Adaptive estimation for inverse problems with applications to cell division.

Summary This thesis is divided into two independent parts. In the first part, we consider a stochastic individual-centered model in continuous time describing a population structured by size. The population is represented by a point measure evolving according to a deterministic piecewise random process. We study here the non-parametric estimation of the kernel governing the splits, under two different observation schemes. First, in the case where we obtain the entire tree of splits, we construct a kernel estimator with data-dependent adaptive window selection. We obtain an oracle inequality and optimal exponential convergence speeds. Second, in the case where the splitting tree is not completely observed, we show that the renormalized microscopic process describing the evolution of the population converges to the weak solution of a partial differential equation. We propose an estimator of the division kernel using Fourier techniques. We show the consistency of the estimator. In the second part, we consider the non-parametric regression model with errors on the variables in the multidimensional context. Our objective is to estimate the unknown multivariate regression function. We propose an adaptive estimator based on projection kernels founded on a multi-index wavelet basis and a deconvolution operator. The resolution level of the wavelets is obtained by the Goldenshluger-Lepski method. We obtain an oracle inequality and optimal convergence speeds on anisotropic Hölder spaces.