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We investigate path-wise observables in experiments on driven colloids in a periodic optical field to dissect selected intricate transport features, kinetics, and transition-path time statistics out of thermodynamic equilibrium. These observables directly reflect the properties of individual paths in contrast to the properties of an ensemble of particles, such as radial distribution functions or mean-squared displacements. In particular, we present two distinct albeit equivalent formulations of the underlying stochastic equation of motion, highlight their respective practical relevance, and show how to interchange between them. We discuss conceptually different notions of local velocities and interrogate one- and two-sided first-passage and transition-path-time statistics in and out of equilibrium. We emphasize how path-wise observables may be employed to systematically assess the quality of experimental data and demonstrate that, given sufficient control and sampling, one may quantitatively verify subtle theoretical predictions.

Autonomous Vehicles (AVs) need an accurate and up-to-date representation of the environment for safe navigation. Traditional methods, which often rely on detailed environmental representations constructed offline, struggle in dynamically changing environments or when dealing with outdated maps. Consequently, there is a pressing need for real-time solutions that can integrate diverse data sources and adapt to the current situation. An existing framework that addresses these challenges is SDS (situation-aware drivable space). However, SDS faces several limitations, including its use of a non-standard output representation, its choice of encoding objects as points, restricting representation of more complex geometries like road lanes, and the fact that its methodology has been validated only with simulated or heavily post-processed data. This work builds upon SDS and introduces SDS++, designed to overcome SDS's shortcomings while preserving its benefits. SDS++ has been rigorously validated not only in simulations but also with unrefined vehicle data, and it is integrated with a model predictive control (MPC)-based planner to verify its advantages for the planning task. The results demonstrate that SDS++ significantly enhances trajectory planning capabilities, providing increased robustness against localization noise, and enabling the planning of trajectories that adapt to the current driving context.

We design quasi-interpolation operators based on piecewise polynomial weight functions of degree less than or equal to $p$ that map into the space of continuous piecewise polynomials of degree less than or equal to $p+1$. We show that the operators have optimal approximation properties, i.e., of order $p+2$. This can be exploited to enhance the accuracy of finite element approximations provided that they are sufficiently close to the orthogonal projection of the exact solution on the space of piecewise polynomials of degree less than or equal to $p$. Such a condition is met by various numerical schemes, e.g., mixed finite element methods and discontinuous Petrov--Galerkin methods. Contrary to well-established postprocessing techniques which also require this or a similar closeness property, our proposed method delivers a conforming postprocessed solution that does not rely on discrete approximations of derivatives nor local versions of the underlying PDE. In addition, we introduce a second family of quasi-interpolation operators that are based on piecewise constant weight functions, which can be used, e.g., to postprocess solutions of hybridizable discontinuous Galerkin methods. Another application of our proposed operators is the definition of projection operators bounded in Sobolev spaces with negative indices. Numerical examples demonstrate the effectiveness of our approach.

Scatter plots are widely recognized as fundamental tools for illustrating the relationship between two numerical variables. Despite this, based on solid theoretical foundations, scatter plots generated from pairs of continuous random variables may not serve as reliable tools for assessing dependence. Sklar's Theorem implies that scatter plots created from ranked data are preferable for such analysis as they exclusively convey information pertinent to dependence. This is in stark contrast to conventional scatter plots, which also encapsulate information about the variables' marginal distributions. Such additional information is extraneous to dependence analysis and can obscure the visual interpretation of the variables' relationship. In this article, we delve into the theoretical underpinnings of these ranked data scatter plots, hereafter referred to as rank plots. We offer insights into interpreting the information they reveal and examine their connections with various association measures, including Pearson's and Spearman's correlation coefficients, as well as Schweizer-Wolff's measure of dependence. Furthermore, we introduce a novel graphical combination for dependence analysis, termed a dplot, and demonstrate its efficacy through real data examples., Comment: 10 pages, 11 figures, 1 table

