This paper presents an algorithm to generate a new kind of polygonal mesh obtained from triangulations. Each polygon is built from a terminal-edge region surrounded by edges that are not the longest-edge of any of the two triangles that share them. The algorithm is divided into three phases. The first phase consists of labeling each edge and triangle of the input triangulation according to its size; the second phase builds polygons (simple or not) from terminal-edges regions using the label system; and the third phase transforms each non simple polygon into simple ones. The final mesh contains polygons with convex and nonconvex shape. Since Voronoi based meshes are currently the most used polygonal meshes, we compare some geometric properties of our meshes against constrained Voronoi meshes. Several experiments are run to compare the shape and size of polygons, the number of final mesh points and polygons. Finally, we validate these polygonal meshes by solving a Laplace equation on an L-shaped domain using the Virtual Element Method (VEM) and show the optimal convergence rate of the numerical solution.
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In this paper, we propose a depth-first search (DFS) algorithm for searching maximum matchings in general graphs. Unlike blossom shrinking algorithms, which store all possible alternative alternating paths in the super-vertices shrunk from blossoms, the newly proposed algorithm does not involve blossom shrinking. The basic idea is to deflect the alternating path when facing blossoms. The algorithm maintains detour information in an auxiliary stack to minimize the redundant data structures. A benefit of our technique is to avoid spending time on shrinking and expanding blossoms. This DFS algorithm can determine a maximum matching of a general graph with $m$ edges and $n$ vertices in $O(mn)$ time with space complexity $O(n)$.
We investigate the feature compression of high-dimensional ridge regression using the optimal subsampling technique. Specifically, based on the basic framework of random sampling algorithm on feature for ridge regression and the A-optimal design criterion, we first obtain a set of optimal subsampling probabilities. Considering that the obtained probabilities are uneconomical, we then propose the nearly optimal ones. With these probabilities, a two step iterative algorithm is established which has lower computational cost and higher accuracy. We provide theoretical analysis and numerical experiments to support the proposed methods. Numerical results demonstrate the decent performance of our methods.
Many existing algorithms for streaming geometric data analysis have been plagued by exponential dependencies in the space complexity, which are undesirable for processing high-dimensional data sets. In particular, once $d\geq\log n$, there are no known non-trivial streaming algorithms for problems such as maintaining convex hulls and L\"owner-John ellipsoids of $n$ points, despite a long line of work in streaming computational geometry since [AHV04]. We simultaneously improve these results to $\mathrm{poly}(d,\log n)$ bits of space by trading off with a $\mathrm{poly}(d,\log n)$ factor distortion. We achieve these results in a unified manner, by designing the first streaming algorithm for maintaining a coreset for $\ell_\infty$ subspace embeddings with $\mathrm{poly}(d,\log n)$ space and $\mathrm{poly}(d,\log n)$ distortion. Our algorithm also gives similar guarantees in the \emph{online coreset} model. Along the way, we sharpen results for online numerical linear algebra by replacing a log condition number dependence with a $\log n$ dependence, answering a question of [BDM+20]. Our techniques provide a novel connection between leverage scores, a fundamental object in numerical linear algebra, and computational geometry. For $\ell_p$ subspace embeddings, we give nearly optimal trade-offs between space and distortion for one-pass streaming algorithms. For instance, we give a deterministic coreset using $O(d^2\log n)$ space and $O((d\log n)^{1/2-1/p})$ distortion for $p>2$, whereas previous deterministic algorithms incurred a $\mathrm{poly}(n)$ factor in the space or the distortion [CDW18]. Our techniques have implications in the offline setting, where we give optimal trade-offs between the space complexity and distortion of subspace sketch data structures. To do this, we give an elementary proof of a "change of density" theorem of [LT80] and make it algorithmic.
We consider M-estimation problems, where the target value is determined using a minimizer of an expected functional of a Levy process. With discrete observations from the Levy process, we can produce a "quasi-path" by shuffling increments of the Levy process, we call it a quasi-process. Under a suitable sampling scheme, a quasi-process can converge weakly to the true process according to the properties of the stationary and independent increments. Using this resampling technique, we can estimate objective functionals similar to those estimated using the Monte Carlo simulations, and it is available as a contrast function. The M-estimator based on these quasi-processes can be consistent and asymptotically normal.
There has been an arising trend of adopting deep learning methods to study partial differential equations (PDEs). This article is to propose a Deep Learning Galerkin Method (DGM) for the closed-loop geothermal system, which is a new coupled multi-physics PDEs and mainly consists of a framework of underground heat exchange pipelines to extract the geothermal heat from the geothermal reservoir. This method is a natural combination of Galerkin Method and machine learning with the solution approximated by a neural network instead of a linear combination of basis functions. We train the neural network by randomly sampling the spatiotemporal points and minimize loss function to satisfy the differential operators, initial condition, boundary and interface conditions. Moreover, the approximate ability of the neural network is proved by the convergence of the loss function and the convergence of the neural network to the exact solution in L^2 norm under certain conditions. Finally, some numerical examples are carried out to demonstrate the approximation ability of the neural networks intuitively.
