Contents
Part I Electrostatics
1 Dielectric Constant and Fluctuation Formulae for Molecular Dynamics 3
1.1 Electrostatics of Charges and Dipoles 3
1.2 Polarization P and Displacement Flux D 5
1.2.1 Bound Charges Induced by Polarization 6
1.2.2 Electric Field Epoi(r) of a Polarization Density P(r) 8
1.2.3 Singular Integral Expressions of Epo/ (r) Inside Dielectrics 10
1.3 Clausius-Mossotti and Onsager Formulae for Dielectric Constant 11
1.3.1 Clausius-Mossotti Formula for Non-polar Dielectrics 11
1.3.2 Onsager Dielectric Theory for Dipolar Liquids 13
1.4 Statistical Molecular Theory and Dielectric Fluctuation Formulae 18
1.4.1 Statistical Methods for Polarization Density Change AP 20
1.4.2 Classical Electrostatics for Polarization Density Change AP 22
1.4.3 Fluctuation Formulae for Dielectric Constant 24
1.5 Appendices 26
1.5.1 Appendix 1: Average Field of a Charge in a Dielectric Sphere 26
1.5.2 Appendix 2: Electric Field Due to a Uniformly Polarized Sphere 26
1.6 Summary 28
References 28
2 Poisson-Boltzmann Electrostatics and Analytical Approximations 31
2.1 Poisson-Boltzmann (PB) Model for Electrostatic Solvation 31
2.1.1 Debye—Hiickel Poisson-Boltzmann Theory 33
2.1.2 Helmholtz Double Layer and Ion-Size Effect 36
2.1.3 Electrostatic Solvation Energy 40
2.2 Generalized Bom (GB) Approximations of Solvation Energy 42
2.2.1 Still’s Generalized Bom Formulism 42
2.2.2 Integral Expression for Born Radii 43
2.2.3 FFT-Based Algorithm for the Bom Radii 45
2.3 Method of Images for Reaction Fields 51
2.3.1 Methods of Images for Simple Geometries 52
2.3.2 Image Methods for Dielectric Spheres 54
2.3.3 Image Methods for Dielectric Spheres in Ionic Solvent 61
2.3.4 Image Methods for Multi-layered Media 63
2.4 Summary 68
References 68
3 Numerical Methods for Poisson-Boltzmann Equations 73
3.1 Boundary Element Methods (BEMs) 73
3.1.1 Cauchy Principal Value (CPV) and Hadamard Finite Part (p.f.) 74
3.1.2 Surface Integral Equations for the PB Equation 79
3.1.3 Computations of CPV and Hadamard p.f.and Collocation BEMs 86
3.2 Finite Element Methods (FEMs) 99
3.3 Immersed Interface Methods (IIMs) 102
3.4 Summary 105
References 106
4 Random Walk Stochastic Methods for PDE Boundary Value Problems 109
4.1 Brownian Motion， It6 Integral， and It6，s Formula 109
4.1.1 Probability Space， Random Variable，and Conditional Expectation 109
4.1.2 Brownian Motion， Ito Integral of Stochastic Processes， and It6，s Formula 113
4.2 Feynman-Kac Formula for Mixed BVPs 117
4.2.1 Dynkin’s Formula and Feynman-Kac Formula for Dirichlet BVPs of Elliptic PDEs 117
4.2.2 Local Time L{t) and Feynman-Kac Formulae for Neumann and Robin BVPs of the Schrodinger Equation 121
4.3 Pardoux-Peng s Nonlinear Feynman-Kac Formula for Quasilinear Parabolic PDEs 123
4.4 Walk-On-Spheres (WOS) and Local Time L(t) for Reflecting Brownian Motion 125
4.4.1 WOS Method and Dirichlet BVP125
4.4.2 Reflecting Brownian Motion and Its Local Time L(t) 127
4.5 Electrical Impedance Tomography (EIT).129
4.5.1 Feynman-Kac Formula for Mixed BVPs 129
4.5.2 Local EIT Solution with the WOS Method 130
4.6 Boundary Integral Equation and WOS (BIE-WOS) Method for the Dirichlet BVP of the Laplace Equation 132
