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 références expérimentales sont disponibles sur sa capacité à contribuer aux enjeux de la sécurité alimentaire au Cameroun. De ce fait, la question de sa performance par rapport à l'agriculture conventionnelle reste cruciale pour sa reconnaissance institutionnelle. L'objectif est d'identifier des situations (régions et espèces cultivées) où les performances de l'agriculture naturelle sont proches ou supérieures à celle de l'agriculture conventionnelle. Pour atteindre cet objectif, nous avons mobilisé l'élicitation probabiliste des jugements d'experts. Cette méthode est appliquée auprès de 56 experts dans la région du Centre Cameroun. Les résultats montrent que les rendements en agriculture naturelle sont inférieurs de 28% à ceux conventionnelle tandis que pour les prix bord champ des spéculations en agriculture naturelle sont sensiblement identiques de celles en conventionnelle. De plus, le temps de travail exprimé en nombre d'heures/jour est élevé dans le système naturel par rapport au système conventionnel. Par ailleurs, la quantité de travail exprimé en nombre d'hommes/jour est plus faible en système 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