Monte Carlo sampling of solutions to inverse problems

@article{Mosegaard1995MonteCS,
  title={Monte Carlo sampling of solutions to inverse problems},
  author={Klaus Mosegaard and Albert Tarantola},
  journal={Journal of Geophysical Research},
  year={1995},
  volume={100},
  pages={12431-12447}
}
Probabilistic formulation of inverse problems leads to the definition of a probability distribution in the model space. This probability distribution combines a priori information with new information obtained by measuring some observable parameters (data). As, in the general case, the theory linking data with model parameters is nonlinear, the a posteriori probability in the model space may not be easy to describe (it may be multimodal, some moments may not be defined, etc.). When analyzing an… 
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References

SHOWING 1-10 OF 37 REFERENCES
Inverse problems = Quest for information
TLDR
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.
Nonlinear inversion, statistical mechanics, and residual statics estimation
TLDR
The fully nonlinear approach presented is rooted in statistical mechanics and formulate inversion as a problem of Bayesian estimation, in which the prior probability distribution is the Gibbs distribution of statistical mechanics.
Inference from Inadequate and Inaccurate Data, III.
  • G. Backus
  • Mathematics
    Proceedings of the National Academy of Sciences of the United States of America
  • 1970
TLDR
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.
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
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
Inference from inadequate and inaccurate data, I.
  • G. Backus
  • Mathematics
    Proceedings of the National Academy of Sciences of the United States of America
  • 1970
TLDR
This paper describes how one can proceed when E is adequately described by one member m(E) of a Hilbert space [unk] of possible models of E, when he believes that the Hilbert norm of m( E) is very likely rather smaller than some known number M, and (except for section 6) when all the observed and sought-after properties of E are continuous linear functionals on [unk].
Monte Carlo estimation and resolution analysis of seismic background velocities
The complete solution to an inverse problem, including information on accuracy and resolution, is given by the a posteriori probability density in the model space. By running a modified simulated
Probabilistic Solution of Ill-Posed Problems in Computational Vision
TLDR
This work derives efficient algorithms and describes parallel implementations on digital parallel SIMD architectures, as well as a new class of parallel hybrid computers that mix digital with analog components.
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
Stochastic Relaxation, Gibbs Distributions, and the Bayesian Restoration of Images
  • S. Geman, D. Geman
  • Physics
    IEEE Transactions on Pattern Analysis and Machine Intelligence
  • 1984
TLDR
The analogy between images and statistical mechanics systems is made and the analogous operation under the posterior distribution yields the maximum a posteriori (MAP) estimate of the image given the degraded observations, creating a highly parallel ``relaxation'' algorithm for MAP estimation.
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