Inverse problems: A Bayesian perspective

By Andrew M. Stuart. Acta Numerica, 2010

2,043 citations recorded by the source corpus.

Subfields: Artificial Intelligence, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/a136f1140f2686edf51eb3cf5b44f36c51ef730a

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Citations in the corpus

Bayesian Recovery of the Initial Condition for the Heat Equation (2011, background): “n, 2008; Cavalier, 2008, 2011; Golubev and Khas′minski˘ı, 1999; Mair, 1994; Mair and Ruymgaart, 1996) and has also been addressed in Bayesian framework (with additional assumptions on the noise), cf. Stuart (2010). For more background on how this backward heat conduction problem arises in practical ”
Polynomial chaos expansions for dependent random variables (2019, background): “1 Bayesian inference Bayesian inference [57] is often used to infer densities of random variables conditional on available observational data.”
Modern regularization methods for inverse problems (2018, background): “This topic is beyond the scope of this survey, we refer to [234, 296, 344, 248, 105, 106, 239, 73, 297] for further details.”
The statistical finite element method (statFEM) for coherent synthesis of observation data and model predictions (2019, methodology): “ty of dierent ﬁnite element models by computing their Bayes factors [6,20]. The posteriors are, depending on the level, either analytically approximated or numerically sampled with MCMC. We refer to [21] on mathematical foundations of Bayesian inverse problems for the necessary theory in deﬁning the”
Bayes procedures for adaptive inference in inverse problems for the white noise model (2012, background): “The paper [34] provides a nice overview and many additional references.”
Statistical guarantees for Bayesian uncertainty quantification in nonlinear inverse problems with Gaussian process priors (2020, background): “Various methods aiming to ‘quantify inferential uncertainty’ for inverse problems involving PDEs are now available, particularly based on Bayesian posterior distributions arising from Gaussian process (and other) priors for θ, as advocated in influential work by A. Stuart [58, 15].”
Bayesian inference with optimal maps (2011, background): “This posterior density q is associated with a posterior measure μ on R, such that dμ(z) = q(z)dz and the likelihood function is proportional to the Radon-Nikodym derivative of μ with respect to μ0, L(z) ∝ dμ/dμ0 [1].”
Gaussian Processes and Kernel Methods: A Review on Connections and Equivalences (2018, background): “…an infinite dimensional space of functions, Bayes’ rule is more involved and thus does not produce the expressions (16) and (17) directly (see e.g. Stuart 2010, Theorem 6.31); (ii) When dealing with the noise-free setting where σ2 = 0, Bayes’ rule cannot be used because the likelihood function…”
Nonparametric Bayesian posterior contraction rates for discretely observed scalar diffusions (2015, background): “As pointed out by Stuart [27] and van Zanten [35], obtaining theoretical performance guarantees for Bayesian algorithms in nonlinear inverse problems is, however, of key importance if such methods are to be used in scientific applications.”
Adaptive multi-fidelity polynomial chaos approach to Bayesian inference in inverse problems (2018, positive): “The Bayesian approach can provide a systematic framework for quantifying the uncertainty in the parameter estimation for inverse problems [17, 37].”
MCMC Methods for Functions: ModifyingOld Algorithms to Make Them Faster (2012, background): “These assumptions arise naturally in many Bayesian inverse problems where the data is finite dimensional [49].”
Data Assimilation: A Mathematical Introduction (2015, methodology): “This idea is used for data assimilation problems arising in fluid mechanics in [28], for inverse problems arising in subsurface geophysics in [36, 34] and described more generally in [106].”
A survey of applications of the MFS to inverse problems (2011, background): “been some recent interest in using statistical Bayesian inference for solving inverse problems [38,39].”
Hidden physics models: Machine learning of nonlinear partial differential equations (2017, background): “They can be viewed as a prior on one-layer feed-forward Bayesian neural networks with an inﬁnite number of hidden units [7].”
Learning Parameters and Constitutive Relationships with Physics Informed Deep Neural Networks (2018, methodology): “…are a number of established methods for parameter estimation in (closed-form) PDE models, such as Bayesian inference and maximum a posteriori probability (MAP) estimation [21, 6], existing approaches for learning unknown physics from partially known models and data are few and not fully mature.”
Probabilistic Meshless Methods for Partial Differential Equations and Bayesian Inverse Problems (2016, methodology): “These techniques combine to produce guarantees on the extent of numerical error and, hence, lead to statistically valid inference on the parameter θ [Stuart, 2010].”