587 research outputs found
A variational Bayesian method for inverse problems with impulsive noise
We propose a novel numerical method for solving inverse problems subject to
impulsive noises which possibly contain a large number of outliers. The
approach is of Bayesian type, and it exploits a heavy-tailed t distribution for
data noise to achieve robustness with respect to outliers. A hierarchical model
with all hyper-parameters automatically determined from the given data is
described. An algorithm of variational type by minimizing the Kullback-Leibler
divergence between the true posteriori distribution and a separable
approximation is developed. The numerical method is illustrated on several one-
and two-dimensional linear and nonlinear inverse problems arising from heat
conduction, including estimating boundary temperature, heat flux and heat
transfer coefficient. The results show its robustness to outliers and the fast
and steady convergence of the algorithm.Comment: 20 pages, to appear in J. Comput. Phy
Bregman Cost for Non-Gaussian Noise
One of the tasks of the Bayesian inverse problem is to find a good estimate
based on the posterior probability density. The most common point estimators
are the conditional mean (CM) and maximum a posteriori (MAP) estimates, which
correspond to the mean and the mode of the posterior, respectively. From a
theoretical point of view it has been argued that the MAP estimate is only in
an asymptotic sense a Bayes estimator for the uniform cost function, while the
CM estimate is a Bayes estimator for the means squared cost function. Recently,
it has been proven that the MAP estimate is a proper Bayes estimator for the
Bregman cost if the image is corrupted by Gaussian noise. In this work we
extend this result to other noise models with log-concave likelihood density,
by introducing two related Bregman cost functions for which the CM and the MAP
estimates are proper Bayes estimators. Moreover, we also prove that the CM
estimate outperforms the MAP estimate, when the error is measured in a certain
Bregman distance, a result previously unknown also in the case of additive
Gaussian noise
Convergence Rates for Inverse Problems with Impulsive Noise
We study inverse problems F(f) = g with perturbed right hand side g^{obs}
corrupted by so-called impulsive noise, i.e. noise which is concentrated on a
small subset of the domain of definition of g. It is well known that
Tikhonov-type regularization with an L^1 data fidelity term yields
significantly more accurate results than Tikhonov regularization with classical
L^2 data fidelity terms for this type of noise. The purpose of this paper is to
provide a convergence analysis explaining this remarkable difference in
accuracy. Our error estimates significantly improve previous error estimates
for Tikhonov regularization with L^1-fidelity term in the case of impulsive
noise. We present numerical results which are in good agreement with the
predictions of our analysis
Convergence Rates for Exponentially Ill-Posed Inverse Problems with Impulsive Noise
This paper is concerned with exponentially ill-posed operator equations with
additive impulsive noise on the right hand side, i.e. the noise is large on a
small part of the domain and small or zero outside. It is well known that
Tikhonov regularization with an data fidelity term outperforms Tikhonov
regularization with an fidelity term in this case. This effect has
recently been explained and quantified for the case of finitely smoothing
operators. Here we extend this analysis to the case of infinitely smoothing
forward operators under standard Sobolev smoothness assumptions on the
solution, i.e. exponentially ill-posed inverse problems. It turns out that high
order polynomial rates of convergence in the size of the support of large noise
can be achieved rather than the poor logarithmic convergence rates typical for
exponentially ill-posed problems. The main tools of our analysis are Banach
spaces of analytic functions and interpolation-type inequalities for such
spaces. We discuss two examples, the (periodic) backwards heat equation and an
inverse problem in gradiometry.Comment: to appear in SIAM J. Numer. Ana
Expectation Propagation for Nonlinear Inverse Problems -- with an Application to Electrical Impedance Tomography
In this paper, we study a fast approximate inference method based on
expectation propagation for exploring the posterior probability distribution
arising from the Bayesian formulation of nonlinear inverse problems. It is
capable of efficiently delivering reliable estimates of the posterior mean and
covariance, thereby providing an inverse solution together with quantified
uncertainties. Some theoretical properties of the iterative algorithm are
discussed, and the efficient implementation for an important class of problems
of projection type is described. The method is illustrated with one typical
nonlinear inverse problem, electrical impedance tomography with complete
electrode model, under sparsity constraints. Numerical results for real
experimental data are presented, and compared with that by Markov chain Monte
Carlo. The results indicate that the method is accurate and computationally
very efficient.Comment: Journal of Computational Physics, to appea
Bayesian Estimation for Continuous-Time Sparse Stochastic Processes
We consider continuous-time sparse stochastic processes from which we have
only a finite number of noisy/noiseless samples. Our goal is to estimate the
noiseless samples (denoising) and the signal in-between (interpolation
problem).
By relying on tools from the theory of splines, we derive the joint a priori
distribution of the samples and show how this probability density function can
be factorized. The factorization enables us to tractably implement the maximum
a posteriori and minimum mean-square error (MMSE) criteria as two statistical
approaches for estimating the unknowns. We compare the derived statistical
methods with well-known techniques for the recovery of sparse signals, such as
the norm and Log (- relaxation) regularization
methods. The simulation results show that, under certain conditions, the
performance of the regularization techniques can be very close to that of the
MMSE estimator.Comment: To appear in IEEE TS
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