4.7 Iterative BIE-WOS Method for the Mixed BVP of the Laplace Equation 134
4.8 Summary 136
References 136
5 Stochastic Spectral Methods and Uncertainty Quantification of Microchip Interconnects 139
5.1 Deterministic Spectral Methods 139
5.1.1 Fourier， Chebyshev， Legendre， and Hermite Spectral Approximations 139
5.1.2 Spectral Methods for BVPs of Differential Equations 145
5.2 Stochastic Spectral Methods for Stochastic Differential and Integral Equations 146
5.2.1 Wiener Polynomial Chaos (PC) and Generalized PC (gPC) Expansions 147
5.2.2 Random Variable Dimension Reduction 149
5.2.3 Spectral Galerkin Methods 150
5.2.4 Spectral Collocation Methods and Sparse Grids 153
5.3 Uncertainty Quantification of Rough Microchip Interconnects 155
5.4 Summary 159
References 159
6 Deep Neural Network Learning for PDE Solutions 161
6.1 Deep Neural Network (DNN).161
6.2 Optimization with Stochastic Gradient Descent Method 163
6.3 DeepRitz Method for the Poisson-Boltzmann Equation of Biomolecular Solvation 164
6.4 Physics-Informed Neural Network (PINN) 167
6.5 Galerkin Weak Form Learning 167
6.6 Stochastic Differential Equation (SDE) Based DNN Methods 168
6.6.1 DeepBSDE for Quasilinear Parabolic Equations via a Terminal Condition Loss 169
6.6.2 Backward SDE Based Algorithms for Quasilinear Parabolic Equations with a Path wise Loss 169
6.6.3 Martingale Based Learning Algorithm for BVPs 171
6.6.4 Diffusion Monte Carlo DNN Eigensolver 174
6.7 DNNs for Oscillatory Problems and Their Spectral Bias 175
6.7.1 Phase-Shift DNN (PhaseDNN) 175
6.7.2 Multiscale D
Part I Electrostatics
1 Dielectric Constant and Fluctuation Formulae for Molecular Dynamics 3
1.1 Electrostatics of Charges and Dipoles 3
1.2 Polarization P and Displacement Flux D 5
1.2.1 Bound Charges Induced by Polarization 6
1.2.2 Electric Field Epoi(r) of a Polarization Density P(r) 8
1.2.3 Singular Integral Expressions of Epo/ (r) Inside Dielectrics 10
1.3 Clausius-Mossotti and Onsager Formulae for Dielectric Constant 11
1.3.1 Clausius-Mossotti Formula for Non-polar Dielectrics 11
1.3.2 Onsager Dielectric Theory for Dipolar Liquids 13
1.4 Statistical Molecular Theory and Dielectric Fluctuation Formulae 18
1.4.1 Statistical Methods for Polarization Density Change AP 20
1.4.2 Classical Electrostatics for Polarization Density Change AP 22
1.4.3 Fluctuation Formulae for Dielectric Constant 24
1.5 Appendices 26
1.5.1 Appendix 1: Average Field of a Charge in a Dielectric Sphere 26
1.5.2 Appendix 2: Electric Field Due to a Uniformly Polarized Sphere 26
1.6 Summary 28
References 28
2 Poisson-Boltzmann Electrostatics and Analytical Approximations 31
2.1 Poisson-Boltzmann (PB) Model for Electrostatic Solvation 31
2.1.1 Debye—Hiickel Poisson-Boltzmann Theory 33
2.1.2 Helmholtz Double Layer and Ion-Size Effect 36
2.1.3 Electrostatic Solvation Energy 40
2.2 Generalized Bom (GB) Approximations of Solvation Energy 42
2.2.1 Still’s Generalized Bom Formulism 42
2.2.2 Integral Expression for Born Radii 43
2.2.3 FFT-Based Algorithm for the Bom Radii 45
2.3 Method of Images for Reaction Fields 51
2.3.1 Methods of Images for Simple Geometries 52
