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21
Sequential Monte Carlo methods for highdimensional inverse problems: A case study for the NavierStokes equations
, 2013
"... ar ..."
A random map implementation of implicit filters
"... Implicit particle filters for data assimilation generate highprobability samples by representing each particle location as a separate function of a common reference variable. This representation requires that a certain underdetermined equation be solved for each particle and at each time an observa ..."
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Cited by 23 (13 self)
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Implicit particle filters for data assimilation generate highprobability samples by representing each particle location as a separate function of a common reference variable. This representation requires that a certain underdetermined equation be solved for each particle and at each time an observation becomes available. We present a new implementation of implicit filters in which we find the solution of the equation via a random map. As examples, we assimilate data for a stochastically driven Lorenz system with sparse observations and for a stochastic KuramotoSivashinski equation with observations that are sparse in both space and time.
Bayesian inference with optimal maps
 Journal of Computational Physics
"... We present a new approach to Bayesian inference that entirely avoids Markov chain simulation, by constructing a map that pushes forward the prior measure to the posterior measure. Existence and uniqueness of a suitable measurepreserving map is established by formulating the problem in the context o ..."
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We present a new approach to Bayesian inference that entirely avoids Markov chain simulation, by constructing a map that pushes forward the prior measure to the posterior measure. Existence and uniqueness of a suitable measurepreserving map is established by formulating the problem in the context of optimal transport theory. We discuss various means of explicitly parameterizing the map and computing it efficiently through solution of an optimization problem, exploiting gradient information from the forward model when possible. The resulting algorithm overcomes many of the computational bottlenecks associated with Markov chain Monte Carlo. Advantages of a mapbased representation of the posterior include analytical expressions for posterior moments and the ability to generate arbitrary numbers of independent posterior samples without additional likelihood evaluations or forward solves. The optimization approach also provides clear convergence criteria for posterior approximation and facilitates model selection through automatic evaluation of the marginal likelihood. We demonstrate the accuracy and efficiency of the approach on nonlinear inverse problems of varying dimension, involving the inference of parameters appearing in ordinary and partial differential equations.
Implicit particle methods and their connection with variational data assimilation
 Monthly Weather Review
"... The implicit particle filter is a sequential Monte Carlo method for data assimilation that guides the particles to the highprobability regions via a sequence of steps that includes minimizations. We present a new and more general derivation of this approach and extend the method to particle smoothi ..."
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Cited by 5 (5 self)
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The implicit particle filter is a sequential Monte Carlo method for data assimilation that guides the particles to the highprobability regions via a sequence of steps that includes minimizations. We present a new and more general derivation of this approach and extend the method to particle smoothing as well as to data assimilation for perfect models. We show that the minimizations required by implicit particle methods are similar to the ones one encounters in variational data assimilation and explore the connection of implicit particle methods with variational data assimilation. In particular, we argue that existing variational codes can be converted into implicit particle methods at a low cost, often yielding better estimates, that are also equipped with quantitative measures of the uncertainty. A detailed example is presented. 1
Conditions for successful data assimilation
"... We show that numerical data assimilation is feasible in principle for an idealized model only if an effective dimension of the noise is bounded; this effective dimension is bounded when the noises in model and data satisfy a certain natural balance condition. If this balance condition is satisfied, ..."
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We show that numerical data assimilation is feasible in principle for an idealized model only if an effective dimension of the noise is bounded; this effective dimension is bounded when the noises in model and data satisfy a certain natural balance condition. If this balance condition is satisfied, data assimilation is feasible even if the number of variables in the problem is huge. We then analyze several data assimilation algorithms, including particle filters and variational data assimilation. We show that a particle filter can successfully solve most of the data assimilation problems which are feasible in principle, provided the particle filter is well designed. We also compare the conditions under which variational data assimilation can be successful with the conditions for successful particle filtering. We draw conclusions from our analysis and discuss its limitations. 1
A Survey of Implicit Particle Filters for Data Assimilation
"... Abstract The implicit particle filter is a sequential Monte Carlo method for data assimilation. The idea is to focus the particles onto the high probability regions of the target probability density function (pdf) so that the number of particles required for a good approximation of this pdf remains ..."
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Abstract The implicit particle filter is a sequential Monte Carlo method for data assimilation. The idea is to focus the particles onto the high probability regions of the target probability density function (pdf) so that the number of particles required for a good approximation of this pdf remains manageable, even if the dimension of the state space is large. We explain how this idea is implemented, discuss special cases of practical importance, and work out the relations of the implicit particle filter with other data assimilation methods. We illustrate the theory with four examples. 1
Path Integral Formulation of Stochastic Optimal Control with Generalized Costs?
"... Abstract: Path integral control solves a class of stochastic optimal control problems with a Monte Carlo (MC) method for an associated HamiltonJacobiBellman (HJB) equation. The MC approach avoids the need for a global grid of the domain of the HJB equation and, therefore, path integral control is ..."
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Abstract: Path integral control solves a class of stochastic optimal control problems with a Monte Carlo (MC) method for an associated HamiltonJacobiBellman (HJB) equation. The MC approach avoids the need for a global grid of the domain of the HJB equation and, therefore, path integral control is in principle applicable to control problems of moderate to large dimension. The class of problems path integral control can solve, however, is defined by requirements on the cost function, the noise covariance matrix and the control input matrix. We relax the requirements on the cost function by introducing a new state that represents an augmented running cost. In our new formulation the cost function can contain stochastic integral terms and linear control costs, which are important in applications in engineering, economics and finance. We find an efficient numerical implementation of our gridfree MC approach and demonstrate its performance and usefulness in examples from hierarchical electric load management. The dimension of one of our examples is large enough to make classical gridbased HJB solvers impractical. 1.
Implicit sampling for path integral control, Monte Carlo localization and online SLAM. in review
, 2013
"... The applicability and usefulness of implicit sampling in stochastic optimal control, stochastic localization, and simultaneous localization and mapping (SLAM), is explored; implicit sampling is a recentlydeveloped variationallyenhanced sampling method. The theory is illustrated with examples, and ..."
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The applicability and usefulness of implicit sampling in stochastic optimal control, stochastic localization, and simultaneous localization and mapping (SLAM), is explored; implicit sampling is a recentlydeveloped variationallyenhanced sampling method. The theory is illustrated with examples, and it is found that implicit sampling is significantly more efficient than current Monte Carlo methods in test problems for all three applications. 1
Smallnoise analysis and symmetrization of implicit Monte Carlo samplers
, 2014
"... Implicit samplers are algorithms for producing independent, weighted samples from multivariate probability distributions. These are often applied in Bayesian data assimilation algorithms. We use Laplace asymptotic expansions to analyze two implicit samplers in the small noise regime. Our analysis s ..."
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Implicit samplers are algorithms for producing independent, weighted samples from multivariate probability distributions. These are often applied in Bayesian data assimilation algorithms. We use Laplace asymptotic expansions to analyze two implicit samplers in the small noise regime. Our analysis suggests a symmetrization of the algorithms that leads to improved (implicit) sampling schemes at a relatively small additional cost. Computational experiments confirm the theory and show that symmetrization is effective for small noise sampling problems. 1