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Felix Lindner

Publications and source records attributed to Felix Lindner.

At least 19 recordsLinked to original sources
arXiv2026

Data-driven subsampling rates for diffusion parameter estimation of SDEs

We study the problem of diffusion parameter estimation for stochastic differential equation (SDE) models in scenarios where data and model are compatible only on specific scales that have yet to be determined. We introduce a simple and efficient method for selecting suitable rates at which given time series data should be subsampled in order to ensure that the statistical structure of the subsampled data is consistent with the behavior of the SDE model on an infinitesimal scale. Our approach is based on analyzing the statistics of the lengths of monotonically increasing or decreasing segments in the subsampled data sequence, which we refer to as monotone runs. As an analytical foundation, we prove for a large class of SDEs with additive noise that the lengths of monotone runs at an infinitesimal scale are approximately geometrically distributed with success probability $1/2$. This universal characterization is employed to derive an automated method for selecting appropriate subsampling rates for given time series data that is directly applicable in real-world scenarios and does not rely on an asymptotic framework of multiscale diffusions. The approach is demonstrated using an application from industrial mathematics concerning surrogate models for fiber lay-down curves in production processes of nonwoven textiles.

math.PR↗
arXiv2025

Random field reconstruction of inhomogeneous turbulence. Part II: Numerical approximation and simulation

A novel random field model or the reconstruction of turbulent velocity fluctuations from inhomogeneous characteristic flow quantities in terms of stochastic Fourier-type integrals has recently been introduced and analyzed by the authors. This article concerns the numerical discretization and implementation of the model and discusses its key features by means of numerical simulations. We present a suitable discretization scheme that combines a randomized quadrature method for stochastic integrals with a local linearization of the non-uniform advection of the turbulent structures by the mean flow. The convergence of the scheme towards the continuous model is verified analytically. Moreover, we describe an efficient algorithmic implementation that allows for flexible local evaluations of the simulated turbulence field. The main features of the model are illustrated by a variety of simulation results, each highlighting specific aspects such as the influence of the inhomogeneous model parameters on the generated fluctuations, spatio-temporal ergodicity properties under inhomogeneous flow conditions, and the validity of Kolmogorov's two-thirds law in dependence on the local turbulence Reynolds number.

physics.flu-dyn↗
arXiv2023

Random field reconstruction of inhomogeneous turbulence. Part I: Modeling and analysis

We develop and analyze a random field model for the reconstruction of turbulent velocity fluctuations from inhomogeneous characteristic flow quantities provided by RANS simulations that is accessible to both a rigorous analytical validation of the model properties and efficient numerical simulation. The model is fully continuous and based on an explicit representation formula in terms of stochastic integrals combining moving average and Fourier-type representations in time and space, respectively. The structure of the model is systematically derived from spectral representations of homogeneous fields by means of suitable stochastic integral transformations that ensure the preservation of consistency properties when progressing to the case of inhomogeneous flow characteristics. Moreover, we employ a two-scale approach that separates a macro scale related to the variations of the characteristic flow quantities from a representative turbulence scale on which the fluctuations are modeled, allowing to assess the model properties by asymptotic analysis w.r.t. the scale ratio. In particular, a novel inhomogeneous ergodicity result establishes the recovery of the inhomogeneous characteristic flow quantities by means of local averages of a single sample path in time and space.

math.PR↗
arXiv2022

Blow up phenomena for gradient descent optimization methods in the training of artificial neural networks

In this article we investigate blow up phenomena for gradient descent optimization methods in the training of artificial neural networks (ANNs). Our theoretical analysis is focused on shallow ANNs with one neuron on the input layer, one neuron on the output layer, and one hidden layer. For ANNs with ReLU activation and at least two neurons on the hidden layer we establish the existence of a target function such that there exists a lower bound for the risk values of the critical points of the associated risk function which is strictly greater than the infimum of the image of the risk function. This allows us to demonstrate that every gradient flow trajectory with an initial risk smaller than this lower bound diverges. Furthermore, we analyze and compare various popular types of activation functions with regard to the divergence of gradient flow trajectories and gradient descent trajectories in the training of ANNs and with regard to the closely related question concerning the existence of global minimum points of the risk function.

math.OC↗
arXiv2022

Understanding a Robot's Guiding Ethical Principles via Automatically Generated Explanations

