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Journal articles on the topic 'Gradient Smoothing'

Author: Grafiati

Published: 30 June 2021

Last updated: 29 July 2025

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1

Matsuoka, Ryo, та Masahiro Okuda. "Beyond Staircasing Effect: Robust Image Smoothing via ℓ0 Gradient Minimization and Novel Gradient Constraints". Signals 4, № 4 (2023): 669–86. http://dx.doi.org/10.3390/signals4040037.

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In this paper, we propose robust image-smoothing methods based on ℓ0 gradient minimization with novel gradient constraints to effectively suppress pseudo-edges. Simultaneously minimizing the ℓ0 gradient, i.e., the number of nonzero gradients in an image, and the ℓ2 data fidelity results in a smooth image. However, this optimization often leads to undesirable artifacts, such as pseudo-edges, known as the “staircasing effect”, and halos, which become more visible in image enhancement tasks, like detail enhancement and tone mapping. To address these issues, we introduce two types of gradient cons

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2

Zhou, Zhengyong, and Qi Yang. "An Active Set Smoothing Method for Solving Unconstrained Minimax Problems." Mathematical Problems in Engineering 2020 (June 24, 2020): 1–25. http://dx.doi.org/10.1155/2020/9108150.

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In this paper, an active set smoothing function based on the plus function is constructed for the maximum function. The active set strategy used in the smoothing function reduces the number of gradients and Hessians evaluations of the component functions in the optimization. Combing the active set smoothing function, a simple adjustment rule for the smoothing parameters, and an unconstrained minimization method, an active set smoothing method is proposed for solving unconstrained minimax problems. The active set smoothing function is continuously differentiable, and its gradient is locally Lip

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3

Fang, Shuai, Zhenji Yao, and Jing Zhang. "Scale and Gradient Aware Image Smoothing." IEEE Access 7 (2019): 166268–81. http://dx.doi.org/10.1109/access.2019.2953550.

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4

Wang, Dongdong, Jiarui Wang, and Junchao Wu. "Superconvergent gradient smoothing meshfree collocation method." Computer Methods in Applied Mechanics and Engineering 340 (October 2018): 728–66. http://dx.doi.org/10.1016/j.cma.2018.06.021.

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5

Zhang, Cong, Limin Sun, and Ya Xiao. "A Generalized Projetion Gradient Algorithm for Mathematical Programs with Complementary Constraints." Journal of Physics: Conference Series 2289, no. 1 (2022): 012019. http://dx.doi.org/10.1088/1742-6596/2289/1/012019.

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Abstract Against the shortcomings that many existing algorithms for solving the standard smoothing nonlinear programming would fail if they were used directly to solve the mathematical programs with complementary constraints( MPCC). By using a complementarity function and the idea of smoothing approximation method, the MPCC problem was transformed into a smoothing nonlinear programming. Combined with the supermemory gradient idea, a generalized projection gradient algorithm is proposed and its global convergence is obtained.

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Xu, Li, Cewu Lu, Yi Xu, and Jiaya Jia. "Image smoothing via L 0 gradient minimization." ACM Transactions on Graphics 30, no. 6 (2011): 1–12. http://dx.doi.org/10.1145/2070781.2024208.

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7

Burke, James V., Tim Hoheisel, and Christian Kanzow. "Gradient Consistency for Integral-convolution Smoothing Functions." Set-Valued and Variational Analysis 21, no. 2 (2013): 359–76. http://dx.doi.org/10.1007/s11228-013-0235-6.

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8

Yao, Jianyao, Weimin Wu, Kun Zhang, et al. "Development of Three-Dimensional GSM-CFD Solver for Compressible Flows." International Journal of Computational Methods 14, no. 04 (2017): 1750037. http://dx.doi.org/10.1142/s0219876217500372.

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A three-dimensional (3D) Computational Fluid Dynamics (CFD) solver based on the gradient smoothing method (GSM) is developed for compressible flows based on previous research. The piecewise constant smoothing function with one-point integration scheme is implemented for gradient approximation of field variables and convective fluxes. The matrix-based method for gradient approximations is also developed to improve the numerical efficiency. Numerical examples of gradient approximations of several given functions have shown that the proposed GSM is more accurate and robust to mesh distortion. A t

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9

Chuang, Ming, Szymon Rusinkiewicz, and Misha Kazhdan. "Gradient-Domain Processing of Meshes." Journal of Computer Graphics Techniques 5, no. 4 (2016): 44–55. https://doi.org/10.5281/zenodo.7953865.

