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10. Probabilistic approach for granular media equations in the non uniformly convex case

Author

Cattiaux, Patrick, Arnaud, Guillin, and Florent, Malrieu

Subjects
Mathematics - Probability, 65C35, 35K55, 65C05, 82C22, 26D10, 60E15
Abstract

We use here a particle system to prove a convergence result as well as a deviation inequality for solutions of granular media equation when the confinement potential and the interaction potential are no more uniformly convex. Proof is straightforward, simplifying deeply proofs of Carrillo-McCann-Villani \cite{CMV,CMV2} and completing results of Malrieu \cite{malrieu03} in the uniformly convex case. It relies on an uniform propagation of chaos property and a direct control in Wasserstein distance of solutions starting with different initial measures. The deviation inequality is obtained via a $T\_1$ transportation cost inequality replacing the logarithmic Sobolev inequality which is no more clearly dimension free.

Published
2006

12. PDMP Characterisation of Event-Chain Monte Carlo Algorithms for Particle Systems

Author

Athina Monemvassitis, Arnaud Guillin, Manon Michel, Michel, Manon, Développer des algorithmes stochastiques scalables par non-réversibilité, parallélisation et schémas adaptifs - - SuSa2020 - ANR-20-CE46-0007 - AAPG2020 - VALID, Entropie, flots, inégalités - - EFI2017 - ANR-17-CE40-0030 - AAPG2017 - VALID, CAP 20-25 - - CAP 20-252016 - ANR-16-IDEX-0001 - IDEX - VALID, Laboratoire de Mathématiques Blaise Pascal (LMBP), Centre National de la Recherche Scientifique (CNRS)-Université Clermont Auvergne (UCA), Université Clermont Auvergne [2017-2020] (UCA [2017-2020])-Centre National de la Recherche Scientifique (CNRS), ANR-20-CE46-0007,SuSa,Développer des algorithmes stochastiques scalables par non-réversibilité, parallélisation et schémas adaptifs(2020), ANR-17-CE40-0030,EFI,Entropie, flots, inégalités(2017), and ANR-16-IDEX-0001,CAP 20-25,CAP 20-25(2016)

Subjects
[PHYS]Physics [physics], Particle systems, Statistical Mechanics (cond-mat.stat-mech), Event-chain algorithm, Probability (math.PR), Ergodicity, FOS: Physical sciences, Markov-chain Monte Carlo, Statistical and Nonlinear Physics, [MATH] Mathematics [math], [PHYS] Physics [physics], Piecewise deterministic Markov Process, FOS: Mathematics, [MATH]Mathematics [math], Condensed Matter - Statistical Mechanics, Mathematics - Probability, Mathematical Physics
Abstract

International audience; Monte Carlo simulations of systems of particles such as hard spheres or soft spheres with singular kernels can display around a phase transition prohibitively long convergence times when using traditional Hasting-Metropolis reversible schemes. Efficient algorithms known as event-chain Monte Carlo were then developed to reach necessary accelerations. They are based on non-reversible continuoustime Markov processes. Proving invariance and ergodicity for such schemes cannot be done as for discrete-time schemes and a theoretical framework to do so was lacking, impeding the generalisation of ECMC algorithms to more sophisticated systems or processes. In this work, we characterize the Markov processes generated in ECMC as piecewise deterministic Markov processes. It first allows us to propose more general schemes, for instance regarding the direction refreshment. We then prove the invariance of the correct stationary distribution. Finally, we show the ergodicity of the processes in soft-and hard-sphere systems, with a density condition for the latter.

Published
2023
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13. A generic deep-learning based defect segmentation model for electron micrographs for automatic defect inspection

Author

Julien Baderot, Ali Hallal, Martin Jacob, Vincent Barra, Arnaud Guillin, Sergio Martinez, Johann Foucher, BARRA, Vincent, POLLEN METROLOGY, Laboratoire d'Informatique, de Modélisation et d'Optimisation des Systèmes (LIMOS), Ecole Nationale Supérieure des Mines de St Etienne (ENSM ST-ETIENNE)-Centre National de la Recherche Scientifique (CNRS)-Université Clermont Auvergne (UCA)-Institut national polytechnique Clermont Auvergne (INP Clermont Auvergne), Université Clermont Auvergne (UCA)-Université Clermont Auvergne (UCA), Laboratoire de Mathématiques Blaise Pascal (LMBP), and Centre National de la Recherche Scientifique (CNRS)-Université Clermont Auvergne (UCA)

Subjects
[INFO.INFO-AI] Computer Science [cs]/Artificial Intelligence [cs.AI], [INFO.INFO-AI]Computer Science [cs]/Artificial Intelligence [cs.AI]
Published
2023

15. Long-runout pyroclastic density currents: Analysis and implications

Author

Olivier Roche, Christopher D. Henry, Nourddine Azzaoui, Arnaud Guillin, Laboratoire Magmas et Volcans (LMV), Institut national des sciences de l'Univers (INSU - CNRS)-Institut de Recherche pour le Développement et la société-Centre National de la Recherche Scientifique (CNRS)-Université Clermont Auvergne (UCA)-Observatoire de Physique du Globe de Clermont-Ferrand (OPGC), Institut national des sciences de l'Univers (INSU - CNRS)-Centre National de la Recherche Scientifique (CNRS)-Université Clermont Auvergne (UCA)-Institut national des sciences de l'Univers (INSU - CNRS)-Centre National de la Recherche Scientifique (CNRS)-Université Clermont Auvergne (UCA), Nevada Bureau of Mines and Geology, University of Nevada, Laboratoire de Mathématiques Blaise Pascal (LMBP), Centre National de la Recherche Scientifique (CNRS)-Université Clermont Auvergne (UCA), ANR-10-LABX-0006,CLERVOLC,Clermont-Ferrand centre for research on volcanism(2010), and ANR-16-IDEX-0001,CAP 20-25,CAP 20-25(2016)

Subjects
[SDU.STU.VO]Sciences of the Universe [physics]/Earth Sciences/Volcanology, Geology
Abstract

Pyroclastic density currents are hazardous, ground-hugging, hot mixtures of gas and solid particles produced by volcanoes. Currents generated by the largest explosive eruptions have traveled distances on the order of 100 km, and their devastating impact has repeatedly marked Earth's geologic history. We show that pyroclastic density currents from super-eruptions during the Oligocene in the Great Basin of the United States had exceptional runout distances that may have exceeded 300 km. We present a quantitative analysis of the data from these currents and consider, in particular, the areal extent of their deposits (ignimbrites) as well as the relationship between their runout and the eruption mass discharge rate. The ignimbrites have elliptical distributions characterized by axis length ratios of ~2–6, in contrast to common subcircular ignimbrites, while the maximum runouts versus the discharge rates are clearly outside of the prediction intervals defined by other eruption data. We argue that the long runouts resulted from the channeling of concentrated currents in regional paleovalleys whose gentle slope worked to lengthen the travel distance. Our study demonstrates that these additional factors should be considered in assessing hazards posed by future super-eruptions.

Published
2022
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16. Low-Lying Eigenvalues and Convergence to the Equilibrium of Some Piecewise Deterministic Markov Processes Generators in the Small Temperature Regime

Author

Boris Nectoux, Arnaud Guillin, Laboratoire de Mathématiques Blaise Pascal (LMBP), Université Blaise Pascal - Clermont-Ferrand 2 (UBP)-Centre National de la Recherche Scientifique (CNRS), and Université Clermont Auvergne [2017-2020] (UCA [2017-2020])-Centre National de la Recherche Scientifique (CNRS)

Subjects
Nuclear and High Energy Physics, Work (thermodynamics), Spectral theory, Markov process, 01 natural sciences, metastability, symbols.namesake, small temperature regime, [MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph], Metastability, 0103 physical sciences, Convergence (routing), [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Statistical physics, [MATH]Mathematics [math], 0101 mathematics, Mathematical Physics, Eigenvalues and eigenvectors, Mathematics, 010102 general mathematics, Statistical and Nonlinear Physics, spectral theory, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], Piecewise, symbols, Condensed Matter::Strongly Correlated Electrons, 010307 mathematical physics, semiclassical analysis, Piecewise Deterministic Markov Processes, [MATH.MATH-SP]Mathematics [math]/Spectral Theory [math.SP]
Abstract

International audience; In this work we study the number of small eigenvalues and the convergence to the equilibrium of the Bouncy Particle Sampler process and the Zig-Zag process generators in the small temperature regime. Such processes, which fall in the class of Piecewise Deterministic Markov Processes, are non diffusive and non reversible.

Published
2020
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18. Tephra Fallout Probabilistic Hazard Maps for Cotopaxi and Guagua Pichincha Volcanoes (Ecuador) With Uncertainty Quantification

Author

Andrea Bevilacqua, Benjamin Bernard, Arnaud Guillin, Silvana Hidalgo, Olivier Roche, Mathieu Gouhier, Alessandro Tadini, Nourddine Azzaoui, Pablo Samaniego, Laboratoire Magmas et Volcans (LMV), Institut national des sciences de l'Univers (INSU - CNRS)-Institut de Recherche pour le Développement et la société-Centre National de la Recherche Scientifique (CNRS)-Université Clermont Auvergne (UCA)-Observatoire de Physique du Globe de Clermont-Ferrand (OPGC), Institut national des sciences de l'Univers (INSU - CNRS)-Centre National de la Recherche Scientifique (CNRS)-Université Clermont Auvergne (UCA)-Institut national des sciences de l'Univers (INSU - CNRS)-Centre National de la Recherche Scientifique (CNRS)-Université Clermont Auvergne (UCA), Laboratoire de Mathématiques Blaise Pascal (LMBP), Centre National de la Recherche Scientifique (CNRS)-Université Clermont Auvergne (UCA), Instituto Geofísico, Escuela Politécnica Nacional, Escuela Politécnica Nacional (EPN), Istituto Nazionale di Geofisica e Vulcanologia – Sezione di Pisa (INGV), Istituto Nazionale di Geofisica e Vulcanologia, ANR-10-LABX-0006,CLERVOLC,Clermont-Ferrand centre for research on volcanism(2010), and ANR-16-IDEX-0001,CAP 20-25,CAP 20-25(2016)

Subjects
geography, geography.geographical_feature_category, Cotopaxi, uncertainty quantification, Probabilistic logic, Numerical models, Hazard analysis, Hazard, Tephra fallout, Guagua Pichincha, Geophysics, Volcano, Space and Planetary Science, Geochemistry and Petrology, volcanic hazard assessment, Earth and Planetary Sciences (miscellaneous), [SDU.STU.VO]Sciences of the Universe [physics]/Earth Sciences/Volcanology, Environmental science, Physical geography, Uncertainty quantification, Tephra
Abstract

co-auteur étranger; International audience; Tephra fallout hazard assessment is undertaken with probabilistic maps that rely on numerical models. Regarding maps production, the input parameters of the model (including atmospheric conditions), the physical approximations of the numerical simulations, and the probabilities of occurrence of different eruption types in specific time frames are among the most critical sources of uncertainty. We therefore present a tephra fallout hazard assessment study for two active volcanoes (Cotopaxi and Guagua Pichincha) in Ecuador. We utilize PLUME-MoM/HYSPLIT models, and a procedure for uncertainty quantification where: i) the uncertainty on eruptive source parameters and eruption type occurrence is quantified through expert elicitation; ii) we implement a new procedure for correlations between the different parameters, and iii) we use correction coeffients to take into account the uncertainty of the numerical model. Maps of exceedance probability given a deposit thickness threshold, and thickness maps given a robability of exceedence, are produced 1) for two eruptive scenarios (sub-plinian and Plinian) and 2) as a combination of these scenarios in case the next eruption will be sub-Plinian or Plinian. These maps are described according to the uncertainty distribution of eruption type occurrence probabilities, considering their 5th percentile, mean, and 95th percentile values. We finally present hazard curves describing exceeding probabilities in 10 sensitive sites within the city of Quito. Additional information includes the areal extent and the population potentially affected by different isolines of tephra accumulation. This work indicates that full uncertainty quantification helps in providing more robust scientific information, improving the hazard assessment reliability.

