• Corpus ID: 88524103

# Inverse Problems and Data Assimilation.

@article{Stuart2018InversePA,
title={Inverse Problems and Data Assimilation.},
author={Andrew M. Stuart and Armeen Taeb},
journal={arXiv: Methodology},
year={2018}
}
• Published 15 October 2018
• Mathematics
• arXiv: Methodology
These notes are designed with the aim of providing a clear and concise introduction to the subjects of Inverse Problems and Data Assimilation, and their inter-relations, together with citations to some relevant literature in this area. The first half of the notes is dedicated to studying the Bayesian framework for inverse problems. Techniques such as importance sampling and Markov Chain Monte Carlo (MCMC) methods are introduced; these methods have the desirable property that in the limit of an…

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## References

SHOWING 1-10 OF 112 REFERENCES
The Bayesian Approach to Inverse Problems
• Mathematics
• 2017
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
Ergodicity and Accuracy of Optimal Particle Filters for Bayesian Data Assimilation
• Mathematics
Chinese Annals of Mathematics, Series B
• 2019
For particle filters and ensemble Kalman filters it is of practical importance to understand how and why data assimilation methods can be effective when used with a fixed small number of particles,
Variational Characterizations of Local Entropy and Heat Regularization in Deep Learning
• Mathematics, Computer Science
Entropy
• 2019
Variational characterizations that naturally suggest a two-step scheme for local entropy and heat regularized loss are introduced, based on the iterative shift of a probability density and the calculation of a best Gaussian approximation in Kullback–Leibler divergence.
Continuum Limits of Posteriors in Graph Bayesian Inverse Problems
• Mathematics, Computer Science
SIAM J. Math. Anal.
• 2018
A graph-based Bayesian inverse problem is introduced, and it is shown that the graph-posterior measures over functions in $M_n$ converge, in the large $n$ limit, to a posterior over Functions in M that solves a Bayesian inverted problem with known domain.
Data assimilation in the geosciences: An overview of methods, issues, and perspectives
• Physics, Mathematics
WIREs Climate Change
• 2018
We commonly refer to state-estimation theory in geosciences as data assimilation. This term encompasses the entire sequence of operations that, starting from the observations of a system, and from
Importance Sampling and Necessary Sample Size: An Information Theory Approach
• D. Sanz-Alonso
• Mathematics, Computer Science
SIAM/ASA J. Uncertain. Quantification
• 2018
A general bound is derived that needs to hold for importance sampling to be successful, and relates the f-divergence between the target and the proposal to the sample size, which is deduced from a new and simple information theory paradigm for the study of importance sampling.
Adaptive importance sampling Monte Carlo simulation for general multivariate probability laws
• R. Kawai
• Mathematics, Computer Science
J. Comput. Appl. Math.
• 2017
A parametric adaptive importance sampling variance reduction method for general multivariate probability laws and establishes the asymptotic normality of the estimator of the desired mean and of the importance sampling parameter as the number of observations tends to infinity.
Analysis of the Ensemble Kalman Filter for Inverse Problems
• Computer Science, Mathematics
SIAM J. Numer. Anal.
• 2017
The goal of this paper is to analyze the EnKF when applied to inverse problems with fixed ensemble size, and to demonstrate that the conclusions of the analysis extend beyond the linear inverse problem setting.
Bernstein–von Mises theorems for statistical inverse problems I: Schrödinger equation
The inverse problem of determining the unknown potential $f>0$ in the partial differential equation \frac{\Delta}{2} u - fu =0 \text{ on } \mathcal O ~~\text{s.t. } u = g \text { on } \partial
Existence and Uniqueness for Four-Dimensional Variational Data Assimilation in Discrete Time
• J. Bröcker
• Computer Science, Mathematics
SIAM J. Appl. Dyn. Syst.
• 2017
Variational techniques for data assimilation, i.e., estimating orbits of dynamical models from observations, are revisited. It is shown that under mild hypotheses a solution to this variational