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In numerical simulations such as computational fluid dynamics simulations or finite element analyses, mesh quality affects simulation accuracy directly and significantly. Smoothing is one of the most widely adopted methods to improve unstructured mesh quality in mesh generation practices. Compared with the optimization-based smoothing method, heuristic smoothing methods are efficient but yield lower mesh quality. The balance between smoothing efficiency and mesh quality has been pursued in previous studies. In this paper, we propose a new smoothing method that combines the advantages of the heuristic Laplacian method and the optimization-based method based on the deep reinforcement learning method under the Deep Deterministic Policy Gradient framework. Within the framework, the actor artificial neural network predicts the optimal position of each interior free node with its surrounding ring nodes. At the same time, a critic-network is established and takes the mesh quality as input and outputs the reward of the action taken by the actor-network. Training of the networks will maximize the cumulative long-term reward, which ends up maximizing the mesh quality. Training and validation of the proposed method are presented both on 2-dimensional triangular meshes and 3-dimensional surface meshes, which demonstrates the efficiency and mesh quality of the proposed method. Finally, numerical simulations on perturbed meshes and smoothed meshes are carried out and compared which prove the influence of mesh quality on the simulation accuracy. [ABSTRACT FROM AUTHOR]

Polygonal finite elements remain an attractive option in finite element analysis due to their flexibility in modeling arbitrary shapes compared to triangles. In this study, a pentagonal membrane element was developed with the strain approach for the first time. The element possesses invariance, and the equilibrium constraint was applied to the assumed strain field using corrective coefficients. Inspired by the advancing front technique, a pentagonal mesh was generated, and the mesh quality was enhanced with Laplacian smoothing. The performance of the developed pentagonal element was assessed in a few numerical tests, and the results revealed its suitability in modeling the bending of beams. Besides, the numerical results are enhanced when pentagonal elements are used in mesh transitions along boundaries to smoothen curved edges and capture distributed loads.

Loss functions with a large number of saddle points are one of the major obstacles for training modern machine learning (ML) models efficiently. First-order methods such as gradient descent (GD) are usually the methods of choice for training ML models. However, these methods converge to saddle points for certain choices of initial guesses. In this paper, we propose a modification of the recently proposed Laplacian smoothing gradient descent (LSGD) [Osher et al., arXiv:1806.06317 ], called modified LSGD (mLSGD), and demonstrate its potential to avoid saddle points without sacrificing the convergence rate. Our analysis is based on the attraction region, formed by all starting points for which the considered numerical scheme converges to a saddle point. We investigate the attraction region's dimension both analytically and numerically. For a canonical class of quadratic functions, we show that the dimension of the attraction region for mLSGD is $\lfloor (n-1)/2\rfloor$ , and hence it is significantly smaller than that of GD whose dimension is $n-1$. [ABSTRACT FROM AUTHOR]

Attribute graph clustering requires joint modeling of both graph structure and node properties, which is challenging. In recent years, graph neural networks have been utilized to mine deep information on attribute graphs through feature aggregation, learning node embeddings, and using traditional methods to obtain clustering results, exhibiting excellent clustering performance. However, these approaches often face the following issues: the original graph structure and node features contain noise, and the quality dramatically affects the clustering results; the two-step framework of first learning node embeddings and then clustering is often suboptimal as it is not target-oriented and prone to producing suboptimal results. Through research, we propose an attribute graph clustering method called FK-SENet based on a self-supervised spectral embedding network. It utilizes Laplacian smoothing filters to smooth and denoise node features. It optimizes the initial graph structure by leveraging shared neighbor information to improve the quality of the original data, thereby enhancing clustering performance. Soft labels are generated from the node embeddings themselves to achieve self-supervision, and they jointly guide the clustering process with spectral clustering loss, iteratively optimizing the clustering results. The effectiveness of this model has been demonstrated through extensive experiments and comparisons with baseline methods.

