• Corpus ID: 16323001

Inverse problems = Quest for information

@article{Tarantola1982InverseP,
  title={Inverse problems = Quest for information},
  author={Albert Tarantola and Bernard Valette},
  journal={Journal of geophysics},
  year={1982},
  volume={50},
  pages={159-170}
}
We examine the general non-linear inverse problem with a nite number of parameters. In order to permit the incorporation of any a priori information about parameters and any distribution of data (not only of gaussian type) we propose to formulate the problem not using single quantities (such as bounds, means, etc.) but using probability density functions for data and parameters. We also want our formulation to allow for the incorporation of theoretical errors, i.e. non-exact theoretical… 

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References

SHOWING 1-10 OF 23 REFERENCES
Interpretation of Inaccurate, Insufficient and Inconsistent Data
Sumntary Many problems in physical science involve the estimation of a number of unknown parameters which bear a linear or quasi-linear relationship to a set of experimental data. The data may be
The general linear inverse problem - Implication of surface waves and free oscillations for earth structure.
The discrete general linear inverse problem reduces to a set of m equations in n unknowns. There is generally no unique solution, but we can find k linear combinations of parameters for which
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, 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].
Prior Probabilities
  • E. Jaynes
  • Mathematics
    Encyclopedia of Machine Learning
  • 1968
TLDR
It is shown that in many problems, including some of the most important in practice, this ambiguity can be removed by applying methods of group theoretical reasoning which have long been used in theoretical physics.
The Resolving Power of Gross Earth Data
A gross Earth datum is a single measurable number describing some property of the whole Earth, such as mass, moment of interia, or the frequency of oscillation of some identified
Uniqueness in the inversion of inaccurate gross Earth data
  • G. Backus, F. Gilbert
  • Mathematics, Geology
    Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences
  • 1970
A gross Earth datum is a single measurable number describing some property of the whole Earth, such as mass, moment of inertia, or the frequency of oscillation of some identified
Robust Modeling With Erratic Data
An attractive alternative to least‐squares data modeling techniques is the use of absolute value error criteria. Unlike the least‐squares techniques the inclusion of some infinite blunders along with
Scientific Inference
SCIENTISTS generally care so little for scientific principles that the title of this book may repel as many as its author's name attracts. Let it be, therefore, stated at once that it is not a formal
Well-posed stochastic extensions of ill-posed linear problems☆
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