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The design of particle simulation methods for collisional plasma physics has always represented a challenge due to the unbounded total collisional cross section, which prevents a natural extension of the classical Direct Simulation Monte Carlo (DSMC) method devised for the Boltzmann equation. One way to overcome this problem is to consider the design of Monte Carlo algorithms that are robust in the so-called grazing collision limit. In the first part of this manuscript, we will focus on the construction of collision algorithms for the Landau-Fokker-Planck equation based on the grazing collision asymptotics and which avoids the use of iterative solvers. Subsequently, we discuss problems involving uncertainties and show how to develop a stochastic Galerkin projection of the particle dynamics which permits to recover spectral accuracy for smooth solutions in the random space. Several classical numerical tests are reported to validate the present approach.

In this work we propose tailored model order reduction for varying boundary optimal control problems governed by parametric partial differential equations. With varying boundary control, we mean that a specific parameter changes where the boundary control acts on the system. This peculiar formulation might benefit from model order reduction. Indeed, fast and reliable simulations of this model can be of utmost usefulness in many applied fields, such as geophysics and energy engineering. However, varying boundary control features very complicated and diversified parametric behaviour for the state and adjoint variables. The state solution, for example, changing the boundary control parameter, might feature transport phenomena. Moreover, the problem loses its affine structure. It is well known that classical model order reduction techniques fail in this setting, both in accuracy and in efficiency. Thus, we propose reduced approaches inspired by the ones used when dealing with wave-like phenomena. Indeed, we compare standard proper orthogonal decomposition with two tailored strategies: geometric recasting and local proper orthogonal decomposition. Geometric recasting solves the optimization system in a reference domain simplifying the problem at hand avoiding hyper-reduction, while local proper orthogonal decomposition builds local bases to increase the accuracy of the reduced solution in very general settings (where geometric recasting is unfeasible). We compare the various approaches on two different numerical experiments based on geometries of increasing complexity.

Language is compositional; an instruction can express multiple relation constraints to hold among objects in a scene that a robot is tasked to rearrange. Our focus in this work is an instructable scene-rearranging framework that generalizes to longer instructions and to spatial concept compositions never seen at training time. We propose to represent language-instructed spatial concepts with energy functions over relative object arrangements. A language parser maps instructions to corresponding energy functions and an open-vocabulary visual-language model grounds their arguments to relevant objects in the scene. We generate goal scene configurations by gradient descent on the sum of energy functions, one per language predicate in the instruction. Local vision-based policies then re-locate objects to the inferred goal locations. We test our model on established instruction-guided manipulation benchmarks, as well as benchmarks of compositional instructions we introduce. We show our model can execute highly compositional instructions zero-shot in simulation and in the real world. It outperforms language-to-action reactive policies and Large Language Model planners by a large margin, especially for long instructions that involve compositions of multiple spatial concepts. Simulation and real-world robot execution videos, as well as our code and datasets are publicly available on our website: //ebmplanner.github.io.

Collision avoidance (CA) has always been the foremost task for autonomous vehicles (AVs) under safety criteria. And path planning is directly responsible for generating a safe path to accomplish CA while satisfying other commands. Due to the real-time computation and simple structure, the potential field (PF) has emerged as one of the mainstream path-planning algorithms. However, the current PF is primarily simulated in ideal CA scenarios, assuming complete obstacle information while disregarding occlusion issues where obstacles can be partially or entirely hidden from the AV's sensors. During the occlusion period, the occluded obstacles do not possess a PF. Once the occlusion is over, these obstacles can generate an instantaneous virtual force that impacts the ego vehicle. Therefore, we propose an occlusion-aware path planning (OAPP) with the responsibility-sensitive safety (RSS)-based PF to tackle the occlusion problem for non-connected AVs. We first categorize the detected and occluded obstacles, and then we proceed to the RSS violation check. Finally, we can generate different virtual forces from the PF for occluded and non-occluded obstacles. We compare the proposed OAPP method with other PF-based path planning methods via MATLAB/Simulink. The simulation results indicate that the proposed method can eliminate instantaneous lateral oscillation or sway and produce a smoother path than conventional PF methods.

Path planning is critical for autonomous vehicles (AVs) to determine the optimal route while considering constraints and objectives. The potential field (PF) approach has become prevalent in path planning due to its simple structure and computational efficiency. However, current PF methods used in AVs focus solely on the path generation of the ego vehicle while assuming that the surrounding obstacle vehicles drive at a preset behavior without the PF-based path planner, which ignores the fact that the ego vehicle's PF could also impact the path generation of the obstacle vehicles. To tackle this problem, we propose a PF-based path planning approach where local paths are shared among ego and obstacle vehicles via vehicle-to-vehicle (V2V) communication. Then by integrating this shared local path into an objective function, a new optimization function called interactive speed optimization (ISO) is designed to allow driving safety and comfort for both ego and obstacle vehicles. The proposed method is evaluated using MATLAB/Simulink in the urgent merging scenarios by comparing it with conventional methods. The simulation results indicate that the proposed method can mitigate the impact of other AVs' PFs by slowing down in advance, effectively reducing the oscillations for both ego and obstacle AVs.

