Bayesian model inversion using stochastic spectral embedding

Wagner, Paul-Remo; Marelli, Stefano; Sudret, Bruno

doi:10.1016/j.jcp.2021.110141

Cited by 15 publications

(14 citation statements)

References 38 publications

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“…Leveraging the flexibility of the recently proposed stochastic spectral embedding formalism , we show that the adaptive sequential partitioning approach introduced in (Wagner et al, 2021) can be efficiently modified to an active learning reliability method.…”

Section: Discussionmentioning

confidence: 99%

“…In this paper we propose a novel active learning reliability method that utilizes the recently proposed stochastic spectral embedding method (SSE, ) and more specifically the active learning sequential partitioning approach developed for Bayesian inverse problems in Wagner et al (2021). The proposed approach, termed stochastic spectral embedding-based reliability (SSER), benefits from a handful of modifications to the original SSE, namely new refinement domain selection, partitioning and sample enrichment schemes.…”

Section: Introductionmentioning

confidence: 99%

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Rare event estimation using stochastic spectral embedding

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Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Rare event estimation using stochastic spectral embedding

et al. 2022

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“…However, when the forward model is more non-linear, constructing a globally-valid surrogate applicable for any realization of the prior can be challenging. In these cases, the accuracy of the surrogate can be refined in regions of high posterior probability (Li and Marzouk, 2014;Wagner et al, 2021a). Even if we considered borehole GPR applications herein, the the strategy could be easily adapted to other imaging problems such as active or passive seismic tomography at different scales (Bodin and Sambridge, 2009;Galetti et al, 2017).…”

Section: Discussionmentioning

confidence: 99%

“…To further decrease the computational costs associated with Markov chain Monte Carlo (MCMC) inversion, we employ unbiased surrogate modelling to approximate standard travel-time forward solvers (Xiu and Karniadakis, 2002). Various classes of surrogate models, such as those based on Gaussian process models or kriging (Sacks et al, 1989;Rasmussen, 2003) and polynomial chaos expansions (PCEs) (Xiu and Karniadakis, 2002;Blatman and Sudret, 2011), can be employed in Bayesian inverse problems (Nagel, 2019;Higdon et al, 2015;Marzouk and Xiu, 2009;Marzouk et al, 2007;Wagner et al, 2020Wagner et al, , 2021a. In this contribution, we rely on regression-based sparse PCE due to their extrapolation capabilities and robustness with respect to noise (Blatman and Sudret, 2011;Lüthen et al, 2021b;Marelli et al, 2021b).…”

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confidence: 99%

Bayesian tomography with prior-knowledge-based parametrization and surrogate modeling

Meles,

Linde,

Marelli

2022

Preprint

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We present a Bayesian tomography framework operating with prior-knowledge-based parametrization that is accelerated by surrogate models. Standard high-fidelity forward solvers (e.g., Finite-Difference Time-Domain schemes) solve wave equations with natural spatial parametrizations based on fine discretization. Similar parametrizations, typically involving tens of thousand of variables, are usually employed to parameterize the subsurface in tomography applications. When the data do not allow to resolve details at such finely parameterized scales, it is often beneficial to instead rely on a priorknowledge-based parametrization defined on a lower dimension domain (or manifold). Due to the increased identifiability in the reduced domain, the concomitant inversion is better constrained and generally faster. We illustrate the potential of a prior-knowledge-based approach by considering ground penetrating radar (GPR) travel-time tomography in a crosshole configuration. An effective parametrization of the input (i.e., the permittivity distributions determining the slowness field) and compression in the output (i.e., the travel-time gathers) spaces are achieved via data-driven principal component decomposition based on random realizations of the prior Gaussian-process model with a truncation determined by the performances of the standard solver on the full and reduced model domains. To accelerate the inversion process, we employ a high-fidelity polynomial chaos expansion (PCE) surrogate model. We investigate the impact of the size of the training set on the performance of the PCE and show that a few hundreds design data sets is sufficient to provide reliable Markov chain Monte Carlo inversion at a fraction of the cost associated with a standard approach involving a fine discretization and physics-based forward solvers. Appropriate uncertainty quantification is achieved by reintroducing the truncated higher-order principle components in the original model space after inversion on the manifold and by adapting a likelihood function that accounts for the fact that the truncated higher-order components are not completely located in the null-space.

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“…Even if one can formulate the problem in an infinite dimensional setting (section 2) and discretization should take place as late as possible (Stuart, 2010), when solving inverse problems in practice there is always some form of discretization involved, be it through quadrature methods (Hansen, 2010) or through basis expansion (Wagner et al, 2021). It turns out that, regardless of the type of discretization used, one quickly encounters computational bottlenecks arising from memory limitations when trying to scale inversion to man real-world problems.…”

Section: Computing the Posterior Undermentioning

confidence: 99%

Uncertainty Quantification and Experimental Design for Large-Scale Linear Inverse Problems under Gaussian Process Priors

Cédric¹,

Ginsbourger²,

Linde³

2021

Preprint

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We consider the use of Gaussian process (GP) priors for solving inverse problems in a Bayesian framework. As is well known, the computational complexity of GPs scales cubically in the number of datapoints. We here show that in the context of inverse problems involving integral operators, one faces additional difficulties that hinder inversion on large grids. Furthermore, in that context, covariance matrices can become too large to be stored. By leveraging results about sequential disintegrations of Gaussian measures, we are able to introduce an implicit representation of posterior covariance matrices that reduces the memory footprint by only storing low rank intermediate matrices, while allowing individual elements to be accessed on-the-fly without needing to build full posterior covariance matrices. Moreover, it allows for fast sequential inclusion of new observations. These features are crucial when considering sequential experimental design tasks. We demonstrate our approach by computing sequential data collection plans for excursion set recovery for a gravimetric inverse problem, where the goal is to provide fine resolution estimates of high density regions inside the Stromboli volcano, Italy. Sequential data collection plans are computed by extending the weighted integrated variance reduction (wIVR) criterion to inverse problems. Our results show that this criterion is able to significantly reduce the uncertainty on the excursion volume, reaching close to minimal levels of residual uncertainty. Overall, our techniques allow the advantages of probabilistic models to be brought to bear on large-scale inverse problems arising in the natural sciences. Particularly, applying the latest developments in Bayesian sequential experimental design on realistic large-scale problems opens new venues of research at a crossroads between mathematical modelling of natural phenomena, statistical data science and active learning.

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Bayesian model inversion using stochastic spectral embedding

Cited by 15 publications

References 38 publications

Rare event estimation using stochastic spectral embedding

Rare event estimation using stochastic spectral embedding

Bayesian tomography with prior-knowledge-based parametrization and surrogate modeling

Uncertainty Quantification and Experimental Design for Large-Scale Linear Inverse Problems under Gaussian Process Priors

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