Published
2022
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19. Statistical Analysis of Absenteeism in a University Hospital Center between 2007 and 2019

Author

Charlène Millot, Bruno Pereira, Sophie Miallaret, Maëlys Clinchamps, Luc Vialatte, Arnaud Guillin, Yan Bailly, Ukadike Chris Ugbolue, Valentin Navel, Julien Steven Baker, Jean-Baptiste Bouillon-Minois, Frédéric Dutheil, LAPSCO, HAL, Service Santé Travail Environnement [CHU Clermont-Ferrand], CHU Gabriel Montpied [Clermont-Ferrand], CHU Clermont-Ferrand-CHU Clermont-Ferrand, Unité de Biostatistiques [CHU Clermont-Ferrand], Direction de la recherche clinique et de l’innovation [CHU Clermont-Ferrand] (DRCI), Laboratoire de Mathématiques Blaise Pascal (LMBP), Centre National de la Recherche Scientifique (CNRS)-Université Clermont Auvergne (UCA), Laboratoire de Psychologie Sociale et Cognitive (LAPSCO), University of the West of Scotland (UWS), Génétique, Reproduction et Développement (GReD), Institut National de la Santé et de la Recherche Médicale (INSERM)-Centre National de la Recherche Scientifique (CNRS)-Université Clermont Auvergne (UCA), CHU Clermont-Ferrand, and Hong Kong Baptist University (HKBU)

Subjects
[SDV.MHEP] Life Sciences [q-bio]/Human health and pathology, Health, Toxicology and Mutagenesis, sociodemographic factors, Public Health, Environmental and Occupational Health, compressible absences, Middle Aged, Hospitals, absenteeism, [SDV.SPEE] Life Sciences [q-bio]/Santé publique et épidémiologie, Humans, Female, [SDV.SPEE]Life Sciences [q-bio]/Santé publique et épidémiologie, occupational factors, Occupations, hospital, Child, Workplace, Occupational Health, [SDV.MHEP]Life Sciences [q-bio]/Human health and pathology
Abstract

Objectives: To estimate the evolution of compressible absenteeism in a hospital center and identify the professional and sociodemographic factors that influence absenteeism. Method: All hospital center employees have been included over a period of twelve consecutive years (2007 to 2019). Compressible absences and occupational and sociodemographic factors were analyzed using Occupational Health data. Since the distribution of the data did not follow a normal distribution, the number of days of absence was presented as a median (interquartile range (IQR): 1st quartile–3rd quartile), and comparisons were made using non-parametric tests followed by a negative binomial model with zero inflation (ZINB). Results: A total of 16,413 employees were included, for a total of 2,828,599 days of absence, of which 2,081,553 were compressible absences (73.6% of total absences). Overall, 42% of employees have at least one absence per year. Absent employees had a median of 15 (IQR 5–53) days of absence per year, with an increase of a factor of 1.9 (CI95 1.8–2.1) between 2007 and 2019 (p < 0.001). Paramedical staff were most at risk of absence (p < 0.001 vs. all other occupational categories). Between 2007 and 2019, the number of days of absence was multiplied by 2.4 (CI95 1.8–3.1) for administrative staff, 2.1 (CI95 1.9–2.3) for tenured, 1.7 (CI95 1.5–2.0) for those living more than 12 km from the workplace, 1.8 (CI95 1.6–2.0) among women, 2.1 (CI95 1.8–2.6) among those over 50 years of age, 2.4 (CI95 1.8–3.0) among “separated” workers, and 2.0 (CI95 1.8–2.2) among those with at least one child. Conclusions: Paramedical personnel are most at risk of absenteeism. Meanwhile, absenteeism is increasing steadily, and overall, the increase is major for administrative staff. The profile of an employee at risk of absenteeism is a titular employee, living at distance from work, probably female, over 50 years old, separated, and with children. Identifying professionals at risk of absenteeism is essential to propose adapted and personalized preventive measures.

Published
2022
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20. Piecewise deterministic Markov processes and their invariant measures

Author

Arnaud Guillin, Alain Durmus, Pierre Monmarché, CB - Centre Borelli - UMR 9010 (CB), Service de Santé des Armées-Institut National de la Santé et de la Recherche Médicale (INSERM)-Université Paris-Saclay-Centre National de la Recherche Scientifique (CNRS)-Ecole Normale Supérieure Paris-Saclay (ENS Paris Saclay)-Université Paris Cité (UPCité), Laboratoire de Mathématiques Blaise Pascal (LMBP), Centre National de la Recherche Scientifique (CNRS)-Université Clermont Auvergne (UCA), Laboratoire Jacques-Louis Lions (LJLL (UMR_7598)), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité), Laboratoire de chimie théorique (LCT), Institut de Chimie du CNRS (INC)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS), ANR-17-CE40-0030,EFI,Entropie, flots, inégalités(2017), Centre de Mathématiques et de Leurs Applications (CMLA), École normale supérieure - Cachan (ENS Cachan)-Centre National de la Recherche Scientifique (CNRS), Université Blaise Pascal - Clermont-Ferrand 2 (UBP)-Centre National de la Recherche Scientifique (CNRS), Laboratoire Jacques-Louis Lions (LJLL), Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS), Durmus, Alain, and Université Clermont Auvergne [2017-2020] (UCA [2017-2020])-Centre National de la Recherche Scientifique (CNRS)

Subjects
FOS: Computer and information sciences, Statistics and Probability, [MATH.MATH-PR] Mathematics [math]/Probability [math.PR], Markov process, Statistics - Computation, 01 natural sciences, 010104 statistics & probability, symbols.namesake, FOS: Mathematics, Applied mathematics, 0101 mathematics, Invariant (mathematics), [MATH]Mathematics [math], Computation (stat.CO), Mathematics, Probability measure, Probability (math.PR), 010102 general mathematics, Markov chain Monte Carlo, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], Flow (mathematics), symbols, Piecewise, Invariant measure, Statistics, Probability and Uncertainty, Marginal distribution, Mathematics - Probability
Abstract

International audience; Piecewise Deterministic Markov Processes (PDMPs) are studied in a general framework. First, different constructions are proven to be equivalent. Second, we introduce a coupling between two PDMPs following the same differential flow which implies quantitative bounds on the total variation between the marginal distributions of the two processes. Finally two results are established regarding the invariant measures of PDMPs. A practical condition to show that a probability measure is invariant for the associated PDMP semi-group is presented. In a second time, a bound on the invariant probability measures in V-norm of two PDMPs following the same differential flow is established. This last result is then applied to study the asymptotic bias of some non-exact PDMP MCMC methods.; Nous établissons dans cette étude quelques propriétés générales des processus de Markov déterministes par morceaux (PDMP). Tout d’abord, plusieurs constructions de tels processus sont décrites. Ensuite, nous introduisons un couplage entre deux PDMP suivant la même flot différentiel impliquant des bornes quantitatives en variation totale entre les distributions marginales des deux processus. Enfin, nous nous intéressons aux PDMP ergodiques. Nous introduisons des conditions permettant de montrer que certaines classes de fonctions sont des cores pour le générateur d’un PDMP, ce qui permet de vérifier aisément qu’une distribution est stationnaire pour le PDMP considéré. De plus, nous établissons sous certaines hypothèses une borne quantitative sur la V-norme entre les mesures stationnaires de deux PDMP suivant le même flot différentiel. Ce dernier résultat est ensuite appliqué à l’étude du biais asymptotique de certaines méthodes de Monte Carlo par Processus de Markov inexactes.

Published
2021
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21. THE KINETIC FOKKER-PLANCK EQUATION WITH MEAN FIELD INTERACTION

Author

Arnaud Guillin, Wei Liu, Liming Wu, Chaoen Zhang, Laboratoire de Mathématiques Blaise Pascal (LMBP), Université Clermont Auvergne [2017-2020] (UCA [2017-2020])-Centre National de la Recherche Scientifique (CNRS), and Wuhan University [China]

Subjects
Lyapunov function, General Mathematics, Type (model theory), Kinetic energy, mean field interaction, [MATH.MATH-FA]Mathematics [math]/Functional Analysis [math.FA], 01 natural sciences, 010104 statistics & probability, symbols.namesake, Hypocoercivity, Mathematics - Analysis of PDEs, FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Limit (mathematics), 0101 mathematics, logarithmic Sobolev inequality, Mathematics, Lyapunov conditions, Applied Mathematics, 010102 general mathematics, Mathematical analysis, Probability (math.PR), Poincaré inequalities, Functional Analysis (math.FA), Sobolev space, Mathematics - Functional Analysis, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], Rate of convergence, Mean field theory, symbols, Fokker–Planck equation, Mathematics - Probability, Analysis of PDEs (math.AP)
Abstract

We study the long time behavior of the kinetic Fokker-Planck equation with mean field interaction, whose limit is often called Vlasov-Fokker-Planck equation. We prove a uniform (in the number of particles) exponential convergence to equilibrium for the solutions in the weighted Sobolev space H 1 ( μ ) with a rate of convergence which is explicitly computable and independent of the number of particles. The originality of the proof relies on functional inequalities and hypocoercivity with Lyapunov type conditions, usually not suitable to provide adimensional results.

