arXiv subjects

Maximilian Siebel

Publications and source records attributed to Maximilian Siebel.

4 recordsLinked to original sources

arXiv2026

Posterior contraction under misspecification and heteroscedasticity in non-linear inverse problems

In many practical and numerical inverse problems, the exact data log-likelihood is not fully accessible, motivating the use of surrogate models. We study heteroscedastic nonparametric nonlinear regression problems with Gaussian errors and establish contraction results for posterior distributions arising from a surrogate log-likelihood constructed from proxy error variances, an approximate forward map, and an appropriate Gaussian process prior. Under general assumptions on the approximation quality, we show that the resulting surrogate posterior is statistically reliable and contracts about the true parameter at rates comparable to those of the exact posterior. The analysis leverages consistency properties of the (penalised) MLE to effectively handle heteroscedastic noise and to control the impact of likelihood approximation errors. We apply the framework to PDE-constrained inverse problems for a reaction-diffusion equation and the two-dimensional Navier-Stokes equation. In the latter case, we consider misspecified viscosity and forcing terms as well as Oseen-type linearization models, highlighting the relevance of our results for numerical analysis applications.

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arXiv2025

Convergence Rates for the Maximum A Posteriori Estimator in PDE-Regression Models with Random Design

We consider the statistical inverse problem of recovering a parameter $θ\in H^α$ from data arising from the Gaussian regression problem \begin{equation*} Y = \mathscr{G}(θ)(Z)+\varepsilon \end{equation*} with nonlinear forward map $\mathscr{G}:\mathbb{L}^2\to\mathbb{L}^2$, random design points $Z$ and Gaussian noise $\varepsilon$. The estimation strategy is based on a least squares approach under $\Vert\cdot\Vert_{H^α}$-constraints. We establish the existence of a least squares estimator $\hatθ$ as a maximizer for a given functional under Lipschitz-type assumptions on the forward map $\mathscr{G}$. A general concentration result is shown, which is used to prove consistency and upper bounds for the prediction error. The corresponding rates of convergence reflect not only the smoothness of the parameter of interest but also the ill-posedness of the underlying inverse problem. We apply the general model to the Darcy problem, where the recovery of an unknown coefficient function $f$ of a PDE is of interest. For this example, we also provide corresponding rates of convergence for the prediction and estimation errors. Additionally, we briefly discuss the applicability of the general model to other problems.

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arXiv2024

Lower Bounds for Nonparametric Estimation of Ordinary Differential Equations

We noisily observe solutions of an ordinary differential equation $\dot u = f(u)$ at given times, where $u$ lives in a $d$-dimensional state space. The model function $f$ is unknown and belongs to a Hölder-type smoothness class with parameter $β$. For the nonparametric problem of estimating $f$, we provide lower bounds on the error in two complementary model specifications: the snake model with few, long observed solutions and the stubble model with many short ones. The lower bounds are minimax optimal in some settings. They depend on various parameters, which in the optimal asymptotic regime leads to the same rate for the squared error in both models: it is characterized by the exponent $-2β/(2(β+1)+d)$ for the total number of observations $n$. To derive these results, we establish a master theorem for lower bounds in general nonparametric regression problems, which makes the proofs more comparable and seems to be a useful tool for future use.

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arXiv2023

Multiplicative deconvolution under unknown error distribution

We consider a multiplicative deconvolution problem, in which the density $f$ or the survival function $S^X$ of a strictly positive random variable $X$ is estimated nonparametrically based on an i.i.d. sample from a noisy observation $Y = X\cdot U$ of $X$. The multiplicative measurement error $U$ is supposed to be independent of $X$. The objective of this work is to construct a fully data-driven estimation procedure when the error density $f^U$ is unknown. We assume that in addition to the i.i.d. sample from $Y$, we have at our disposal an additional i.i.d. sample drawn independently from the error distribution. The proposed estimation procedure combines the estimation of the Mellin transformation of the density $f$ and a regularisation of the inverse of the Mellin transform by a spectral cut-off. The derived risk bounds and oracle-type inequalities cover both - the estimation of the density $f$ as well as the survival function $S^X$. The main issue addressed in this work is the data-driven choice of the cut-off parameter using a model selection approach. We discuss conditions under which the fully data-driven estimator can attain the oracle-risk up to a constant without any previous knowledge of the error distribution. We compute convergences rates under classical smoothness assumptions. We illustrate the estimation strategy by a simulation study with different choices of distributions.

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