# A04 – Nonlinear statistical inverse problems with random observations

## Objectives

How can a doctor adapt a medical treatment, in particular drug dosage, precisely to an individual patient? In order for the drug to be effective while avoiding side effects or toxicity, the drug levels have to be maintained in a certain therapeutic window.

A critical task in pharmacology is thus the identification of factors, so-called covariates that influence the drug levels in the body. Mathematical models, like physiologically based pharmacokinetic (PBPK) models, are used to predict the concentration-time profiles in the blood and the tissues of a patient. The model parameters have a mechanistic interpretation (e.g., blood flows, organ volumes, enzymatic activity), and depend on the patient, e.g., its body weight, body mass index, genetic disposition. Typically, however, it is not possible to measure them directly for a given patient.

While covariate selection criteria have been studied extensively, the choice of the actual functional relationship between model parameters and covariates has received less attention. In contrast to the above physiologically based pharmacokinetic models, modeling this relationship is rather empirical; in many cases the choice of a linear, exponential or power functional relationship is largely driven by convenience.

This project takes this problem as a motivation to study non-parametric estimation of covariate effects on the parameters of a time-dependent process. The process is modeled as a function G(θ,t) whose form is known (e.g. by a physical model), and depends on an unknown parameter vector theta. We assume that we observe different realizations Y_i of this process (contaminated with observation noise) at the same fixed time points, along with covariates X_i having an influence on the unknown model parameters θ_i (both of which randomly change across realizations). Estimating from observed data the functional relationship between X and θ constitutes a non-linear inverse problem.

The project will concentrate on two theoretical aspects:

- Statistical properties of the inverse problem and application to covariate modelling, including estimation by reproducing kernel methods, confidence regions for the model prediction and goodness-of-fit tests.
- Numerical algorithms with statistical guarantees: investigation of the performance of local iterative procedures to numerically compute the estimator with controlled statistical error, construction of statistically adaptive early stopping rules.

Furthermore, we will evaluate the performance of nonparametric covariate modeling against simulated data from a PBPK model and design specific kernels for the application field.

**Position:** A post-doctoral position hosted at the University of Potsdam and supervised by Gilles Blanchard and Wilhelm Huisinga will be focusing on aspect 1 including methodological aspects of the inverse problem and applications in pharmacology. Suitable candidates for the postdoctoral research position are expected to have prior expertise in nonparametric statistics, inverse problems or optimization.

## Preprints

Duval, C. and Mariucci, E. (2020).

*Non-asymptotic control of the cumulative distribution function of Lévy processes.*arXiv 2003.09281Rastogi, A. (2020).

*Tikhonov regularization with oversmoothing penalty for nonlinear statistical inverse problems.*arXiv:2002.01303Blanchard, G., Mathé, P. and Mücke, N. (2019).

*Lepskii Principle in Supervised Learning.*arXiv: 1905.10764Wahl, M. (2019).

*A note on the prediction error of principal component regression.*arXiv: 1811.02998Carpentier, A., Duval, C. and Mariucci, E. (2019).

*Total variation distance for discretely observed Lévy processes: a Gaussian approximation of the small jumps.*arXiv: 1810.02998Reiß, M. and Wahl, M. (2019).

*Non-asymptotic upper bounds for the reconstruction error of PCA.*Annals of Statistics, to appear. arXiv 1609.03779Rastogi, A., Blanchard, G. and Mathé, P. (2019).

*Convergence analysis of Tikhonov regularization for non-linear statistical inverse learning problems.*arXiv 1902.05404Duval, C. and Mariucci, E. (2019).

*Compound Poisson approximation to estimate the Lévy density.*arXiv: 1702.08787Jirak, M. and Wahl, M. (2018).

*Perturbation bounds for eigenspaces under a relative gap condition.*arXiv: 1803.03868Jirak, M. and Wahl, M. (2018).

*Relative perturbation bounds with applications to empirical covariance operators.*arXiv: 1802.02869

## Publications

Gugushvili, S., Mariucci, E. and Meulen, van der F. (2019).

*Decompounding discrete distributions: A non-parametric Bayesian approach.*To appear in Scandinavian Journal of Statistics. arXiv: 1903.11142Blanchard, G., Neuvial, P. and Roquain, E. (2019).

*Post hoc inference via joint family-wise error rate control. (t*o appear in*Annals of Statistics)*arXiv: 1703.02307Blanchard, G., Hoffmann, M. and Reiß, M. (2018).

*Early stopping for statistical inverse problems via truncated SVD estimation.*Electron. J. Statist. 12 (2): 3204-3231. arXiv 1710.07278; doi: 10.1214/18-EJS1482Blanchard, G., Hoffmann, M. and Reiß, M. (2018).

*Optimal adaptation for early stopping in statistical inverse problems.*SIAM/ASA Journal on Uncertainty Quantification 6(3): 1043-1075. arXiv 1606.07702; doi:10.1137/17M1154096Bachoc, F., Blanchard, G. and Neuvial, P. (2018).

*On the post selection inference constant under restricted isometry properties.*Electron. J. Statist. 12(2): 3736-3757. doi: 10.1214/18-EJS1490