Published
2021
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22. Penalised least square in sparse setting with convex penalty and non gaussian errors

Author

Nourddine Azzaoui, Guillaume Le Mailloux, Arnaud Guillin, Doualeh Abdillahi-Ali, Tomoko Matsui, Laboratoire de Mathématiques Blaise Pascal (LMBP), Centre National de la Recherche Scientifique (CNRS)-Université Clermont Auvergne (UCA), Ecole Normale Supérieure Paris-Saclay (ENS Paris Saclay), CB - Centre Borelli - UMR 9010 (CB), Service de Santé des Armées-Institut National de la Santé et de la Recherche Médicale (INSERM)-Université Paris-Saclay-Centre National de la Recherche Scientifique (CNRS)-Ecole Normale Supérieure Paris-Saclay (ENS Paris Saclay)-Université de Paris (UP), The Institute of Statistical Mathematics (Tokyo ), ANR-17-CE40-0030,EFI,Entropie, flots, inégalités(2017), Guillin, Arnaud, Entropie, flots, inégalités - - EFI2017 - ANR-17-CE40-0030 - AAPG2017 - VALID, Université Clermont Auvergne [2017-2020] (UCA [2017-2020])-Centre National de la Recherche Scientifique (CNRS), and Service de Santé des Armées-Institut National de la Santé et de la Recherche Médicale (INSERM)-Université Paris-Saclay-Centre National de la Recherche Scientifique (CNRS)-Ecole Normale Supérieure Paris-Saclay (ENS Paris Saclay)-Université Paris Cité (UPCité)

Subjects
Statistics::Theory, General Mathematics, Gaussian, Regular polygon, MathematicsofComputing_NUMERICALANALYSIS, General Physics and Astronomy, Estimator, Least squares, Noise, symbols.namesake, Lasso (statistics), [MATH.MATH-ST]Mathematics [math]/Statistics [math.ST], symbols, Applied mathematics, Spectral gap, [MATH.MATH-ST] Mathematics [math]/Statistics [math.ST], Mathematics, Probability measure
Abstract

International audience; This paper considers the penalized least squares estimators with convex penalties or regularisation norms. We provide sparsity oracles inequalities for the prediction error for a general convex penalty and for the particular cases of Lasso and Group Lasso estimators in a regression setting. The main contributions are that our oracle inequalities are established for the more general case where the observations noise is issued from probability measures that satisfy a weak spectral gap (or Poincaré) inequality instead of gaussian distributions, and five easier to verify bounds on compatibility. We Illustrate our results on a heavy tailed example and a sub gaussian one; we especially give the explicit bounds of the oracle inequalities for these two special examples.

Published
2021
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23. Eruption type probability and eruption source parameters at Cotopaxi and Guagua Pichincha volcanoes (Ecuador) with uncertainty quantification

Author

Andrea Bevilacqua, Silvana Hidalgo, Antonio Proaño, Nourddine Azzaoui, Hugo Yepes, Pablo Samaniego, Benjamin Bernard, Olivier Roche, Mia Pique, Augusto Neri, Elizabeth Gaunt, Alessandro Tadini, Willy Aspinall, Mattia de' Michieli Vitturi, Marjorie Encalada, Arnaud Guillin, Raffaello Cioni, Julia Eychenne, Marco Pistolesi, Mathieu Gouhier, Silvia Vallejo, Laboratoire Magmas et Volcans (LMV), Institut national des sciences de l'Univers (INSU - CNRS)-Institut de Recherche pour le Développement et la société-Centre National de la Recherche Scientifique (CNRS)-Université Clermont Auvergne (UCA)-Observatoire de Physique du Globe de Clermont-Ferrand (OPGC), Institut national des sciences de l'Univers (INSU - CNRS)-Centre National de la Recherche Scientifique (CNRS)-Université Clermont Auvergne (UCA)-Institut national des sciences de l'Univers (INSU - CNRS)-Centre National de la Recherche Scientifique (CNRS)-Université Clermont Auvergne (UCA), Instituto Geofísico, Escuela Politécnica Nacional, Laboratoire de Mathématiques Blaise Pascal (LMBP), Centre National de la Recherche Scientifique (CNRS)-Université Clermont Auvergne (UCA), Istituto Nazionale di Geofisica e Vulcanologia – Sezione di Pisa (INGV), Istituto Nazionale di Geofisica e Vulcanologia, School of Earth Sciences [Bristol], University of Bristol [Bristol], Dipartimento di Scienze della Terra [Firenze] (DST), Università degli Studi di Firenze = University of Florence [Firenze] (UNIFI), Dipartimento di Scienze della Terra [Pisa], University of Pisa - Università di Pisa, School of Earth and Environmental Sciences, University of Queensland, University of Queensland [Brisbane], ANR-10-LABX-0006,CLERVOLC,Clermont-Ferrand centre for research on volcanism(2010), ANR-16-IDEX-0001,CAP 20-25,CAP 20-25(2016), and Università degli Studi di Firenze = University of Florence (UniFI)

Subjects
Volcanic hazards, 010504 meteorology & atmospheric sciences, 010502 geochemistry & geophysics, 01 natural sciences, Geochemistry and Petrology, Joint probability distribution, Rhyolite, [SDU.STU.VO]Sciences of the Universe [physics]/Earth Sciences/Volcanology, Tephra, Uncertainty quantification, 0105 earth and related environmental sciences, Event (probability theory), Guagua Pichincha volcano, geography, geography.geographical_feature_category, Andesite, Cotopaxi volcano, Expert elicitation, Volcanic hazard, Scoring methods, Elicitation, Volcano, 13. Climate action, Seismology, Geology
Abstract

Co-auteur étranger; International audience; Future occurrence of explosive eruptive activity at Cotopaxi and Guagua Pichincha volcanoes, Ecuador, is assessed probabilistically, utilizing expert elicitation. Eight eruption types were considered for each volcano. Type event probabilities were evaluated for the next eruption at each volcano and for at least one of each type within the next 100 years. For each type, we elicited relevant eruption source parameters (duration, average plume height, and total tephra mass). We investigated the robustness of these elicited evaluations by deriving probability uncertainties using three expert scoring methods. For Cotopaxi, we considered both rhyolitic and andesitic magmas. Elicitation findings indicate that the most probable next eruption type is an andesitic hydrovolcanic/ash-emission (~ 26–44% median probability), which has also the highest median probability of recurring over the next 100 years. However, for the next eruption at Cotopaxi, the average joint probabilities for sub-Plinian or Plinian type eruption is of order 30–40%—a significant chance of a violent explosive event. It is inferred that any Cotopaxi rhyolitic eruption could involve a longer duration and greater erupted mass than an andesitic event, likely producing a prolonged emergency. For Guagua Pichincha, future eruption types are expected to be andesitic/dacitic, and a vulcanian event is judged most probable for the next eruption (median probability ~40–55%); this type is expected to be most frequent over the next 100 years, too. However, there is a substantial probability (possibly >40% in average) that the next eruption could be sub-Plinian or Plinian, with all that implies for hazard levels.

Published
2021
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24. Mathematical Modeling of the Evolution of Absenteeism in a University Hospital over 12 Years

Author

Luc Vialatte, Bruno Pereira, Arnaud Guillin, Sophie Miallaret, Julien Steven Baker, Rémi Colin-Chevalier, Anne-Françoise Yao-Lafourcade, Nourddine Azzaoui, Maëlys Clinchamps, Jean-Baptiste Bouillon-Minois, Frédéric Dutheil, Service Santé Travail Environnement [CHU Clermont-Ferrand], CHU Gabriel Montpied [Clermont-Ferrand], CHU Clermont-Ferrand-CHU Clermont-Ferrand, Unité de Biostatistiques [CHU Clermont-Ferrand], Direction de la recherche clinique et de l’innovation [CHU Clermont-Ferrand] (DRCI), Laboratoire de Mathématiques Blaise Pascal (LMBP), Centre National de la Recherche Scientifique (CNRS)-Université Clermont Auvergne (UCA), Hong Kong Baptist University (HKBU), and Laboratoire de Psychologie Sociale et Cognitive (LAPSCO)

Subjects
Adult, Male, sociodemographic factors, Health, Toxicology and Mutagenesis, statistical model, Public Health, Environmental and Occupational Health, Middle Aged, absenteeism, hospital, occupational factors, Hospitals, University, [MATH.MATH-ST]Mathematics [math]/Statistics [math.ST], Child, Preschool, Absenteeism, Humans, [SDV.SPEE]Life Sciences [q-bio]/Santé publique et épidémiologie, Female, Occupations, Child
Abstract

International audience; Increased absenteeism in health care institutions is a major problem, both economically and health related. Our objectives were to understand the general evolution of absenteeism in a university hospital from 2007 to 2019 and to analyze the professional and sociodemographic factors influencing this issue. An initial exploratory analysis was performed to understand the factors that most influence absences. The data were then transformed into time series to analyze the evolution of absences over time. We performed a temporal principal components analysis (PCA) of the absence proportions to group the factors. We then created profiles with contributions from each variable. We could then observe the curves of these profiles globally but also compare the profiles by period. Finally, a predictive analysis was performed on the data using a VAR model. Over the 13 years of follow-up, there were 1,729,097 absences for 14,443 different workers (73.8% women; 74.6% caregivers). Overall, the number of absences increased logarithmically. The variables contributing most to the typical profile of the highest proportions of absences were having a youngest child between 4 and 10 years old (6.44% of contribution), being aged between 40 and 50 years old (5.47%), being aged between 30 and 40 years old (5.32%), working in the administrative field (4.88%), being tenured (4.87%), being a parent (4.85%), being in a coupled relationship (4.69%), having a child over the age of 11 (4.36%), and being separated (4.29%). The forecasts predict a stagnation in the proportion of absences for the profiles of the most absent factors over the next 5 years including annual peaks. During this study, we looked at the sociodemographic and occupational factors that led to high levels of absenteeism. Being aware of these factors allows health companies to act to reduce absenteeism, which represents real financial and public health threats for hospitals.

Published
2022
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25. Particles systems and numerical schemes for mean reflected stochastic differential equations

Author

Arnaud Guillin, Philippe Briand, Paul-Eric Chaudru de Raynal, Céline Labart, Laboratoire de Mathématiques (LAMA), Université Savoie Mont Blanc (USMB [Université de Savoie] [Université de Chambéry])-Centre National de la Recherche Scientifique (CNRS), Laboratoire de Mathématiques Blaise Pascal (LMBP), Université Blaise Pascal - Clermont-Ferrand 2 (UBP)-Centre National de la Recherche Scientifique (CNRS), ANR-12-BS01-0019,STAB,Stabilité du comportement asymptotique d'EDP, de processus stochastiques et de leurs discrétisations.(2012), Guillin, Arnaud, BLANC - Stabilité du comportement asymptotique d'EDP, de processus stochastiques et de leurs discrétisations. - - STAB2012 - ANR-12-BS01-0019 - BLANC - VALID, and Centre National de la Recherche Scientifique (CNRS)-Université Savoie Mont Blanc (USMB [Université de Savoie] [Université de Chambéry])

Subjects
Statistics and Probability, Particle system, [MATH.MATH-PR] Mathematics [math]/Probability [math.PR], Probability (math.PR), 010102 general mathematics, Context (language use), particles system, 01 natural sciences, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], Constraint (information theory), 010104 statistics & probability, Stochastic differential equation, Scheme (mathematics), McKean–Vlasov SDE, FOS: Mathematics, Applied mathematics, mean-reflected, 60H10, 0101 mathematics, Statistics, Probability and Uncertainty, Focus (optics), particle system, reflected stochastic differential equations, Mathematics - Probability, Mathematics
Abstract

This paper is devoted to the study of reflected Stochastic Differential Equations when the constraint is not on the paths of the solution but acts on its law. These reflected equations have been introduced recently in a backward form by Briand, Elie and Hu (Ann. Appl. Probab. 28 (2018) 482–510) in the context of risk measures. We here focus on the forward version of such reflected equations. Our main objective is to provide an approximation of the solutions with the help of interacting particles systems. This approximation allows to design a numerical scheme for this kind of equations.

