arXiv subjects

Hendrik Weber

Publications and source records attributed to Hendrik Weber.

At least 37 records · Page 2Linked to original sources

arXiv2021

The ConScenD Dataset: Concrete Scenarios from the highD Dataset According to ALKS Regulation UNECE R157 in OpenX

With Regulation UNECE R157 on Automated Lane-Keeping Systems, the first framework for the introduction of passenger cars with Level 3 systems has become available in 2020. In accordance with recent research projects including academia and the automotive industry, the Regulation utilizes scenario based testing for the safety assessment. The complexity of safety validation of automated driving systems necessitates system-level simulations. The Regulation, however, is missing the required parameterization necessary for test case generation. To overcome this problem, we incorporate the exposure and consider the heterogeneous behavior of the traffic participants by extracting concrete scenarios according to Regulation's scenario definition from the established naturalistic highway dataset highD. We present a methodology to find the scenarios in real-world data, extract the parameters for modeling the scenarios and transfer them to simulation. In this process, more than 340 scenarios were extracted. OpenSCENARIO files were generated to enable an exemplary transfer of the scenarios to CARLA and esmini. We compare the trajectories to examine the similarity of the scenarios in the simulation to the recorded scenarios. In order to foster research, we publish the resulting dataset called ConScenD together with instructions for usage with both simulation tools. The dataset is available online at https://www.levelXdata.com/scenarios.

cs.RO↗

arXiv2021

6-Layer Model for a Structured Description and Categorization of Urban Traffic and Environment

Verification and validation of automated driving functions impose large challenges. Currently, scenario-based approaches are investigated in research and industry, aiming at a reduction of testing efforts by specifying safety relevant scenarios. To define those scenarios and operate in a complex real-world design domain, a structured description of the environment is needed. Within the PEGASUS research project, the 6-Layer Model (6LM) was introduced for the description of highway scenarios. This paper refines the 6LM and extends it to urban traffic and environment. As defined in PEGASUS, the 6LM provides the possibility to categorize the environment and, therefore, functions as a structured basis for subsequent scenario description. The model enables a structured description and categorization of the general environment, without incorporating any knowledge or anticipating any functions of actors. Beyond that, there is a variety of other applications of the 6LM, which are elaborated in this paper. The 6LM includes a description of the road network and traffic guidance objects, roadside structures, temporary modifications of the former, dynamic objects, environmental conditions and digital information. The work at hand specifies each layer by categorizing its items. Guidelines are formulated and explanatory examples are given to standardize the application of the model for an objective environment description. In contrast to previous publications, the model and its design are described in far more detail. Finally, the holistic description of the 6LM presented includes remarks on possible future work when expanding the concept to machine perception aspects.

cs.OH↗

arXiv2020

Phase transitions for $ϕ^4_3$

We establish a surface order large deviation estimate for the magnetisation of low temperature $ϕ^4_3$. As a byproduct, we obtain a decay of spectral gap for its Glauber dynamics given by the $ϕ^4_3$ singular stochastic PDE. Our main technical contributions are contour bounds for $ϕ^4_3$, which extends 2D results by Glimm, Jaffe, and Spencer (1975). We adapt an argument by Bodineau, Velenik, and Ioffe (2000) to use these contour bounds to study phase segregation. The main challenge to obtain the contour bounds is to handle the ultraviolet divergences of $ϕ^4_3$ whilst preserving the structure of the low temperature potential. To do this, we build on the variational approach to ultraviolet stability for $ϕ^4_3$ developed recently by Barashkov and Gubinelli (2019).

