Non-Gaussian statistical inverse problems. Part I: Posterior distributions

Lasanen, Sari

doi:10.3934/ipi.2012.6.215

Cited by 41 publications

(55 citation statements)

References 94 publications

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“…First each particle v .n/ j is updated by proposing a new candidate particle O v .n/ j C1 according to the Markov kernel P j ; this corresponds to (63a) and creates an approximation to O j C1 : (See the last two parts of Remark 8 for a discussion on the role of P j in the algorithm.) We can think of this approximation as a prior distribution for application of Bayes' rule in the form (63b), or equivalently (64). Secondly, each new particle is re-weighted according to the desired distribution j C1 given by (64).…”

Section: Sequential Monte Carlo Methodsmentioning

confidence: 99%

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The Bayesian Approach to Inverse Problems

Dashti

Stuart

2015

Handbook of Uncertainty Quantification

143

222

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These lecture notes highlight the mathematical and computational structure relating to the formulation of, and development of algorithms for, the Bayesian approach to inverse problems in differential equations. This approach is fundamental in the quantification of uncertainty within applications involving the blending of mathematical models with data. The finite dimensional situation is described first, along with some motivational examples. Then the development of probability measures on separable Banach space is undertaken, using a random series over an infinite set of functions to construct draws; these probability measures are used as priors in the Bayesian approach to inverse problems. Regularity of draws from the priors is studied in the natural Sobolev or Besov spaces implied by the choice of functions in the random series construction, and the Kolmogorov continuity theorem is used to extend regularity considerations to the space of Hölder continuous functions. Bayes' theorem is derived in this prior setting, and here interpreted as finding conditions under which the posterior is absolutely continuous with respect to the prior, and determining a formula for the Radon-Nikodym derivative in terms of the likelihood of the data. Having established the form of the posterior, we then describe various properties common to it in the infinite dimensional setting. These properties include well-posedness, approximation theory, and the existence of maximum a posteriori estimators. We then describe measure-preserving dynamics, again on the infinite dimensional space, including Markov chain-Monte Carlo and sequential Monte Carlo methods, and measure-preserving reversible stochastic differential equations. By formulating the theory and algorithms on the underlying infinite dimensional space, we obtain a framework suitable for rigorous analysis of the accuracy of reconstructions, of computational complexity, as well as naturally constructing algorithms which perform well under mesh refinement, since they are inherently well-defined in infinite dimensions.

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Section: Sequential Monte Carlo Methodsmentioning

confidence: 99%

“…General development of Bayes' Theorem for inverse problems on function space, along the lines described here, may be found in [17,92]. The reader is also directed to the papers [61,62] for earlier related material and to [63][64][65] for recent developments. • Section 3.3.…”

Section: Bibliographic Notesmentioning

confidence: 99%

The Bayesian Approach to Inverse Problems

Dashti

Stuart

2015

Handbook of Uncertainty Quantification

143

222

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“…The functions log κ † 1 and log κ † 2 are draws from the Gaussian measures N (m 1 , C 1 ) and N (m 2 , C 2 ) that we use to define the prior (26). However, in order to avoid the inverse crime, these priors that we use to generate log κ † 1 and log κ † 2 (and so u † ) are defined on a discretized domain (of 100 × 100 grids) that is finer than the one used for the inversion.…”

Section: Example Two-layer Model With Spatially-varying Permeabilitiesmentioning

confidence: 99%

“…The corresponding variances are σ a = 1.03 × 10 −2 and σ b = 9.5 × 10 −3 , respectively. The permeability (26) corresponding to the meanû is displayed in Figure 10 (top-middle).…”

Section: Example Two-layer Model With Spatially-varying Permeabilitiesmentioning

confidence: 99%

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Well-posed Bayesian geometric inverse problems arising in subsurface flow

2014

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Abstract. In this paper, we consider the inverse problem of determining the permeability of the subsurface from hydraulic head measurements, within the framework of a steady Darcy model of groundwater flow. We study geometrically defined prior permeability fields, which admit layered, fault and channel structures, in order to mimic realistic subsurface features; within each layer we adopt either constant or continuous function representation of the permeability. This prior model leads to a parameter identification problem for a finite number of unknown parameters determining the geometry, together with either a finite number of permeability values (in the constant case) or a finite number of fields (in the continuous function case). We adopt a Bayesian framework showing existence and well-posedness of the posterior distribution. We also introduce novel Markov Chain-Monte Carlo (MCMC) methods, which exploit the different character of the geometric and permeability parameters, and build on recent advances in function space MCMC. These algorithms provide rigorous estimates of the permeability, as well as the uncertainty associated with it, and only require forward model evaluations. No adjoint solvers are required and hence the methodology is applicable to black-box forward models. We then use these methods to explore the posterior and to illustrate the methodology with numerical experiments.

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Anomaly Detection in Random Heterogeneous Media

Simón¹

2015

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Non-Gaussian statistical inverse problems. Part I: Posterior distributions

Cited by 41 publications

References 94 publications

The Bayesian Approach to Inverse Problems

The Bayesian Approach to Inverse Problems

Well-posed Bayesian geometric inverse problems arising in subsurface flow

Anomaly Detection in Random Heterogeneous Media

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