Published
2020
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26. An analysis of the relationship between mass eruption rate and runout distance of pyroclastic density currents

Author

Olivier Roche, Azzaoui, N., Arnaud Guillin, Laboratoire Magmas et Volcans (LMV), Centre National de la Recherche Scientifique (CNRS)-Institut de Recherche pour le Développement et la société-Institut national des sciences de l'Univers (INSU - CNRS)-Observatoire de Physique du Globe de Clermont-Ferrand (OPGC), Centre National de la Recherche Scientifique (CNRS)-Université Clermont Auvergne [2017-2020] (UCA [2017-2020])-Institut national des sciences de l'Univers (INSU - CNRS)-Centre National de la Recherche Scientifique (CNRS)-Université Clermont Auvergne [2017-2020] (UCA [2017-2020])-Institut national des sciences de l'Univers (INSU - CNRS)-Université Clermont Auvergne [2017-2020] (UCA [2017-2020])-Université Jean Monnet [Saint-Étienne] (UJM), Laboratoire de Mathématiques Blaise Pascal (LMBP), Université Clermont Auvergne [2017-2020] (UCA [2017-2020])-Centre National de la Recherche Scientifique (CNRS), Université Clermont Auvergne [2017-2020] (UCA [2017-2020]), Institut national des sciences de l'Univers (INSU - CNRS)-Université Jean Monnet - Saint-Étienne (UJM)-Institut de Recherche pour le Développement et la société-Université Clermont Auvergne [2017-2020] (UCA [2017-2020])-Centre National de la Recherche Scientifique (CNRS)-Observatoire de Physique du Globe de Clermont-Ferrand (OPGC), Institut national des sciences de l'Univers (INSU - CNRS)-Université Clermont Auvergne [2017-2020] (UCA [2017-2020])-Centre National de la Recherche Scientifique (CNRS)-Institut national des sciences de l'Univers (INSU - CNRS)-Université Clermont Auvergne [2017-2020] (UCA [2017-2020])-Centre National de la Recherche Scientifique (CNRS), and Jouhannel, Sylvaine

Subjects
[SDU.STU.VO] Sciences of the Universe [physics]/Earth Sciences/Volcanology, [SDU.STU.VO]Sciences of the Universe [physics]/Earth Sciences/Volcanology, ComputingMilieux_MISCELLANEOUS
Abstract

International audience

Published
2020

27. Uniform long-time and propagation of chaos estimates for mean field kinetic particles in non-convex landscapes

Author

Pierre Monmarché, Arnaud Guillin, Laboratoire de Mathématiques Blaise Pascal (LMBP), Université Clermont Auvergne [2017-2020] (UCA [2017-2020])-Centre National de la Recherche Scientifique (CNRS), Laboratoire Jacques-Louis Lions (LJLL), Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS), Laboratoire de chimie théorique (LCT), Institut de Chimie du CNRS (INC)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS), and Monmarché, Pierre

Subjects
[MATH.MATH-PR] Mathematics [math]/Probability [math.PR], Particle number, Logarithm, 01 natural sciences, Sobolev inequality, 010104 statistics & probability, Mathematics - Analysis of PDEs, Convergence (routing), FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], 0101 mathematics, [MATH.MATH-AP] Mathematics [math]/Analysis of PDEs [math.AP], Mathematical Physics, Physics, Probability (math.PR), 010102 general mathematics, Mathematical analysis, Regular polygon, Statistical and Nonlinear Physics, 60J60, 35K58, 82B40, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], Nonlinear system, Mean field theory, Rate of convergence, Mathematics - Probability, Analysis of PDEs (math.AP)
Abstract

Combining the results of Guillin (Uniform Poincare and logarithmic Sobolev inequalities for mean field particles systems, 2019) and Monmarche (Stoch Process Appl 127(6):1721–1737, 2017), the trend to equilibrium in large time is studied for a large particle system associated to a Vlasov–Fokker–Planck equation. Under some conditions (that allow non-convex confining potentials) the convergence rate is proven to be independent from the number of particles. From this are derived uniform in time propagation of chaos estimates and an exponentially fast convergence for the nonlinear equation itself.

Published
2020
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28. Quantitative Harris-type theorems for diffusions and McKean–Vlasov processes

Author

Raphael Zimmer, Andreas Eberle, and Arnaud Guillin

Subjects
Lyapunov function, Applied Mathematics, General Mathematics, 010102 general mathematics, Ergodicity, Type (model theory), Lipschitz continuity, 01 natural sciences, Small set, 010104 statistics & probability, symbols.namesake, Nonlinear system, Bounded function, symbols, Applied mathematics, Ergodic theory, 0101 mathematics, Mathematics
Abstract

We consider ℝd-valued diffusion processes of type dXt = b(Xt)dt+dBt. Assuming a geometric drift condition, we establish contractions of the transitions kernels in Kantorovich (L1 Wasserstein) distances with explicit constants. Our results are in the spirit of Hairer and Mattingly's extension of Harris' Theorem. In particular, they do not rely on a small set condition. Instead we combine Lyapunov functions with reflection coupling and concave distance functions. We retrieve constants that are explicit in parameters which can be computed with little effort from one-sided Lipschitz conditions for the drift coefficient and the growth of a chosen Lyapunov function. Consequences include exponential convergence in weighted total variation norms, gradient bounds, bounds for ergodic averages, and Kantorovich contractions for nonlinear McKean-Vlasov diffusions in the case of sufficiently weak but not necessarily bounded nonlinearities. We also establish quantitative bounds for sub-geometric ergodicity assuming a sub-geometric drift condition.

Published
2018
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29. Equivalence between dimensional contractions in Wasserstein distance and the curvature-dimension condition

Author

Ivan Gentil, Arnaud Guillin, François Bolley, Kazumasa Kuwada, Laboratoire de Probabilités, Statistique et Modélisation (LPSM (UMR_8001)), Université Paris Diderot - Paris 7 (UPD7)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS), Département de Mathématiques et Applications - ENS Paris (DMA), École normale supérieure - Paris (ENS-PSL), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS), Institut Camille Jordan (ICJ), École Centrale de Lyon (ECL), Université de Lyon-Université de Lyon-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Institut National des Sciences Appliquées de Lyon (INSA Lyon), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Jean Monnet - Saint-Étienne (UJM)-Centre National de la Recherche Scientifique (CNRS), Équations aux dérivées partielles, analyse (EDPA), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Jean Monnet - Saint-Étienne (UJM)-Centre National de la Recherche Scientifique (CNRS)-École Centrale de Lyon (ECL), Laboratoire de Mathématiques Blaise Pascal (LMBP), Université Blaise Pascal - Clermont-Ferrand 2 (UBP)-Centre National de la Recherche Scientifique (CNRS), Tokyo Institute of Technology [Tokyo] (TITECH), ANR-12-BS01-0019,STAB,Stabilité du comportement asymptotique d'EDP, de processus stochastiques et de leurs discrétisations.(2012), ANR-10-LABX-0070,MILYON,Community of mathematics and fundamental computer science in Lyon(2010), Laboratoire de Probabilités, Statistiques et Modélisations (LPSM (UMR_8001)), Institut Camille Jordan [Villeurbanne] (ICJ), Université de Lyon-Université Jean Monnet [Saint-Étienne] (UJM)-Institut National des Sciences Appliquées de Lyon (INSA Lyon), Institut National des Sciences Appliquées (INSA)-Université de Lyon-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS), and Institut National des Sciences Appliquées (INSA)-Université de Lyon-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS)-École Centrale de Lyon (ECL)

Subjects
Mathematics - Differential Geometry, Pure mathematics, Curvature-dimension condition, Metric measure space, Curvature, 01 natural sciences, Hamilton–Jacobi equation, Theoretical Computer Science, Sobolev inequality, 010104 statistics & probability, Mathematics - Analysis of PDEs, Mathematics (miscellaneous), Quadratic equation, Converse, FOS: Mathematics, Optimal transport, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], 0101 mathematics, Mathematics, Semigroup, Probability (math.PR), 010102 general mathematics, Riemannian manifold, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], Differential Geometry (math.DG), [MATH.MATH-DG]Mathematics [math]/Differential Geometry [math.DG], Variational inequality, Mathematics::Differential Geometry, Mathematics - Probability, Markov diffusion semigroup, Analysis of PDEs (math.AP)
Abstract

International audience; The curvature-dimension condition is a generalization of the Bochner inequality to weighted Riemannian manifolds and general metric measure spaces. It is now known to be equivalent to evolution variational inequalities for the heat semigroup, and quadratic Wasserstein distance contraction properties at different times. On the other hand, in a compact Riemannian manifold, it implies a same-time Wasserstein contraction property for this semigroup. In this work we generalize the latter result to metric measure spaces and more importantly prove the converse: contraction inequalities are equivalent to curvature-dimension conditions. Links with functional inequalities are also investigated.