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  This paper describes an implementation of gradient-domain processing for editing the geometry of triangle meshes in 3D. We show applications to mesh smoothing and sharpening and describe anisotropic extensions that enable edge-aware processing.

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10

Xu, Jingyan, and Frederic Noo. "Efficient gradient computation for optimization of hyperparameters." Physics in Medicine & Biology 67, no. 3 (2022): 03NT01. http://dx.doi.org/10.1088/1361-6560/ac4442.

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Abstract We are interested in learning the hyperparameters in a convex objective function in a supervised setting. The complex relationship between the input data to the convex problem and the desirable hyperparameters can be modeled by a neural network; the hyperparameters and the data then drive the convex minimization problem, whose solution is then compared to training labels. In our previous work (Xu and Noo 2021 Phys. Med. Biol. 66 19NT01), we evaluated a prototype of this learning strategy in an optimization-based sinogram smoothing plus FBP reconstruction framework. A question arising

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11

CUI, X. Y., G. Y. LI, and G. R. LIU. "AN EXPLICIT SMOOTHED FINITE ELEMENT METHOD (SFEM) FOR ELASTIC DYNAMIC PROBLEMS." International Journal of Computational Methods 10, no. 01 (2013): 1340002. http://dx.doi.org/10.1142/s0219876213400021.

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This paper presents an explicit smoothed finite element method (SFEM) for elastic dynamic problems. The central difference method for time integration will be used in presented formulations. A simple but general contact searching algorithm is used to treat the contact interface and an algorithm for the contact force is presented. In present method, the problem domain is first divided into elements as in the finite element method (FEM), and the elements are further subdivided into several smoothing cells. Cell-wise strain smoothing operations are used to obtain the stresses, which are constants

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12

Liu, Wenli, Xiaoni Chi, Qili Yang, and Ranran Cui. "Jacobian Consistency of a Smoothing Function for the Weighted Second-Order Cone Complementarity Problem." Mathematical Problems in Engineering 2021 (January 23, 2021): 1–11. http://dx.doi.org/10.1155/2021/6674520.

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In this paper, a weighted second-order cone (SOC) complementarity function and its smoothing function are presented. Then, we derive the computable formula for the Jacobian of the smoothing function and show its Jacobian consistency. Also, we estimate the distance between the subgradient of the weighted SOC complementarity function and the gradient of its smoothing function. These results will be critical to achieve the rapid convergence of smoothing methods for weighted SOC complementarity problems.

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13

NGUYEN-THOI, T., P. PHUNG-VAN, T. RABCZUK, H. NGUYEN-XUAN, and C. LE-VAN. "AN APPLICATION OF THE ES-FEM IN SOLID DOMAIN FOR DYNAMIC ANALYSIS OF 2D FLUID–SOLID INTERACTION PROBLEMS." International Journal of Computational Methods 10, no. 01 (2013): 1340003. http://dx.doi.org/10.1142/s0219876213400033.

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An edge-based smoothed finite element method (ES-FEM-T3) using triangular elements was recently proposed to improve the accuracy and convergence rate of the existing standard finite element method (FEM) for the solid mechanics analyses. In this paper, the ES-FEM-T3 is further extended to the dynamic analysis of 2D fluid–solid interaction problems based on the pressure-displacement formulation. In the present coupled method, both solid and fluid domain is discretized by triangular elements. In the fluid domain, the standard FEM is used, while in the solid domain, we use the ES-FEM-T3 in which t

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14

Avrashi, Jacob. "High order gradient smoothing towards improved C1 eigenvalues." Engineering Computations 12, no. 6 (1995): 513–28. http://dx.doi.org/10.1108/02644409510799749.

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15

Pinilla, Samuel, Tamir Bendory, Yonina C. Eldar, and Henry Arguello. "Frequency-Resolved Optical Gating Recovery via Smoothing Gradient." IEEE Transactions on Signal Processing 67, no. 23 (2019): 6121–32. http://dx.doi.org/10.1109/tsp.2019.2951192.

