Results 1 
8 of
8
Sequential Monte Carlo methods for highdimensional inverse problems: A case study for the NavierStokes equations
, 2013
"... ar ..."
IMPLEMENTATION OF THE DAUMHUANG EXACTFLOW PARTICLE FILTER
"... Several versions of the DaumHuang (DH) filter have been introduced recently to address the task of discretetime nonlinear filtering. The filters propagate a particle set over time to track the system state, but, in contrast to conventional particle filters, there is no proposal density or importa ..."
Abstract

Cited by 2 (0 self)
 Add to MetaCart
(Show Context)
Several versions of the DaumHuang (DH) filter have been introduced recently to address the task of discretetime nonlinear filtering. The filters propagate a particle set over time to track the system state, but, in contrast to conventional particle filters, there is no proposal density or importance sampling involved. Particles are smoothly migrated using a particle flow derived from a loghomotopy relating the prior and the posterior. Impressive performance has been demonstrated for a wide range of systems, but the implemented algorithms rely on an extended/unscented Kalman filter (EKF/UKF) that is executed in parallel. We illustrate through simulation that the performance of the exact flow DH filter can be compromised when the UKF and EKF fail. By introducing simple but important modifications to the exact flow DH filter implementation, the performance can be improved dramatically. Index Terms — DaumHuang filter, loghomotopy, particle filter, particle flow, exact flow
GENERALISED PARTICLE FILTERS WITH GAUSSIAN MEASURES
"... The stochastic filtering problem deals with the estimation of the posterior distribution of the current state of a signal process X = {Xt}t≥0 given the information supplied by an associate process Y = {Yt}t≥0. The scope and range of its applications includes the control of engineering systems, gl ..."
Abstract

Cited by 2 (0 self)
 Add to MetaCart
The stochastic filtering problem deals with the estimation of the posterior distribution of the current state of a signal process X = {Xt}t≥0 given the information supplied by an associate process Y = {Yt}t≥0. The scope and range of its applications includes the control of engineering systems, global data assimilation in meteorology, volatility estimation in financial markets, computer vision and vehicle tracking. A massive scientific and computational effort is dedicated to the development of viable tools for approximating the solution of the filtering problem. Classical PDE methods can be successful, particularly if the state space has low dimensions. In higher dimensions, a class of numerical methods called particle filters have proved the most successful methods todate. These methods produce an approximations of the posterior distribution by using the empirical distribution of a cloud of particles that explore the signal’s state space. We discuss here a more general class of numerical methods which involve generalised particles, that is, particles that evolve through larger spaces. Such generalised particles include Gaussian measures, wavelets, and finite elements in addition to the classical particle methods. We will construct the approximating particle system under the Gaussian measure framework and prove the corresponding convergence result. 1. THE FILTERING FRAMEWORK Let (Ω,F,P) be a probability space on which we have defined a process X = {Xt}t≥0 called the signal and an associate process Y = {Yt}t≥0 called the observation. The process X is the solution of a ddimensional stochastic differential equation driven by a pdimensional Brownian motion V, that is: Xt = X0+ ∫ t
Online Static Parameter Estimation for ABC Approximation of Hidden Markov Models
, 2012
"... In this article we focus on Maximum Likelihood estimation (MLE) for the static parameters of hidden Markov models (HMMs). We will consider the case where one cannot or does not want to compute the conditional likelihood density of the observation given the hidden state because of increased computati ..."
Abstract

Cited by 2 (1 self)
 Add to MetaCart
(Show Context)
In this article we focus on Maximum Likelihood estimation (MLE) for the static parameters of hidden Markov models (HMMs). We will consider the case where one cannot or does not want to compute the conditional likelihood density of the observation given the hidden state because of increased computational complexity or analytical intractability. Instead we will assume that one may obtain samples from this conditional likelihood and hence use approximate Bayesian computation (ABC) approximations of the original HMM. ABC approximations are biased, but the bias can be controlled to arbitrary precision via a parameter > 0; the bias typically goes to zero as ↘ 0. We first establish that the bias in the loglikelihood and gradient of the loglikelihood of the ABC approximation, for a fixed batch of data, is no worse than O(n), n being the number of data; hence, for computational reasons, one might expect reasonable parameter estimates using such an ABC approximation. Turning to the computational problem of estimating θ, we propose, using the ABCsequential Monte Carlo (SMC) algorithm in [18], an approach based upon simultaneous perturbation stochastic approximation (SPSA). Our method is investigated on two numerical examples.
A hybrid particleensemble Kalman filter for high dimensional Lagrangian data assimilation
"... Abstract. We apply the recently proposed hybrid particleensemble Kalman filter to assimilate Lagrangian data into a nonlinear, highdimensional quasigeostrophic ocean model. Effectively the hybrid filter applies a particle filter to the highly nonlinear, lowdimensional Lagrangian instrument va ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
(Show Context)
Abstract. We apply the recently proposed hybrid particleensemble Kalman filter to assimilate Lagrangian data into a nonlinear, highdimensional quasigeostrophic ocean model. Effectively the hybrid filter applies a particle filter to the highly nonlinear, lowdimensional Lagrangian instrument variables while applying an ensemble Kalman type update to the highdimensional Eulerian flow field. We present some initial results from this hybrid filter and compare those to results from a standard ensemble Kalman filter and an ensemble run without assimilation. 1
SIAM/ASA J. UNCERTAINTY QUANTIFICATION c ⃝ xxxx Society for Industrial and Applied Mathematics Vol. xx, pp. x x{x Sequential Monte Carlo Methods for HighDimensional Inverse Problems: A
"... case study for the NavierStokes equations ..."
(Show Context)
Recommended for Acceptance
, 2014
"... The goal of filtering theory is to compute the filter distribution, that is, the conditional distribution of a stochastic model given observed data. While exact computations are rarely possible, sequential Monte Carlo algorithms known as particle filters have been successfully applied to approximate ..."
Abstract
 Add to MetaCart
(Show Context)
The goal of filtering theory is to compute the filter distribution, that is, the conditional distribution of a stochastic model given observed data. While exact computations are rarely possible, sequential Monte Carlo algorithms known as particle filters have been successfully applied to approximate the filter distribution, providing estimates whose error is uniform in time. However, the number of Monte Carlo samples needed to approximate the filter distribution is typically exponential in the number of degrees of freedom of the model. This issue, known as curse of dimensionality, has rendered sequential Monte Carlo algorithms largely useless in highdimensional applications such as multitarget tracking, weather prediction, and oceanography. While over the past twenty years many heuristics have been suggested to run particle filters in high dimension, no principled approach has ever been proposed to address the core of the problem. In this thesis we develop a novel framework to investigate highdimensional filtering models and to design algorithms that can avoid the curse of dimensionality. Using