Published
2018
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30. Corrigendum to 'Discharge rate of explosive volcanic eruption controls runout distance of pyroclastic density currents' [Earth Planet. Sci. Lett. 568 (2021) 117017]

Author

Nourddine Azzaoui, Olivier Roche, Arnaud Guillin, Observatoire de Physique du Globe de Clermont-Ferrand (OPGC), Institut national des sciences de l'Univers (INSU - CNRS)-Centre National de la Recherche Scientifique (CNRS)-Université Clermont Auvergne (UCA), Laboratoire Magmas et Volcans (LMV), Institut national des sciences de l'Univers (INSU - CNRS)-Institut de Recherche pour le Développement et la société-Centre National de la Recherche Scientifique (CNRS)-Université Clermont Auvergne (UCA)-Observatoire de Physique du Globe de Clermont-Ferrand (OPGC), Institut national des sciences de l'Univers (INSU - CNRS)-Centre National de la Recherche Scientifique (CNRS)-Université Clermont Auvergne (UCA)-Institut national des sciences de l'Univers (INSU - CNRS)-Centre National de la Recherche Scientifique (CNRS)-Université Clermont Auvergne (UCA), Laboratoire de Mathématiques Blaise Pascal (LMBP), and Centre National de la Recherche Scientifique (CNRS)-Université Clermont Auvergne (UCA)

Subjects
Vulcanian eruption, Explosive material, Pyroclastic rock, Geophysics, Discharge rate, [SDU]Sciences of the Universe [physics], Space and Planetary Science, Geochemistry and Petrology, Planet, [SDU.STU.VO]Sciences of the Universe [physics]/Earth Sciences/Volcanology, Earth and Planetary Sciences (miscellaneous), ComputingMilieux_MISCELLANEOUS, Geology, Earth (classical element)
Abstract

International audience

Published
2021
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31. Estimation of the realized (co-)volatility vector: Large deviations approach

Author

Hacène Djellout, Yacouba Samoura, and Arnaud Guillin

Subjects
Statistics and Probability, Stochastic volatility, Realized variance, Applied Mathematics, 010102 general mathematics, Estimator, 01 natural sciences, 010104 statistics & probability, Law of large numbers, Modeling and Simulation, Statistics, Consistent estimator, Econometrics, Forward volatility, Large deviations theory, 0101 mathematics, Volatility (finance), Mathematics
Abstract

Realized statistics based on high frequency returns have become very popular in financial economics. In recent years, different non-parametric estimators of the variation of a log-price process have appeared. Among them are the realized quadratic (co-)variation which is perhaps the most well known example, providing a consistent estimator of the integrated (co-)volatility when the logarithmic price process is continuous. In this paper, we propose to study the large deviation properties of realized (co-)volatility. Our main motivation is to improve upon the existing limit theorems such as the weak law of large numbers or the central limit theorem which have been proved in different contexts. Our large deviations results can be used to evaluate and approximate tail probabilities of realized (co-)volatility. As an application we provide the large deviations for the standard dependence measures between the two assets returns such as the realized regression coefficients or the realized correlation. Our study should contribute to the recent trend of research on the (co-)variance estimation problems, which are quite often discussed in high-frequency financial data analysis.

Published
2017
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32. Quantifying the Uncertainty of a Coupled Plume and Tephra Dispersal Model: PLUME‐MOM/HYSPLIT Simulations Applied to Andean Volcanoes

Author

Arnaud Guillin, Olivier Roche, Pablo Samaniego, M. de' Michieli Vitturi, Silvana Hidalgo, Alessandro Tadini, Federica Pardini, Benjamin Bernard, Mathieu Gouhier, Nourddine Azzaoui, Julia Eychenne, J. L. Le Pennec, Istituto Nazionale di Geofisica e Vulcanologia – Sezione di Pisa (INGV), Istituto Nazionale di Geofisica e Vulcanologia, Laboratoire Magmas et Volcans (LMV), Institut national des sciences de l'Univers (INSU - CNRS)-Université Jean Monnet - Saint-Étienne (UJM)-Institut de Recherche pour le Développement et la société-Université Clermont Auvergne [2017-2020] (UCA [2017-2020])-Centre National de la Recherche Scientifique (CNRS)-Observatoire de Physique du Globe de Clermont-Ferrand (OPGC), Institut national des sciences de l'Univers (INSU - CNRS)-Université Clermont Auvergne [2017-2020] (UCA [2017-2020])-Centre National de la Recherche Scientifique (CNRS)-Institut national des sciences de l'Univers (INSU - CNRS)-Université Clermont Auvergne [2017-2020] (UCA [2017-2020])-Centre National de la Recherche Scientifique (CNRS), Instituto Geofísico, Escuela Politécnica Nacional, Quito, Ecuador, Laboratoire de Mathématiques Blaise Pascal (LMBP), Université Clermont Auvergne [2017-2020] (UCA [2017-2020])-Centre National de la Recherche Scientifique (CNRS), Institut national des sciences de l'Univers (INSU - CNRS)-Institut de Recherche pour le Développement et la société-Centre National de la Recherche Scientifique (CNRS)-Université Clermont Auvergne (UCA)-Observatoire de Physique du Globe de Clermont-Ferrand (OPGC), Institut national des sciences de l'Univers (INSU - CNRS)-Centre National de la Recherche Scientifique (CNRS)-Université Clermont Auvergne (UCA)-Institut national des sciences de l'Univers (INSU - CNRS)-Centre National de la Recherche Scientifique (CNRS)-Université Clermont Auvergne (UCA), ANR-16-IDEX-0001,CAP 20-25,CAP 20-25(2016), ANR-10-LABX-0006,CLERVOLC,Clermont-Ferrand centre for research on volcanism(2010), Institut national des sciences de l'Univers (INSU - CNRS)-Université Jean Monnet [Saint-Étienne] (UJM)-Institut de Recherche pour le Développement et la société-Université Clermont Auvergne [2017-2020] (UCA [2017-2020])-Centre National de la Recherche Scientifique (CNRS)-Observatoire de Physique du Globe de Clermont-Ferrand (OPGC), and Centre National de la Recherche Scientifique (CNRS)-Université Clermont Auvergne [2017-2020] (UCA [2017-2020])-Institut national des sciences de l'Univers (INSU - CNRS)-Centre National de la Recherche Scientifique (CNRS)-Université Clermont Auvergne [2017-2020] (UCA [2017-2020])-Institut national des sciences de l'Univers (INSU - CNRS)

Subjects
geography, geography.geographical_feature_category, 010504 meteorology & atmospheric sciences, Advection, Magnitude (mathematics), Atmospheric sciences, 01 natural sciences, Plume, Geophysics, Volcano, 13. Climate action, Space and Planetary Science, Geochemistry and Petrology, [SDU.STU.VO]Sciences of the Universe [physics]/Earth Sciences/Volcanology, Earth and Planetary Sciences (miscellaneous), HYSPLIT, Environmental science, Biological dispersal, Uncertainty quantification, Tephra, 0105 earth and related environmental sciences
Abstract

Co-auteur étranger; International audience; Numerical modelling of tephradispersaland depositionis essential forevaluation of volcanichazards. Manymodels consider reasonable physical approximations in order toreducecomputational times, but this may introduce a certain degree of uncertainty in the simulation outputs.The important step of uncertainty quantification is dealt in this paper with respect to a coupled version of a plume model (PLUME-MoM) and a tephra dispersal model (HYSPLIT). The performances of this model are evaluated through simulationsoffourpasteruptions of different magnitudesand styles from three Andean volcanoes, and the uncertainty is quantified by evaluating the differences between modeled and observed data of plume height (at different time steps above the vent)as well asmass loading and grain sizeat givenstratigraphic sections.Different meteorological datasets were alsotestedand hada sensible influence on the model outputs. Other results highlight that the modeltendsto underestimate plume heights whileoverestimatingmass loading values, especially for higher magnitude eruptions.Moreover, the advective part of HYSPLIT seems to work more efficiently than the diffusive part.Finally, though the coupledPLUME-MoM/HYSPLIT model generally is less efficient inreproducing deposit grain sizes, we propose itmay be used for hazard maps production for higher magnitude eruptions (sub-Plinian or Plinian)for what concern mass loading.

Published
2020
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33. On the Poincaré Constant of Log-Concave Measures

Author

Arnaud Guillin, Patrick Cattiaux, Institut de Mathématiques de Toulouse UMR5219 (IMT), Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT)-Université de Toulouse (UT)-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Institut National des Sciences Appliquées (INSA)-Université Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS), Laboratoire de Mathématiques Blaise Pascal (LMBP), Université Blaise Pascal - Clermont-Ferrand 2 (UBP)-Centre National de la Recherche Scientifique (CNRS), and Université Clermont Auvergne [2017-2020] (UCA [2017-2020])-Centre National de la Recherche Scientifique (CNRS)

Subjects
010102 general mathematics, Poincaré inequality, log-concave measure, [MATH.MATH-FA]Mathematics [math]/Functional Analysis [math.FA], 01 natural sciences, Measure (mathematics), Cheeger inequality, transference principle, mollification procedure, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], Section (fiber bundle), 010104 statistics & probability, symbols.namesake, total variation, Poincaré conjecture, symbols, Applied mathematics, Wasserstein distance, 0101 mathematics, Constant (mathematics), Link (knot theory), Mathematics
Abstract

The goal of this paper is to push forward the study of those properties of log-concave measures that help to estimate their Poincare constant. First we revisit E. Milman’s result (Invent Math 177:1–43, 2009) on the link between weak (Poincare or concentration) inequalities and Cheeger’s inequality in the log-concave cases, in particular extending localization ideas and a result of Latala, as well as providing a simpler proof of the nice Poincare (dimensional) bound in the unconditional case. Then we prove alternative transference principle by concentration or using various distances (total variation, Wasserstein). A mollification procedure is also introduced enabling, in the log-concave case, to reduce to the case of the Poincare inequality for the mollified measure. We finally complete the transference section by the comparison of various probability metrics (Fortet-Mourier, bounded-Lipschitz, …) under a log-concavity assumption.

Published
2020
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34. Quantitative approximation of the discrete Moran process by a Wright-Fisher diffusion

Author

Arnaud Guillin, Gorgui Gackou, and Arnaud Personne

Subjects
Markov process, 01 natural sciences, 010305 fluids & plasmas, Diffusion, 03 medical and health sciences, symbols.namesake, Dimension (vector space), 0103 physical sciences, Moran process, Quantitative Biology::Populations and Evolution, Statistical physics, Limit (mathematics), Diffusion (business), Selection, Genetic, Selection (genetic algorithm), 030304 developmental biology, Mathematics, 0303 health sciences, Models, Genetic, Applied Mathematics, Agricultural and Biological Sciences (miscellaneous), Genetics, Population, Diffusion process, Modeling and Simulation, symbols, Jump process
Abstract

The Moran discrete process and the Wright–Fisher model are the most popular models in population genetics. The Wright–Fisher diffusion is commonly used as an approximation in order to understand the dynamics of population genetics models. Here, we give a quantitative large-population limit of the error occurring by using the approximating diffusion in the presence of weak selection and weak immigration in one dimension. The approach is robust enough to consider the case where selection and immigration are Markovian processes, whose large-population limit is either a finite state jump process, or a diffusion process.

