Particle system

Abstract: This paper introduces particle systems--a method for modeling fuzzy objects such as fire, clouds, and water. Particle systems model an object as a cloud of primitive particles that define its volume. Over a period of time, particles are generated into the system, move and change form within the system, and die from the system. The resulting model is able to represent motion, changes of form, and dynamics that are not possible with classical surface-based representations. The particles can easily be motion blurred, and therefore do not exhibit temporal aliasing or strobing. Stochastic processes are used to generate and control the many particles within a particle system. The application of particle systems to the wall of fire element from the Genesis Demo sequence of the film Star Trek H: The Wrath of Khan [10] is presented.

Abstract: The evaluation of Coulombic or gravitational interactions in large-scale ensembles of particles is an integral part of the numerical simulation of a large number of physical processes. Examples include celestial mechanics, plasma physics, the vortex method in fluid dynamics, molecular dynamics, and the solution of the Laplace equation via potential theory. In a typical application, a numerical model follows the trajectories of a number of particles moving in accordance with Newton's second law of motion in a field generated by the whole ensemble. In many situations, in order to be of physical interest, the simulation has to involve thousands of particles (or more), and the fields have to be evaluated for a large number of configurations. Unfortunately, an amount of work of the order $O(N\sp 2)$ has traditionally been required to evaluate all pairwise interactions in a system of N particles, unless some approximation or truncation method is used. As a result, large-scale simulations have been extremely expensive in some cases, and prohibitive in others. We present an algorithm for the rapid evaluation of the potential and force fields in large-scale systems of particles. In order to evaluate all pairwise Coulombic interactions of N particles to within round-off error, the algorithm requires an amount of work proportional to N, and this estimate does not depend on the statistics of the distribution. Both two and three dimensional versions of the algorithm have been constructed, and we will discuss their applications to several problems in physics, chemistry, biology, and numerical complex analysis.

Abstract: This paper presents new algorithms to trace objects represented by densities within a volume grid, e.g. clouds, fog, flames, dust, particle systems. We develop the light scattering equations, discuss previous methods of solution, and present a new approximate solution to the full three-dimensional radiative scattering problem suitable for use in computer graphics. Additionally we review dynamical models for clouds used to make an animated movie.

