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

Statistical analysis of the individual variability of 1D protein profiles as a tool in ecology: an application to parasitoid venom

International audienceUnderstanding the forces that shape eco-evolutionary patterns often requires linking phenotypes to genotypes, allowing characterization of these patterns at the molecular level. DNA-based markers are less informative in this aim compared to markers associated with gene expression and, more specifically, with protein quantities. The characterization of eco-evolutionary patterns also usually requires the analysis of large sample sizes to accurately estimate interindividual variability. However, the methods used to characterize and compare protein samples are generally expensive and time-consuming, which constrains the size of the produced data sets to few individuals. We present here a method that estimates the interindividual variability of protein quantities based on a global, semi-automatic analysis of 1D electrophoretic profiles, opening the way to rapid analysis and comparison of hundreds of individuals. The main original features of the method are the in silico normalization of sample protein quantities using pictures of electrophoresis gels at different staining levels, as well as a new method of analysis of electrophoretic profiles based on a median profile. We demonstrate that this method can accurately discriminate between species and between geographically distant or close populations, based on interindividual variation in venom protein profiles from three endoparasitoid wasps of two different genera (Psyttalia concolor, Psyttalia lounsburyi and Leptopili-na boulardi). Finally, we discuss the experimental designs that would benefit from the use of this method

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

Quinuclidine compounds differently act as agonists of Kenyon cell nicotinic acetylcholine receptors and induced distinct effect on insect ganglionic depolarizations

We have recently demonstrated that a new quinuclidine benzamide compound named LMA10203 acted as an agonist of insect nicotinic acetylcholine receptors. Its specific pharmacological profile on cockroach dorsal unpaired median neurons (DUM) helped to identify alpha-bungarotoxin-insensitive nAChR2 receptors. In the present study, we tested its effect on cockroach Kenyon cells. We found that it induced an inward current demonstrating that it bounds to nicotinic acetylcholine receptors expressed on Kenyon cells. Interestingly, LMA10203-induced currents were completely blocked by the nicotinic antagonist alpha-bungarotoxin. We suggested that LMA10203 effect occurred through the activation of alpha-bungarotoxin-sensitive receptors and did not involve alpha-bungarotoxin-insensitive nAChR2, previously identified in DUM neurons. In addition, we have synthesized two new compounds, LMA10210 and LMA10211, and compared their effects on Kenyon cells. These compounds were members of the 3-quinuclidinyl benzamide or benzoate families. Interestingly, 1 mM LMA10210 was not able to induce an inward current on Kenyon cells compared to LMA10211. Similarly, we did not find any significant effect of LMA10210 on cockroach ganglionic depolarization, whereas these three compounds were able to induce an effect on the central nervous system of the third instar M. domestica larvae. Our data suggested that these three compounds could bind to distinct cockroach nicotinic acetylcholine receptors

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 $F(u)=g$ in Banach spaces where the Newton equations $F'(u_n;u_{n+1}-u_n) = g-F(u_n)$ 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

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