Published
2019

35. Entropic multipliers method for langevin diffusion and weighted log sobolev inequalities

Author

Arnaud Guillin, Chaoen Zhang, Patrick Cattiaux, Pierre Monmarché, Institut de Mathématiques de Toulouse UMR5219 (IMT), Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS), Laboratoire de Mathématiques Blaise Pascal (LMBP), Université Blaise Pascal - Clermont-Ferrand 2 (UBP)-Centre National de la Recherche Scientifique (CNRS), Université Paris-Est Marne-la-Vallée (UPEM), Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique (CERMICS), École des Ponts ParisTech (ENPC), ANR-12-BS01-0019,STAB,Stabilité du comportement asymptotique d'EDP, de processus stochastiques et de leurs discrétisations.(2012), ANR-17-CE40-0030,EFI,Entropie, flots, inégalités(2017), Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT)-Université de Toulouse (UT)-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Institut National des Sciences Appliquées (INSA)-Université Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS), Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), and Institut National des Sciences Appliquées (INSA)-Université Fédérale Toulouse Midi-Pyrénées-Institut National des Sciences Appliquées (INSA)-Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse III - Paul Sabatier (UT3)

Subjects
Hessian matrix, weighted logarithmic Sobolev inequality, entropic convergence, Langevin diffusion, [MATH.MATH-FA]Mathematics [math]/Functional Analysis [math.FA], 01 natural sciences, Sobolev inequality, symbols.namesake, Mathematics - Analysis of PDEs, 0103 physical sciences, FOS: Mathematics, Applied mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], 0101 mathematics, Mathematics, Lyapunov conditions, Probability (math.PR), 010102 general mathematics, hypocercivity, Functional Analysis (math.FA), Mathematics - Functional Analysis, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], symbols, 010307 mathematical physics, Analysis, Mathematics - Probability, Analysis of PDEs (math.AP), Logarithmic sobolev inequality
Abstract

International audience; In his work about hypocercivity, Villani [18] considers in particular convergence to equilibrium for the kinetic Langevin process. While his convergence results in L 2 are given in a quite general setting, convergence in entropy requires some boundedness condition on the Hessian of the Hamiltonian. We will show here how to get rid of this assumption in the study of the hypocoercive entropic relaxation to equilibrium for the Langevin diffusion. Our method relies on a generalization to entropy of the multipliers method and an adequate functional inequality. As a byproduct, we also give tractable conditions for this functional inequality, which is a particular instance of a weighted logarithmic Sobolev inequality, to hold.

Published
2019
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36. Author Correction: Low efficiency of large volcanic eruptions in transporting very fine ash into the atmosphere

Author

Antonio Costa, Nourddine Azzaoui, Arnaud Guillin, Mathieu Gouhier, Mathieu Deslandes, Julia Eychenne, Philippe Husson, Matthieu Poret, Laboratoire Magmas et Volcans (LMV), Institut national des sciences de l'Univers (INSU - CNRS)-Institut de Recherche pour le Développement et la société-Centre National de la Recherche Scientifique (CNRS)-Université Clermont Auvergne (UCA)-Observatoire de Physique du Globe de Clermont-Ferrand (OPGC), Institut national des sciences de l'Univers (INSU - CNRS)-Centre National de la Recherche Scientifique (CNRS)-Université Clermont Auvergne (UCA)-Institut national des sciences de l'Univers (INSU - CNRS)-Centre National de la Recherche Scientifique (CNRS)-Université Clermont Auvergne (UCA), Laboratoire de Mathématiques Blaise Pascal (LMBP), Centre National de la Recherche Scientifique (CNRS)-Université Clermont Auvergne (UCA), and Météo-France

Subjects
geography, Multidisciplinary, geography.geographical_feature_category, 010504 meteorology & atmospheric sciences, Earth science, lcsh:R, lcsh:Medicine, 010502 geochemistry & geophysics, 01 natural sciences, Computer Science::Digital Libraries, Atmosphere, Volcano, [SDU]Sciences of the Universe [physics], ComputingMethodologies_DOCUMENTANDTEXTPROCESSING, [SDU.STU.VO]Sciences of the Universe [physics]/Earth Sciences/Volcanology, lcsh:Q, lcsh:Science, Author Correction, Geology, 0105 earth and related environmental sciences
Abstract

Volcanic ash clouds are common, often unpredictable, phenomena generated during explosive eruptions. Mainly composed of very fine ash particles, they can be transported in the atmosphere at great distances from the source, having detrimental socio-economic impacts. However, proximal settling processes controlling the proportion (ε) of the very fine ash fraction distally transported in the atmosphere are still poorly understood. Yet, for the past two decades, some operational meteorological agencies have used a default value of ε = 5% as input for forecast models of atmospheric ash cloud concentration. Here we show from combined satellite and field data of sustained eruptions that ε actually varies by two orders of magnitude with respect to the mass eruption rate. Unexpectedly, we demonstrate that the most intense eruptions are in fact the least efficient (with ε = 0.1%) in transporting very fine ash through the atmosphere. This implies that the amount of very fine ash distally transported in the atmosphere is up to 50 times lower than previously anticipated. We explain this finding by the efficiency of collective particle settling in ash-rich clouds which enhance early and en masse fallout of very fine ash. This suggests that proximal sedimentation during powerful eruptions is more controlled by the concentration of ash than by the grain size. This has major consequences for decision-makers in charge of air traffic safety regulation, as well as for the understanding of proximal settling processes. Finally, we propose a new statistical model for predicting the source mass eruption rate with an unprecedentedly low level of uncertainty.

Published
2019
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37. From a Kac-like particle system to the Landau equation for hard potentials and Maxwell molecules

Author

Nicolas Fournier, Arnaud Guillin, Laboratoire de Probabilités, Statistique et Modélisation (LPSM (UMR_8001)), Université Paris Diderot - Paris 7 (UPD7)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS), Laboratoire de Mathématiques Blaise Pascal (LMBP), Université Blaise Pascal - Clermont-Ferrand 2 (UBP)-Centre National de la Recherche Scientifique (CNRS), ANR-12-BS01-0019,STAB,Stabilité du comportement asymptotique d'EDP, de processus stochastiques et de leurs discrétisations.(2012), ANR-17-CE40-0030,EFI,Entropie, flots, inégalités(2017), and Laboratoire de Probabilités, Statistiques et Modélisations (LPSM (UMR_8001))

Subjects
Physics, Particle system, Class (set theory), Kac's particle system, General Mathematics, Probability (math.PR), 010102 general mathematics, 82C40, 60K35, 65C05, 16. Peace & justice, 01 natural sciences, Measure (mathematics), [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], Nonlinear system, Stochastic particle systems, Rate of convergence, Homogeneous, 0103 physical sciences, Convergence (routing), FOS: Mathematics, 010307 mathematical physics, 0101 mathematics, Propagation of Chaos, Mathematics - Probability, Independence (probability theory), Mathematical physics
Abstract

International audience; We prove a quantitative result of convergence of a conservative stochastic particle system to the solution of the homogeneous Landau equation for hard potentials. There are two main difficulties: (i) the known stability results for this class of Landau equations concern regular solutions and seem difficult to extend to study the rate of convergence of some empirical measures; (ii) the conservativeness of the particle system is an obstacle for (approximate) independence. To overcome (i), we prove a new stability result for the Landau equation for hard potentials concerning very general measure solutions. Due to (ii), we have to couple, our particle system with some non independent nonlinear processes, of which the law solves, in some sense, the Landau equation. We then prove that these nonlinear processes are not so far from being independent. Using finally some ideas of Rousset [25], we show that in the case of Maxwell molecules, the convergence of the particle system is uniform in time.

Published
2017
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38. Couplings and quantitative contraction rates for Langevin dynamics

Author

Andreas Eberle, Raphael Zimmer, Arnaud Guillin, Guillin, Arnaud, BLANC - Stabilité du comportement asymptotique d'EDP, de processus stochastiques et de leurs discrétisations. - - STAB2012 - ANR-12-BS01-0019 - BLANC - VALID, Instutute for Applied Mathematics [Bonn], Rheinische Friedrich-Wilhelms-Universität Bonn, Laboratoire de Mathématiques Blaise Pascal (LMBP), Université Blaise Pascal - Clermont-Ferrand 2 (UBP)-Centre National de la Recherche Scientifique (CNRS), and ANR-12-BS01-0019,STAB,Stabilité du comportement asymptotique d'EDP, de processus stochastiques et de leurs discrétisations.(2012)

Subjects
Statistics and Probability, Lyapunov function, [MATH.MATH-PR] Mathematics [math]/Probability [math.PR], 60J60, 60H10, 35Q84, 35B40, Langevin diffusion, hypocoercivity, Kinetic energy, 01 natural sciences, Upper and lower bounds, convergence to equilibrium, 010104 statistics & probability, symbols.namesake, Mathematics - Analysis of PDEs, 35Q84, FOS: Mathematics, Wasserstein distance, 0101 mathematics, Langevin dynamics, 60J60, Mathematics, Lyapunov functions, stochastic Hamiltonian dynamics, Coupling, reflection coupling, quantitative bounds, Probability (math.PR), 35B40, 010102 general mathematics, Mathematical analysis, stochastic Hamil- tonian dynamics, Langevin equation, Maxima and minima, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], kinetic Fokker-Planck equation, Brownian dynamics, symbols, 60H10, Statistics, Probability and Uncertainty, Mathematics - Probability, Analysis of PDEs (math.AP), kinetic Fokker–Planck equation
Abstract

We introduce a new probabilistic approach to quantify convergence to equilibrium for (kinetic) Langevin processes. In contrast to previous analytic approaches that focus on the associated kinetic Fokker-Planck equation, our approach is based on a specific combination of reflection and synchronous coupling of two solutions of the Langevin equation. It yields contractions in a particular Wasserstein distance, and it provides rather precise bounds for convergence to equilibrium at the borderline between the overdamped and the underdamped regime. In particular, we are able to recover kinetic behavior in terms of explicit lower bounds for the contraction rate. For example, for a rescaled double-well potential with local minima at distance $a$, we obtain a lower bound for the contraction rate of order $\Omega (a^{-1})$ provided the friction coefficient is of order $\Theta (a^{-1})$.

Published
2019

39. Séminaire de probabilités XLIX

Author

P Petit, Jean Brossard, Patrick Cattiaux, Antoine Lejay, Mireille Capitaine, Peter Kratz, Benedikt Wilbertz, Emmanuel Boissard, Rajeev Bhaskaran, Denis Villemonais, Florian Bouguet, Paul McGill, Laurent Miclo, Arnaud Guillin, Henri Elad Altman, Koléhé Abdoulaye Coulibaly-Pasquier, Gilles Pagès, Hiroshi Tsukada, Etienne Pardoux, Laurent Serlet, Nicolas Champagnat, Christophe Leuridan, Institut de Mathématiques de Toulouse UMR5219 (IMT), Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT)-Université de Toulouse (UT)-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Institut National des Sciences Appliquées (INSA)-Université Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS), Laboratoire de Mathématiques Blaise Pascal (LMBP), Université Clermont Auvergne [2017-2020] (UCA [2017-2020])-Centre National de la Recherche Scientifique (CNRS), Université de Lorraine (UL), Biology, genetics and statistics (BIGS), Inria Nancy - Grand Est, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Institut Élie Cartan de Lorraine (IECL), Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS)-Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS), Institut Fourier (IF ), Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes [2016-2019] (UGA [2016-2019]), Institut Élie Cartan de Lorraine (IECL), Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS), TO Simulate and CAlibrate stochastic models (TOSCA), Inria Sophia Antipolis - Méditerranée (CRISAM), Laboratoire de Probabilités, Statistique et Modélisation (LPSM (UMR_8001)), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité), Aix-Marseille Université - Faculté des Sciences (AMU SCI), Aix Marseille Université (AMU), Université Claude Bernard Lyon 1 (UCBL), Université de Lyon, Université Paris Diderot - Paris 7 (UPD7)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS), Indian Statistical Institute [Bangalore] (ISI), Graduate School of Engineering Science [Toyonaka, Osaka], Osaka University, Donati-Martin, Catherine, Lejay, Antoine, Rouault, Alain, Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS), Sorbonne Université - Faculté de Médecine (SU FM), Sorbonne Université (SU), Laboratoire de Probabilités, Statistiques et Modélisations (LPSM (UMR_8001)), Graduate School of Engineering Science [Osaka], Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Toulouse 1 Capitole (UT1)-Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS), Laboratoire de Mathématiques Blaise Pascal - Clermont Auvergne (LMBP), Université Clermont Auvergne (UCA)-Centre National de la Recherche Scientifique (CNRS), Institut Fourier (IF), Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA), Graduate School of Engineering Science, and Osaka University [Osaka]

