A04 – Nonlinear statistical inverse problems with random observations

Many mathematical models of time-dependent processes are used to make predictions. These models often come in the form of ordinary differential equation initial value problems, where the vector field depends on a finite-dimensional vector of parameters. Estimating the parameters is important for making predictions. However, in some practical problems, the parameter cannot be observed directly. Instead, it is known that the parameter is related to an  vector of "covariates", which can be observed directly. For example, in pharmacology, the model parameters may be blood flow rates, organ volumes, or enzymatic activity, and the covariates may be body weight, body mass index, or genetic disposition. The relationship is encoded in a so-called “covariate-to-parameter map”. Inferring the unknown covariate-to-parameter map using  pairs of observed covariates and the associated measurements of the time-dependent process constitutes an ill-posed nonlinear inverse problem. This project aims to develop nonparametric statistical methods for solving this inverse problem, with a focus on problems that feature the constraint of random design, i.e. where the observed covariates are i.i.d. copies of a random variable.

In the second funding period, we will focus on the problems of adaptivity and inference, and tackle the inverse problem from both the frequentist and Bayesian points of view. Adaptivity means that regularisation parameters, such as the penalty in the Tikhonov functional or the iteration number in gradient descent, are chosen in a data-driven way without prior smoothness assumptions on the covariate-to-parameter map. For the frequentist part, we will study methods that perform adaptive regularisation by early stopping. We will also study the construction of honest and optimal confidence sets for early-stopping estimators. For the Bayesian part, we shall study adaptive methods that work under the random design constraint and the posterior concentration properties of these methods. We will also investigate Bayesian credible sets and their frequentist coverage properties. We aim to test the developed methods on some problems from pharmacology.

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