Bayesian Field Theory Applied to Scattered Data Interpolation and Inverse Problems

Farmer, Chris L.

doi:10.1007/978-3-540-46551-5_12

Cited by 15 publications

(23 citation statements)

References 18 publications

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“…This leads to a Bayesian perspective on data assimilation in which observations are used to convert a prior distribution on velocity fields into a posterior. See [10,11], for example, in the context of the atmospheric sciences and, in the context of applications to oil reservoir simulation, see [1].…”

Section: Statistical Perspectivementioning

confidence: 99%

“…It is essential to do so in any fields that are data rich and for which well-founded predictive mathematical models exist. Geophysical applications [1], the atmospheric sciences [2] and oceanography [3] provide important application areas of this type. Here, we adopt a Bayesian view of data assimilation in which prior information (background velocity field and model error) is combined with data to provide a posterior distribution [4].…”

Section: Introductionmentioning

confidence: 99%

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Data assimilation: Mathematical and statistical perspectives

Apte

Jones

Stuart

et al. 2007

Numerical Methods in Fluids

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SUMMARYThe bulk of this paper contains a concise mathematical overview of the subject of data assimilation, highlighting three primary ideas: (i) the standard optimization approaches of 3DVAR, 4DVAR and weak constraint 4DVAR are described and their interrelations explained; (ii) statistical analogues of these approaches are then introduced, leading to filtering (generalizing 3DVAR) and a form of smoothing (generalizing 4DVAR and weak constraint 4DVAR) and the optimization methods are shown to be maximum a posteriori estimators for the probability distributions implied by these statistical approaches; and (iii) by taking a general dynamical systems perspective on the subject it is shown that the incorporation of Lagrangian data can be handled by a straightforward extension of the preceding concepts.We argue that the smoothing approach to data assimilation, based on statistical analogues of 4DVAR and weak constraint 4DVAR, provides the optimal solution to the assimilation of space-time distributed data into a model. The optimal solution obtained is a probability distribution on the relevant class of functions (initial conditions or time-dependent solutions). The approach is a useful one in the first instance because it clarifies the notion of what is the optimal solution, thereby providing a benchmark against which existing approaches can be evaluated. In the longer term it also provides the potential for new methods to create ensembles of solutions to the model, incorporating the available data in an optimal fashion.Two examples are given illustrating this approach to data assimilation, both in the context of Lagrangian data, one based on statistical 4DVAR and the other on weak constraint statistical 4DVAR. The former is compared with the ensemble Kalman filter, which is thereby shown to be inaccurate in a variety of scenarios.

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Section: Statistical Perspectivementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Data assimilation: Mathematical and statistical perspectives

Apte

Jones

Stuart

et al. 2007

Numerical Methods in Fluids

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show abstract

“…We have derived relative errors by dividing the sum-of-squares error ( P N i=1 (y obs i (t k ) − y i (t k )) 2 ) resulting from Inferelator 2.0 by the equivalent sum-of-squares error resulting from Inferelator 1.0. Box plots are shown for time-interval bins of [1,5), [10,20), [20, 50) and [50, 60] minutes.…”

Section: Methodsmentioning

confidence: 99%

“…This means that the example minimization procedure discussed in IV-B is likely to get trapped in local minima of E(β). Thus, it is preferable to return to (13) and proceed with an actual sampling of this posterior distribution for β [5], [8].…”

Section: Full Posterior Sampling Via Importance Sampling Mcmcmentioning

confidence: 99%

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The inferelator 2.0: A scalable framework for reconstruction of dynamic regulatory network models

Madar

Greenfield

Ostrer

et al. 2009

2009 Annual International Conference of the IEEE Engineering in Medicine and Biology Society

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Current methods for reconstructing biological networks often learn either the topology of large networks or the kinetic parameters of smaller networks with a well-characterized topology. We have recently described a network reconstruction algorithm, the Inferelator 1.0, that given a set of genome-wide measurements as input, simultaneously learns both topology and kinetic-parameters. Specifically, it learns a system of ordinary differential equations (ODEs) that describe the rate of change in transcription of each gene or gene-cluster, as a function of environmental and transcription factors. In order to scale to large networks, in Inferelator 1.0 we have approximated the system of ODEs to be uncoupled, and have solved each ODE using a one-step finite difference approximation. Naturally, these approximations become crude as the simulated time-interval increases. Here we present, implement, and test a new Markov-Chain-Monte-Carlo (MCMC) dynamical modeling method, Inferelator 2.0, that works in tandem with Inferelator 1.0 and is designed to relax these approximations. We show results for the prokaryote Halobacterium that demonstrate a marked improvement in our predictive performance in modeling the regulatory dynamics of the system over longer time-scales.

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Random Fields Based on Local Interactions

Hristopulos

2020

Advances in Geographic Information Science

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Bayesian Field Theory Applied to Scattered Data Interpolation and Inverse Problems

Cited by 15 publications

References 18 publications

Data assimilation: Mathematical and statistical perspectives

Data assimilation: Mathematical and statistical perspectives

The inferelator 2.0: A scalable framework for reconstruction of dynamic regulatory network models

Random Fields Based on Local Interactions

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