math.PR↗

arXiv2019

A priori bounds for the $Φ^4$ equation in the full sub-critical regime

We derive a priori bounds for the $Φ^4$ equation in the full sub-critical regime using Hairer's theory of regularity structures. The equation is formally given by \begin{equation} \label{e}(\partial_t-Δ)ϕ= -ϕ^3 + \infty ϕ+ξ, \tag{$\star$} \end{equation} where the term $+\infty ϕ$ represents infinite terms that have to be removed in a renormalisation procedure. We emulate fractional dimensions $d<4$ by adjusting the regularity of the noise term $ξ$, choosing $ξ\in C^{-3+δ}$. Our main result states that if $ϕ$ satisfies this equation on a space-time cylinder $P= (0,1) \times \{ |x| \leq 1 \}$, then away from the boundary $\partial P$ the solution $ϕ$ can be bounded in terms of a finite number of explicit polynomial expressions in $ξ$, and this bound holds uniformly over all possible choices of boundary data for $ϕ$. The derivation of this bound makes full use of the super-linear damping effect of the non-linear term $-ϕ^3$. A key part of our analysis consists of an appropriate re-formulation of the theory of regularity structures in the specific context of \eqref{e}, which allows to couple the small scale control one obtains from this theory with a suitable large scale argument. Along the way we make several new observations and simplifications. Instead of a model $(Π_x)_x$ and the family of translation operators $(Γ_{x,y})_{x,y}$ we work with just a single object $(\mathbb{X}_{x, y})$ which acts on itself for translations, very much in the spirit of Gubinelli's theory of branched rough paths. Furthermore, we show that in the specific context of \eqref{e} the hierarchy of continuity conditions which constitute Hairer's definition of a \emph{modelled distribution} can be reduced to the single continuity condition on the "coefficient on the constant level".

math.AP↗

arXiv2018

Space-time localisation for the dynamic $Φ^4_3$ model

We prove an a priori bound for solutions of the dynamic $Φ^4_3$ equation. This bound provides a control on solutions on a compact space-time set only in terms of the realisation of the noise on an enlargement of this set, and it does not depend on any choice of space-time boundary conditions. We treat the large and small scale behaviour of solutions with completely different arguments. For small scales we use bounds akin to those presented in Hairer's theory of regularity structures. We stress immediately that our proof is fully self-contained, but we give a detailed explanation of how our arguments relate to Hairer's. For large scales we use a PDE argument based on the maximum principle. Both regimes are connected by a solution-dependent regularisation procedure. The fact that our bounds do not depend on space-time boundary conditions makes them useful for the analysis of large scale properties of solutions. They can for example be used in a compactness argument to construct solutions on the full space and their invariant measures.

math.AP↗

arXiv2018

Quasilinear SPDEs via rough paths

We are interested in (uniformly) parabolic PDEs with a nonlinear dependance of the leading-order coefficients, driven by a rough right hand side. For simplicity, we consider a space-time periodic setting with a single spatial variable: \begin{equation*} \partial_2u -P( a(u)\partial_1^2u - σ(u)f ) =0 \end{equation*} where $P$ is the projection on mean-zero functions, and $f$ is a distribution and only controlled in the low regularity norm of $ C^{α-2}$ for $α> \frac{2}{3}$ on the parabolic Hölder scale. The example we have in mind is a random forcing $f$ and our assumptions allow, for example, for an $f$ which is white in the time variable $x_2$ and only mildly coloured in the space variable $x_1$; any spatial covariance operator $(1 + |\partial_1|)^{-λ_1 }$ with $λ_1 > \frac13$ is admissible. On the deterministic side we obtain a $C^α$-estimate for $u$, assuming that we control products of the form $v\partial_1^2v$ and $vf$ with $v$ solving the constant-coefficient equation $\partial_2 v-a_0\partial_1^2v=f$. As a consequence, we obtain existence, uniqueness and stability with respect to $(f, vf, v \partial_1^2v)$ of small space-time periodic solutions for small data. We then demonstrate how the required products can be bounded in the case of a random forcing $f$ using stochastic arguments. For this we extend the treatment of the singular product $σ(u)f$ via a space-time version of Gubinelli's notion of controlled rough paths to the product $a(u)\partial_1^2u$, which has the same degree of singularity but is more nonlinear since the solution $u$ appears in both factors. The PDE ingredient mimics the (kernel-free) Krylov-Safanov approach to ordinary Schauder theory.

