400 research outputs found
Global Saturation of Regularization Methods for Inverse Ill-Posed Problems
In this article the concept of saturation of an arbitrary regularization
method is formalized based upon the original idea of saturation for spectral
regularization methods introduced by A. Neubauer in 1994. Necessary and
sufficient conditions for a regularization method to have global saturation are
provided. It is shown that for a method to have global saturation the total
error must be optimal in two senses, namely as optimal order of convergence
over a certain set which at the same time, must be optimal (in a very precise
sense) with respect to the error. Finally, two converse results are proved and
the theory is applied to find sufficient conditions which ensure the existence
of global saturation for spectral methods with classical qualification of
finite positive order and for methods with maximal qualification. Finally,
several examples of regularization methods possessing global saturation are
shown.Comment: 29 page
Generalized Qualification and Qualification Levels for Spectral Regularization Methods
The concept of qualification for spectral regularization methods for inverse
ill-posed problems is strongly associated to the optimal order of convergence
of the regularization error. In this article, the definition of qualification
is extended and three different levels are introduced: weak, strong and
optimal. It is shown that the weak qualification extends the definition
introduced by Mathe and Pereverzev in 2003, mainly in the sense that the
functions associated to orders of convergence and source sets need not be the
same. It is shown that certain methods possessing infinite classical
qualification, e.g. truncated singular value decomposition (TSVD), Landweber's
method and Showalter's method, also have generalized qualification leading to
an optimal order of convergence of the regularization error. Sufficient
conditions for a SRM to have weak qualification are provided and necessary and
sufficient conditions for a given order of convergence to be strong or optimal
qualification are found. Examples of all three qualification levels are
provided and the relationships between them as well as with the classical
concept of qualification and the qualification introduced by Mathe and
Perevezev are shown. In particular, spectral regularization methods having
extended qualification in each one of the three levels and having zero or
infinite classical qualification are presented. Finally several implications of
this theory in the context of orders of convergence, converse results and
maximal source sets for inverse ill-posed problems, are shown.Comment: 20 pages, 1 figur
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
Error bounds for computing the expectation by Markov chain Monte Carlo
We study the error of reversible Markov chain Monte Carlo methods for
approximating the expectation of a function. Explicit error bounds with respect
to different norms of the function are proven. By the estimation the well known
asymptotical limit of the error is attained, i.e. there is no gap between the
estimate and the asymptotical behavior. We discuss the dependence of the error
on a burn-in of the Markov chain. Furthermore we suggest and justify a specific
burn-in for optimizing the algorithm
Regularization of statistical inverse problems and the Bakushinskii veto
In the deterministic context Bakushinskii's theorem excludes the existence of
purely data driven convergent regularization for ill-posed problems. We will
prove in the present work that in the statistical setting we can either
construct a counter example or develop an equivalent formulation depending on
the considered class of probability distributions. Hence, Bakushinskii's
theorem does not generalize to the statistical context, although this has often
been assumed in the past. To arrive at this conclusion, we will deduce from the
classic theory new concepts for a general study of statistical inverse problems
and perform a systematic clarification of the key ideas of statistical
regularization.Comment: 20 page
General regularization schemes for signal detection in inverse problems
The authors discuss how general regularization schemes, in particular linear regularization schemes and projection schemes, can be used to design tests for signal detection in statistical inverse problems. It is shown that such tests can attain the minimax separation rates when the regularization parameter is chosen appropriately. It is also shown how to modify these tests in order to obtain (up to a factor) a test which adapts to the unknown smoothness in the alternative. Moreover, the authors discuss how the so-called \emph{direct} and \emph{indirect} tests are related via interpolation properties
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
Self-energy limited ion transport in sub-nanometer channels
The current-voltage characteristics of the alpha-Hemolysin protein pore
during the passage of single-stranded DNA under varying ionic strength, C, are
studied experimentally. We observe strong blockage of the current, weak
super-linear growth of the current as a function of voltage, and a minimum of
the current as a function of C. These observations are interpreted as the
result of the ion electrostatic self-energy barrier originating from the large
difference in the dielectric constants of water and the lipid bilayer. The
dependence of DNA capture rate on C also agrees with our model.Comment: more experimental material is added. 4 pages, 7 figure
Digging into acceptor splice site prediction : an iterative feature selection approach
Feature selection techniques are often used to reduce data dimensionality, increase classification performance, and gain insight into the processes that generated the data. In this paper, we describe an iterative procedure of feature selection and feature construction steps, improving the classification of acceptor splice sites, an important subtask of gene prediction.
We show that acceptor prediction can benefit from feature selection, and describe how feature selection techniques can be used to gain new insights in the classification of acceptor sites. This is illustrated by the identification of a new, biologically motivated feature: the AG-scanning feature.
The results described in this paper contribute both to the domain of gene prediction, and to research in feature selection techniques, describing a new wrapper based feature weighting method that aids in knowledge discovery when dealing with complex datasets
Innovation et gouvernance territoriale : une analyse par les dispositifs
International audienceCette communication vise à présenter les outils méthodologiques d'analyse/évaluation de la gouvernance territoriale élaborés dans le cadre du projet de recherche PSDR Gouv.Innov sur les innovations organisationnelles relatives à la gouvernance territoriale. Il s'agit d'étudier les transformations introduites par les politiques de développement durable au niveau des dispositifs de gouvernance territoriale visant à favoriser une gestion intégrée des espaces ruraux. Dans un contexte de recomposition de l'action publique où les procédures d'aménagement sont plutôt normées par des représentations urbaines, l'accent est mis sur la question des modalités de représentation des activités rurales, pour lesquelles nous faisons l'hypothèse qu'elles sont sous représentées. Les premiers résultats méthodologiques permettent, dans une première partie, de proposer une définition générique et pragmatique de la gouvernance territoriale et de préciser la notion de dispositifs de gouvernance comme objet d'observation. A partir de cette définition une grille d'analyse permettant d'appréhender l'ensemble des dimensions en jeu dans les processus de gouvernance territoriale est élaborée. Dans la deuxième partie nous explorons l'intérêt de la notion de dispositif pour observer les processus de gouvernance et proposons une grille de collecte et de structuration des informations pour constituer des chroniques des dispositifs étudiés. / This paper seeks to present the methodological tools used in the analysis/evaluation of territorial governance developed during research undertaken within the framework of the PSDR Gouv-Innov project on organisational innovations with respect to territorial governance. It involves the study of the changes resulting from the emergence of sustainable development policies in territorial governance systems aiming to implement the integrated management of rural areas. In a context of changes in the way in which public action is implemented (for instance, public-private partnerships), where development procedures are usually standardised by urban representations, the emphasis is placed on the issue of the representation of rural activities, the hypothesis being that they are under-represented. The initial methodological results lead, in the first instance, to a proposition for a generic and pragmatic definition of territorial governance and a clarification of the notion of governance systems as objects of observation. This definition can be used as the basis of a framework to address all the analytical dimensions of the territorial governance process. In the second part, the authors explore the idea of system in order to observe governance processes and suggest a matrix for the collection and structuring of information in order to report on the studied systems
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