Parameter Estimation via Conditional Expectation --- A Bayesian Inversion

When a mathematical or computational model is used to analyse some system, it is usual that some parameters resp.\ functions or fields in the model are not known, and hence uncertain. These parametric quantities are then identified by actual observations of the response of the real system. In a probabilistic setting, Bayes's theory is the proper mathematical background for this identification process. The possibility of being able to compute a conditional expectation turns out to be crucial for this purpose. We show how this theoretical background can be used in an actual numerical procedure, and shortly discuss various numerical approximations

Experimental control of natural perturbations in channel flow

International audienceA combined approach using system identification and feed-forward control design has been applied to experimental laminar channel flow in an effort to reduce the naturally occurring disturbance level. A simple blowing/suction strategy was capable of reducing the standard deviation of the measured sensor signal by 45 %, which markedly exceeds previously obtained results under comparable conditions. A comparable reduction could be verified over a significant streamwise extent, implying an improvement over previous, more localized disturbance control. The technique is effective, flexible, and robust, and the obtained results encourage further explorations of experimental control of convection-dominated flows

Determination of a nonlinear source term in a reaction-diffusion equation by using finite element method and radial basis functions method

In this paper, two numerical methods are presented to solve a nonlinear inverse parabolic problem of determining the unknown reaction term in the scalar reactiondiffusion equation. In the first method, the finite element method will be used to discretize the variational form of the problem and in the second method, we use the radial basis functions (RBFs) method for spatial discretization and finite-difference for time discretization. Usually, the matrices obtained from the discretization of the equations are ill-conditioned, especially in higher-dimensional problems. To overcome such difficulties, we use Tikhonov regularization method. In fact, this work considers a comparative study between the finite element method and radial basis functions method. As we will see, these methods are very useful and convenient tools for approximation problems and they are stable with respect to small perturbation in the input data. The effectiveness of the proposed methods are illustrated by numerical examples.Publisher's Versio