math.AP↗

arXiv2018

Local bounds for stochastic reaction diffusion equations

We prove a priori bounds for solutions of stochastic reaction diffusion equations with super-linear damping in the reaction term. These bounds provide a control on the supremum of solutions on any compact space-time set which only depends on the specific realisation of the noise on a slightly larger set and which holds uniformly over all possible space-time boundary values. This constitutes a space-time version of the so-called 'coming down from infinity' property. Bounds of this type are very useful to control the large scale behaviour of solutions effectively and can be used, for example, to construct solutions on the full space even if the driving noise term has no decay at infinity. Our method shows the interplay of the large scale behaviour, dictated by the non-linearity, and the small scale oscillations, dictated by the rough driving noise. As a by-product we show that there is a close relation between the regularity of the driving noise term and the integrability of solutions.

math.AP↗

arXiv2018

Exponential loss of memory for the 2-dimensional Allen-Cahn equation with small noise

We prove an asymptotic coupling theorem for the $2$-dimensional Allen--Cahn equation perturbed by a small space-time white noise. We show that with overwhelming probability two profiles that start close to the minimisers of the potential of the deterministic system contract exponentially fast in a suitable topology. In the $1$-dimensional case a similar result was shown in \cite{MS88,MOS89}. It is well-known that in more than one dimension solutions of this equation are distribution-valued, and the equation has to be interpreted in a renormalised sense. Formally, this renormalisation corresponds to moving the minima of the potential infinitely far apart and making them infinitely deep. We show that despite this renormalisation, solutions behave like perturbations of the deterministic system without renormalisation: they spend large stretches of time close to the minimisers of the (un-renormalised) potential and the exponential contraction rate of different profiles is given by the second derivative of the potential in these points. As an application we prove an Eyring--Kramers law for the transition times between the stable solutions of the deterministic system for fixed initial conditions.

math.PR↗

arXiv2018

Glauber dynamics of 2D Kac-Blume-Capel model and their stochastic PDE limits

We study the Glauber dynamics of a two dimensional Blume-Capel model (or dilute Ising model) with Kac potential parametrized by $(β,θ)$ - the "inverse temperature" and the "chemical potential". We prove that the locally averaged spin field rescales to the solution of the dynamical $Φ^4$ equation near a curve in the $(β,θ)$ plane and to the solution of the dynamical $Φ^6$ equation near one point on this curve. Our proof relies on a discrete implementation of Da Prato-Debussche method as in a result by Mourrat-Weber but an additional coupling argument is needed to show convergence of the linearized dynamics.

math.PR↗

arXiv2018

Parabolic equations with rough coefficients and singular forcing

This article focuses on parabolic equations with rough diffusion coefficients which are ill-posed in the classical sense of distributions due to the presence of a singular forcing. Inspired by the philosophy of rough paths and regularity structures, we introduce a notion of modelled distribution which is suitable in this context. We prove two general tools for reconstruction and integration, as well as a product lemma which is tailor made for the reconstruction of the rough diffusion operator. This yields a partially automated deterministic theory, which we apply to obtain an existence and uniqueness theory for parabolic equations with rough diffusion coefficients and a singular forcing in the negative parabolic Hölder space of order larger than $-\frac{3}{2}$.

math.AP↗

arXiv2017

The dynamic $Φ^4_3$ model comes down from infinity

We prove an a priori bound for the dynamic $Φ^4_3$ model on the torus wich is independent of the initial condition. In particular, this bound rules out the possibility of finite time blow-up of the solution. It also gives a uniform control over solutions at large times, and thus allows to construct invariant measures via the Krylov-Bogoliubov method. It thereby provides a new dynamic construction of the Euclidean $Φ^4_3$ field theory on finite volume. Our method is based on the local-in-time solution theory developed recently by Gubinelli, Imkeller, Perkowski and Catellier, Chouk. The argument relies entirely on deterministic PDE arguments (such as embeddings of Besov spaces and interpolation), which are combined to derive energy inequalities.

math.AP↗

arXiv2017

Stochastic PDEs, Regularity Structures, and Interacting Particle Systems

These lecture notes grew out of a series of lectures given by the second named author in short courses in Toulouse, Matsumoto, and Darmstadt. The main aim is to explain some aspects of the theory of "Regularity structures" developed recently by Hairer in arXiv:1303.5113 . This theory gives a way to study well-posedness for a class of stochastic PDEs that could not be treated previously. Prominent examples include the KPZ equation as well as the dynamic $Φ^4_3$ model. Such equations can be expanded into formal perturbative expansions. Roughly speaking the theory of regularity structures provides a way to truncate this expansion after finitely many terms and to solve a fixed point problem for the "remainder". The key ingredient is a new notion of "regularity" which is based on the terms of this expansion.