Abstract: 1 Introduction- 11 On the Origins of Feynman-Kac and Particle Models- 12 Notation and Conventions- 13 Feynman-Kac Path Models- 131 Path-Space and Marginal Models- 132 Nonlinear Equations- 14 Motivating Examples- 141 Engineering Science- 142 Bayesian Methodology- 143 Particle and Statistical Physics- 144 Biology- 145 Applied Probability and Statistics- 15 Interacting Particle Systems- 151 Discrete Time Models- 152 Continuous Time Models- 16 Sequential Monte Carlo Methodology- 17 Particle Interpretations- 18 A Contents Guide for the Reader- 2 Feynman-Kac Formulae- 21 Introduction- 22 An Introduction to Markov Chains- 221 Canonical Probability Spaces- 222 Path-Space Markov Models- 223 Stopped Markov chains- 224 Examples- 23 Description of the Models- 24 Structural Stability Properties- 241 Path Space and Marginal Models- 242 Change of Reference Probability Measures- 243 Updated and Prediction Flow Models- 25 Distribution Flows Models- 251 Killing Interpretation- 252 Interacting Process Interpretation- 253 McKean Models- 254 Kalman-Bucy filters- 26 Feynman-Kac Models in Random Media- 261 Quenched and Annealed Feynman-Kac Flows- 262 Feynman-Kac Models in Distribution Space- 27 Feynman-Kac Semigroups- 271 Prediction Semigroups- 272 Updated Semigroups- 3 Genealogical and Interacting Particle Models- 31 Introduction- 32 Interacting Particle Interpretations- 33 Particle models with Degenerate Potential- 34 Historical and Genealogical Tree Models- 341 Introduction- 342 A Rigorous Approach and Related Transport Problems- 343 Complete Genealogical Tree Models- 35 Particle Approximation Measures- 351 Some Convergence Results- 352 Regularity Conditions- 4 Stability of Feynman-Kac Semigroups- 41 Introduction- 42 Contraction Properties of Markov Kernels- 421 h-relative Entropy- 422 Lipschitz Contractions- 43 Contraction Properties of Feynman-Kac Semigroups- 431 Functional Entropy Inequalities- 432 Contraction Coefficients- 433 Strong Contraction Estimates- 434 Weak Regularity Properties- 44 Updated Feynman-Kac Models- 45 A Class of Stochastic Semigroups- 5 Invariant Measures and Related Topics- 51 Introduction- 52 Existence and Uniqueness- 53 Invariant Measures and Feynman-Kac Modeling- 54 Feynman-Kac and Metropolis-Hastings Models- 55 Feynman-Kac-Metropolis Models- 551 Introduction- 552 The Genealogical Metropolis Particle Model- 553 Path Space Models and Restricted Markov Chains- 554 Stability Properties- 6 Annealing Properties- 61 Introduction- 62 Feynman-Kac-Metropolis Models- 621 Description of the Model- 622 Regularity Properties- 623 Asymptotic Behavior- 63 Feynman-Kac Trapping Models- 631 Description of the Model- 632 Regularity Properties- 633 Asymptotic Behavior- 634 Large-Deviation Analysis- 635 Concentration Levels- 7 Asymptotic Behavior- 71 Introduction- 72 Some Preliminaries- 721 McKean Interpretations- 722 Vanishing Potentials- 73 Inequalities for Independent Random Variables- 731 Lp and Exponential Inequalities- 732 Empirical Processes- 74 Strong Law of Large Numbers- 741 Extinction Probabilities- 742 Convergence of Empirical Processes- 743 Time-Uniform Estimates- 8 Propagation of Chaos- 81 Introduction- 82 Some Preliminaries- 83 Outline of Results- 84 Weak Propagation of Chaos- 85 Relative Entropy Estimates- 86 A Combinatorial Transport Equation- 87 Asymptotic Properties of Boltzmann-Gibbs Distributions- 88 Feynman-Kac Semigroups- 881 Marginal Models- 882 Path-Space Models- 89 Total Variation Estimates- 9 Central Limit Theorems- 91 Introduction- 92 Some Preliminaries- 93 Some Local Fluctuation Results- 94 Particle Density Profiles- 941 Unnormalized Measures- 942 Normalized Measures- 943 Killing Interpretations and Related Comparisons- 95 A Berry-Esseen Type Theorem- 96 A Donsker Type Theorem- 97 Path-Space Models- 98 Covariance Functions- 10 Large-Deviation Principles- 101 Introduction- 102 Some Preliminary Results- 1021 Topological Properties- 1022 Idempotent Analysis- 1023 Some Regularity Properties- 103 Cramer's Method- 104 Laplace-Varadhan's Integral Techniques- 105 Dawson-Gartner Projective Limits Techniques- 106 Sanov's Theorem- 1061 Introduction- 1062 Topological Preliminaries- 1063 Sanov's Theorem in the r-Topology- 107 Path-Space and Interacting Particle Models- 1071 Proof of Theorem 1011- 1072 Sufficient Conditions- 108 Particle Density Profile Models- 1081 Introduction- 1082 Strong Large-Deviation Principles- 11 Feynman-Kac and Interacting Particle Recipes- 111 Introduction- 112 Interacting Metropolis Models- 1121 Introduction- 1122 Feynman-Kac-Metropolis and Particle Models- 1123 Interacting Metropolis and Gibbs Samplers- 113 An Overview of some General Principles- 114 Descendant and Ancestral Genealogies- 115 Conditional Explorations- 116 State-Space Enlargements and Path-Particle Models- 117 Conditional Excursion Particle Models- 118 Branching Selection Variants- 1181 Introduction- 1182 Description of the Models- 1183 Some Branching Selection Rules- 1184 Some L2-mean Error Estimates- 1185 Long Time Behavior- 1186 Conditional Branching Models- 119 Exercises- 12 Applications- 121 Introduction- 122 Random Excursion Models- 1221 Introduction- 1222 Dirichlet Problems with Boundary Conditions- 1223 Multilevel Feynman-Kac Formulae- 1224 Dirichlet Problems with Hard Boundary Conditions- 1225 Rare Event Analysis- 1226 Asymptotic Particle Analysis of Rare Events- 1227 Fluctuation Results and Some Comparisons- 1228 Exercises- 123 Change of Reference Measures- 1231 Introduction- 1232 Importance Sampling- 1233 Sequential Analysis of Probability Ratio Tests- 1234 A Multisplitting Particle Approach- 1235 Exercises- 124 Spectral Analysis of Feynman-Kac-Schrodinger Semigroups- 1241 Lyapunov Exponents and Spectral Radii- 1242 Feynman-Kac Asymptotic Models- 1243 Particle Lyapunov Exponents- 1244 Hard, Soft and Repulsive Obstacles- 1245 Related Spectral Quantities- 1246 Exercises- 125 Directed Polymers Simulation- 1251 Feynman-Kac and Boltzmann-Gibbs Models- 1252 Evolutionary Particle Simulation Methods- 1253 Repulsive Interaction and Self-Avoiding Markov Chains- 1254 Attractive Interaction and Reinforced Markov Chains- 1255 Particle Polymerization Techniques- 1256 Exercises- 126 Filtering/Smoothing and Path estimation- 1261 Introduction- 1262 Motivating Examples- 1263 Feynman-Kac Representations- 1264 Stability Properties of the Filtering Equations- 1265 Asymptotic Properties of Log-likelihood Functions- 1266 Particle Approximation Measures- 1267 A Partially Linear/Gaussian Filtering Model- 1268 Exercises- References