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In smoothed particle hydrodynamics (SPH) method, the particle-based approximations are implemented via kernel functions, and the evaluation of performance involves two key criteria: numerical accuracy and computational efficiency. In the SPH community, the Wendland kernel reigns as the prevailing choice due to its commendable accuracy and reasonable computational efficiency. Nevertheless, there exists an urgent need to enhance the computational efficiency of numerical methods while upholding accuracy. In this paper, we employ a truncation approach to limit the compact support of the Wendland kernel to 1.6h. This decision is based on the observation that particles within the range of 1.6h to 2h make negligible contributions, practically approaching zero, to the SPH approximation. To address integration errors stemming from the truncation, we incorporate the Laguerre-Gauss kernel for particle relaxation due to the fact that this kernel has been demonstrated to enable the attainment of particle distributions with reduced residue and integration errors \cite{wang2023fourth}. Furthermore, we introduce the kernel gradient correction to rectify numerical errors from the SPH approximation of kernel gradient and the truncated compact support. A comprehensive set of numerical examples including fluid dynamics in Eulerian formulation and solid dynamics in total Lagrangian formulation are tested and have demonstrated that truncated and standard Wendland kernels enable achieve the same level accuracy but the former significantly increase the computational efficiency.

Biological nervous systems constitute important sources of inspiration towards computers that are faster, cheaper, and more energy efficient. Neuromorphic disciplines view the brain as a coevolved system, simultaneously optimizing the hardware and the algorithms running on it. There are clear efficiency gains when bringing the computations into a physical substrate, but we presently lack theories to guide efficient implementations. Here, we present a principled computational model for neuromorphic systems in terms of spatio-temporal receptive fields, based on affine Gaussian kernels over space and leaky-integrator and leaky integrate-and-fire models over time. Our theory is provably covariant to spatial affine and temporal scaling transformations, and with close similarities to the visual processing in mammalian brains. We use these spatio-temporal receptive fields as a prior in an event-based vision task, and show that this improves the training of spiking networks, which otherwise is known as problematic for event-based vision. This work combines efforts within scale-space theory and computational neuroscience to identify theoretically well-founded ways to process spatio-temporal signals in neuromorphic systems. Our contributions are immediately relevant for signal processing and event-based vision, and can be extended to other processing tasks over space and time, such as memory and control.

An additive Runge-Kutta method is used for the time stepping, which integrates the linear stiff terms by an explicit singly diagonally implicit Runge-Kutta (ESDIRK) method and the nonlinear terms by an explicit Runge-Kutta (ERK) method. In each time step, the implicit solve is performed by the recently developed Hierarchical Poincar\'e-Steklov (HPS) method. This is a fast direct solver for elliptic equations that decomposes the space domain into a hierarchical tree of subdomains and builds spectral collocation solvers locally on the subdomains. These ideas are naturally combined in the presented method since the singly diagonal coefficient in ESDIRK and a fixed time-step ensures that the coefficient matrix in the implicit solve of HPS remains the same for all time stages. This means that the precomputed inverse can be efficiently reused, leading to a scheme with complexity (in two dimensions) $\mathcal{O}(N^{1.5})$ for the precomputation where the solution operator to the elliptic problems is built, and then $\mathcal{O}(N \log N)$ for the solve in each time step. The stability of the method is proved for first order in time and any order in space, and numerical evidence substantiates a claim of stability for a much broader class of time discretization methods. Numerical experiments supporting the accuracy of efficiency of the method in one and two dimensions are presented.

The notion of Laplacian of a graph can be generalized to simplicial complexes and hypergraphs, and contains information on the topology of these structures. Even for a graph, the consideration of associated simplicial complexes is interesting to understand its shape. Whereas the Laplacian of a graph has a simple probabilistic interpretation as the generator of a continuous time Markov chain on the graph, things are not so direct when considering simplicial complexes. We define here new Markov chains on simplicial complexes. For a given order~$k$, the state space is the set of $k$-cycles that are chains of $k$-simplexes with null boundary. This new framework is a natural generalization of the canonical Markov chains on graphs. We show that the generator of our Markov chain is the upper Laplacian defined in the context of algebraic topology for discrete structure. We establish several key properties of this new process: in particular, when the number of vertices is finite, the Markov chain is positive recurrent. This result is not trivial, since the cycles can loop over themselves an unbounded number of times. We study the diffusive limits when the simplicial complexes under scrutiny are a sequence of ever refining triangulations of the flat torus. Using the analogy between singular and Hodge homologies, we express this limit as valued in the set of currents. The proof of tightness and the identification of the limiting martingale problem make use of the flat norm and carefully controls of the error terms in the convergence of the generator. Uniqueness of the solution to the martingale problem is left open. An application to hole detection is carried.