Highly correlated structures appear in various fields, such as biology, biochemistry, and finance, with challenges of dimensionality and sparse estimation. To solve this problem, we propose an algorithm called local linear approximation with the Laplacian smoothing penalty (LLA-LSP). This method produces an accurate and smooth estimate that incorporates the correlation structure among predictors. We compare and discuss the difference between the Laplacian smoothing penalty and the total variance penalty. We prove that this algorithm converges to the oracle solution in a few iterations with a large probability. Numerical results show that the LLA-LSP has good performance in both variable selection and estimation. We apply the proposed algorithm to two biological datasets, a gene expression dataset and a chemical protein dataset, and provide meaningful insights. [ABSTRACT FROM AUTHOR]

Federated learning aims to protect data privacy by collaboratively learning a model without sharing private data among users. However, an adversary may still be able to infer the private training data by attacking the released model. Differential privacy provides a statistical protection against such attacks at the price of significantly degrading the accuracy or utility of the trained models. In this paper, we investigate a utility enhancement scheme based on Laplacian smoothing for differentially private federated learning (DP-Fed-LS), to improve the statistical precision of parameter aggregation with injected Gaussian noise without losing privacy budget. Our key observation is that the aggregated gradients in federated learning often enjoy a type of smoothness, i.e. sparsity in a graph Fourier basis with polynomial decays of Fourier coefficients as frequency grows, which can be exploited by the Laplacian smoothing efficiently. Under a prescribed differential privacy budget, convergence error bounds with tight rates are provided for DP-Fed-LS with uniform subsampling of heterogeneous non-iid data, revealing possible utility improvement of Laplacian smoothing in effective dimensionality and variance reduction, among others. Experiments over MNIST, SVHN, and Shakespeare datasets show that the proposed method can improve model accuracy with DP-guarantee and membership privacy under both uniform and Poisson subsampling mechanisms. [ABSTRACT FROM AUTHOR]

We propose a class of very simple modifications of gradient descent and stochastic gradient descent leveraging Laplacian smoothing. We show that when applied to a large variety of machine learning problems, ranging from logistic regression to deep neural nets, the proposed surrogates can dramatically reduce the variance, allow to take a larger step size, and improve the generalization accuracy. The methods only involve multiplying the usual (stochastic) gradient by the inverse of a positive definitive matrix (which can be computed efficiently by FFT) with a low condition number coming from a one-dimensional discrete Laplacian or its high-order generalizations. Given any vector, e.g., gradient vector, Laplacian smoothing preserves the mean and increases the smallest component and decreases the largest component. Moreover, we show that optimization algorithms with these surrogates converge uniformly in the discrete Sobolev H σ p sense and reduce the optimality gap for convex optimization problems. The code is available at: https://github.com/BaoWangMath/LaplacianSmoothing-GradientDescent. [ABSTRACT FROM AUTHOR]

Many machine learning tasks such as structured sparse coding and multi-task learning can be converted into an equality constrained optimization problem. The stochastic alternating direction method of multipliers (SADMM) is a popular algorithm to solve such large-scale problems, and has been successfully used in many real-world applications. However, existing SADMMs fail to take into consideration an important issue in their designs, i.e., protecting sensitive information. To address this challenging issue, this paper proposes a novel differential privacy stochastic ADMM framework for solving equality constrained machine learning problems. In particular, to further lift the utility in privacy-preserving equality constrained optimization, a Laplacian smoothing operation is also introduced into our differential privacy ADMM framework, and it can smooth out the Gaussian noise used in the Gaussian mechanism. Then we propose an efficient differentially private variance reduced stochastic ADMM (DP-VRADMM) algorithm with Laplacian smoothing for both strongly convex and general convex objectives. As a by-product, we also present a new differentially private stochastic ADMM algorithm with DP guarantees. In theory, we provide both private guarantees and utility guarantees for the proposed algorithms, which show that Laplacian smoothing can improve the utility bounds of our algorithms. Experimental results on real-world datasets verify our theoretical results and the effectiveness of our algorithms. [ABSTRACT FROM AUTHOR]

The potential of additive layer manufacturing (ALM) is high, with a whole new set of manufacturable parts with unseen complexity being offered. Moreover, the combination of topology optimization (TO) with ALM has brought mutual advantages. However, the transition between TO and ALM is a nontrivial step that requires a robust methodology. Thus, the purpose of this work is to evaluate the capabilities of adopting the commonly used Laplacian smoothing methodology as the bridging tool between TO and ALM. Several algorithms are presented and compared in terms of efficiency and performance. Most importantly, a different concept of Laplacian smoothing is presented as well as a set of metrics to evaluate the performance of the algorithms, with the advantages and disadvantages of each algorithm being discussed. In the end, the proposed mutable diffusion Laplacian algorithm is presented and exhibits less volume shrinkage and shows better preservation of some geometrical features such as thin members and edges. Moreover, a new volume constraint is presented, decreasing the resulting structural changes in the presented geometry and improving the final mesh quality. [ABSTRACT FROM AUTHOR]

