Institut de Mathématiques de Toulouse

On considère l’opérateur elliptique <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"> <mml:mrow> <mml:mi>L</mml:mi> <mml:mi>u</mml:mi> <mml:mo>=</mml:mo> <mml:mo>-</mml:mo> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>a</mml:mi> <mml:mrow> <mml:mi>i</mml:mi> <mml:mi>j</mml:mi> </mml:mrow> </mml:msub> <mml:msub> <mml:mi>u</mml:mi> <mml:msub> <mml:mi>x</mml:mi> <mml:mi>i</mml:mi> </mml:msub> </mml:msub> <mml:mo>+</mml:mo> <mml:msub> <mml:mi>d</mml:mi> <mml:mi>j</mml:mi> </mml:msub> <mml:mi>u</mml:mi> <mml:msub> <mml:mo>)</mml:mo> <mml:msub> <mml:mi>x</mml:mi> <mml:mi>j</mml:mi> </mml:msub> </mml:msub> <mml:mo>+</mml:mo> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>b</mml:mi> <mml:mi>i</mml:mi> </mml:msub> <mml:msub> <mml:mi>u</mml:mi> <mml:msub> <mml:mi>x</mml:mi> <mml:mi>i</mml:mi> </mml:msub> </mml:msub> <mml:mo>+</mml:mo> <mml:mi>c</mml:mi> <mml:mi>u</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> où les coefficients sont des fonctions mesurables appartenant à des espaces <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mi>L</mml:mi> <mml:mo>*</mml:mo> </mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>Ω</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> convenables dans un ouvert borné <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Ω</mml:mi> </mml:math> de <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi mathvariant="bold">R</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> </mml:math> . Le but principal est d’étendre un résultat [par W. Littman, G. Stampacchia et H. Weinberger] sur les points réguliers pour le problème de Dirichlet à des équations plus générales (§10). Le paragraphe 3 contient aussi un principe de maximum pour les solutions faibles. Le paragraphe 4 contient des majorations a priori dans <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mi>L</mml:mi> <mml:mi>p</mml:mi> </mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>Ω</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> des solutions. Les paragraphes 5 et 7 sont consacrés à l’extension du théorème de Giorgi sur la continuité höldérienne des solutions ; l’extension au bord est aussi envisagée. La généralisation d’un théorème de Moser sur l’inégalité de Harnack est considérée dans le paragraphe 8. Le paragraphe 9 contient l’étude de la fonction de Green.

We consider the Dirichlet Laplacian operator<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M1"><mml:mrow><mml:mo>−</mml:mo><mml:mi mathvariant="normal">Δ</mml:mi></mml:mrow></mml:math>on a curved quantum guide in<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M2"><mml:mrow><mml:msup><mml:mi>ℝ </mml:mi><mml:mi>n</mml:mi></mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mi>n</mml:mi><mml:mo>=</mml:mo><mml:mn>2</mml:mn><mml:mo>,</mml:mo><mml:mn>3</mml:mn></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:math>with an asymptotically straight reference curve. We give uniqueness results for the inverse problem associated to the reconstruction of the curvature by using either observations of spectral data or a boot-strapping method.

Many important applications in global optimization, algebra, probability and statistics, applied mathematics, control theory, financial mathematics, inverse problems, etc. can be modeled as a particular instance of the Generalized Moment Problem (GMP) . This book introduces a new general methodology to solve the GMP when its data are polynomials and basic semi-algebraic sets. This methodology combines semidefinite programming with recent results from real algebraic geometry to provide a hierarchy of semidefinite relaxations converging to the desired optimal value. Applied on appropriate cones,

The stochastic multi-armed bandit model is a simple abstraction that has proven useful in many different contexts in statistics and machine learning. Whereas the achievable limit in terms of regret minimization is now well known, our aim is to contribute to a better understanding of the performance in terms of identifying the m best arms. We introduce generic notions of complexity for the two dominant frameworks considered in the literature: fixed-budget and fixed-confidence settings. In the fixed-confidence setting, we provide the first known distribution-dependent lower bound on the complexity that involves information-theoretic quantities and holds when m is larger than 1 under general assumptions. In the specific case of two armed-bandits, we derive refined lower bounds in both the fixed-confidence and fixed-budget settings, along with matching algorithms for Gaussian and Bernoulli bandit models. These results show in particular that the complexity of the fixed-budget setting may be smaller than the complexity of the fixed-confidence setting, contradicting the familiar behavior observed when testing fully specified alternatives. In addition, we also provide improved sequential stopping rules that have guaranteed error probabilities and shorter average running times. The proofs rely on two technical results that are of independent interest : a deviation lemma for self-normalized sums (Lemma 19) and a novel change of measure inequality for bandit models (Lemma 1).