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16

Wang, Bao, Difan Zou, Quanquan Gu, and Stanley J. Osher. "Laplacian Smoothing Stochastic Gradient Markov Chain Monte Carlo." SIAM Journal on Scientific Computing 43, no. 1 (2021): A26—A53. http://dx.doi.org/10.1137/19m1294356.

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17

Lin, Qihang, Xi Chen, and Javier Peña. "A smoothing stochastic gradient method for composite optimization." Optimization Methods and Software 29, no. 6 (2014): 1281–301. http://dx.doi.org/10.1080/10556788.2014.891592.

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18

He, Liangtian, та Yilun Wang. "Image smoothing via truncated ℓ 0 gradient regularisation". IET Image Processing 12, № 2 (2018): 226–34. http://dx.doi.org/10.1049/iet-ipr.2017.0533.

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19

Liu, Qian, Caiming Zhang, Qiang Guo, and Yuanfeng Zhou. "A nonlocal gradient concentration method for image smoothing." Computational Visual Media 1, no. 3 (2015): 197–209. http://dx.doi.org/10.1007/s41095-015-0012-6.

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20

Chi, Xiaoni, Zhongping Wan, and Zijun Hao. "The Jacobian Consistency of a One-Parametric Class of Smoothing Functions for SOCCP." Abstract and Applied Analysis 2013 (2013): 1–12. http://dx.doi.org/10.1155/2013/965931.

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Second-order cone (SOC) complementarity functions and their smoothing functions have been much studied in the solution of second-order cone complementarity problems (SOCCP). In this paper, we study the directional derivative and B-subdifferential of the one-parametric class of SOC complementarity functions, propose its smoothing function, and derive the computable formula for the Jacobian of the smoothing function. Based on these results, we prove the Jacobian consistency of the one-parametric class of smoothing functions, which will play an important role for achieving the rapid convergence o

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21

Chen, Miao, and Shou-qiang Du. "The Smoothing FR Conjugate Gradient Method for Solving a Kind of Nonsmooth Optimization Problem with l1-Norm." Mathematical Problems in Engineering 2018 (2018): 1–9. http://dx.doi.org/10.1155/2018/5817931.

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We study the method for solving a kind of nonsmooth optimization problems with l1-norm, which is widely used in the problem of compressed sensing, image processing, and some related optimization problems with wide application background in engineering technology. Transformated by the absolute value equations, this kind of nonsmooth optimization problem is rewritten as a general unconstrained optimization problem, and the transformed problem is solved by a smoothing FR conjugate gradient method. Finally, the numerical experiments show the effectiveness of the given smoothing FR conjugate gradie

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22

Sakovich, Nikita, Dmitry Aksenov, Ekaterina Pleshakova, and Sergey Gataullin. "MAMGD: Gradient-Based Optimization Method Using Exponential Decay." Technologies 12, no. 9 (2024): 154. http://dx.doi.org/10.3390/technologies12090154.

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Optimization methods, namely, gradient optimization methods, are a key part of neural network training. In this paper, we propose a new gradient optimization method using exponential decay and the adaptive learning rate using a discrete second-order derivative of gradients. The MAMGD optimizer uses an adaptive learning step, exponential smoothing and gradient accumulation, parameter correction, and some discrete analogies from classical mechanics. The experiments included minimization of multivariate real functions, function approximation using multilayer neural networks, and training neural n

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23

Crisan, D., and M. Ottobre. "Pointwise gradient bounds for degenerate semigroups (of UFG type)." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 472, no. 2195 (2016): 20160442. http://dx.doi.org/10.1098/rspa.2016.0442.

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In this paper, we consider diffusion semigroups generated by second-order differential operators of degenerate type. The operators that we consider do not , in general, satisfy the Hörmander condition and are not hypoelliptic. In particular, instead of working under the Hörmander paradigm, we consider the so-called UFG (uniformly finitely generated) condition, introduced by Kusuoka and Strook in the 1980s. The UFG condition is weaker than the uniform Hörmander condition, the smoothing effect taking place only in certain directions (rather than in every direction, as it is the case when the Hör

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24

Yerokhin, Alexander, Yury Deniskin, Anton Dubrovin, and Alexei Sheryshev. "Airfoil smoothing using unconditional optimization taking into account aerodynamic criteria." E3S Web of Conferences 460 (2023): 07021. http://dx.doi.org/10.1051/e3sconf/202346007021.