In this paper, we conduct an in-depth investigation of the structural intricacies inherent to the Invariant Energy Quadratization (IEQ) method as applied to gradient flows, and we dissect the mechanisms that enable this method to uphold linearity and the conservation of energy simultaneously. Building upon this foundation, we propose two methods: Invariant Energy Convexification and Invariant Energy Functionalization. These approaches can be perceived as natural extensions of the IEQ method. Employing our novel approaches, we reformulate the system connected to gradient flow, construct a semi-discretized numerical scheme, and obtain a commensurate modified energy dissipation law for both proposed methods. Finally, to underscore their practical utility, we provide numerical evidence demonstrating these methods' accuracy, stability, and effectiveness when applied to both Allen-Cahn and Cahn-Hilliard equations.

Modular approaches, which use a different composition of modules for each problem and avoid forgetting by design, have been shown to be a promising direction in continual learning (CL). However, searching through the large, discrete space of possible module compositions is a challenge because evaluating a composition's performance requires a round of neural network training. To address this challenge, we develop a modular CL framework, called PICLE, that accelerates search by using a probabilistic model to cheaply compute the fitness of each composition. The model combines prior knowledge about good module compositions with dataset-specific information. Its use is complemented by splitting up the search space into subsets, such as perceptual and latent subsets. We show that PICLE is the first modular CL algorithm to achieve different types of transfer while scaling to large search spaces. We evaluate it on two benchmark suites designed to capture different desiderata of CL techniques. On these benchmarks, PICLE offers significantly better performance than state-of-the-art CL baselines.

Object rearrangement is a fundamental problem in robotics with various practical applications ranging from managing warehouses to cleaning and organizing home kitchens. While existing research has primarily focused on single-agent solutions, real-world scenarios often require multiple robots to work together on rearrangement tasks. This paper proposes a comprehensive learning-based framework for multi-agent object rearrangement planning, addressing the challenges of task sequencing and path planning in complex environments. The proposed method iteratively selects objects, determines their relocation regions, and pairs them with available robots under kinematic feasibility and task reachability for execution to achieve the target arrangement. Our experiments on a diverse range of environments demonstrate the effectiveness and robustness of the proposed framework. Furthermore, results indicate improved performance in terms of traversal time and success rate compared to baseline approaches.

Finite element methods and kinematically coupled schemes that decouple the fluid velocity and structure's displacement have been extensively studied for incompressible fluid-structure interaction (FSI) over the past decade. While these methods are known to be stable and easy to implement, optimal error analysis has remained challenging. Previous work has primarily relied on the classical elliptic projection technique, which is only suitable for parabolic problems and does not lead to optimal convergence of numerical solutions to the FSI problems in the standard $L^2$ norm. In this article, we propose a new kinematically coupled scheme for incompressible FSI thin-structure model and establish a new framework for the numerical analysis of FSI problems in terms of a newly introduced coupled non-stationary Ritz projection, which allows us to prove the optimal-order convergence of the proposed method in the $L^2$ norm. The methodology presented in this article is also applicable to numerous other FSI models and serves as a fundamental tool for advancing research in this field.

Accurate trajectory prediction of nearby vehicles is crucial for the safe motion planning of automated vehicles in dynamic driving scenarios such as highway merging. Existing methods cannot initiate prediction for a vehicle unless observed for a fixed duration of two or more seconds. This prevents a fast reaction by the ego vehicle to vehicles that enter its perception range, thus creating safety concerns. Therefore, this paper proposes a novel transformer-based trajectory prediction approach, specifically trained to handle any observation length larger than one frame. We perform a comprehensive evaluation of the proposed method using two large-scale highway trajectory datasets, namely the highD and exiD. In addition, we study the impact of the proposed prediction approach on motion planning and control tasks using extensive merging scenarios from the exiD dataset. To the best of our knowledge, this marks the first instance where such a large-scale highway merging dataset has been employed for this purpose. The results demonstrate that the prediction model achieves state-of-the-art performance on highD dataset and maintains lower prediction error w.r.t. the constant velocity across all observation lengths in exiD. Moreover, it significantly enhances safety, comfort, and efficiency in dense traffic scenarios, as compared to the constant velocity model.