Friendly rebuttal to Chen and Mehra: incompressible particle flow for nonlinear filters

We respond to the very thorough analysis of singularities in the incompressible flow for one dimensional nonlinear filters by Chen and Mehra. We emphasize that the singularities occur at a few points in d-dimensional state space, and thus the chance of hitting a singularity is very small for dimensions higher than one. Furthermore, in the unlikely event of hitting a singularity, there is ample room to flow around it in spaces of dimension higher than one. Moreover, the deep mathematical theory of incompressible particle flow that was developed recently by Shnirelman can be used to provide insight into why our particle flow algorithms work so well.

[1]  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.

[2]  Alexander Shnirelman.  Curvature and instability of flows of ideal incompressible , 2006 .

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[29]  Alexander Shnirelman,et al.  Microglobal Analysis of the Euler Equations , 2005 .

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

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

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

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

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

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

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

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

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

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

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