4,362 research outputs found
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
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