Causality, endogeneity and nonparametric estimation

Cette thèse porte sur les problèmes de causalité et d'endogénéité avec estimation non-paramétrique de la fonction d’intérêt. On explore ces problèmes dans deux modèles différents. Dans le cas de données en coupe transversale et iid, on considère l'estimation d'un modèle additif séparable, dans lequel la fonction de régression dépend d'une variable endogène. L'endogénéité est définie, dans ce cas, de manière très générale : elle peut être liée à une causalité inverse (la variable dépendante peut aussi intervenir dans la réalisation des régresseurs), ou à la simultanéité (les résidus contiennent de l'information qui peut influencer la variable indépendante). L'identification et l'estimation de la fonction de régression se font par variables instrumentales. Dans le cas de séries temporelles, on étudie les effets de l'hypothèse d'exogénéité dans un modèle de régression en temps continu. Dans un tel modèle, la variable d'état est fonction de son passé, mais aussi du passé d'autres variables et on s'intéresse à l'estimation nonparamétrique de la moyenne et de la variance conditionnelle. Le premier chapitre traite de ce dernier cas. En particulier, on donne des conditions suffisantes pour qu'on puisse faire de l'inférence statistique dans un tel modèle. On montre que la non-causalité est une condition suffisante pour l'exogénéité, quand on ne veut pas faire d'hypothèses sur les dynamiques du processus des covariables. Cependant, si on est prêt à supposer que le processus des covariables suit une simple équation différentielle stochastique, l'hypothèse de non-causalité devient immatérielle. Les chapitres de deux à quatre se concentrent sur le modèle iid simple. Etant donné que la fonction de régression est solution d'un problème mal-posé, on s'intéresse aux méthodes d'estimation par régularisation. Dans le deuxième chapitre, on considère ce modèle dans le cas d'un régularisation sur la norme L2 de la fonction (régularisation de type Tikhonov). On dérive les propriétés d'un critère de validation croisée pour définir le choix du paramètre de régularisation. Dans le chapitre trois, coécrit avec Jean-Pierre Florens, on étend ce modèle au cas où la variable dépendante n'est pas directement observée mais où on observe seulement une transformation binaire de cette dernière. On montre que le modèle peut être identifié en utilisant la décomposition de la variable dépendante dans l'espace des variables instrumentales et en supposant que les résidus de ce modèle réduit ont une distribution connue. On démontre alors, sous ces hypothèses, qu'on préserve les propriétés de convergence de l'estimateur non-paramétrique. Enfin, le chapitre quatre, coécrit avec Frédérique Fève et Jean-Pierre Florens, décrit une étude numérique, qui compare les propriétés de diverses méthodes de régularisation. En particulier, on discute des critères pour le choix adaptatif des paramètres de lissage et de régularisation et on teste la validité du bootstrap sauvage dans le cas des modèles de régression non-paramétrique avec variables instrumentales.This thesis deals with the broad problem of causality and endogeneity in econometrics when the function of interest is estimated nonparametrically. It explores this problem in two separate frameworks. In the cross sectional, iid setting, it considers the estimation of a nonlinear additively separable model, in which the regression function depends on an endogenous explanatory variable. Endogeneity is, in this case, broadly denned. It can relate to reverse causality (the dependent variable can also affects the independent regressor) or to simultaneity (the error term contains information that can be related to the explanatory variable). Identification and estimation of the regression function is performed using the method of instrumental variables. In the time series context, it studies the implications of the assumption of exogeneity in a regression type model in continuous time. In this model, the state variable depends on its past values, but also on some external covariates and the researcher is interested in the nonparametric estimation of both the conditional mean and the conditional variance functions. This first chapter deals with the latter topic. In particular, we give sufficient conditions under which the researcher can make meaningful inference in such a model. It shows that noncausality is a sufficient condition for exogeneity if the researcher is not willing to make any assumption on the dynamics of the covariate process. However, if the researcher is willing to assume that the covariate process follows a simple stochastic differential equation, then the assumption of noncausality becomes irrelevant. Chapters two to four are instead completely devoted to the simple iid model. The function of interest is known to be the solution of an inverse problem. In the second chapter, this estimation problem is considered when the regularization is achieved using a penalization on the L2-norm of the function of interest (so-called Tikhonov regularization). We derive the properties of a leave-one-out cross validation criterion in order to choose the regularization parameter. In the third chapter, coauthored with Jean-Pierre Florens, we extend this model to the case in which the dependent variable is not directly observed, but only a binary transformation of it. We show that identification can be obtained via the decomposition of the dependent variable on the space spanned by the instruments, when the residuals in this reduced form model are taken to have a known distribution. We finally show that, under these assumptions, the consistency properties of the estimator are preserved. Finally, chapter four, coauthored with Frédérique Fève and Jean-Pierre Florens, performs a numerical study, in which the properties of several regularization techniques are investigated. In particular, we gather data-driven techniques for the sequential choice of the smoothing and the regularization parameters and we assess the validity of wild bootstrap in nonparametric instrumental regressions

Greedy low-rank algorithm for spatial connectome regression

Recovering brain connectivity from tract tracing data is an important computational problem in the neurosciences. Mesoscopic connectome reconstruction was previously formulated as a structured matrix regression problem (Harris et al., 2016), but existing techniques do not scale to the whole-brain setting. The corresponding matrix equation is challenging to solve due to large scale, ill-conditioning, and a general form that lacks a convergent splitting. We propose a greedy low-rank algorithm for connectome reconstruction problem in very high dimensions. The algorithm approximates the solution by a sequence of rank-one updates which exploit the sparse and positive definite problem structure. This algorithm was described previously (Kressner and Sirkovi\'c, 2015) but never implemented for this connectome problem, leading to a number of challenges. We have had to design judicious stopping criteria and employ efficient solvers for the three main sub-problems of the algorithm, including an efficient GPU implementation that alleviates the main bottleneck for large datasets. The performance of the method is evaluated on three examples: an artificial "toy" dataset and two whole-cortex instances using data from the Allen Mouse Brain Connectivity Atlas. We find that the method is significantly faster than previous methods and that moderate ranks offer good approximation. This speedup allows for the estimation of increasingly large-scale connectomes across taxa as these data become available from tracing experiments. The data and code are available online