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The smoothing of airfoils in sections of the surface of an aircraft wing is considered. The issue of using the distribution of pressure and Mach number along the chord of an airfoil as criteria for smoothing efficiency is being studied. Smoothing is considered as a problem of unconditional minimization of a quadratic objective function. The arguments of the objective function are the second derivatives of the tabular function interpolating the contour of the airfoil. Minimization is performed using the gradient descent method.

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25

Meihong, Li, Jin Shuang, and Du Yu. "Adversarial sample attack method based on loss smoothing." Scientific Insights and Discoveries Review 3 (October 11, 2024): 218–28. http://dx.doi.org/10.59782/sidr.v3i1.138.

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Deep neural networks (DNNs) are vulnerable to adversarial examples.Although the existing momentum-based adversarial example generation method can achieve a close 100%white-box attack success rate, it is still not ideal when attacking other models, and the black- box attack success rate is low. To address this, an adversarial example attack method based on loss smoothing is proposed to improve the transferability of adversarial examples. In the iterative process of calculating the gradient at each step, the current gradient is not used directly, but the local average gradient is used to accumul

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26

PRASATH, V. B. SURYA, and ARINDAMA SINGH. "AN ADAPTIVE DIFFUSION SCHEME FOR IMAGE RESTORATION AND SELECTIVE SMOOTHING." International Journal of Image and Graphics 12, no. 01 (2012): 1250003. http://dx.doi.org/10.1142/s0219467812500039.

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Anisotropic partial differential equation (PDE)-based image restoration schemes employ a local edge indicator function typically based on gradients. In this paper, an alternative pixel-wise adaptive diffusion scheme is proposed. It uses a spatial function giving better edge information to the diffusion process. It avoids the over-locality problem of gradient-based schemes and preserves discontinuities coherently. The scheme satisfies scale space axioms for a multiscale diffusion scheme; and it uses a well-posed regularized total variation (TV) scheme along with Perona-Malik type functions. Med

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27

Yoon, Sangpil, Cheng-Tang Wu, Hui-Ping Wang, and Jiun-Shyan Chen. "Efficient Meshfree Formulation for Metal Forming Simulations." Journal of Engineering Materials and Technology 123, no. 4 (2000): 462–67. http://dx.doi.org/10.1115/1.1396349.

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A stabilized conforming (SC) nodal integration method is developed for elastoplastic contact analysis of metal forming processes. In this approach, strain smoothing stabilization is introduced to eliminate spatial instability in collocation meshfree methods. The gradient matrix associated with strain smoothing satisfies the integration constraint (IC) of linear exactness in the Galerkin approximation. Strain smoothing formulation and numerical procedures for history-dependent problems are introduced. Applications to metal forming analysis are presented, with the results demonstrating a signifi

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28

Liu, Guohao, Yonghong Deng, and Zhibin Li. "An Ultra-Precision Smoothing Polishing Model for Optical Surface Fabrication with Morphology Gradient Awareness." Micromachines 16, no. 7 (2025): 734. https://doi.org/10.3390/mi16070734.

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To improve the surface morphology quality of ultra-precision optical components, particularly in the suppression of mid-spatial frequency (MSF) errors, this paper proposes a morphology gradient-aware spatiotemporal coupled smoothing model based on convolutional material removal. By introducing the Laplacian curvature into the surface evolution framework, a curvature-sensitive “peak-priority” mechanism is established to dynamically guide the local dwell time. A nonlinear spatiotemporal coupling equation is constructed, in which the dwell time is adaptively modulated by surface gradient magnitud

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29

Kuprat, Andrew, Denise George, Eldon Linnebur, Harold Trease, and R. Kent Smith. "Moving Adaptive Unstructured 3-D Meshes in Semiconductor Process Modeling Applications." VLSI Design 6, no. 1-4 (1998): 373–78. http://dx.doi.org/10.1155/1998/15828.

