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Journal articles on the topic 'Approximate norm descent methods'

Author: Grafiati

Published: 10 March 2023

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1

Morini, Benedetta, Margherita Porcelli, and Philippe L. Toint. "Approximate norm descent methods for constrained nonlinear systems." Mathematics of Computation 87, no. 311 (2017): 1327–51. http://dx.doi.org/10.1090/mcom/3251.

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Jin, Yang, Li, and Liu. "Sparse Recovery Algorithm for Compressed Sensing Using Smoothed l0 Norm and Randomized Coordinate Descent." Mathematics 7, no. 9 (2019): 834. http://dx.doi.org/10.3390/math7090834.

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Compressed sensing theory is widely used in the field of fault signal diagnosis and image processing. Sparse recovery is one of the core concepts of this theory. In this paper, we proposed a sparse recovery algorithm using a smoothed l0 norm and a randomized coordinate descent (RCD), then applied it to sparse signal recovery and image denoising. We adopted a new strategy to express the (P0) problem approximately and put forward a sparse recovery algorithm using RCD. In the computer simulation experiments, we compared the performance of this algorithm to other typical methods. The results show

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Xu, Kai, and Zhi Xiong. "Nonparametric Tensor Completion Based on Gradient Descent and Nonconvex Penalty." Symmetry 11, no. 12 (2019): 1512. http://dx.doi.org/10.3390/sym11121512.

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Existing tensor completion methods all require some hyperparameters. However, these hyperparameters determine the performance of each method, and it is difficult to tune them. In this paper, we propose a novel nonparametric tensor completion method, which formulates tensor completion as an unconstrained optimization problem and designs an efficient iterative method to solve it. In each iteration, we not only calculate the missing entries by the aid of data correlation, but consider the low-rank of tensor and the convergence speed of iteration. Our iteration is based on the gradient descent met

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4

Ko, Dongnam, and Enrique Zuazua. "Model predictive control with random batch methods for a guiding problem." Mathematical Models and Methods in Applied Sciences 31, no. 08 (2021): 1569–92. http://dx.doi.org/10.1142/s0218202521500329.

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We model, simulate and control the guiding problem for a herd of evaders under the action of repulsive drivers. The problem is formulated in an optimal control framework, where the drivers (controls) aim to guide the evaders (states) to a desired region of the Euclidean space. The numerical simulation of such models quickly becomes unfeasible for a large number of interacting agents, as the number of interactions grows [Formula: see text] for [Formula: see text] agents. For reducing the computational cost to [Formula: see text], we use the Random Batch Method (RBM), which provides a computatio

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Utomo, Rukmono Budi. "METODE NUMERIK STEPEST DESCENT DENGAN DIRECTION DAN NORMRERATA ARITMATIKA." AKSIOMA Journal of Mathematics Education 5, no. 2 (2017): 128. http://dx.doi.org/10.24127/ajpm.v5i2.673.

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This research is investigating ofSteepest Descent numerical method with direction and norm arithmetic mean. This research is begin with try to understand what Steepest Descent Numerical is and its algorithm. After that, we constructing the new Steepest Descent numerical method using another direction and norm called arithmetic mean. This paper also containing numerical counting examples using both of these methods and analyze them self.

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Goh, B. S. "Approximate Greatest Descent Methods for Optimization with Equality Constraints." Journal of Optimization Theory and Applications 148, no. 3 (2010): 505–27. http://dx.doi.org/10.1007/s10957-010-9765-3.

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7

Xiao, Yunhai, Chunjie Wu, and Soon-Yi Wu. "Norm descent conjugate gradient methods for solving symmetric nonlinear equations." Journal of Global Optimization 62, no. 4 (2014): 751–62. http://dx.doi.org/10.1007/s10898-014-0218-7.

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8

Qiu, Yixuan, and Xiao Wang. "Stochastic Approximate Gradient Descent via the Langevin Algorithm." Proceedings of the AAAI Conference on Artificial Intelligence 34, no. 04 (2020): 5428–35. http://dx.doi.org/10.1609/aaai.v34i04.5992.

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We introduce a novel and efficient algorithm called the stochastic approximate gradient descent (SAGD), as an alternative to the stochastic gradient descent for cases where unbiased stochastic gradients cannot be trivially obtained. Traditional methods for such problems rely on general-purpose sampling techniques such as Markov chain Monte Carlo, which typically requires manual intervention for tuning parameters and does not work efficiently in practice. Instead, SAGD makes use of the Langevin algorithm to construct stochastic gradients that are biased in finite steps but accurate asymptotical

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9

Yang, Yin, and Yunqing Huang. "Spectral-Collocation Methods for Fractional Pantograph Delay-Integrodifferential Equations." Advances in Mathematical Physics 2013 (2013): 1–14. http://dx.doi.org/10.1155/2013/821327.

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We propose and analyze a spectral Jacobi-collocation approximation for fractional order integrodifferential equations of Volterra type with pantograph delay. The fractional derivative is described in the Caputo sense. We provide a rigorous error analysis for the collocation method, which shows that the error of approximate solution decays exponentially inL∞norm and weightedL2-norm. The numerical examples are given to illustrate the theoretical results.

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10

Poggio, Tomaso, Andrzej Banburski, and Qianli Liao. "Theoretical issues in deep networks." Proceedings of the National Academy of Sciences 117, no. 48 (2020): 30039–45. http://dx.doi.org/10.1073/pnas.1907369117.

