437 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
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
Evaluation de la contribution de l'agriculture naturelle à la sécurité alimentaire au Cameroun par l'élicitation probabiliste du jugement d'experts
De nombreux agriculteurs Camerounais pratiquent l'agriculture naturelle, un type d'agriculture potentiellement qualifiable de " biologique ". Pourtant, peu de rĂ©fĂ©rences expĂ©rimentales sont disponibles sur sa capacitĂ© Ă contribuer aux enjeux de la sĂ©curitĂ© alimentaire au Cameroun. De ce fait, la question de sa performance par rapport Ă l'agriculture conventionnelle reste cruciale pour sa reconnaissance institutionnelle. L'objectif est d'identifier des situations (rĂ©gions et espĂšces cultivĂ©es) oĂč les performances de l'agriculture naturelle sont proches ou supĂ©rieures Ă celle de l'agriculture conventionnelle. Pour atteindre cet objectif, nous avons mobilisĂ© l'Ă©licitation probabiliste des jugements d'experts. Cette mĂ©thode est appliquĂ©e auprĂšs de 56 experts dans la rĂ©gion du Centre Cameroun. Les rĂ©sultats montrent que les rendements en agriculture naturelle sont infĂ©rieurs de 28% Ă ceux conventionnelle tandis que pour les prix bord champ des spĂ©culations en agriculture naturelle sont sensiblement identiques de celles en conventionnelle. De plus, le temps de travail exprimĂ© en nombre d'heures/jour est Ă©levĂ© dans le systĂšme naturel par rapport au systĂšme conventionnel. Par ailleurs, la quantitĂ© de travail exprimĂ© en nombre d'hommes/jour est plus faible en systĂšme naturel
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
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
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