International audience

Recent biotechnology advances allow for multiple types of omics data, such as transcriptomic, proteomic or metabolomic data sets to be integrated. The problem of feature selection has been addressed several times in the context of classification, but needs to be handled in a specific manner when integrating data. In this study, we focus on the integration of two-block data that are measured on the same samples. Our goal is to combine integration and simultaneous variable selection of the two data sets in a one-step procedure using a Partial Least Squares regression (PLS) variant to facilitate the biologists' interpretation. A novel computational methodology called ;;sparse PLS" is introduced for a predictive analysis to deal with these newly arisen problems. The sparsity of our approach is achieved with a Lasso penalization of the PLS loading vectors when computing the Singular Value Decomposition. Sparse PLS is shown to be effective and biologically meaningful. Comparisons with classical PLS are performed on a simulated data set and on real data sets. On one data set, a thorough biological interpretation of the obtained results is provided. We show that sparse PLS provides a valuable variable selection tool for highly dimensional data sets.

We describe a major update of our Matlab freeware GloptiPoly for parsing generalized problems of moments and solving them numerically with semidefinite programming.

A timely and comprehensive treatment of random field theory with applications across diverse areas of study Level Sets and Extrema of Random Processes and Fields discusses how to understand the properties of the level sets of paths as well as how to compute the probability distribution of its extremal values, which are two general classes of problems that arise in the study of random processes and fields and in related applications. This book provides a unified and accessible approach to these two topics and their relationship to classical theory and Gaussian processes and fields, and the most modern research findings are also discussed. The authors begin with an introduction to the basic concepts of stochastic processes, including a modern review of Gaussian fields and their classical inequalities. Subsequent chapters are devoted to Rice formulas, regularity properties, and recent results on the tails of the distribution of the maximum. Finally, applications of random fields to various areas of mathematics are provided, specifically to systems of random equations and condition numbers of random matrices. Throughout the book, applications are illustrated from various areas of study such as statistics, genomics, and oceanography while other results are relevant to econometrics, engineering, and mathematical physics. The presented material is reinforced by end-of-chapter exercises that range in varying degrees of difficulty. Most fundamental topics are addressed in the book, and an extensive, up-to-date bibliography directs readers to existing literature for further study. Level Sets and Extrema of Random Processes and Fields is an excellent book for courses on probability theory, spatial statistics, Gaussian fields, and probabilistic methods in real computation at the upper-undergraduate and graduate levels. It is also a valuable reference for professionals in mathematics and applied fields such as statistics, engineering, econometrics, mathematical physics, and biology.

In this survey we consider the development and mathematical analysis of numerical methods for kinetic partial differential equations. Kinetic equations represent a way of describing the time evolution of a system consisting of a large number of particles. Due to the high number of dimensions and their intrinsic physical properties, the construction of numerical methods represents a challenge and requires a careful balance between accuracy and computational complexity. Here we review the basic numerical techniques for dealing with such equations, including the case of semi-Lagrangian methods, discrete-velocity models and spectral methods. In addition we give an overview of the current state of the art of numerical methods for kinetic equations. This covers the derivation of fast algorithms, the notion of asymptotic-preserving methods and the construction of hybrid schemes.

The effectiveness of most cancer targeted therapies is short-lived. Tumors often develop resistance that might be overcome with drug combinations. However, the number of possible combinations is vast, necessitating data-driven approaches to find optimal patient-specific treatments. Here we report AstraZeneca's large drug combination dataset, consisting of 11,576 experiments from 910 combinations across 85 molecularly characterized cancer cell lines, and results of a DREAM Challenge to evaluate computational strategies for predicting synergistic drug pairs and biomarkers. 160 teams participated to provide a comprehensive methodological development and benchmarking. Winning methods incorporate prior knowledge of drug-target interactions. Synergy is predicted with an accuracy matching biological replicates for >60% of combinations. However, 20% of drug combinations are poorly predicted by all methods. Genomic rationale for synergy predictions are identified, including ADAM17 inhibitor antagonism when combined with PIK3CB/D inhibition contrasting to synergy when combined with other PI3K-pathway inhibitors in PIK3CA mutant cells.