As an important Markov chain Monte Carlo (MCMC) method, the stochastic gradient Langevin dynamics (SGLD) algorithm has achieved great success in Bayesian learning and posterior sampling. However, SGLD typically suffers from a slow convergence rate due to its large variance caused by the stochastic gradient. In order to alleviate these drawbacks, we leverage the recently developed Laplacian smoothing technique and propose a Laplacian smoothing stochastic gradient Langevin dynamics (LS-SGLD) algorithm. We prove that for sampling from both log-concave and non-log-concave densities, LS-SGLD achieves strictly smaller discretization error in 2-Wasserstein distance, although its mixing rate can be slightly slower. Experiments on both synthetic and real datasets verify our theoretical results and demonstrate the superior performance of LS-SGLD on different machine learning tasks including posterior sampling, Bayesian logistic regression, and training Bayesian convolutional neural networks. The code is available at https://github.com/BaoWangMath/LS-MCMC. [ABSTRACT FROM AUTHOR]

In this paper, Laplacian smoothing, which is an algorithm originally used to smooth polygon meshes in computer graphics (CG), is applied to solve a structural form-finding problem with the proof that the result of such algorithm is equivalent to force density method. Such CG algorithm is used on the design of a new-built suspension footbridge in Shaoxing, China and the algorithm works well. Since Laplacian smoothing is a pure geometric algorithm without any mechanical concept, the algorithm shows the inner relationship between force and shape, more structural applicable CG algorithms are expected to be found in the future. [ABSTRACT FROM AUTHOR]

The mesh quality is of vital importance to obtain the numerical results precisely. Poorly shaped or distorted elements can be produced by automatic mesh generation tools. In this article, the mesh smoothing algorithm based on the Markov chain Monte Carlo method is proposed to improve the quality of the mesh. The movement of nodes position is converted to a stochastic process to seek the best position for the element quality. Compared with the widely known Laplacian smoothing and optimization-based smoothing techniques, the mesh quality by the proposed method is found better than these methods. Examples are performed to illustrate the applicability of the approach. The numerical results show that the proposed algorithm is effective and valuable. [ABSTRACT FROM AUTHOR]

The pole-like object detection is of significance for robot navigation, autonomous driving, road infrastructure inventory, and detailed 3-D map generation. In this letter, we develop a skeleton-based hierarchical method for automatic detection of pole-like objects from mobile LiDAR point clouds. First, coarse extraction of building facades is adopted for the occlusion analysis. Second, slice-based Euclidean clustering algorithm is implemented to derive a set of pole-like object candidates. Third, skeleton-based principal component analysis shape recognition is presented to robustly locate all possible positions of pole-like objects. Finally, a Voronoi-constrained vertical region growing algorithm is proposed to adaptively producing the individual pole-like objects. Experiments were conducted on the public Paris–Lille-3-D data set. Experimental results demonstrate that the proposed method is robust and efficient for extracting the pole-like objects, with average quality of 90.43%. Furthermore, the proposed method outperforms other existing methods, especially for detecting pole-like objects with a large radius. [ABSTRACT FROM AUTHOR]

The safety of the electricity infrastructure significantly affects both our daily life and industrial activities. Timely and accurate monitoring of the safety of electricity network can prevent dangerous situations effectively. Thus, we, in this paper, develop a voxel-based method for automatically extracting the transmission lines from airborne LiDAR point cloud data. The method proposed in this paper uses three-dimensional (3-D) voxels as primitives and consist of the following steps: First, skeleton structure extraction using Laplacian smoothing; second, feature construction of a 3-D voxel using Latent Dirichlet allocation topic model; and third Markov random field model-based extraction for generating locally continuous and globally optimal results. To evaluate the effectiveness and robustness of the proposed method, experiments were conducted on four different types of power line scenes with flat and complex terrains from helicopter-borne LiDAR point cloud data. Experimental results demonstrate that our proposed method is efficient and robust for automatically detecting both the single conductor and the bundled conductors, with precision, recall, and quality of over 96.78%, 98.67%, and 96.66%, respectively. Moreover, compared with other existing methods, our proposed method provides higher detection correctness rate. [ABSTRACT FROM AUTHOR]

