arXiv subjects

Wolfgang Hoegele

Publications and source records attributed to Wolfgang Hoegele.

6 recordsLinked to original sources

arXiv2026

Random Inverse Problems with Structural and Probabilistic Ambiguities

In this paper, we investigate a computational class of random inverse problems that incorporates model uncertainties through random variable parameters nonlinearly in the forward model as well as additive observational uncertainty. Random inverse problems with nonlinear parameter dependencies may arise in engineering, geophysics, image processing or uncertainty quantification. We study a perspective on structural ambiguities due to the non-injectivity of the forward model together with probabilistic ambiguities by assigning mixture model densities with separate components to the parameters, which leads to an in general complex forward model, observation model and posterior. As a result, the mixture-model parameters in the forward model can be viewed as simultaneously describing aspects of both the nonlinear ambiguity and the uncertainty of the inverse problem. The presented solution algorithms are based on Bayesian inversion and Monte Carlo computation for three observation scenarios leading to posterior densities for the input for given output samples or an observed output density. We apply the derived algorithms to analytic 1D and 2D quadratic models, to an epidemiological inverse parameter estimation and to a heat equation for inverse source localization. We numerically demonstrate in which observation scenarios the proposed algorithm can resolve probabilistic ambiguities in the solution and in which they cannot and show the interaction patterns between these two types of ambiguities. The results suggest that the proposed perspective is useful in making residual structural ambiguities visible in highly ambiguous inverse problems, including cases with a finite and an infinite number of solutions.

stat.ME↗

arXiv2026

A Full-Density Approach to Simulating Random Iteration Equations with Applications

The goal of this study is to introduce a unified framework for simulating random iteration equations (RIE), understood as iteration equations containing random variables. The main idea is to propagate approximations of the full state density from one iteration to the next, rather than estimating it from many repeated pathwise Monte Carlo simulations. The presentation of the RIE modeling framework is conceptually simple based on recent work on static random equations and designed to be accessible. The modeling requirements for RIEs allow for potential nonsmooth nonlinearities and stochasticities in the transfer function. Additionally, the RIE computational strategy for full-density propagation is presented based on iterative likelihood / posterior calculations. As results, illustrative applications of nonlinear random and stochastic differential equation simulations, a new full-density gradient descent method (FDGD) for global optimization under uncertainty and examples of chaotic mappings are presented in order to demonstrate the breadth of the utility of this framework. In total, the character of the presentation is explorative and encourages new applications and theoretical studies.

math.DS↗

arXiv2026

Steady State Distribution and Stability Analysis of Random Differential Equations with Uncertainties and Superpositions: Application to a Predator Prey Model

We present a computational framework to investigate steady state distributions and perform stability analysis for random ordinary differential equations driven by parameter uncertainty. Using the nonlinear Rosenzweig McArthur predator prey model as a case study, we characterize the non-trivial equilibrium steady state of the system and investigate its complex distribution when the parameter probability densities are multi-modal mixture models with partially overlapping or separated components. In consequence, this application includes both, uncertainties and superpositions, of the system parameters. In addition, we present the stability analysis of steady states based on the eigenvalue distribution of the system's Jacobian matrix in this stochastic regime. The steady state posterior density and stability metrics are computed with a recently published Monte Carlo based numerical scheme specifically designed for random equation systems (Hoegele, 2026). Particularly, the simplicity of this stochastic extension of dynamic systems combined with a broadly applicable computational approach is demonstrated. Numerical experiments show the emergence of multi-modal steady state distributions of the predator prey model and we calculate their stability regions, illustrating the method's applicability to uncertainty quantification in dynamical systems.

math.DS↗

arXiv2025

Combinatorial Potential of Random Equations with Mixture Models: Modeling and Simulation

The goal of this paper is to demonstrate the general modeling and practical simulation of random equations with mixture model parameter random variables. Random equations, understood as stationary (non-dynamical) equations with parameters as random variables, have a long history and a broad range of applications. The specific novelty of this explorative study lies on the demonstration of the combinatorial complexity of these equations and their solutions with mixture model parameters. In a Bayesian argumentation framework, we derive a likelihood function and posterior density of approximate solutions while avoiding significant restrictions about the type of nonlinearity of the equation or mixture models, and demonstrate their numerically efficient implementation for the applied researcher. In the results section, we are specifically focusing on expressive example simulations showcasing the combinatorial potential of random linear equation systems and nonlinear systems of random conic section equations. Introductory applications to portfolio optimization, stochastic control and random matrix theory are provided in order to show the wide applicability of the presented methodology.

stat.CO↗

arXiv2024

Errors-In-Variables Model Fitting for Partially Unpaired Data Utilizing Mixture Models

We introduce a general framework for regression in the errors-in-variables regime, allowing for full flexibility about the dimensionality of the data, observational error probability density types, the (nonlinear) model type and the avoidance of ad-hoc definitions of loss functions. In this framework, we introduce model fitting for partially unpaired data, i.e. for given data groups the pairing information of input and output is lost (semi-supervised). This is achieved by constructing mixture model densities, which directly model the loss of pairing information allowing inference. In a numerical simulation study linear and nonlinear model fits are illustrated as well as a real data study is presented based on life expectancy data from the world bank utilizing a multiple linear regression model. These results show that high quality model fitting is possible with partially unpaired data, which opens the possibility for new applications with unfortunate or deliberate loss of pairing information in data.

stat.ME↗

arXiv2024

A Stochastic-Geometrical Framework for Object Pose Estimation based on Mixture Models Avoiding the Correspondence Problem

Background: Pose estimation of rigid objects is a practical challenge in optical metrology and computer vision. This paper presents a novel stochastic-geometrical modeling framework for object pose estimation based on observing multiple feature points. Methods: This framework utilizes mixture models for feature point densities in object space and for interpreting real measurements. Advantages are the avoidance to resolve individual feature correspondences and to incorporate correct stochastic dependencies in multi-view applications. First, the general modeling framework is presented, second, a general algorithm for pose estimation is derived, and third, two example models (camera and lateration setup) are presented. Results: Numerical experiments show the effectiveness of this modeling and general algorithm by presenting four simulation scenarios for three observation systems, including the dependence on measurement resolution, object deformations and measurement noise. Probabilistic modeling utilizing mixture models shows the potential for accurate and robust pose estimations while avoiding the correspondence problem.

cs.CV↗