math.AP↗

arXiv2017

Quasilinear SPDEs in divergence-form

We develop a solution theory in Hölder spaces for a quasilinear stochastic PDE driven by an additive noise. The key ingredients are two deterministic PDE Lemmas which establish a priori Hölder bounds for an equation with irregular right hand side written in divergence form. We apply these deterministic bounds to the case of a noise term which is white in time and trace class in space to obtain stretched exponential bounds for the Hölder semi-norms of the solution for the stochastic equation.

math.AP↗

arXiv2017

Gaussian approximations for transition paths in Brownian dynamics

This paper is concerned with transition paths within the framework of the overdamped Langevin dynamics model of chemical reactions. We aim to give an efficient description of typical transition paths in the small temperature regime. We adopt a variational point of view and seek the best Gaussian approximation, with respect to Kullback-Leibler divergence, of the non-Gaussian distribution of the diffusion process. We interpret the mean of this Gaussian approximation as the "most likely path" and the covariance operator as a means to capture the typical fluctuations around this most likely path. We give an explicit expression for the Kullback-Leibler divergence in terms of the mean and the covariance operator for a natural class of Gaussian approximations and show the existence of minimisers for the variational problem. Then the low temperature limit is studied via $Γ$-convergence of the associated variational problem. The limiting functional consists of two parts: The first part only depends on the mean and coincides with the $Γ$-limit of the Freidlin-Wentzell rate functional. The second part depends on both, the mean and the covariance operator and is minimized if the dynamics are given by a time-inhomogenous Ornstein-Uhlenbeck process found by linearization of the Langevin dynamics around the Freidlin-Wentzell minimizer.

math.PR↗

arXiv2017

Gaussian Approximations for Probability Measures on $\mathbf{R}^d$

This paper concerns the approximation of probability measures on $\mathbf{R}^d$ with respect to the Kullback-Leibler divergence. Given an admissible target measure, we show the existence of the best approximation, with respect to this divergence, from certain sets of Gaussian measures and Gaussian mixtures. The asymptotic behavior of such best approximations is then studied in the small parameter limit where the measure concentrates; this asymptotic behaviour is characterized using $Γ$-convergence. The theory developed is then applied to understanding the frequentist consistency of Bayesian inverse problems. For a fixed realization of noise, we show the asymptotic normality of the posterior measure in the small noise limit. Taking into account the randomness of the noise, we prove a Bernstein-Von Mises type result for the posterior measure.

math.PR↗

arXiv2017

Construction of $Φ^4_3$ diagrams for pedestrians

We aim to give a pedagogic and essentially self-contained presentation of the construction of various stochastic objects appearing in the dynamical $Φ^4_3$ model. The construction presented here is based on the use of paraproducts. The emphasis is on describing the stochastic objects themselves rather than introducing a solution theory for the equation.

math.PR↗

arXiv2017

An Eyring-Kramers law for the stochastic Allen-Cahn equation in dimension two

We study spectral Galerkin approximations of an Allen--Cahn equation over the two-dimensional torus perturbed by weak space-time white noise of strength $\sqrt{\varepsilon}$. We introduce a Wick renormalisation of the equation in order to have a system that is well-defined as the regularisation is removed. We show sharp upper and lower bounds on the transition times from a neighbourhood of the stable configuration $-1$ to the stable configuration $1$ in the asymptotic regime $\varepsilon \to 0$. These estimates are uniform in the discretisation parameter $N$, suggesting an Eyring-Kramers formula for the limiting renormalised stochastic PDE. The effect of the "infinite renormalisation" is to modify the prefactor and to replace the ratio of determinants in the finite-dimensional Eyring-Kramers law by a renormalised Carleman-Fredholm determinant.

math.PR↗

arXiv2016

The Schr{ö}dinger equation with spatial white noise potential

We consider the linear and nonlinear Schr{ö}dinger equation with a spatial white noise as a potential in dimension 2. We prove existence and uniqueness of solutions thanks to a change of unknown originally used in [8] and conserved quantities.

math.AP↗