Objectives Dirichlet normal energy (DNE) is a metric of surface topography that has been used to evaluate the relationship between the surface complexity of primate cheek teeth and dietary categories. This study examines the effects of different 3D mesh retriangulation protocols on DNE. We examine how different protocols influence the DNE of a simple geometric shape-a hemisphere-to gain a more thorough understanding than can be achieved by investigating a complex biological surface such as a tooth crown. Materials and Methods We calculate DNE on 3D surface meshes of hemispheres and on primate molars subjected to various retriangulation protocols, including smoothing algorithms, smoothing amounts, target face counts, and criteria for boundary face exclusion. Software used includes R, MorphoTester, Avizo, and MeshLab. DNE was calculated using the R package 'molaR.' Results In all cases, smoothing as performed in Avizo sharply decreases DNE initially, after which DNE becomes stable. Using a broader boundary exclusion criterion or performing additional smoothing (using 'mesh fairing' methods) further decreases DNE. Increasing the mesh face count also results in increased DNE on tooth surfaces. Conclusions Different retriangulation protocols yield different DNE values for the same surfaces, and should not be combined in meta-analyses. Increasing face count will capture surface microfeatures, but at the expense of computational speed. More aggressive smoothing is more likely to alter the essential geometry of the surface. A protocol is proposed that limits potential artifacts created during surface production while preserving pertinent features on the occlusal surface. [ABSTRACT FROM AUTHOR]

An improved grid-based algorithm for the adaptive generation and regeneration of hexahedral element mesh is presented in this paper. The method for the mesh density generation and control is introduced. The refinement field is generated based on the surface curvatures, geometry features, density windows and field variables distribution. To give good description of the geometry features, eight different types of free element facet configurations are given for the mesh matching to the surface of the solid model. Scaled Jacobian and the Condition Number of the Jacobian matrix are used to evaluate the hexahedral element mesh quality. A curvature-based Laplacian smoothing approach is employed to improve the quality of boundary elements and preserve the boundary characters of mesh. To improve the quality of the surface meshes and the interior elements, an optimization approach is proposed by using mesh quality metric as the objective function. By combining the Laplacian smoothing method with the optimization approach, the mesh quality is improved significantly. For the surface meshes, the Condition Number of a set of Jacobian metric associated with the quadrilateral elements is taken as the optimization objective function. For the interior elements, the Condition Number metric associated with the hexahedral elements is employed as the optimization objective function. The effectiveness and robustness of the approaches are demonstrated through two complex three-dimensional models. [ABSTRACT FROM AUTHOR]

The present paper deals with the volume preserving smoothing of triangular isotropic meshes over three-dimensional surfaces. The adopted approach is based on Laplacian smoothing combining in an alternating manner positive and negative weights in consecutive cycles of the smoothing. Since the aim is to improve the shape of individual elements of the mesh rather than to get rid of a noise, the weights are derived in a very simple way using a “do not harm” concept. The paper also extends the smoothing methodology from meshes on closed surfaces to meshes on open surfaces and discusses how the concept can be applied to meshes over surfaces with sharp features and curvature discontinuities. The performance and capabilities of the presented smoothing approach are demonstrated on several examples. [ABSTRACT FROM AUTHOR]

A modified paving algorithm is presented for dynamical generation of all-quadrangle meshes for simulating hydrodynamic interaction forces acting upon ships manoeuvring in restricted waters. The so-called hybrid smoother, combining advantages of the Laplacian and angle-based techniques is proposed to achieve the balance between the computational efficiency and quality of the mesh. Special attention is paid to the accuracy and robustness of the algorithm to guarantee its flawless performance in online simulations. Consistency of the algorithm is demonstrated on numerical examples including those of one or more ships crossing a dredged channel at oblique angle. In addition, possible applications to other problems of ocean and coastal engineering are briefly discussed. [ABSTRACT FROM AUTHOR]