2.3.2 Image Methods for Dielectric Spheres 54
2.3.3 Image Methods for Dielectric Spheres in Ionic Solvent 61
2.3.4 Image Methods for Multi-layered Media 63
2.4 Summary 68
References 68
3 Numerical Methods for Poisson-Boltzmann Equations 73
3.1 Boundary Element Methods (BEMs) 73
3.1.1 Cauchy Principal Value (CPV) and Hadamard Finite Part (p.f.) 74
3.1.2 Surface Integral Equations for the PB Equation 79
3.1.3 Computations of CPV and Hadamard p.f.and Collocation BEMs 86
3.2 Finite Element Methods (FEMs) 99
3.3 Immersed Interface Methods (IIMs) 102
3.4 Summary 105
References 106
4 Random Walk Stochastic Methods for PDE Boundary Value Problems 109
4.1 Brownian Motion， It6 Integral， and It6，s Formula 109
4.1.1 Probability Space， Random Variable，and Conditional Expectation 109
4.1.2 Brownian Motion， Ito Integral of Stochastic Processes， and It6，s Formula 113
4.2 Feynman-Kac Formula for Mixed BVPs 117
4.2.1 Dynkin’s Formula and Feynman-Kac Formula for Dirichlet BVPs of Elliptic PDEs 117
4.2.2 Local Time L{t) and Feynman-Kac Formulae for Neumann and Robin BVPs of the Schrodinger Equation 121
4.3 Pardoux-Peng s Nonlinear Feynman-Kac Formula for Quasilinear Parabolic PDEs 123
4.4 Walk-On-Spheres (WOS) and Local Time L(t) for Reflecting Brownian Motion 125
4.4.1 WOS Method and Dirichlet BVP125
4.4.2 Reflecting Brownian Motion and Its Local Time L(t) 127
4.5 Electrical Impedance Tomography (EIT).129
4.5.1 Feynman-Kac Formula for Mixed BVPs 129
4.5.2 Local EIT Solution with the WOS Method 130
4.6 Boundary Integral Equation and WOS (BIE-WOS) Method for the Dirichlet BVP of the Laplace Equation 132
4.7 Iterative BIE-WOS Method for the Mixed BVP of the Laplace Equation 134
4.8 Summary 136
References 136
5 Stochastic Spectral Methods and Uncertainty Quantification of Microchip Interconnects 139
5.1 Deterministic Spectral Methods 139
5.1.1 Fourier， Chebyshev， Legendre， and Hermite Spectral Approximations 139
5.1.2 Spectral Methods for BVPs of Differential Equations 145
5.2 Stochastic Spectral Methods for Stochastic Differential and Integral Equations 146
5.2.1 Wiener Polynomial Chaos (PC) and Generalized PC (gPC) Expansions 147
5.2.2 Random Variable Dimension Reduction 149
5.2.3 Spectral Galerkin Methods 150
5.2.4 Spectral Collocation Methods and Sparse Grids 153
5.3 Uncertainty Quantification of Rough Microchip Interconnects 155
5.4 Summary 159
References 159
6 Deep Neural Network Learning for PDE Solutions 161
6.1 Deep Neural Network (DNN).161
6.2 Optimization with Stochastic Gradient Descent Method 163
6.3 DeepRitz Method for the Poisson-Boltzmann Equation of Biomolecular Solvation 164
6.4 Physics-Informed Neural Network (PINN) 167
6.5 Galerkin Weak Form Learning 167
6.6 Stochastic Differential Equation (SDE) Based DNN Methods 168
6.6.1 DeepBSDE for Quasilinear Parabolic Equations via a Terminal Condition Loss 169
6.6.2 Backward SDE Based Algorithms for Quasilinear Parabolic Equations with a Path wise Loss 169
6.6.3 Martingale Based Learning Algorithm for BVPs 171
6.6.4 Diffusion Monte Carlo DNN Eigensolver 174
6.7 DNNs for Oscillatory Problems and Their Spectral Bias 175
6.7.1 Phase-Shift DNN (PhaseDNN) 175
6.7.2 Multiscale D