arXiv subjects

Robert Voßhall

Publications and source records attributed to Robert Voßhall.

7 recordsLinked to original sources

arXiv2024

Quantifying Local Model Validity using Active Learning

Real-world applications of machine learning models are often subject to legal or policy-based regulations. Some of these regulations require ensuring the validity of the model, i.e., the approximation error being smaller than a threshold. A global metric is generally too insensitive to determine the validity of a specific prediction, whereas evaluating local validity is costly since it requires gathering additional data.We propose learning the model error to acquire a local validity estimate while reducing the amount of required data through active learning. Using model validation benchmarks, we provide empirical evidence that the proposed method can lead to an error model with sufficient discriminative properties using a relatively small amount of data. Furthermore, an increased sensitivity to local changes of the validity bounds compared to alternative approaches is demonstrated.

stat.ML↗

arXiv2017

Integration by parts on the law of the modulus of the Brownian bridge

We prove an infinite dimensional integration by parts formula on the law of the modulus of the Brownian bridge $BB=(BB_t)_{0 \leq t \leq 1}$ from $0$ to $0$ in use of methods from white noise analysis and Dirichlet form theory. Additionally to the usual drift term, this formula contains a distribution which is constructed in the space of Hida distributions by means of a Wick product with Donsker's delta (which correlates with the local time of $|BB|$ at zero). This additional distribution corresponds to the reflection at zero caused by the modulus.

math.PR↗

arXiv2016

Strong Feller property of sticky reflected distorted Brownian motion

Using Girsanov transformations we construct from sticky reflected Brownian motion on $[0,\infty)$ a conservative diffusion on $E:=[0,\infty)^n$, $n \in \mathbb{N}$, and prove that its transition semigroup possesses the strong Feller property for a specified general class of drift functions. By identifying the Dirichlet form of the constructed process, we characterize it as sticky reflected distorted Brownian motion. In particular, the relations of the underlying analytic Dirichlet form methods to the probabilistic methods of random time changes and Girsanov transformations are presented. Our studies are motivated by its applications to the dynamical wetting model with $δ$-pinning and repulsion.

math.PR↗

arXiv2015

Interacting particle systems with sticky boundary

In this paper, we construct under general assumptions the stochastic dynamics of an interacting particle system in a bounded domain $Ω$ with sticky boundary. Under appropriate conditions on the interaction the constructed process solves the underlying SDE for every starting point in the state space. Moreover, we also obtain a solution for q.e. starting point in the case of singular interactions which generalizes former results. Finally, the setting is applied to the case of particles diffusing in a chromatography tube.

math.PR↗

arXiv2015

Construction and analysis of sticky reflected diffusions

We give a Dirichlet form approach for the construction of distorted Brownian motion in a bounded domain $Ω$ of $\mathbb{R}^d$, $d \geq 1$, with boundary $Γ$, where the behavior at the boundary is sticky. The construction covers the case of a static boundary behavior as well as the case of a diffusion on the hypersurface $Γ$ (for $d \geq 2)$. More precisely, we consider the state space $\overlineΩ=Ω\stackrel{.}{\cup} Γ$, the process is a diffusion process inside $Ω$, the occupation time of the process on the boundary $Γ$ is positive and the process may diffuse on $Γ$ as long as it sticks on the boundary. The problem is formulated in an $L^2$-setting and the construction is formulated under weak assumptions on the coefficients and $Γ$. In order to analyze the process we assume a $C^2$-boundary and some weak differentiability conditions. In this case, we deduce that the process is also a solution to a given SDE for quasi every starting point in $\overlineΩ$ with respect to the underyling Dirichlet form. Under the addtional condition that $\{ \varrho =0 \}$ is of capacity zero, we prove ergodicity of the constructed process and consequently, we verify that the boundary behavior is indeed sticky. Moreover, we show ($\mathcal{L}^p$-)strong Feller properties which allow to characterize the constructed process even for every starting point in $\overlineΩ \backslash \{ \varrho=0\}$.

math.PR↗

arXiv2014

Construction and analysis of a sticky reflected distorted Brownian motion

We give a Dirichlet form approach for the construction of a distorted Brownian motion in $E:=[0,\infty)^n$, $n\in\mathbb{N}$, where the behavior on the boundary is determined by the competing effects of reflection from and pinning at the boundary (sticky boundary behavior). In providing a Skorokhod decomposition of the constructed process we are able to justify that the stochastic process is solving the underlying stochastic differential equation weakly for quasi every starting point with respect to the associated Dirichlet form. That the boundary behavior of the constructed process indeed is sticky, we obtain by proving ergodicity of the constructed process. Therefore, we are able to show that the occupation time on specified parts of the boundary is positive. In particular, our considerations enable us to construct a dynamical wetting model (also known as Ginzburg--Landau dynamics) on a bounded set $D_{\scriptscriptstyle{N}}\subset \mathbb{Z}^d$ under mild assumptions on the underlying pair interaction potential in all dimensions $d\in\mathbb{N}$. In dimension $d=2$ this model describes the motion of an interface resulting from wetting of a solid surface by a fluid.

math.PR↗

arXiv2012

Construction of a finite volume dynamical wetting model with δ-pinning in (d+1)-dimension via Dirichlet forms

We give a Dirichlet form approach for the construction of a distorted Brownian motion in $E := [0;\infty)^n$, $n\in\mathbb{N}$, where the behavior on the boundary is determined by the competing effects of reflection from and pinning at the boundary. The problem is formulated in an $L^2$-setting with underlying measure $μ=\varrho m$. Here $\varrho$ is a positive density, integrable with respect to the measure $m$ and fulfilling the Hamza condition. The measure $m$ is such that the boundary of $E$ is not of $m$-measure zero. A reference measure of this type is needed in order to give meaning to the so-called Wentzell boundary condition which is in literature typical for modeling such kind of boundary behavior. In providing a Skorokhod decomposition of the constructed process we are able to justify that the stochastic process is solving the underlying stochastic differential equation weakly in the sense of N. Ikeda and Sh. Watanabe for quasi every starting point. At the boundary the constructed process indeed is governed by the competing effects of reflection and pinning. In order to obtain the Skorokhod decomposition we need $\varrho$ to be continuously differentiable on $E$, which is equivalent to continuity of the logarithmic derivative of $\varrho$. Furthermore, we assume that the logarithmic derivative of $\varrho$ is square integrable with respect to $μ$. We do not need that the logarithmic derivative of $\varrho$ is Lipschitz continuous. In particular, our considerations enable us to construct a dynamical wetting model (also known as Ginzburg-Landau dynamics) on a bounded set $D_N\subset\mathbb{Z}^d$ under mild assumptions on the underlying pair interaction potentials in all dimensions $d\in\mathbb{N}$. In dimension $d=2$ this model describes the motion of an interface resulting from wetting of a solid surface by a fluid.

math.PR↗