In this article we suggest two discretization methods based on isogeometric analysis (IGA) for planar linear elasticity. On the one hand, we apply the well-known ansatz of weakly imposed symmetry for the stress tensor and obtain a well-posed mixed formulation. Such modified mixed problems have been already studied by different authors. But we concentrate on the exploitation of IGA results to handle also curved boundary geometries. On the other hand, we consider the more complicated situation of strong symmetry, i.e. we discretize the mixed weak form determined by the so-called Hellinger-Reissner variational principle. We show the existence of suitable approximate fields leading to an inf-sup stable saddle-point problem. For both discretization approaches we prove convergence statements and in case of weak symmetry we illustrate the approximation behavior by means of several numerical experiments.
Stability certification and identification of the stabilizable operating region of a dynamical system are two important concerns to ensure its operational safety/security and robustness. With the advent of machine-learning tools, these issues are especially important for systems with machine-learned components in the feedback loop. Here, in presence of unknown discrete variation (DV) of its parameters within a bounded range, a system controlled by a static feedback controller in which the closed-loop (CL) equilibria are subject to variation-induced drift is equivalently represented using a class of time-invariant systems, each with the same control policy. To develop a general theory for stability and stabilizability of such a class of neural-network (NN) controlled nonlinear systems, a Lyapunov-based convex stability certificate is proposed and is further used to devise an estimate of a local Lipschitz upper bound for the NN and a corresponding operating domain in the state space containing an initialization set, starting from where the CL local asymptotic stability of each system in the class is guaranteed, while the trajectory of the original system remains confined to the domain if the DV of the parameters satisfies a certain quasi-stationarity condition. To compute such a robustly stabilizing NN controller, a stability-guaranteed training (SGT) algorithm is also proposed. The effectiveness of the proposed framework is demonstrated using illustrative examples.
We demonstrate that merely analog transmissions and match filtering can realize the function of an edge server in federated learning (FL). Therefore, a network with massively distributed user equipments (UEs) can achieve large-scale FL without an edge server. We also develop a training algorithm that allows UEs to continuously perform local computing without being interrupted by the global parameter uploading, which exploits the full potential of UEs' processing power. We derive convergence rates for the proposed schemes to quantify their training efficiency. The analyses reveal that when the interference obeys a Gaussian distribution, the proposed algorithm retrieves the convergence rate of a server-based FL. But if the interference distribution is heavy-tailed, then the heavier the tail, the slower the algorithm converges. Nonetheless, the system run time can be largely reduced by enabling computation in parallel with communication, whereas the gain is particularly pronounced when communication latency is high. These findings are corroborated via excessive simulations.
Edge intelligence refers to a set of connected systems and devices for data collection, caching, processing, and analysis in locations close to where data is captured based on artificial intelligence. The aim of edge intelligence is to enhance the quality and speed of data processing and protect the privacy and security of the data. Although recently emerged, spanning the period from 2011 to now, this field of research has shown explosive growth over the past five years. In this paper, we present a thorough and comprehensive survey on the literature surrounding edge intelligence. We first identify four fundamental components of edge intelligence, namely edge caching, edge training, edge inference, and edge offloading, based on theoretical and practical results pertaining to proposed and deployed systems. We then aim for a systematic classification of the state of the solutions by examining research results and observations for each of the four components and present a taxonomy that includes practical problems, adopted techniques, and application goals. For each category, we elaborate, compare and analyse the literature from the perspectives of adopted techniques, objectives, performance, advantages and drawbacks, etc. This survey article provides a comprehensive introduction to edge intelligence and its application areas. In addition, we summarise the development of the emerging research field and the current state-of-the-art and discuss the important open issues and possible theoretical and technical solutions.
Recent years have witnessed the enormous success of low-dimensional vector space representations of knowledge graphs to predict missing facts or find erroneous ones. Currently, however, it is not yet well-understood how ontological knowledge, e.g. given as a set of (existential) rules, can be embedded in a principled way. To address this shortcoming, in this paper we introduce a framework based on convex regions, which can faithfully incorporate ontological knowledge into the vector space embedding. Our technical contribution is two-fold. First, we show that some of the most popular existing embedding approaches are not capable of modelling even very simple types of rules. Second, we show that our framework can represent ontologies that are expressed using so-called quasi-chained existential rules in an exact way, such that any set of facts which is induced using that vector space embedding is logically consistent and deductively closed with respect to the input ontology.
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