A well-known boundary observability inequality for the elasticity system establishes that the energy of the system can be estimated from the solution on a sufficiently large part of the boundary for a sufficiently large time. This inequality is relevant in different contexts as the exact boundary controllability, boundary stabilization, or some inverse source problems. Here we show that a corresponding boundary observability inequality for the spectral collocation approximation of the linear elasticity system in a d-dimensional cube also holds, uniformly with respect to the discretization parameter. This property is essential to prove that natural numerical approaches to the previous problems based on replacing the elasticity system by collocation discretization will give successful approximations of the continuous counterparts.
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The elastic energy of a bending-resistant interface depends both on its geometry and its material composition. We consider such a heterogeneous interface in the plane, modeled by a curve equipped with an additional density function. The resulting energy captures the complex interplay between curvature and density effects, resembling the Canham-Helfrich functional. We describe the curve by its inclination angle, so that the equilibrium equations reduce to an elliptic system of second order. After a brief variational discussion, we investigate the associated nonlocal $L^2$-gradient flow evolution, a coupled quasilinear parabolic problem. We analyze the (non)preservation of quantities such as convexity, positivity, and symmetry, as well as the asymptotic behavior of the system. The results are illustrated by numerical experiments.
We consider two-phase fluid deformable surfaces as model systems for biomembranes. Such surfaces are modeled by incompressible surface Navier-Stokes-Cahn-Hilliard-like equations with bending forces. We derive this model using the Lagrange-D'Alembert principle considering various dissipation mechanisms. The highly nonlinear model is solved numerically to explore the tight interplay between surface evolution, surface phase composition, surface curvature and surface hydrodynamics. It is demonstrated that hydrodynamics can enhance bulging and furrow formation, which both can further develop to pinch-offs. The numerical approach builds on a Taylor-Hood element for the surface Navier-Stokes part, a semi-implicit approach for the Cahn-Hilliard part, higher order surface parametrizations, appropriate approximations of the geometric quantities, and mesh redistribution. We demonstrate convergence properties that are known to be optimal for simplified sub-problems.
We propose a many-sorted modal logic for reasoning about knowledge in multi-agent systems. Our logic introduces a clear distinction between participating agents and the environment. This allows to express local properties of agents and global properties of worlds in a uniform way, as well as to talk about the presence or absence of agents in a world. The logic subsumes the standard epistemic logic and is a conservative extension of it. The semantics is given in chromatic hypergraphs, a generalization of chromatic simplicial complexes, which were recently used to model knowledge in distributed systems. We show that the logic is sound and complete with respect to the intended semantics. We also show a further connection of chromatic hypergraphs with neighborhood frames.
The paper addresses an optimal ensemble control problem for nonlocal continuity equations on the space of probability measures. We admit the general nonlinear cost functional, and an option to directly control the nonlocal terms of the driving vector field. For this problem, we design a descent method based on Pontryagin's maximum principle (PMP). To this end, we derive a new form of PMP with a decoupled Hamiltonian system. Specifically, we extract the adjoint system of linear nonlocal balance laws on the space of signed measures and prove its well-posedness. As an implementation of the designed descent method, we propose an indirect deterministic numeric algorithm with backtracking. We prove the convergence of the algorithm and illustrate its modus operandi by treating a simple case involving a Kuramoto-type model of a population of interacting oscillators.
The proliferation of data generation has spurred advancements in functional data analysis. With the ability to analyze multiple variables simultaneously, the demand for working with multivariate functional data has increased. This study proposes a novel formulation of the epigraph and hypograph indexes, as well as their generalized expressions, specifically tailored for the multivariate functional context. These definitions take into account the interrelations between components. Furthermore, the proposed indexes are employed to cluster multivariate functional data. In the clustering process, the indexes are applied to both the data and their first and second derivatives. This generates a reduced-dimension dataset from the original multivariate functional data, enabling the application of well-established multivariate clustering techniques that have been extensively studied in the literature. This methodology has been tested through simulated and real datasets, performing comparative analyses against state-of-the-art to assess its performance.