Predicting the movement of other road users is beneficial for improving automated vehicle (AV) performance. However, the relationship between the time horizon associated with these predictions and AV performance remains unclear. Despite the existence of numerous trajectory prediction algorithms, no studies have been conducted on how varying prediction lengths affect AV safety and other vehicle performance metrics, resulting in undefined horizon requirements for prediction methods. Our study addresses this gap by examining the effects of different prediction horizons on AV performance, focusing on safety, comfort, and efficiency. Through multiple experiments using a state-of-the-art, risk-based predictive trajectory planner, we simulated predictions with horizons up to 20 seconds. Based on our simulations, we propose a framework for specifying the minimum required and optimal prediction horizons based on specific AV performance criteria and application needs. Our results indicate that a horizon of 1.6 seconds is required to prevent collisions with crossing pedestrians, horizons of 7-8 seconds yield the best efficiency, and horizons up to 15 seconds improve passenger comfort. We conclude that prediction horizon requirements are application-dependent, and recommend aiming for a prediction horizon of 11.8 seconds as a general guideline for applications involving crossing pedestrians., Comment: Submitted to IEEE Intelligent Vehicles Symposium. 9 pages. 10 figures. 6 tables

This paper introduces discrete-holomorphic Perfectly Matched Layers (PMLs) specifically designed for high-order finite difference (FD) discretizations of the scalar wave equation. In contrast to standard PDE-based PMLs, the proposed method achieves the remarkable outcome of completely eliminating numerical reflections at the PML interface, in practice achieving errors at the level of machine precision. Our approach builds upon the ideas put forth in a recent publication [Journal of Computational Physics 381 (2019): 91-109] expanding the scope from the standard second-order FD method to arbitrary high-order schemes. This generalization uses additional localized PML variables to accommodate the larger stencils employed. We establish that the numerical solutions generated by our proposed schemes exhibit an exponential decay rate as they propagate within the PML domain. To showcase the effectiveness of our method, we present a variety of numerical examples, including waveguide problems. These examples highlight the importance of employing high-order schemes to effectively address and minimize undesired numerical dispersion errors, emphasizing the practical advantages and applicability of our approach.

Many players in the automotive field support scenario-based assessment of automated vehicles (AVs), where individual traffic situations can be tested and, thus, facilitate concluding on the performance of AVs in different situations. Since a large number of different scenarios can occur in real-world traffic, the question is how to find a finite set of relevant scenarios. Scenarios extracted from large real-world datasets represent real-world traffic since real driving data is used. Extracting scenarios, however, is challenging because (1) the scenarios to be tested should assess the AVs behave safely, which conflicts with the fact that the majority of the data contains scenarios that are not interesting from a safety perspective, and (2) extensive data processing is required, which hinders the utilization of large real-world datasets. In this work, we propose an approach for extracting scenarios from real-world driving data. The first step is data preprocessing to tackle the errors and noise in real-world data by reconstructing the data. The second step performs data tagging to label actors' activities, their interactions with each other and the environment. Finally, the scenarios are extracted by searching for combinations of tags. The proposed approach is evaluated using data simulated with CARLA and applied to a part of a large real-world driving dataset, i.e., the Waymo Open Motion Dataset (WOMD). The code and scenarios extracted from WOMD are open to the research community to facilitate the assessment of the automated driving functions in different scenarios., Comment: 6 pages, accepted by ITSC 2023

Accurate and robust trajectory predictions of road users are needed to enable safe automated driving. To do this, machine learning models are often used, which can show erratic behavior when presented with previously unseen inputs. In this work, two environment-aware models (MotionCNN and MultiPath++) and two common baselines (Constant Velocity and an LSTM) are benchmarked for robustness against various perturbations that simulate functional insufficiencies observed during model deployment in a vehicle: unavailability of road information, late detections, and noise. Results show significant performance degradation under the presence of these perturbations, with errors increasing up to +1444.8\% in commonly used trajectory prediction evaluation metrics. Training the models with similar perturbations effectively reduces performance degradation, with error increases of up to +87.5\%. We argue that despite being an effective mitigation strategy, data augmentation through perturbations during training does not guarantee robustness towards unforeseen perturbations, since identification of all possible on-road complications is unfeasible. Furthermore, degrading the inputs sometimes leads to more accurate predictions, suggesting that the models are unable to learn the true relationships between the different elements in the data.