BACKGROUND: Each omics platform is now able to generate a large amount of data. Genomics, proteomics, metabolomics, interactomics are compiled at an ever increasing pace and now form a core part of the fundamental systems biology framework. Recently, several integrative approaches have been proposed to extract meaningful information. However, these approaches lack of visualisation outputs to fully unravel the complex associations between different biological entities. RESULTS: The multivariate statistical approaches 'regularized Canonical Correlation Analysis' and 'sparse Partial Least Squares regression' were recently developed to integrate two types of highly dimensional 'omics' data and to select relevant information. Using the results of these methods, we propose to revisit few graphical outputs to better understand the relationships between two 'omics' data and to better visualise the correlation structure between the different biological entities. These graphical outputs include Correlation Circle plots, Relevance Networks and Clustered Image Maps. We demonstrate the usefulness of such graphical outputs on several biological data sets and further assess their biological relevance using gene ontology analysis. CONCLUSIONS: Such graphical outputs are undoubtedly useful to aid the interpretation of these promising integrative analysis tools and will certainly help in addressing fundamental biological questions and understanding systems as a whole. AVAILABILITY: The graphical tools described in this paper are implemented in the freely available R package mixOmics and in its associated web application.

We study degenerate complex Monge-Ampère equations of the form <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis omega plus d d Superscript c Baseline phi right-parenthesis Superscript n Baseline equals e Superscript t phi Baseline mu"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi> ω </mml:mi> <mml:mo>+</mml:mo> <mml:mi>d</mml:mi> <mml:msup> <mml:mi>d</mml:mi> <mml:mi>c</mml:mi> </mml:msup> <mml:mi> φ </mml:mi> <mml:msup> <mml:mo stretchy="false">)</mml:mo> <mml:mi>n</mml:mi> </mml:msup> <mml:mo>=</mml:mo> <mml:msup> <mml:mi>e</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>t</mml:mi> <mml:mi> φ </mml:mi> </mml:mrow> </mml:msup> <mml:mi> μ </mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">(\omega +dd^c\varphi )^n = e^{t \varphi }\mu</mml:annotation> </mml:semantics> </mml:math> </inline-formula> where <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="omega"> <mml:semantics> <mml:mi> ω </mml:mi> <mml:annotation encoding="application/x-tex">\omega</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a big semi-positive form on a compact Kähler manifold <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X"> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding="application/x-tex">X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of dimension <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n"> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding="application/x-tex">n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="t element-of double-struck upper R Superscript plus"> <mml:semantics> <mml:mrow> <mml:mi>t</mml:mi> <mml:mo> ∈ </mml:mo> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mo>+</mml:mo> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">t \in \mathbb {R}^+</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="mu equals f omega Superscript n"> <mml:semantics> <mml:mrow> <mml:mi> μ </mml:mi> <mml:mo>=</mml:mo> <mml:mi>f</mml:mi> <mml:msup> <mml:mi> ω </mml:mi> <mml:mi>n</mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">\mu =f\omega ^n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a positive measure with density <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f element-of upper L Superscript p Baseline left-parenthesis upper X comma omega Superscript n Baseline right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo> ∈ </mml:mo> <mml:msup> <mml:mi>L</mml:mi> <mml:mi>p</mml:mi> </mml:msup> <mml:mo stretchy="false">(</mml:mo> <mml:mi>X</mml:mi> <mml:mo>,</mml:mo> <mml:msup> <mml:mi> ω </mml:mi> <mml:mi>n</mml:mi> </mml:msup> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">f\in L^p(X,\omega ^n)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p greater-than 1"> <mml:semantics> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>&gt;</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">p&gt;1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . We prove the existence and unicity of bounded <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="omega"> <mml:semantics> <mml:mi> ω </mml:mi> <mml:annotation encoding="application/x-tex">\omega</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -plurisubharmonic solutions. We also prove that the solution is continuous under a further technical condition. In case <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X"> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding="application/x-tex">X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is projective and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="omega equals psi Superscript asterisk Baseline omega prime"> <mml:semantics> <mml:mrow> <mml:mi> ω </mml:mi> <mml:mo>=</mml:mo> <mml:msup> <mml:mi> ψ </mml:mi>

