191 research outputs found
Pinsker estimators for local helioseismology
A major goal of helioseismology is the three-dimensional reconstruction of
the three velocity components of convective flows in the solar interior from
sets of wave travel-time measurements. For small amplitude flows, the forward
problem is described in good approximation by a large system of convolution
equations. The input observations are highly noisy random vectors with a known
dense covariance matrix. This leads to a large statistical linear inverse
problem.
Whereas for deterministic linear inverse problems several computationally
efficient minimax optimal regularization methods exist, only one
minimax-optimal linear estimator exists for statistical linear inverse
problems: the Pinsker estimator. However, it is often computationally
inefficient because it requires a singular value decomposition of the forward
operator or it is not applicable because of an unknown noise covariance matrix,
so it is rarely used for real-world problems. These limitations do not apply in
helioseismology. We present a simplified proof of the optimality properties of
the Pinsker estimator and show that it yields significantly better
reconstructions than traditional inversion methods used in helioseismology,
i.e.\ Regularized Least Squares (Tikhonov regularization) and SOLA (approximate
inverse) methods.
Moreover, we discuss the incorporation of the mass conservation constraint in
the Pinsker scheme using staggered grids. With this improvement we can
reconstruct not only horizontal, but also vertical velocity components that are
much smaller in amplitude
Image reconstruction by regularized nonlinear inversion - Joint estimation of coil sensitivities and image content.
Convergence rates in expectation for Tikhonov-type regularization of Inverse Problems with Poisson data
In this paper we study a Tikhonov-type method for ill-posed nonlinear
operator equations \gdag = F(
ag) where \gdag is an integrable,
non-negative function. We assume that data are drawn from a Poisson process
with density t\gdag where may be interpreted as an exposure time. Such
problems occur in many photonic imaging applications including positron
emission tomography, confocal fluorescence microscopy, astronomic observations,
and phase retrieval problems in optics. Our approach uses a
Kullback-Leibler-type data fidelity functional and allows for general convex
penalty terms. We prove convergence rates of the expectation of the
reconstruction error under a variational source condition as both
for an a priori and for a Lepski{\u\i}-type parameter choice rule
Iteratively regularized Newton-type methods for general data misfit functionals and applications to Poisson data
We study Newton type methods for inverse problems described by nonlinear
operator equations in Banach spaces where the Newton equations
are regularized variationally using a general
data misfit functional and a convex regularization term. This generalizes the
well-known iteratively regularized Gauss-Newton method (IRGNM). We prove
convergence and convergence rates as the noise level tends to 0 both for an a
priori stopping rule and for a Lepski{\u\i}-type a posteriori stopping rule.
Our analysis includes previous order optimal convergence rate results for the
IRGNM as special cases. The main focus of this paper is on inverse problems
with Poisson data where the natural data misfit functional is given by the
Kullback-Leibler divergence. Two examples of such problems are discussed in
detail: an inverse obstacle scattering problem with amplitude data of the
far-field pattern and a phase retrieval problem. The performence of the
proposed method for these problems is illustrated in numerical examples
In-situ characterization of metal clusters supported on a birefringent substrate using reflectance difference spectroscopy
The Iteratively Regularized Gau{\ss}-Newton Method with Convex Constraints and Applications in 4Pi-Microscopy
This paper is concerned with the numerical solution of nonlinear ill-posed
operator equations involving convex constraints. We study a Newton-type method
which consists in applying linear Tikhonov regularization with convex
constraints to the Newton equations in each iteration step. Convergence of this
iterative regularization method is analyzed if both the operator and the right
hand side are given with errors and all error levels tend to zero. Our study
has been motivated by the joint estimation of object and phase in 4Pi
microscopy, which leads to a semi-blind deconvolution problem with
nonnegativity constraints. The performance of the proposed algorithm is
illustrated both for simulated and for three-dimensional experimental data
Necessary conditions for variational regularization schemes
We study variational regularization methods in a general framework, more
precisely those methods that use a discrepancy and a regularization functional.
While several sets of sufficient conditions are known to obtain a
regularization method, we start with an investigation of the converse question:
How could necessary conditions for a variational method to provide a
regularization method look like? To this end, we formalize the notion of a
variational scheme and start with comparison of three different instances of
variational methods. Then we focus on the data space model and investigate the
role and interplay of the topological structure, the convergence notion and the
discrepancy functional. Especially, we deduce necessary conditions for the
discrepancy functional to fulfill usual continuity assumptions. The results are
applied to discrepancy functionals given by Bregman distances and especially to
the Kullback-Leibler divergence.Comment: To appear in Inverse Problem
Relaxation and reconstruction on (111) surfaces of Au, Pt, and Cu
We have theoretically studied the stability and reconstruction of (111)
surfaces of Au, Pt, and Cu. We have calculated the surface energy, surface
stress, interatomic force constants, and other relevant quantities by ab initio
electronic structure calculations using the density functional theory (DFT), in
a slab geometry with periodic boundary conditions. We have estimated the
stability towards a quasi-one-dimensional reconstruction by using the
calculated quantities as parameters in a one-dimensional Frenkel-Kontorova
model. On all surfaces we have found an intrinsic tensile stress. This stress
is large enough on Au and Pt surfaces to lead to a reconstruction in which a
denser surface layer is formed, in agreement with experiment. The
experimentally observed differences between the dense reconstruction pattern on
Au(111) and a sparse structure of stripes on Pt(111) are attributed to the
details of the interaction potential between the first layer of atoms and the
substrate.Comment: 8 pages, 3 figures, submitted to Physical Review
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