Subjects
Quasi-stationary Distribution, Stochastic process, 010102 general mathematics, Martingale Theory, 01 natural sciences, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], 010104 statistics & probability, Large deviations, Stochastic processes, Large deviations theory, Statistical physics, 0101 mathematics, Random matrix, Diffusion processes, Martingale theory, ComputingMilieux_MISCELLANEOUS, Random Matrices, Mathematics
Abstract

International audience

Published
2018
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40. An Elementary Approach To Uniform In Time Propagation Of Chaos

Author

Raphael Zimmer, Arnaud Guillin, Andreas Eberle, Alain Durmus, Centre de Mathématiques et de Leurs Applications (CMLA), École normale supérieure - Cachan (ENS Cachan)-Centre National de la Recherche Scientifique (CNRS), Instutute for Applied Mathematics [Bonn], Rheinische Friedrich-Wilhelms-Universität Bonn, Laboratoire de Mathématiques Blaise Pascal (LMBP), Université Clermont Auvergne [2017-2020] (UCA [2017-2020])-Centre National de la Recherche Scientifique (CNRS), and ANR-17-CE40-0030,EFI,Entropie, flots, inégalités(2017)

Subjects
Coupling, Physics, Particle system, Applied Mathematics, General Mathematics, Probability (math.PR), 010102 general mathematics, Regular polygon, 01 natural sciences, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], 010104 statistics & probability, Classical mechanics, Reflection (mathematics), Mathematics - Analysis of PDEs, FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], 0101 mathematics, Propagation of chaos, Mathematics - Probability, Analysis of PDEs (math.AP)
Abstract

International audience; Based on a coupling approach, we prove uniform in time propagation of chaos for weakly interacting mean-field particle systems with possibly non-convex confinement and interaction potentials. The approach is based on a combination of reflection and synchronous couplings applied to the individual particles. It provides explicit quantitative bounds that significantly extend previous results for the convex case.

Published
2018

41. Dimensional improvements of the logarithmic Sobolev, Talagrand and Brascamp–Lieb inequalities

Author

Ivan Gentil, François Bolley, Arnaud Guillin, Laboratoire de Probabilités, Statistiques et Modélisations (LPSM (UMR_8001)), Université Paris Diderot - Paris 7 (UPD7)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS), Équations aux dérivées partielles, analyse (EDPA), Institut Camille Jordan [Villeurbanne] (ICJ), École Centrale de Lyon (ECL), Université de Lyon-Université de Lyon-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Université Jean Monnet [Saint-Étienne] (UJM)-Institut National des Sciences Appliquées de Lyon (INSA Lyon), Institut National des Sciences Appliquées (INSA)-Université de Lyon-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS)-École Centrale de Lyon (ECL), Institut National des Sciences Appliquées (INSA)-Université de Lyon-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS), Laboratoire de Mathématiques Blaise Pascal (LMBP), Université Blaise Pascal - Clermont-Ferrand 2 (UBP)-Centre National de la Recherche Scientifique (CNRS), Institut Mittag-Leffler 2016, ANR-12-BS01-0019,STAB,Stabilité du comportement asymptotique d'EDP, de processus stochastiques et de leurs discrétisations.(2012), Laboratoire de Probabilités, Statistique et Modélisation (LPSM (UMR_8001)), Département de Mathématiques et Applications - ENS Paris (DMA), École normale supérieure - Paris (ENS-PSL), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS), Institut Camille Jordan (ICJ), Université de Lyon-Institut National des Sciences Appliquées de Lyon (INSA Lyon), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Jean Monnet - Saint-Étienne (UJM)-Centre National de la Recherche Scientifique (CNRS)-École Centrale de Lyon (ECL), and Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Jean Monnet - Saint-Étienne (UJM)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Statistics and Probability, Scale (ratio), Logarithm, Dimension (graph theory), Logarithmic Sobolev inequality, Brascamp-Lieb inequality, [MATH.MATH-FA]Mathematics [math]/Functional Analysis [math.FA], 01 natural sciences, 010104 statistics & probability, Stochastic differential equation, Convergence (routing), FOS: Mathematics, Fokker–Planck equations, Applied mathematics, 0101 mathematics, Contraction (operator theory), Mathematics, 60J60, Mathematics::Functional Analysis, Brascamp–Lieb inequality, 010102 general mathematics, Probability (math.PR), Talagrand inequality, Functional Analysis (math.FA), Mathematics - Functional Analysis, Sobolev space, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], optimal transport, 60H10, Statistics, Probability and Uncertainty, 60E15, Fokker-Planck equations, Mathematics - Probability
Abstract

International audience; In this work we consider dimensional improvements of the logarithmic Sobolev, Talagrand and Brascamp-Lieb inequalities. For this we use optimal transport methods and the Borell-Brascamp-Lieb inequality. These refinements can be written as a deficit in the classical inequalities. They have the right scale with respect to the dimension. They lead to sharpened concentration properties as well as refined contraction bounds, convergence to equilibrium and short time behaviour for the laws of solutions to stochastic differential equations.

Published
2018
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42. Ornstein-Uhlenbeck Pinball and the Poincaré Inequality in a Punctured Domain

Author

Laurent Miclo, Patrick Cattiaux, Arnaud Guillin, Emmanuel Boissard, Institut de Mathématiques de Toulouse UMR5219 (IMT), Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS), Laboratoire de Mathématiques Blaise Pascal (LMBP), Université Clermont Auvergne [2017-2020] (UCA [2017-2020])-Centre National de la Recherche Scientifique (CNRS), Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT)-Université de Toulouse (UT)-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Institut National des Sciences Appliquées (INSA)-Université Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), and Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS)

Subjects
010102 general mathematics, Open set, Poincaré inequality, Ornstein–Uhlenbeck process, Gaussian measure, 01 natural sciences, Domain (mathematical analysis), Combinatorics, 010104 statistics & probability, symbols.namesake, Disjoint union (topology), Mathematics::Probability, Bounded function, symbols, Hypercube, [MATH]Mathematics [math], 0101 mathematics, ComputingMilieux_MISCELLANEOUS, Mathematics
Abstract

In this paper we study the Poincare constant for the Gaussian measure restricted to \(D={\mathbb R}^d - \mathbb {B}\) where \(\mathbb {B}\) is the disjoint union of bounded open sets. We will mainly look at the case where the obstacles are Euclidean balls B(xi, ri) with radii ri, or hypercubes with vertices of length 2ri, and d ≥ 2. This will explain the asymptotic behavior of a d-dimensional Ornstein-Uhlenbeck process in the presence of obstacles with elastic normal reflections (the Ornstein-Uhlenbeck pinball).

Published
2018
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43. Transportation and concentration inequalities for bifurcating Markov chains

Author

Mikael Escobar-Bach, Arnaud Guillin, S. Valère Bitseki Penda, Institut de Mathématiques de Bourgogne [Dijon] (IMB), Centre National de la Recherche Scientifique (CNRS)-Université de Franche-Comté (UFC), Université Bourgogne Franche-Comté [COMUE] (UBFC)-Université Bourgogne Franche-Comté [COMUE] (UBFC)-Université de Bourgogne (UB), Department of Mathematics and Computer Science [Odense] (IMADA), University of Southern Denmark (SDU), Laboratoire de Mathématiques Blaise Pascal (LMBP), Université Clermont Auvergne [2017-2020] (UCA [2017-2020])-Centre National de la Recherche Scientifique (CNRS), VILLUM FONDEN. Grant number : VKR023480, International project for scientific cooperation. Grant number : PICS-6416, ANR-11-LABX-0056,LMH,LabEx Mathématique Hadamard(2011), Institut de Mathématiques de Bourgogne [Dijon] ( IMB ), Université de Bourgogne ( UB ) -Centre National de la Recherche Scientifique ( CNRS ), Department of Mathematics and Computer Science [Odense] ( IMADA ), University of Southern Denmark, Laboratoire de Mathématiques Blaise Pascal - Clermont Auvergne ( LMBP ), Université Clermont Auvergne ( UCA ) -Centre National de la Recherche Scientifique ( CNRS ), and ANR-11-IDEX-0003-02/11-LABX-0056,LMH,LabEx Mathématique Hadamard ( 2011 )

Subjects
Statistics and Probability, [ MATH ] Mathematics [math], Class (set theory), Inequality, media_common.quotation_subject, MSC: 60J05, 60E15, 62G07, 01 natural sciences, 010104 statistics & probability, Bifurcating Markov chains, [MATH.MATH-ST]Mathematics [math]/Statistics [math.ST], Transportation inequalities, Applied mathematics, [ MATH.MATH-ST ] Mathematics [math]/Statistics [math.ST], Wasserstein distance, 0101 mathematics, [MATH]Mathematics [math], Contraction (operator theory), media_common, Mathematics, Binary tree, Markov chain, Geometric ergodicity, 010102 general mathematics, Mathematical analysis, Function (mathematics), [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], Nonlinear system, Autoregressive model, Deviation inequalities, [ MATH.MATH-PR ] Mathematics [math]/Probability [math.PR]
Abstract

International audience; We investigate the transportation inequality for bifurcating Markov chains which are a class of processes indexed by a regular binary tree. Fitting well models like cell growth when each individual gives birth to exactly two offsprings, we use transportation inequalities to provide useful concentration inequalities. We also study deviation inequalities for the empirical means under relaxed assumptions on the Wasserstein contraction for the Markov kernels. Applications to bifurcating nonlinear autoregressive processes are considered for point-wise estimates of the non-linear autoregressive function.