This manuscript derives locally weighted ensemble Kalman methods from the point of view of ensemble-based function approximation. This is done by using pointwise evaluations to build up a local linear or quadratic approximation of a function, tapering off the effect of distant particles via local weighting. This introduces a candidate method (the locally weighted Ensemble Kalman method for inversion) with the motivation of combining some of the strengths of the particle filter (ability to cope with nonlinear maps and non-Gaussian distributions) and the Ensemble Kalman filter (no filter degeneracy).

First order shape optimization methods, in general, require a large number of iterations until they reach a locally optimal design. While higher order methods can significantly reduce the number of iterations, they exhibit only local convergence properties, necessitating a sufficiently close initial guess. In this work, we present an unregularized shape-Newton method and combine shape optimization with homotopy (or continuation) methods in order to allow for the use of higher order methods even if the initial design is far from a solution. The idea of homotopy methods is to continuously connect the problem of interest with a simpler problem and to follow the corresponding solution path by a predictor-corrector scheme. We use a shape-Newton method as a corrector and arbitrary order shape derivatives for the predictor. Moreover, we apply homotopy methods also to the case of multi-objective shape optimization to efficiently obtain well-distributed points on a Pareto front. Finally, our results are substantiated with a set of numerical experiments.

Successfully addressing a wide variety of tasks is a core ability of autonomous agents, requiring flexibly adapting the underlying decision-making strategies and, as we argue in this work, also adapting the perception modules. An analogical argument would be the human visual system, which uses top-down signals to focus attention determined by the current task. Similarly, we adapt pre-trained large vision models conditioned on specific downstream tasks in the context of multi-task policy learning. We introduce task-conditioned adapters that do not require finetuning any pre-trained weights, combined with a single policy trained with behavior cloning and capable of addressing multiple tasks. We condition the visual adapters on task embeddings, which can be selected at inference if the task is known, or alternatively inferred from a set of example demonstrations. To this end, we propose a new optimization-based estimator. We evaluate the method on a wide variety of tasks from the CortexBench benchmark and show that, compared to existing work, it can be addressed with a single policy. In particular, we demonstrate that adapting visual features is a key design choice and that the method generalizes to unseen tasks given a few demonstrations.

We propose a continuous approach for computing the pseudospectra of linear operators following a 'solve-then-discretize' strategy. Instead of taking a finite section approach or using a finite-dimensional matrix to approximate the operator of interest, the new method employs an operator analogue of the Lanczos process to work directly with operators and functions. The method is shown to be free of spectral pollution and spectral invisibility, fully adaptive, nearly optimal in accuracy, and well-conditioned. The advantages of the method are demonstrated by extensive numerical examples and comparison with the traditional method.

We study the properties of a family of distances between functions of a single variable. These distances are examples of integral probability metrics, and have been used previously for comparing probability measures on the line; special cases include the Earth Mover's Distance and the Kolmogorov Metric. We examine their properties for general signals, proving that they are robust to a broad class of deformations. We also establish corresponding robustness results for the induced sliced distances between multivariate functions. Finally, we establish error bounds for approximating the univariate metrics from finite samples, and prove that these approximations are robust to additive Gaussian noise. The results are illustrated in numerical experiments, which include comparisons with Wasserstein distances.

The emerging behaviors of swarms have fascinated scientists and gathered significant interest in the field of robotics. Traditionally, swarms are viewed as egalitarian, with robots sharing identical roles and capabilities. However, recent findings highlight the importance of hierarchy for deploying robot swarms more effectively in diverse scenarios. Despite nature's preference for hierarchies, the robotics field has clung to the egalitarian model, partly due to a lack of empirical evidence for the conditions favoring hierarchies. Our research demonstrates that while egalitarian swarms excel in environments proportionate to their collective sensing abilities, they struggle in larger or more complex settings. Hierarchical swarms, conversely, extend their sensing reach efficiently, proving successful in larger, more unstructured environments with fewer resources. We validated these concepts through simulations and physical robot experiments, using a complex radiation cleanup task. This study paves the way for developing adaptable, hierarchical swarm systems applicable in areas like planetary exploration and autonomous vehicles. Moreover, these insights could deepen our understanding of hierarchical structures in biological organisms.