Probabilistic Approach to Inverse Problems

  title={Probabilistic Approach to Inverse Problems},
  author={Klaus Mosegaard and Albert Tarantola},
  journal={International Geophysics},
Inverse problems: From regularization to Bayesian inference
Inverse problems deal with the quest for unknown causes of observed consequences, based on predictive models, known as the forward models, that associate the former quantities to the latter in the
Efficient probabilistic inversion using the rejection sampler—exemplified on airborne EM data
A novel approach, for sampling the posterior distribution is suggested based on using pre-calculated lookup tables with the extended rejection sampler, which is fast, fast, generates independent realizations of the posterior, and does not get stuck in local minima.
Analysis of the Impact of Model Nonlinearities in Inverse Problem Solving
In this study, the relationship between nonlinear model properties and inverse problem solutions is analyzed using a numerical technique based on the inverse problem theory formulated by Mosegaard
Posterior population expansion for solving inverse problems
A new ensemble‐based exploration scheme for geostatistical prior models generated by a multiple‐point statistics (MPS) tool is proposed, to expand an existing set of models by using posterior facies information for conditioning new MPS realizations.
Multiple Scenario Generation of Subsurface Models: Consistent Integration of Information from Geophysical and Geological Data throuh Combination of Probabilistic Inverse Problem Theory and Geostatistics
Markov chain Monte Carlo methods such as the Gibbs sampler and the Metropolis algorithm can be used to sample the solutions to non-linear inverse problems. In principle these methods allow
1 Uncertainty Quantification in Inverse Problems : Model-Based versus Prediction-Focused Inversion
Most formulations and solutions to inverse modeling in the Earth Sciences consider the relationship between data and model. In this paper we consider an additional element to these formulations,
A review of Markov Chain Monte Carlo and information theory tools for inverse problems in subsurface flow
Parameter identification is one of the key elements in the construction of models in geosciences. However, inherent difficulties such as the instability of ill-posed problems or the presence of
Accounting for imperfect forward modeling in geophysical inverse problems — Exemplified for crosshole tomography
For a specific forward problem, computation of crosshole tomographic first-arrival traveltimes, it is evaluated how the modeling error, given several different approximate forward models, can be more than an order of magnitude larger than the measurement uncertainty.


Monte Carlo sampling of solutions to inverse problems
In inverse problems, obtaining a maximum likelihood model is usually not sucient, as the theory linking data with model parameters is nonlinear and the a posteriori probability in the model space may not be easy to describe.
Probabilistic analysis of implicit inverse problems
Many inverse problems are most naturally formulated as implicit problems where data cannot be expressed in closed form as a function of the unknown model parameters. Notable examples are cases where
Inference from Inadequate and Inaccurate Data, II.
  • G. Backus
  • Mathematics
    Proceedings of the National Academy of Sciences of the United States of America
  • 1970
A technique often available for reducing to this tractable case problems in which the functionals are discontinuous and are defined only on parts of [unk], and [unk] is an arbitrary real linear space.
Inverse problems = Quest for information
The inverse problem may be formulated as a problem of combination of information: the experimental information about data, the a priori information about parameters, and the theoretical information, and it is shown that the general solution of the non-linear inverse problem is unique and consistent.
Inverse problem theory : methods for data fitting and model parameter estimation
Part 1. Discrete Inverse Problems. 1. The General Discrete Inverse Problem. 2. The Trial and Error Method. 3. Monte Carlo Methods. 4. The Least-Squares (l 2 -norm) Criterion. 5. The Least-Absolute
Linear inverse problems for generalised random variables
In a statistical inverse theory both the unknown quantity and the measurement are random variables. The solution of the inverse problem is then the conditional distribution of the unknown variable
The use of a priori data to resolve non‐uniqueness in linear inversion
Summary. The recent, but by now classical method for dealing with non-uniqueness in geophysical inverse problems is to construct linear averages of the unknown function whose values are uniquely
Generalized Nonlinear Inverse Problems Solved Using the Least Squares Criterion (Paper 1R1855)
We attempt to give a general definition of the nonlinear least squares inverse problem. First, we examine the discrete problem (finite number of data and unknowns), setting the problem in its fully
Inference from Inadequate and Inaccurate Data, III.
  • G. Backus
  • Mathematics
    Proceedings of the National Academy of Sciences of the United States of America
  • 1970
The present paper gives estimates when it is likely that the authors can guess an upper bound M on the Hilbert norm not of h(E), the model representing E in some Hilbert space, but of the orthogonal projection of h (E) onto a sufficiently large subspace.