Small curvature particle flow for nonlinear filters

We derive five new particle flow algorithms for nonlinear filters based on the small curvature approximation inspired by fluid dynamics. We find it extremely interesting that this physically motivated approximation generalizes two of our previous exact flow algorithms, namely incompressible flow and Gaussian flow. We derive a new algorithm to compute the inverse of the sum of two linear differential operators using a second homotopy, similar to Feynman's perturbation theory for quantum electrodynamics as well as Gromov's h-principle.

[1]  Fred Daum,et al.  Hollywood log-homotopy: movies of particle flow for nonlinear filters , 2011, Defense + Commercial Sensing.

[2]  Lingji Chen,et al.  A study of nonlinear filters with particle flow induced by log-homotopy , 2010, Defense + Commercial Sensing.

[3]  Fred Daum,et al.  Zero curvature particle flow for nonlinear filters , 2012, Defense, Security, and Sensing.

[4]  Fred Daum,et al.  Numerical experiments for Coulomb's law particle flow for nonlinear filters , 2011, Optical Engineering + Applications.

[5]  R. Feynman.  Space - time approach to quantum electrodynamics , 1949 .

[6]  M. Gromov,et al.  Partial Differential Relations , 1986 .

[7]  Sorin Mardare.  On systems of first order linear partial differential equations with $L^p$ coefficients , 2007, Advances in Differential Equations.

[8]  F. Tröltzsch.  Optimal Control of Partial Differential Equations: Theory, Methods and Applications , 2010 .

[9]  Robert D. Russell,et al.  Adaptivity with moving grids , 2009, Acta Numerica.

[10]  Fred Daum,et al.  A fresh perspective on research for nonlinear filters , 2010, Defense + Commercial Sensing.

[11]  C. Villani.  Topics in Optimal Transportation , 2003 .

[12]  Christian Musso,et al.  Improving Regularised Particle Filters , 2001, Sequential Monte Carlo Methods in Practice.

[13]  Tao Ding,et al.  Implementation of the Daum-Huang exact-flow particle filter , 2012, 2012 IEEE Statistical Signal Processing Workshop (SSP).

[14]  J. Lions.  Optimal Control of Systems Governed by Partial Differential Equations , 1971 .

[15]  Robert D. Russell,et al.  Optimal mass transport for higher dimensional adaptive grid generation , 2011, J. Comput. Phys..

[16]  A. Shnirelman,et al.  Evolution of singularities, generalized Liapunov function and generalized integral for an ideal incompressible fluid , 1997 .

[17]  Emmanuel Villermaux,et al.  The diffusive strip method for scalar mixing in two dimensions , 2010, Journal of Fluid Mechanics.

[18]  Allen R. Tannenbaum,et al.  An Efficient Numerical Method for the Solution of the L2 Optimal Mass Transfer Problem , 2010, SIAM J. Sci. Comput..

[19]  Fred Daum,et al.  Numerical experiments for nonlinear filters with exact particle flow induced by log-homotopy , 2010, Defense + Commercial Sensing.

[20]  C. Villani.  The founding fathers of optimal transport , 2009 .

[21]  Frederick Daum.  A new nonlinear filtering formula for discrete time measurements , 1985, 1985 24th IEEE Conference on Decision and Control.

[22]  Fred Daum,et al.  Coulomb's law particle flow for nonlinear filters , 2011, Optical Engineering + Applications.

[23]  Mark R. Morelande,et al.  Optimal parameterization of posterior densities using homotopy , 2011, 14th International Conference on Information Fusion.

[24]  Branko Ristic,et al.  Beyond the Kalman Filter: Particle Filters for Tracking Applications , 2004 .

[25]  F. Daum.  Nonlinear filters: beyond the Kalman filter , 2005, IEEE Aerospace and Electronic Systems Magazine.

[26]  A. Doucet,et al.  A Tutorial on Particle Filtering and Smoothing: Fifteen years later , 2008 .

[27]  Fred Daum.  Dimensional interpolation for nonlinear filters , 2005, SPIE Optics + Photonics.

[28]  Martin Raussen,et al.  Interview with John Milnor , 2012 .

[29]  Frederick Daum.  A new nonlinear filtering formula non-Gaussian discrete time measurements , 1986, 1986 25th IEEE Conference on Decision and Control.

[30]  Fred Daum,et al.  Particle degeneracy: root cause and solution , 2011, Defense + Commercial Sensing.

[31]  Peter K. Jimack,et al.  Moving mesh methods for solving parabolic partial differential equations , 2011 .

[32]  Fred Daum,et al.  Seventeen dubious methods to approximate the gradient for nonlinear filters with particle flow , 2009, Optical Engineering + Applications.

[33]  Fred Daum,et al.  Exact particle flow for nonlinear filters , 2010, Defense + Commercial Sensing.

[34]  Fred Daum,et al.  Exact particle flow for nonlinear filters: Seventeen dubious solutions to a first order linear underdetermined PDE , 2010, 2010 Conference Record of the Forty Fourth Asilomar Conference on Signals, Systems and Computers.

[35]  Uwe D. Hanebeck,et al.  Progressive Bayes: a new framework for nonlinear state estimation , 2003, SPIE Defense + Commercial Sensing.

[36]  M. Pitt,et al.  Filtering via Simulation: Auxiliary Particle Filters , 1999 .

[37]  Michio Kaku,et al.  Quantum Field Theory: A Modern Introduction , 1993 .

[38]  H. Rue,et al.  Approximate Bayesian inference for latent Gaussian models by using integrated nested Laplace approximations , 2009 .

[39]  Giuseppe Buttazzo,et al.  Calculus of Variations and Partial Differential Equations , 1988 .

[40]  R. Chartrand,et al.  A Gradient Descent Solution to the Monge-Kantorovich Problem , 2009 .

[41]  Tim B. Swartz,et al.  Approximating Integrals Via Monte Carlo and Deterministic Methods , 2000 .

[42]  S. Coleman,et al.  Aspects of Symmetry , 1985 .

[43]  A. Beskos,et al.  Error Bounds and Normalizing Constants for Sequential Monte Carlo in High Dimensions , 2011, 1112.1544.

[44]  Fred Daum,et al.  Nonlinear filters with log-homotopy , 2007, SPIE Optical Engineering + Applications.

[45]  Gian Luca Delzanno,et al.  The fluid dynamic approach to equidistribution methods for grid generation and adaptation , 2009 .

[46]  Simon J. Godsill,et al.  Improvement Strategies for Monte Carlo Particle Filters , 2001, Sequential Monte Carlo Methods in Practice.

[47]  Xiao-Li Meng,et al.  Simulating Normalizing Constants: From Importance Sampling to Bridge Sampling to Path Sampling , 1998 .

[48]  M. Girolami,et al.  Riemann manifold Langevin and Hamiltonian Monte Carlo methods , 2011, Journal of the Royal Statistical Society: Series B (Statistical Methodology).

[49]  Fred Daum,et al.  Generalized particle flow for nonlinear filters , 2010, Defense + Commercial Sensing.

[50]  Richard Bellman,et al.  Green’s Functions for Partial Differential Equations , 1985 .

[51]  V. Arnold,et al.  Topological methods in hydrodynamics , 1998 .

[52]  R. Courant,et al.  Methods of Mathematical Physics , 1962 .