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We consider optimal sequential allocation in the context of the so-called stochastic multi-armed bandit model. We describe a generic index policy, in the sense of Gittins [J. R. Stat. Soc. Ser. B Stat. Methodol. 41 (1979) 148–177], based on upper confidence bounds of the arm payoffs computed using the Kullback–Leibler divergence. We consider two classes of distributions for which instances of this general idea are analyzed: the kl-UCB algorithm is designed for one-parameter exponential families and the empirical KL-UCB algorithm for bounded and finitely supported distributions. Our main contribution is a unified finite-time analysis of the regret of these algorithms that asymptotically matches the lower bounds of Lai and Robbins [Adv. in Appl. Math. 6 (1985) 4–22] and Burnetas and Katehakis [Adv. in Appl. Math. 17 (1996) 122–142], respectively. We also investigate the behavior of these algorithms when used with general bounded rewards, showing in particular that they provide significant improvements over the state-of-the-art.

International audience

Canonical correlations analysis (CCA) is an exploratory statistical method to highlight correlations between two data sets acquired on the same experimental units. The <code>cancor()</code> function in R (R Development Core Team 2007) performs the core of computations but further work was required to provide the user with additional tools to facilitate the interpretation of the results. We implemented an R package, CCA, freely available from the Comprehensive R Archive Network (CRAN, http://CRAN.R-project.org/), to develop numerical and graphical outputs and to enable the user to handle missing values. The CCA package also includes a regularized version of CCA to deal with data sets with more variables than units. Illustrations are given through the analysis of a data set coming from a nutrigenomic study in the mouse.

Canonical correlations analysis (CCA) is an exploratory statistical method to highlight correlations between two data sets acquired on the same experimental units. The cancor() function in R (R Development Core Team 2007) performs the core of computations but further work was required to provide the user with additional tools to facilitate the interpretation of the results. We implemented an R package, CCA, freely available from the Comprehensive R Archive Network (CRAN, http://CRAN.R-project.org/), to develop numerical and graphical outputs and to enable the user to handle missing values. The CCA package also includes a regularized version of CCA to deal with data sets with more variables than units. Illustrations are given through the analysis of a data set coming from a nutrigenomic study in the mouse.

We prove local existence of smooth solutions for large data and global smooth solutions for small data to the incompressible, resistive, viscous or inviscid Hall-MHD model. We also show a Liouville theorem for the stationary solutions.

Abstract The Surface Water and Ocean Topography (SWOT) satellite mission planned for launch in 2020 will map river elevations and inundated area globally for rivers &gt;100 m wide. In advance of this launch, we here evaluated the possibility of estimating discharge in ungauged rivers using synthetic, daily “remote sensing” measurements derived from hydraulic models corrupted with minimal observational errors. Five discharge algorithms were evaluated, as well as the median of the five, for 19 rivers spanning a range of hydraulic and geomorphic conditions. Reliance upon a priori information, and thus applicability to truly ungauged reaches, varied among algorithms: one algorithm employed only global limits on velocity and depth, while the other algorithms relied on globally available prior estimates of discharge. We found at least one algorithm able to estimate instantaneous discharge to within 35% relative root‐mean‐squared error (RRMSE) on 14/16 nonbraided rivers despite out‐of‐bank flows, multichannel planforms, and backwater effects. Moreover, we found RRMSE was often dominated by bias; the median standard deviation of relative residuals across the 16 nonbraided rivers was only 12.5%. SWOT discharge algorithm progress is therefore encouraging, yet future efforts should consider incorporating ancillary data or multialgorithm synergy to improve results.

Many mathematical models involve input parameters, which are not precisely known. Global sensitivity analysis aims to identify the parameters whose uncertainty has the largest impact on the variability of a quantity of interest (output of the model). One of the statistical tools used to quantify the influence of each input variable on the output is the Sobol sensitivity index. We consider the statistical estimation of this index from a finite sample of model outputs: we present two estimators and state a central limit theorem for each. We show that one of these estimators has an optimal asymptotic variance. We also generalize our results to the case where the true output is not observable, and is replaced by a noisy version.