The production of heavy mass resonances has been widely studied theoretically and experimentally. Several extensions of the standard model (SM) of particle physics, naturally give rise to a new resonance, with neutral electric charge, commonly referred to as the $\textrm{Z}^{\prime}$ boson. The nature, mass, couplings, and associated quantum numbers of this hypothetical particle are yet to be determined. We present a feasibility study on the production of a vector like $\textrm{Z}^{\prime}$ boson at the LHC, with preferential couplings to third generation fermions, considering proton-proton collisions at $\sqrt{s} = 13$ $\mathrm{TeV}$ and 14 TeV. We work under two simplified phenomenological frameworks where the $\mathrm{Z}^{\prime}$ masses and couplings to the SM particles are free parameters, and consider final states of the $\textrm{Z}^{\prime}$ decaying to a pair of $\mathrm{b}$ quarks. The analysis is performed using machine learning techniques in order to maximize the experimental sensitivity. The proposed search methodology can be a key mode for discovery, complementary to the existing search strategies considered in literature, and extends the LHC sensitivity to the $\mathrm{Z}^{\prime}$ parameter space., Comment: 13 pages, 15 figures

To operate safely, an automated vehicle (AV) must anticipate how the environment around it will evolve. For that purpose, it is important to know which prediction models are most appropriate for every situation. Currently, assessment of prediction models is often performed over a set of trajectories without distinction of the type of movement they capture, resulting in the inability to determine the suitability of each model for different situations. In this work we illustrate how standardized evaluation methods result in wrong conclusions regarding a model's predictive capabilities, preventing a clear assessment of prediction models and potentially leading to dangerous on-road situations. We argue that following evaluation practices in safety assessment for AVs, assessment of prediction models should be performed in a scenario-based fashion. To encourage scenario-based assessment of prediction models and illustrate the dangers of improper assessment, we categorize trajectories of the Waymo Open Motion dataset according to the type of movement they capture. Next, three different models are thoroughly evaluated for different trajectory types and prediction horizons. Results show that common evaluation methods are insufficient and the assessment should be performed depending on the application in which the model will operate., Comment: To be published in IEEE Intelligent Transportation Systems Conference 2022

A frequent task in exploratory data analysis consists in examining pairwise dependencies between data variables. Popular approaches include visualizing correlation or scatter plot matrices. However, both methods can be misleading. The former is primarily limited because it reports a single value for a pair of random variables. Furthermore, scatter plots can fail to convey the dependency structure between variables properly. In this paper we discuss these shortcomings and present alternative and richer visualizations based on copula functions, which fully determine the dependency between continuous random variables. Since copulas seldom appear in the data visualization literature we first review essential theory, and propose alternative scatter plots and several heatmaps for assessing the statistical association between two continuous random variables. These visualizations not only allow users to detect independence, but also increasing and/or decreasing trends in the data through a color coding, which can also be applied in other methods such as parallel coordinates., Comment: 10 pages, 7 figures, 2 tables

We study the 2D motion of colloidal dimers by single-particle tracking and compare the experimental observations obtained by bright-field microscopy to theoretical predictions for anisotropic diffusion. The comparison is based on the mean-square displacements in the laboratory and particle frame as well as generalizations of the self-intermediate scattering functions, which provide insights into the rotational dynamics of the dimer. The diffusional anisotropy leads to a measurable translational-rotational coupling that becomes most prominent by aligning the coordinate system with the initial orientation of the particles. In particular, we find a splitting of the time-dependent diffusion coefficients parallel and perpendicular to the long axis of the dimer which decays over the orientational relaxation time. Deviations of the self-intermediate scattering functions from pure exponential relaxation are small but can be resolved experimentally. The theoretical predictions and experimental results agree quantitatively.