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The next generation of semiconductor process and device modeling codes will require 3-D mesh capabilities including moving volume and surface grids, adaptive mesh refinement and adaptive mesh smoothing. To illustrate the value of these techniques, a time dependent process simulation model was constructed using analytic functions to return time dependent dopant concentration and time dependent SiO2 volume and surface velocities. Adaptive mesh refinement and adaptive mesh smoothing techniques were used to resolve the moving boron dopant diffusion front in the Si substrate. The adaptive mesh smoo

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30

De Silva, Kushani, Carlo Cafaro, and Adom Giffin. "Gradient Profile Estimation Using Exponential Cubic Spline Smoothing in a Bayesian Framework." Entropy 23, no. 6 (2021): 674. http://dx.doi.org/10.3390/e23060674.

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Attaining reliable gradient profiles is of utmost relevance for many physical systems. In many situations, the estimation of the gradient is inaccurate due to noise. It is common practice to first estimate the underlying system and then compute the gradient profile by taking the subsequent analytic derivative of the estimated system. The underlying system is often estimated by fitting or smoothing the data using other techniques. Taking the subsequent analytic derivative of an estimated function can be ill-posed. This becomes worse as the noise in the system increases. As a result, the uncerta

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31

Tian, Ying, and Haodi Ma. "L0 Gradient based Image Smoothing Method for Ear Identification." International Journal of Signal Processing, Image Processing and Pattern Recognition 8, no. 6 (2015): 61–68. http://dx.doi.org/10.14257/ijsip.2015.8.6.08.

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32

Yang, Wei, Lili Pan, and Jinhui Wan. "Smoothing gradient descent algorithm for the composite sparse optimization." AIMS Mathematics 9, no. 12 (2024): 33401–22. http://dx.doi.org/10.3934/math.20241594.

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<p>Composite sparsity generalizes the standard sparsity that considers the sparsity on a linear transformation of the variables. In this paper, we study the composite sparse optimization problem consisting of minimizing the sum of a nondifferentiable loss function and the $ {\mathcal{\ell}_0} $ penalty term of a matrix times the coefficient vector. First, we consider an exact continuous relaxation problem with a capped-$ {\mathcal{\ell}_1} $ penalty that has the same optimal solution as the primal problem. Specifically, we propose the lifted stationary point of the relaxation problem and

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33

G R, Byra Reddy, and Prasanna Kumar H. "Smoothing of Mammogram Using an Improved Gradient based Technique." Advanced Biomedical Engineering 9 (2020): 202–8. http://dx.doi.org/10.14326/abe.9.202.

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34

Chen, Xi, Qihang Lin, Seyoung Kim, Jaime G. Carbonell, and Eric P. Xing. "Smoothing proximal gradient method for general structured sparse regression." Annals of Applied Statistics 6, no. 2 (2012): 719–52. http://dx.doi.org/10.1214/11-aoas514.

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35

Henderson, Daniel J., Qi Li, Christopher F. Parmeter, and Shuang Yao. "Gradient-based smoothing parameter selection for nonparametric regression estimation." Journal of Econometrics 184, no. 2 (2015): 233–41. http://dx.doi.org/10.1016/j.jeconom.2014.09.007.

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36

Wu, Caiying, and Guoqing Chen. "A smoothing conjugate gradient algorithm for nonlinear complementarity problems." Journal of Systems Science and Systems Engineering 17, no. 4 (2008): 460–72. http://dx.doi.org/10.1007/s11518-008-5091-9.

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37

Xu, Mengwei, Soon-Yi Wu, and Jane J. Ye. "Solving semi-infinite programs by smoothing projected gradient method." Computational Optimization and Applications 59, no. 3 (2014): 591–616. http://dx.doi.org/10.1007/s10589-014-9654-z.

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38

Liu, G. R., and George X. Xu. "A gradient smoothing method (GSM) for fluid dynamics problems." International Journal for Numerical Methods in Fluids 58, no. 10 (2008): 1101–33. http://dx.doi.org/10.1002/fld.1788.

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39

Tang, Anping, Jan Heiland, and Guang-Da Hu. "Smoothing gradient method for group-structured sparse feedback stabilization." Systems & Control Letters 202 (August 2025): 106117. https://doi.org/10.1016/j.sysconle.2025.106117.