Published
2017
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44. New sharp Gagliardo-Nirenberg-Sobolev inequalities and an improved Borell-Brascamp-Lieb inequality

Author

François Bolley, Dario Cordero-Erausquin, Yasuhiro Fujita, Arnaud Guillin, Ivan Gentil, Laboratoire de Probabilités et Modèles Aléatoires (LPMA), Centre National de la Recherche Scientifique (CNRS)-Université Paris Diderot - Paris 7 (UPD7)-Université Pierre et Marie Curie - Paris 6 (UPMC), Institut de Mathématiques de Jussieu - Paris Rive Gauche (IMJ-PRG), University of Toyama, Équations aux dérivées partielles, analyse (EDPA), Institut Camille Jordan [Villeurbanne] (ICJ), École Centrale de Lyon (ECL), Université de Lyon-Université de Lyon-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Université Jean Monnet [Saint-Étienne] (UJM)-Institut National des Sciences Appliquées de Lyon (INSA Lyon), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS)-École Centrale de Lyon (ECL), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS), Laboratoire de Mathématiques Blaise Pascal (LMBP), Université Clermont Auvergne [2017-2020] (UCA [2017-2020])-Centre National de la Recherche Scientifique (CNRS), ANR-12-BS01-0019,STAB,Stabilité du comportement asymptotique d'EDP, de processus stochastiques et de leurs discrétisations.(2012), ANR-17-CE40-0030,EFI,Entropie, flots, inégalités(2017), Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS), Département de Mathématiques et Applications - ENS Paris (DMA), École normale supérieure - Paris (ENS-PSL), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS), Institut Camille Jordan (ICJ), Université de Lyon-Institut National des Sciences Appliquées de Lyon (INSA Lyon), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Jean Monnet - Saint-Étienne (UJM)-Centre National de la Recherche Scientifique (CNRS)-École Centrale de Lyon (ECL), and Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Jean Monnet - Saint-Étienne (UJM)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Kantorovich inequality, Pure mathematics, General Mathematics, Borell–Brascamp–Lieb inequality, Sobolev inequality, Ky Fan inequality, Mathematics::Analysis of PDEs, Poincaré inequality, 01 natural sciences, symbols.namesake, FOS: Mathematics, Mathematics::Metric Geometry, Hopf-Lax solution, 0101 mathematics, Gagliardo-Nirenberg inequality, Mathematics, Young's inequality, Mathematics::Functional Analysis, Probability (math.PR), 010102 general mathematics, Mathematical analysis, Hamilton-Jacobi equation, Brunn-Minkowski inequality, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], symbols, Rearrangement inequality, Isoperimetric inequality, Mathematics - Probability
Abstract

We propose a new Borell–Brascamp–Lieb inequality that leads to novel sharp Euclidean inequalities such as Gagliardo–Nirenberg–Sobolev inequalities in $ {\mathbb{R}}^n$ and in the half-space $ {\mathbb{R}}^n_+$. This gives a new bridge between the geometric point of view of the Brunn–Minkowski inequality and the functional point of view of the Sobolev-type inequalities. In this way we unify, simplify, and generalize results by S. Bobkov–M. Ledoux, M. del Pino–J. Dolbeault, and B. Nazaret.

Published
2017
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45. On the rate of convergence in Wasserstein distance of the empirical measure

Author

Arnaud Guillin, Nicolas Fournier, Laboratoire de Probabilités et Modèles Aléatoires (LPMA), Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS), Laboratoire de Mathématiques Blaise Pascal (LMBP), and Université Blaise Pascal - Clermont-Ferrand 2 (UBP)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Statistics and Probability, Mathematics - Statistics Theory, Statistics Theory (math.ST), Mixing (mathematics), [MATH.MATH-ST]Mathematics [math]/Statistics [math.ST], Quantization, FOS: Mathematics, Wasserstein distance, Empirical measure, Mathematics, 60F25, 60F10, 65C05, 60E15, 65D32, Discrete mathematics, Particle system, Sequence of i.i.d. random variables, Markov chains, Markov chain, Mathematical finance, Probability (math.PR), Order (ring theory), [STAT.TH]Statistics [stat]/Statistics Theory [stat.TH], [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], Rate of convergence, Probability distribution, Concentration inequalities, Statistics, Probability and Uncertainty, Mathematics - Probability, Analysis
Abstract

Let $$\mu _N$$ be the empirical measure associated to a $$N$$ -sample of a given probability distribution $$\mu $$ on $$\mathbb {R}^d$$ . We are interested in the rate of convergence of $$\mu _N$$ to $$\mu $$ , when measured in the Wasserstein distance of order $$p>0$$ . We provide some satisfying non-asymptotic $$L^p$$ -bounds and concentration inequalities, for any values of $$p>0$$ and $$d\ge 1$$ . We extend also the non asymptotic $$L^p$$ -bounds to stationary $$\rho $$ -mixing sequences, Markov chains, and to some interacting particle systems.

Published
2014
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46. Bernstein-type Concentration Inequalities for Symmetric Markov Processes

Author

Arnaud Guillin, Liming Wu, and Fuqing Gao

Subjects
Statistics and Probability, Discrete mathematics, Young's inequality, Ky Fan inequality, Bernstein inequalities, Markov process, symbols.namesake, Mathematics::Probability, Bernstein's inequality, Markov's inequality, symbols, Second moment method, Statistics, Probability and Uncertainty, Cauchy–Schwarz inequality, Mathematics
Abstract

Using the method of transportation-information inequality introduced in [A. Guillin et al., Probab. Theory Related Fields, 144 (2009), pp. 669--695], we establish Bernstein-type concentration inequalities for empirical means of functions of the Markov process $(\int_0^t g(X_s)\,ds)/t$ where $g$ is an unbounded observable of the symmetric Markov process $(X_t)$. Three approaches are proposed: a functional inequalities approach, a Lyapunov function method, and an approach through the Lipschitzian norm of the solution to the Poisson equation. Several applications and examples are studied.

Published
2014
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47. LARGE DEVIATIONS OF THE REALIZED (CO-)VOLATILITY VECTOR

Author

Hacène Djellout, Arnaud Guillin, Yacouba Samoura, Laboratoire de Mathématiques Blaise Pascal (LMBP), Université Blaise Pascal - Clermont-Ferrand 2 (UBP)-Centre National de la Recherche Scientifique (CNRS), Institut Universitaire de France (IUF), and Ministère de l'Education nationale, de l’Enseignement supérieur et de la Recherche (M.E.N.E.S.R.)

Subjects
[QFIN.GN]Quantitative Finance [q-fin]/General Finance [q-fin.GN], Probability (math.PR), diffusion, Mathematics - Statistics Theory, Statistics Theory (math.ST), discrete-time observation, large deviations, FOS: Economics and business, Realised Volatility and covolatility, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], large deviations,diffusion,discrete-time observation,Realised Volatility and covolatility, [MATH.MATH-ST]Mathematics [math]/Statistics [math.ST], FOS: Mathematics, General Finance (q-fin.GN), Quantitative Finance - General Finance, Mathematics - Probability
Abstract

International audience; Realized statistics based on high frequency returns have become very popular in financial economics. In recent years, different non-parametric estimators of the variation of a log-price process have appeared. These were developed by many authors and were motivated by the existence of complete records of price data. Among them are the realized quadratic (co-)variation which is perhaps the most well known example, providing a consistent estimator of the integrated (co-)volatility when the logarithmic price process is continuous. Limit results such as the weak law of large numbers or the central limit theorem have been proved in different contexts. In this paper, we propose to study the large deviation properties of realized (co-)volatility (i.e., when the number of high frequency observations in a fixed time interval increases to infinity. More specifically, we consider a bivariate model with synchronous observation schemes and correlated Brownian motions of the following form: $dX_{\ell,t} = \sigma_{\ell,t}dB_{\ell,t}+b_{\ell}(t,\omega)dt$ for $\ell=1,2$, where $X_{\ell}$ denotes the log-price, we are concerned with the large deviation estimation of the vector $V_t^n(X)=\left(Q_{1,t}^n(X), Q_{2,t}^n(X), C_{t}^n(X)\right)$ where $Q_{\ell,t}^n(X)$ and $C_{t}^n(X)$ represente the estimator of the quadratic variational processes $Q_{\ell,t}=\int_0^t\sigma_{\ell,s}^2ds$ and the integrated covariance $C_t=\int_0^t\sigma_{1,s}\sigma_{2,s}\rho_sds$ respectively, with $\rho_t=cov(B_{1,t}, B_{2,t})$. Our main motivation is to improve upon the existing limit theorems. Our large deviations results can be used to evaluate and approximate tail probabilities of realized (co-)volatility. As an application we provide the large deviation for the standard dependence measures between the two assets returns such as the realized regression coefficients up to time $t$, or the realized correlation. Our study should contribute to the recent trend of research on the (co-)variance estimation problems, which are quite often discussed in high-frequency financial data analysis.

Published
2017

48. Supplement to: 'Spectral Characterization of the Family α-Stable Processes that Generalize Gaussian Process Models.'

Author

Nourddine Azzaoui, Gareth W. Peters, Malcolm Egan, and Arnaud Guillin

Subjects
Pure mathematics, symbols.namesake, Econometrics, symbols, Characterization (mathematics), Mathematical proof, Gaussian process, Mathematics
Abstract

Proofs and Technical Appendix for paper: Spectral characterization of the non-independent increment family of Alpha-Stable processes that generalize Gaussian process models.

Published
2017
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49. Spectral Characterization of the Non-Independent Increment Family of Alpha-Stable Processes that Generalize Gaussian Process Models

Author

Nourddine Azzaoui, Malcolm Egan, Gareth W. Peters, and Arnaud Guillin

Subjects
symbols.namesake, Covariance function, Kriging, Kernel (statistics), symbols, Econometrics, Applied mathematics, Regression analysis, Statistical model, Covariance, Gaussian process, Independence (probability theory), Mathematics
Abstract

The characterization of spatial or temporal processes by a family of sufficient functions such as via a unique spectral representation and a mean function forms the basis of a large number of statistical modelling approaches. For instance, in Gaussian process regression modelling, the mean and covariance function specify uniquely the properties of the resulting statistical model. One can therefore parameterize such regression models and understand their structure and attributes generally via a specification of the covariance kernel. In this paper we generalize significantly the class of available spatial and temporal processes that may be used in statistical applications like regressions to allow for non-stationarity and heavy-tailedness. Importantly, we are able to demonstrate for the first time how to achieve this through a novel formulation no longer requiring independence of the increments in the stochastic construction of the process and statistical model. Furthermore, we achieve this in a manner akin to the covariance kernel specification in a Gaussian process model, we develop a novel covariation spectral representation of some non-stationary and non-indepenent increments symmetric Alpha-stable processes (SalphaS). Such a representation is based on a weaker covariation pseudo additivity condition which is more general than the condition of independence and should allow a very wide class of statistical regression models to be subsequently developed. We present a general framework for sufficient conditions to characterize such processes and develop general constructive approaches to building models satisfying these conditions.

Published
2017
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50. Erratum: Optimal linear drift for the speed of convergence of an hypoelliptic diffusion

Author

Arnaud Guillin and Pierre Monmarché

Subjects
Statistics and Probability, 65C05, irreversibility, optimal sampling, 35K10, hypocoercivity, Statistics, Probability and Uncertainty, Ornstein-Uhlenbeck process, 60J60
Abstract

Erratum for Optimal linear drift for the speed of convergence of an hypoelliptic diffusion, A. Guillin, and P. Monmarché, Electron. Commun. Probab. 21 (2016), paper no. 74, 14 pp. doi:10.1214/16-ECP25.

Published
2017

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