Transport Map sampling with PGD model reduction for fast dynamical Bayesian data assimilation

Rubio, Paul-Baptiste; Louf, François; Chamoin, Ludovic

doi:10.1002/nme.6143

Cited by 8 publications

(4 citation statements)

References 54 publications

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“…Since in general the likelihood function is dependent on the model, the issue arises that also the derivative of the model with respect to θ is needed. In [4,5] this was solved by using the TM approach in conjunction with model order reduction methods based on polynomial functions, which naturally allow for the calculation of gradient and Hessian.…”

Section: Transport Mapsmentioning

confidence: 99%

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Transport map Bayesian parameter estimation for dynamical systems

Grashorn,

Urrea-Quintero,

Broggi

et al. 2023

Proc Appl Math and Mech

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Accurate online state and parameter estimation of uncertain non-linear dynamical systems is a demanding task that has been traditionally handled by adopting non-linear Kalman Filters or particle filters. However, in case of Kalman filters the system needs to be linearised and for particle filters the computational demand can be high. Recent advances in optimal transport theory and the application to Bayesian model updating pave the way for other approaches to system and parameter identification. They also provide a way of formulating the problem in such a way that efficient online estimation for complex systems is possible. In this work, we investigate the properties of the transport map approach when compared to standard Markov Chain Monte Carlo in an off-line setting as a first step towards on-line parameter estimation. We apply both approaches to an analytical exponential model and a dynamical system with seven unknown parameters subjected to ground displacement. Details on the theory of transport maps and on the used MCMC algorithm are also given.

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Section: Transport Mapsmentioning

confidence: 99%

“…The problem of finding this map is solved by optimization. Previous works have implemented transport map (TM) approximation in various use-cases, using synergies of this approach with model order reduction techniques to speed up the process [4,5].…”

Section: Introductionmentioning

confidence: 99%

Transport map Bayesian parameter estimation for dynamical systems

Grashorn,

Urrea-Quintero,

Broggi

et al. 2023

Proc Appl Math and Mech

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“…The application of PGD models to reduce the computational effort of sampling-based approaches, can be found in the context of reliability analysis in [27,28] and for process calibration of additive manufacturing in [29]. Furthermore, Rubio et al [30] used a PGD forward model for real-time identification and model updating based on Monte Carlo sampling as well as based on a transport map sampling [31]. Djatouti et al [32] proposed a goal-oriented inverse method, coupling a PGD model and a variant of the modified constitutive relation error approach, which automatically identifies and updates only the model parameters with the highest influence.…”

Section: Proper Generalized Decomposition (Pgd)mentioning

confidence: 99%

Bayesian inference for random field parameters with a goal-oriented quality control of the PGD forward model’s accuracy

et al. 2022

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Numerical models built as virtual-twins of a real structure (digital-twins) are considered the future of monitoring systems. Their setup requires the estimation of unknown parameters, which are not directly measurable. Stochastic model identification is then essential, which can be computationally costly and even unfeasible when it comes to real applications. Efficient surrogate models, such as reduced-order method, can be used to overcome this limitation and provide real time model identification. Since their numerical accuracy influences the identification process, the optimal surrogate not only has to be computationally efficient, but also accurate with respect to the identified parameters. This work aims at automatically controlling the Proper Generalized Decomposition (PGD) surrogate’s numerical accuracy for parameter identification. For this purpose, a sequence of Bayesian model identification problems, in which the surrogate’s accuracy is iteratively increased, is solved with a variational Bayesian inference procedure. The effect of the numerical accuracy on the resulting posteriors probability density functions is analyzed through two metrics, the Bayes Factor (BF) and a criterion based on the Kullback-Leibler (KL) divergence. The approach is demonstrated by a simple test example and by two structural problems. The latter aims to identify spatially distributed damage, modeled with a PGD surrogate extended for log-normal random fields, in two different structures: a truss with synthetic data and a small, reinforced bridge with real measurement data. For all examples, the evolution of the KL-based and BF criteria for increased accuracy is shown and their convergence indicates when model refinement no longer affects the identification results.

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“…This representation is computed in an offline phase with controlled accuracy [8] before being evaluated at low cost in the online phase. It is shown in the paper that the PGD technique (i) facilitates the computation of the likelihood function involved in the Bayesian inference framework [3,26]; (ii) can be effectively coupled with Transport Map sampling for the calculation of the maps, as it directly provides information on solution derivatives [27,28]; (iii) is a particularly effective tool for performing uncertainty propagation through the forward model as well as command law synthesis. A particular focus is made here on the latter point dealing with effective command in a stochastic framework; this has been investigated in very few works of the literature, even though it is a major aspect of the DDDAS procedure.…”

Section: Introductionmentioning

confidence: 99%

Real-time data assimilation and control on mechanical systems under uncertainties

Rubio

Chamoin

Louf

2021

Adv. Model. and Simul. in Eng. Sci.

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This research work deals with the implementation of so-called Dynamic Data-Driven Application Systems (DDDAS) in structural mechanics activities. It aims at designing a real-time numerical feedback loop between a physical system of interest and its numerical simulator, so that (i) the simulation model is dynamically updated from sequential and in situ observations on the system; (ii) the system is appropriately driven and controlled in service using predictions given by the simulator. In order to build such a feedback loop and take various uncertainties into account, a suitable stochastic framework is considered for both data assimilation and control, with the propagation of these uncertainties from model updating up to command synthesis by using a specific and attractive sampling technique. Furthermore, reduced order modeling based on the Proper Generalized Decomposition (PGD) technique is used all along the process in order to reach the real-time constraint. This permits fast multi-query evaluations and predictions, by means of the parametrized physics-based model, in the online phase of the feedback loop. The control of a fusion welding process under various scenarios is considered to illustrate the proposed methodology and to assess the performance of the associated numerical architecture.

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Transport Map sampling with PGD model reduction for fast dynamical Bayesian data assimilation

Cited by 8 publications

References 54 publications

Transport map Bayesian parameter estimation for dynamical systems

Transport map Bayesian parameter estimation for dynamical systems

Bayesian inference for random field parameters with a goal-oriented quality control of the PGD forward model’s accuracy

Real-time data assimilation and control on mechanical systems under uncertainties

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