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40

Mao, Zirui, and G. R. Liu. "A 3D Lagrangian gradient smoothing method framework with an adaptable gradient smoothing domain‐constructing algorithm for simulating large deformation free surface flows." International Journal for Numerical Methods in Engineering 121, no. 6 (2019): 1268–96. http://dx.doi.org/10.1002/nme.6265.

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41

Li, Wei, Yingbin Chai, Xiangyu You, and Qifan Zhang. "An Edge-Based Smoothed Finite Element Method for Analyzing Stiffened Plates." International Journal of Computational Methods 16, no. 06 (2019): 1840031. http://dx.doi.org/10.1142/s0219876218400315.

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In this paper, an edge-based smoothed finite element method with the discrete shear gap using triangular elements (ES-DSG3) is presented for static, free vibration and sound radiation analyses of plates stiffened by eccentric and concentric stiffeners. In the present model, the ES-DSG3 for the plate element with the isoparametric thick-beam element is employed to formulate stiffened plate structures. The deflections and rotations of the plates and the stiffeners are connected at tying positions. By using Rayleigh integral, sound radiation of stiffened plates subjected to a point load can be ob

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42

LIU, G. R. "A GENERALIZED GRADIENT SMOOTHING TECHNIQUE AND THE SMOOTHED BILINEAR FORM FOR GALERKIN FORMULATION OF A WIDE CLASS OF COMPUTATIONAL METHODS." International Journal of Computational Methods 05, no. 02 (2008): 199–236. http://dx.doi.org/10.1142/s0219876208001510.

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This paper presents a generalized gradient smoothing technique, the corresponding smoothed bilinear forms, and the smoothed Galerkin weakform that is applicable to create a wide class of efficient numerical methods with special properties including the upper bound properties. A generalized gradient smoothing technique is first presented for computing the smoothed strain fields of displacement functions with discontinuous line segments, by "rudely" enforcing the Green's theorem over the smoothing domain containing these discontinuous segments. A smoothed bilinear form is then introduced for Gal

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43

Yao, Gang, Nuno V. da Silva, Vladimir Kazei, Di Wu, and Chenhao Yang. "Extraction of the tomography mode with nonstationary smoothing for full-waveform inversion." GEOPHYSICS 84, no. 4 (2019): R527—R537. http://dx.doi.org/10.1190/geo2018-0586.1.

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Full-waveform inversion (FWI) includes migration and tomography modes. The tomographic component of the gradient from reflection data is usually much weaker than the migration component. To use the tomography mode to fix background velocity errors, it is necessary to extract the tomographic component from the gradient. Otherwise, the inversion will be dominated by the migration mode. We have developed a method based on nonstationary smoothing to extract the tomographic component from the raw gradient. By analyzing the characteristics of the scattering angle filtering, the wavenumber of the tom

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44

Jayadi, Akhmad, Dwi Handoko, Rizka Permata, Atika Arpan, and Dian Meilantika. "IMPLEMENTASI PATH SMOOTHING MENGGUNAKAN ALGORITMA GRADIENT DESCENT MENGGUNAKAN SIMULASI V-REP." HOAQ (High Education of Organization Archive Quality) : Jurnal Teknologi Informasi 15, no. 2 (2024): 107–15. https://doi.org/10.52972/hoaq.vol15no2.p107-115.

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Perkembangan teknologi robot mobile telah mengubah berbagai sektor industri, dan salah satu tantangan utama dalam pengoperasian robot mobile adalah perencanaan jalur (path planning). Jalur yang direncanakan sering kali tidak mempertimbangkan gerakan robot, terutama saat menghadapi belokan tajam yang dapat menyebabkan inefisiensi gerak robot, untuk mengatasi hal ini, teknik path smoothing digunakan untuk mereduksi ketajaman belokan, memungkinkan robot bergerak lebih lancar dan efisien. Salah satu algoritma yang digunakan dalam path smoothing adalah Gradient Descent, yang secara iteratif memperb

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45

Miyazaki, Kazuyuki, and Toshiki Iwasaki. "The Gradient Genesis of Stratospheric Trace Species in the Subtropics and around the Polar Vortex." Journal of the Atmospheric Sciences 65, no. 2 (2008): 490–508. http://dx.doi.org/10.1175/2007jas2403.1.