The continued development of robots has enabled their wider usage in human surroundings. Robots are more trusted to make increasingly important decisions with potentially critical outcomes. Therefore, it is essential to consider the ethical principles under which robots operate. In this paper we examine how contrastive and non-contrastive explanations can be used in understanding the ethics of robot action plans. We build upon an existing ethical framework to allow users to make suggestions about plans and receive automatically generated contrastive explanations. Results of a user study indicate that the generated explanations help humans to understand the ethical principles that underlie a robot's plan.

cs.AI↗
arXiv2022

Stochastic aspects of crack deflection and crack path prediction in short fiber reinforced polymer matrix composites

Owing to the production process, short fiber reinforced composites exhibit a pronounced anisotropy of both elastic properties and crack growth resistance. In particular the latter issue has a major impact on crack deflection and inevitably has to be taken into account for the sake of an accurate prediction of crack paths. The perpendicular axes of transverse isotropy are associated with the fiber orientations, whereupon a crack in transverse direction encounters the largest fracture toughness. While the local mean fiber orientations in polymer matrix composites are determined by the injection molding process, their statistical fluctuations can approximately be described in terms of Gaussian random field models. Furthermore, statistical variations of the volume fraction of fibers and the fiber-matrix adhesion give rise to stochasticity of the local ratio of fracture toughness anisotropy. Influencing factors of prediction regions of crack paths obtained from finite element simulations with an adapted J-integral deflection criterion are investigated, just as stochastic aspects of bifurcation phenomena of crack deflection observed under mode-I loading.

cond-mat.mtrl-sci↗
arXiv2021

Accelerating the Learning of TAMER with Counterfactual Explanations

The capability to interactively learn from human feedback would enable agents in new settings. For example, even novice users could train service robots in new tasks naturally and interactively. Human-in-the-loop Reinforcement Learning (HRL) combines human feedback and Reinforcement Learning (RL) techniques. State-of-the-art interactive learning techniques suffer from slow learning speed, thus leading to a frustrating experience for the human. We approach this problem by extending the HRL framework TAMER for evaluative feedback with the possibility to enhance human feedback with two different types of counterfactual explanations (action and state based). We experimentally show that our extensions improve the speed of learning.

cs.AI↗
arXiv2020

Towards Contrastive Explanations for Comparing the Ethics of Plans

The development of robotics and AI agents has enabled their wider usage in human surroundings. AI agents are more trusted to make increasingly important decisions with potentially critical outcomes. It is essential to consider the ethical consequences of the decisions made by these systems. In this paper, we present how contrastive explanations can be used for comparing the ethics of plans. We build upon an existing ethical framework to allow users to make suggestions to plans and receive contrastive explanations.

cs.AI↗
arXiv2019

Strong convergence rates on the whole probability space for space-time discrete numerical approximation schemes for stochastic Burgers equations

The main result of this article establishes strong convergence rates on the whole probability space for explicit space-time discrete numerical approximations for a class of stochastic evolution equations with possibly non-globally monotone coefficients such as stochastic Burgers equations with additive trace-class noise. The key idea in the proof of our main result is (i) to bring the classical Alekseev-Gr\"obner formula from deterministic analysis into play and (ii) to employ uniform exponential moment estimates for the numerical approximations.

math.PR↗
arXiv2019

Strong and weak divergence of exponential and linear-implicit Euler approximations for stochastic partial differential equations with superlinearly growing nonlinearities

The explicit Euler scheme and similar explicit approximation schemes (such as the Milstein scheme) are known to diverge strongly and numerically weakly in the case of one-dimensional stochastic ordinary differential equations with superlinearly growing nonlinearities. It remained an open question whether such a divergence phenomenon also holds in the case of stochastic partial differential equations with superlinearly growing nonlinearities such as stochastic Allen-Cahn equations. In this work we solve this problem by proving that full-discrete exponential Euler and full-discrete linear-implicit Euler approximations diverge strongly and numerically weakly in the case of stochastic Allen-Cahn equations. This article also contains a short literature overview on existing numerical approximation results for stochastic differential equations with superlinearly growing nonlinearities.

math.NA↗
arXiv2019

Weak convergence rates for temporal numerical approximations of stochastic wave equations with multiplicative noise

