1. Data assimilation - the collaborative research centre SFB1294
  2. Research
  3. Research Area A
  4. A04

A04 – Nonlinear statistical inverse problems with random observations

The project deals with nonlinear statistical inverse problems. The goal is to estimate from
random observations the functional relation between observable covariates and intrinsic
(unobservable) parameters of a system. The system depends on the intrinsic parameters through
a specific model, e.g. a differential equation. Mechanistic modelling approaches from
pharmacokinetics serve as a specific application. We will develop frequentist and Bayesian
nonparametric estimators of the unknown functional relation. Properties of the estimators, e.g.
computational efficiency and uncertainty quantification, will be analysed.

  • Lie, H. C. (2024): Bayesian inference of covariate-parameter relationships for population modelling. ArXiv 2407.09640

  • Rastogi, A. and Mathé, P. (2020): Inverse learning in Hilbert scales.arXiv 2002.10208

  • Celisse, A. and Wahl, M. (2020): Analyzing the discrepancy principle for kernelized spectral filter learning algorithms.arXiv: 2004.08436

  • Duval, C. and Mariucci, E. (2020): Non-asymptotic control of the cumulative distribution function of Lévy processes. arXiv 2003.09281

  • Blanchard, G., Mathé, P., and Mücke, N. (2019): Lepskii Principle in Supervised Learning. arXiv: 1905.10764 

  • Wahl, M. (2019): A note on the prediction error of principal component regression.arXiv: 1811.02998

  • Carpentier, A., Duval, C., and Mariucci, E. (2019): Total variation distance for discretely observed Lévy processes: a Gaussian approximation of the small jumps. arXiv: 1810.02998

  • Duval, C. and Mariucci, E. (2019): Compound Poisson approximation to estimate the Lévy density. arXiv: 1702.08787 

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

  • Jirak, M. and Wahl, M. (2018): Relative perturbation bounds with applications to empirical covariance operators.arXiv: 1802.02869

  • Kretschmann, R. and Werner, F. (2025). Maximum a posteriori testing in statistical inverse problems.Inverse Problems and Imaging 19 (6): 1268-1301. doi: 10.3934/ipi.2025015

  • Cvetkovic, N. and Lie, H. C. and Bansal, H. and Veroy-Grepl, K. (2024): Choosing observation operators to mitigate model error in Bayesian inverse problems. SIAM/ASA Journal of Uncertainty Quantification 12 (3):723-758. ArXiv 2301.04863, doi: 10.1137/23M1602140

  • Stankewitz, B. (2024): Early stopping for L2-boosting in high-dimensional linear models. Annals of Statistics 52 (2):491-518, arXiv:2210.07850.

  • Lie, H. C. and Rudolf, D. and Sprungk, B. and Sullivan, T. J. (2023). Dimension-independent Markov chain Monte Carlo on the sphere. Scandinavian Journal of Statistics 50 (4):1818-1858. ArXiv: 2112.12185  

  • Stankewitz, B. and Mücke, N. and Rosasco, L. (2023). From inexact optimization to learning via gradient concentration. Computational Optimization and Applications 84:265-294. arXiv:2106.05397.

  • Lie, H. C. and Stahn, M. and Sullivan, T.J. (2022). Randomised one-step time integration methods for deterministic operator differential equations. Calcolo, Volume 59, Number 13, ArXiv 2103.16506, doi: 10.1007/s10092-022-00457-6.

  • Hartung, N., Wahl, M., Rastogi, A., and Huisinga, W. (2021). Nonparametric goodness-of-fit tests for parametric covariate models in pharmacometric analyses. CPT Pharmacometrics & Systems Pharmacology 10: 564-576. ArXiv 2011.07539 DOI

  • Rastogi, A. (2020): Tikhonov regularization with oversmoothing penalty for nonlinear statistical inverse problems. Communications on Pure & Applied Analysis 19(8): 4111-4126. ArXiv 2002.01303DOI

  • Rastogi, A., Blanchard, G. and Mathé, P. (2020): Convergence analysis of Tikhonov regularization for non-linear statistical inverse learning problems. Electronic Journal of Statistics 14(2): 2798-2841. ArXiv 1902.05404v2DO

  • Milbradt, C. and Wahl, M. (2020). High-probability bounds for the reconstruction error of PCA. Statist. Probab. Lett. 161. ArXiv 1909.10787

  • Reiß, M. and Wahl, M. (2020). Non-asymptotic upper bounds for the reconstruction error of PCA. Ann. Stat. 48(2): 1098-1123. arXiv 1609.03779

  • Rastogi, A. (2019). Tikhonov regularization with oversmoothing penalty for linear statistical inverse learning problems. AIP Conference Proceedings 2183(1): 110004 AIP Publishing LLC. DOI

  • 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: 1903.11142 

  • Blanchard, G., Neuvial, P. and Roquain, E. (2019). Post hoc inference via joint family-wise error rate control. (to appear in Annals of Statistics) arXiv: 1703.02307

  • Blanchard, 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-EJS1482

  • Blanchard, 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:10.1137/17M1154096

  • Bachoc, 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