Finding a solution to the linear system $Ax = b$ with various minimization properties arises from many engineering and computer science applications, including compressed sensing, image processing, and machine learning. In the age of big data, the scalability of stochastic optimization algorithms has made it increasingly important to solve problems of unprecedented sizes. This paper focuses on the problem of minimizing a strongly convex objective function subject to linearly constraints. We consider the dual formulation of this problem and adopt the stochastic coordinate descent to solve it. The proposed algorithmic framework, called fast stochastic dual coordinate descent, utilizes an adaptive variation of Polyak's heavy ball momentum and user-defined distributions for sampling. Our adaptive heavy ball momentum technique can efficiently update the parameters by using iterative information, overcoming the limitation of the heavy ball momentum method where prior knowledge of certain parameters, such as singular values of a matrix, is required. We prove that, under strongly admissible of the objective function, the propose method converges linearly in expectation. By varying the sampling matrix, we recover a comprehensive array of well-known algorithms as special cases, including the randomized sparse Kaczmarz method, the randomized regularized Kaczmarz method, the linearized Bregman iteration, and a variant of the conjugate gradient (CG) method. Numerical experiments are provided to confirm our results.
Which technological linkages affect the sector's ability to innovate? How do these effects transmit through the technology space? This paper answers these two key questions using novel methods of text mining and network analysis. We examine technological interdependence across sectors over a period of half a century (from 1976 to 2021) by analyzing the text of 6.5 million patents granted by the United States Patent and Trademark Office (USPTO), and applying network analysis to uncover the full spectrum of linkages existing across technology areas. We demonstrate that patent text contains a wealth of information often not captured by traditional innovation metrics, such as patent citations. By using network analysis, we document that indirect linkages are as important as direct connections and that the former would remain mostly hidden using more traditional measures of indirect linkages, such as the Leontief inverse matrix. Finally, based on an impulse-response analysis, we illustrate how technological shocks transmit through the technology (network-based) space, affecting the innovation capacity of the sectors.
Poisson process models are defined in terms of their rates for outage and restore processes in power system resilience events. These outage and restore processes easily yield the performance curves that track the evolution of resilience events, and the area, nadir, and duration of the performance curves are standard resilience metrics. This letter analyzes typical resilience events by analyzing the area, nadir, and duration of mean performance curves. Explicit and intuitive formulas for these metrics are derived in terms of the Poisson process model parameters, and these parameters can be estimated from utility data. This clarifies the calculation of metrics of typical resilience events, and shows what they depend on. The metric formulas are derived with lognormal, exponential, or constant rates of restoration. The method is illustrated with a typical North American transmission event. Similarly nice formulas are obtained for the area metric for empirical power system data.
Fine assembly tasks such as electrical connector insertion have tight tolerances and sensitive components, requiring compensation of alignment errors while applying sufficient force in the insertion direction, ideally at high speeds and while grasping a range of components. Vision, tactile, or force sensors can compensate alignment errors, but have limited bandwidth, limiting the safe assembly speed. Passive compliance such as silicone-based fingers can reduce collision forces and grasp a range of components, but often cannot provide the accuracy or assembly forces required. To support high-speed mechanical search and self-aligning insertion, this paper proposes monolithic additively manufactured fingers which realize a moderate, structured compliance directly proximal to the gripped object. The geometry of finray-effect fingers are adapted to add form-closure features and realize a directionally-dependent stiffness at the fingertip, with a high stiffness to apply insertion forces and lower transverse stiffness to support alignment. Design parameters and mechanical properties of the fingers are investigated with FEM and empirical studies, analyzing the stiffness, maximum load, and viscoelastic effects. The fingers realize a remote center of compliance, which is shown to depend on the rib angle, and a directional stiffness ratio of $14-36$. The fingers are applied to a plug insertion task, realizing a tolerance window of $7.5$ mm and approach speeds of $1.3$ m/s.
Linear inverse problems arise in diverse engineering fields especially in signal and image reconstruction. The development of computational methods for linear inverse problems with sparsity is one of the recent trends in this field. The so-called optimal $k$-thresholding is a newly introduced method for sparse optimization and linear inverse problems. Compared to other sparsity-aware algorithms, the advantage of optimal $k$-thresholding method lies in that it performs thresholding and error metric reduction simultaneously and thus works stably and robustly for solving medium-sized linear inverse problems. However, the runtime of this method is generally high when the size of the problem is large. The purpose of this paper is to propose an acceleration strategy for this method. Specifically, we propose a heavy-ball-based optimal $k$-thresholding (HBOT) algorithm and its relaxed variants for sparse linear inverse problems. The convergence of these algorithms is shown under the restricted isometry property. In addition, the numerical performance of the heavy-ball-based relaxed optimal $k$-thresholding pursuit (HBROTP) has been evaluated, and simulations indicate that HBROTP admits robustness for signal and image reconstruction even in noisy environments.
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