In this work we establish weak convergence rates for temporal discretisations of stochastic wave equations with multiplicative noise, in particular, for the hyperbolic Anderson model. For this class of stochastic partial differential equations the weak convergence rates we obtain are indeed twice the known strong rates. To the best of our knowledge, our findings are the first in the scientific literature which provide essentially sharp weak convergence rates for temporal discretisations of stochastic wave equations with multiplicative noise. Key ideas of our proof are a sophisticated splitting of the error and applications of the recently introduced mild It\^{o} formula. We complement our analytical findings by means of numerical simulations in Python for the decay of the weak approximation error for SPDEs for four different test functions.

math.PR↗
arXiv2018

Exponential moment bounds and strong convergence rates for tamed-truncated numerical approximations of stochastic convolutions

In this article we establish exponential moment bounds, moment bounds in fractional order smoothness spaces, a uniform H\"older continuity in time, and strong convergence rates for a class of fully discrete exponential Euler-type numerical approximations of infinite dimensional stochastic convolution processes. The considered approximations involve specific taming and truncation terms and are therefore well suited to be used in the context of SPDEs with non-globally Lipschitz continuous nonlinearities.

math.PR↗
arXiv2018

On the Alekseev-Gr\"obner formula in Banach spaces

The Alekseev-Gr\"obner formula is a well known tool in numerical analysis for describing the effect that a perturbation of an ordinary differential equation (ODE) has on its solution. In this article we provide an extension of the Alekseev-Gr\"obner formula for Banach space valued ODEs under, loosely speaking, mild conditions on the perturbation of the considered ODEs.

math.CA↗
arXiv2018

Malliavin regularity and weak approximation of semilinear SPDE with L\'evy noise

We investigate the weak order of convergence for space-time discrete approximations of semilinear parabolic stochastic evolution equations driven by additive square-integrable L\'evy noise. To this end, the Malliavin regularity of the solution is analyzed and recent results on refined Malliavin-Sobolev spaces from the Gaussian setting are extended to a Poissonian setting. For a class of path-dependent test functions, we obtain that the weak rate of convergence is twice the strong rate.

math.PR↗
arXiv2018

A Formalization of Kant's Second Formulation of the Categorical Imperative

We present a formalization and computational implementation of the second formulation of Kant's categorical imperative. This ethical principle requires an agent to never treat someone merely as a means but always also as an end. Here we interpret this principle in terms of how persons are causally affected by actions. We introduce Kantian causal agency models in which moral patients, actions, goals, and causal influence are represented, and we show how to formalize several readings of Kant's categorical imperative that correspond to Kant's concept of strict and wide duties towards oneself and others. Stricter versions handle cases where an action directly causally affects oneself or others, whereas the wide version maximizes the number of persons being treated as an end. We discuss limitations of our formalization by pointing to one of Kant's cases that the machinery cannot handle in a satisfying way.

cs.AI↗
arXiv2017

Strong convergence of a half-explicit Euler scheme for constrained stochastic mechanical systems

This paper is concerned with the numerical approximation of stochastic mechanical systems with nonlinear holonomic constraints. Such systems are described by second order stochastic differential-algebraic equations involving an implicitly given Lagrange multiplier process. The explicit representation of the Lagrange multiplier leads to an underlying stochastic ordinary differential equation, the drift coefficient of which is typically not globally one-sided Lipschitz continuous. We investigate a half-explicit drift-truncated Euler scheme which fulfills the constraint exactly. Pathwise uniform $L_p$-convergence is established. The proof is based on a suitable decomposition of the discrete Lagrange multipliers and on norm estimates for the single components, enabling the verification of consistency, semi-stability and moment growth properties of the scheme. To the best of our knowledge, the presented result is the first strong convergence result for a constraint-preserving scheme in the considered setting.

math.PR↗
arXiv2017

Poisson Malliavin calculus in Hilbert space with an application to SPDE

In this paper we introduce a Hilbert space-valued Malliavin calculus for Poisson random measures. It is solely based on elementary principles from the theory of point processes and basic moment estimates, and thus allows for a simple treatment of the Malliavin operators. The main part of the theory is developed for general Poisson random measures, defined on a $\sigma$-finite measure space, with minimal conditions. The theory is shown to apply to a space-time setting, suitable for studying stochastic partial differential equations. As an application, we analyze the weak order of convergence of space-time approximations for a class of linear equations with $\alpha$-stable noise, $\alpha\in(1,2)$. For a suitable class of test functions, the weak order of convergence is found to be $\alpha$ times the strong order.

math.PR↗