Finance and Systemic Risk; 2.6 Life and Environmental Sciences; 2.7 Number Theory and Algebraic Geometry in Cryptography and Other Applications; 2.8 Sustainability and Cooperation; 3 Conclusions; References; Part II Large Deviation Theory and Random Perturbations of Dynamical Systems with Applications; Long Term Effects of Small Random Perturbations on Dynamical Systems: Theoretical and Computational Tools; 1 Introduction; 2 Freidlin-Wentzell Large Deviation Theory (LDT); 2.1 Some Key Concepts in LDT; 2.2 Geometric Action Functional
3 Numerical Minimization of the Geometric Action3.1 Geometric Minimum Action Method; 3.2 A Simplified gMAM; 3.3 Connection to gMAM; 3.4 Simplifications for SDEs with Additive Noise; 3.5 Simplifications for General SDEs (Multiplicative Noise); 3.6 Comments on Improving the Numerical Efficiency; 3.7 SPDEs with Additive Noise; 4 Illustrative Applications; 4.1 Maier-Stein Model; 4.2 Allen-Cahn/Cahn-Hilliard System; 4.2.1 Reduced Allen-Cahn/Cahn-Hilliard Model; 4.2.2 Full Allen-Cahn/Cahn-Hilliard Model; 4.3 Burgers-Huxley Model; 4.4 Noise-Induced Transitions Between Climate Regimes
4.4.1 Metastable Climate Regimes in Egger's Model4.4.2 Metastable Climate Regimes in the Charney-DeVore Model; 4.5 Generalized Voter/Ising Model; 4.6 Bi-Stable Reaction-Diffusion Model; 4.7 Slow-Fast Systems; 5 Concluding Remarks; References; Part III Nonlinear Waves, Hyperbolic Problems, and their Applications; Long Time Dynamics and Coherent States in NonlinearWave Equations; 1 Introduction; 2 General Hamiltonian Formulation; 3 Coherent States; 3.1 Variational Methods: Existence and Stability of Ground States; 3.2 Bifurcation Methods; 3.3 Orbital Stability
4 Asymptotic Stability of Coherent States5 Conclusions; References; About Non Linear Stabilization for Scalar Hyperbolic Problems; 1 Introduction; 2 Formulation of Residual Distribution Schemes; 2.1 Definition, Connection to Finite Element Methods; 2.2 Structure Conditions; 2.2.1 Conservation; 2.2.2 Accuracy; 3 Construction of Monotonicity Preserving Arbitrary Accurate Schemes; 3.1 A Preliminary Remark; 3.2 Explicit Construction; 3.3 Filtering; 3.3.1 Streamline Filtering; 3.3.2 Jump Filtering; 4 Numerical Examples; 5 Conclusions; References