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Abstract Mechanisms that control the formation and decay of meridional gradients in stratospheric trace species in the subtropics and around the polar vortex are investigated using a gradient genesis equation that uses mass-weighted isentropic zonal means. Application of this method to global nitrous oxide (N2O) data output from a global chemical transport model shows that mean vertical transport increases the meridional tracer gradient from the subtropics to midlatitudes through the shearing deformation, particularly related to overturning of the Brewer–Dobson circulation. Mean meridional tra

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46

Alashqar, B. A., A. V. Gasnikov, D. M. Dvinskikh, and A. V. Lobanov. "Gradient-free Federated Learning Methods with l1 and l2-randomization for Non-smooth Convex Stochastic Optimization Problems." Журнал вычислительной математики и математической физики 63, no. 9 (2023): 1458–512. http://dx.doi.org/10.31857/s0044466923090028.

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This paper studies non-smooth problems of convex stochastic optimization. Using the smoothing technique based on the replacement of the function value at the considered point by the averaged function value over a ball (in l1-norm or l2-norm) of a small radius centered at this point, and then the original problem is reduced to a smooth problem (whose Lipschitz constant of the gradient is inversely proportional to the radius of the ball). An essential property of the smoothing used is the possibility of calculating an unbiased estimation of the gradient of a smoothed function based only on reali

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47

Yang, Bohan, Hao Ren, Tongxing Zuo, and Zhenyu Liu. "A Stream Function Smoothing Method for the Design of MRI Gradient Coils on Non-Developable Surfaces." Sensors 23, no. 18 (2023): 7912. http://dx.doi.org/10.3390/s23187912.

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Insert gradient coils with similar imaging body shapes typically have smaller dimensions and higher spatial efficiency. This often allows the gradient coils the achievement of stronger and faster gradient fields. Thus, improving existing methods to make them applicable to the design of MRI gradient coils on complex surfaces has also become a challenge. This article proposes an algorithm that smooths the implicitly expressed stream function based on the intrinsic surface Laplace–Beltrami operator. This algorithm can be used to simplify the design procedure of MRI gradient coils on non-developab

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48

Chipot, M., R. March, M. Rosati, and G. Vergara Caffarelli. "Analysis of a Nonconvex Problem Related to Signal Selective Smoothing." Mathematical Models and Methods in Applied Sciences 07, no. 03 (1997): 313–28. http://dx.doi.org/10.1142/s0218202597000189.

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Abstract:

We study some properties of a nonconvex variational problem. We fail to attain the infimum of the functional that has to be minimized. Instead, minimizing sequences develop gradient oscillations which allow them to reduce the value of the functional. We show an existence result for a perturbed nonconvex version of the problem, and we study the qualitative properties of the corresponding minimizer. The pattern of the gradient oscillations for the original nonperturbed problem is analyzed numerically.

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49

Yang, Dakun, and Wei Wu. "A Smoothing Interval Neural Network." Discrete Dynamics in Nature and Society 2012 (2012): 1–25. http://dx.doi.org/10.1155/2012/456919.

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Abstract:

In many applications, it is natural to use interval data to describe various kinds of uncertainties. This paper is concerned with an interval neural network with a hidden layer. For the original interval neural network, it might cause oscillation in the learning procedure as indicated in our numerical experiments. In this paper, a smoothing interval neural network is proposed to prevent the weights oscillation during the learning procedure. Here, by smoothing we mean that, in a neighborhood of the origin, we replace the absolute values of the weights by a smooth function of the weights in the

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50

Wang, Mingquan, Xiaohua Ma, Xinrui Wang, Jun Wang, Xiuqing Zhou, and Qibing Gao. "Smoothing Estimation of Parameters in Censored Quantile Linear Regression Model." Mathematics 13, no. 2 (2025): 192. https://doi.org/10.3390/math13020192.

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Abstract:

In this paper, we propose a smoothing estimation method for censored quantile regression models. The method associates the convolutional smoothing estimation with the loss function, which is quadratically derivable and globally convex by using a non-negative kernel function. Thus, the parameters of the regression model can be computed by using the gradient-based iterative algorithm. We demonstrate the convergence speed and asymptotic properties of the smoothing estimation for large samples in high dimensions. Numerical simulations show that the smoothing estimation method for censored quantile

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