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While deep learning is successful in a number of applications, it is not yet well understood theoretically. A theoretical characterization of deep learning should answer questions about their approximation power, the dynamics of optimization, and good out-of-sample performance, despite overparameterization and the absence of explicit regularization. We review our recent results toward this goal. In approximation theory both shallow and deep networks are known to approximate any continuous functions at an exponential cost. However, we proved that for certain types of compositional functions, de

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11

Boykov, Ilya, Vladimir Roudnev, and Alla Boykova. "Approximate Methods for Solving Linear and Nonlinear Hypersingular Integral Equations." Axioms 9, no. 3 (2020): 74. http://dx.doi.org/10.3390/axioms9030074.

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We propose an iterative projection method for solving linear and nonlinear hypersingular integral equations with non-Riemann integrable functions on the right-hand sides. We investigate hypersingular integral equations with second order singularities. Today, hypersingular integral equations of this type are widely used in physics and technology. The convergence of the proposed method is based on the Lyapunov stability theory of solutions of ordinary differential equation systems. The advantage of the method for linear equations is in simplicity of unique solvability verification for the approx

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12

Shi, Dongyang, and Zhiyun Yu. "Low-Order Nonconforming Mixed Finite Element Methods for Stationary Incompressible Magnetohydrodynamics Equations." Journal of Applied Mathematics 2012 (2012): 1–21. http://dx.doi.org/10.1155/2012/825609.

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The nonconforming mixed finite element methods (NMFEMs) are introduced and analyzed for the numerical discretization of a nonlinear, fully coupled stationary incompressible magnetohydrodynamics (MHD) problem in 3D. A family of the low-order elements on tetrahedra or hexahedra are chosen to approximate the pressure, the velocity field, and the magnetic field. The existence and uniqueness of the approximate solutions are shown, and the optimal error estimates for the corresponding unknown variables inL2-norm are established, as well as those in a brokenH1-norm for the velocity and the magnetic f

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13

Wei, Yunxia, and Yanping Chen. "Convergence Analysis of the Spectral Methods for Weakly Singular Volterra Integro-Differential Equations with Smooth Solutions." Advances in Applied Mathematics and Mechanics 4, no. 1 (2012): 1–20. http://dx.doi.org/10.4208/aamm.10-m1055.

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AbstractThe theory of a class of spectral methods is extended to Volterra integro-differential equations which contain a weakly singular kernel (t - s)->* with 0< μ <1. In this work, we consider the case when the underlying solutions of weakly singular Volterra integro-differential equations are sufficiently smooth. We provide a rigorous error analysis for the spectral methods, which shows that both the errors of approximate solutions and the errors of approximate derivatives of the solutions decay exponentially inL°°-norm and weightedL2-norm. The numerical examples are given to illus

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14

HENSHALL, JOHN M., and BRUCE TIER. "An algorithm for sampling descent graphs in large complex pedigrees efficiently." Genetical Research 81, no. 3 (2003): 205–12. http://dx.doi.org/10.1017/s0016672303006232.

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No exact method for determining genotypic and identity-by-descent probabilities is available for large complex pedigrees. Approximate methods for such pedigrees cannot be guaranteed to be unbiased. A new method is proposed that uses the Metropolis–Hastings algorithm to sample a Markov chain of descent graphs which fit the pedigree and known genotypes. Unknown genotypes are determined from each descent graph. Genotypic probabilities are estimated as their means. The algorithm is shown to be unbiased for small complex pedigrees and feasible and consistent for moderately large complex pedigrees.

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15

Yang, Long, Yu Zhang, Gang Zheng, et al. "Policy Optimization with Stochastic Mirror Descent." Proceedings of the AAAI Conference on Artificial Intelligence 36, no. 8 (2022): 8823–31. http://dx.doi.org/10.1609/aaai.v36i8.20863.

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Improving sample efficiency has been a longstanding goal in reinforcement learning. This paper proposes VRMPO algorithm: a sample efficient policy gradient method with stochastic mirror descent. In VRMPO, a novel variance-reduced policy gradient estimator is presented to improve sample efficiency. We prove that the proposed VRMPO needs only O(ε−3) sample trajectories to achieve an ε-approximate first-order stationary point, which matches the best sample complexity for policy optimization. Extensive empirical results demonstrate that VRMP outperforms the state-of-the-art policy gradient methods

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He, Linjie, Yumin Chen, Caiming Zhong, and Keshou Wu. "Granular Elastic Network Regression with Stochastic Gradient Descent." Mathematics 10, no. 15 (2022): 2628. http://dx.doi.org/10.3390/math10152628.

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Linear regression is the use of linear functions to model the relationship between a dependent variable and one or more independent variables. Linear regression models have been widely used in various fields such as finance, industry, and medicine. To address the problem that the traditional linear regression model is difficult to handle uncertain data, we propose a granule-based elastic network regression model. First we construct granules and granular vectors by granulation methods. Then, we define multiple granular operation rules so that the model can effectively handle uncertain data. Fur

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17

Yang, Yin, Xinfa Yang, Jindi Wang, and Jie Liu. "The numerical solution of the time-fractional non-linear Klein-Gordon equation via spectral collocation method." Thermal Science 23, no. 3 Part A (2019): 1529–37. http://dx.doi.org/10.2298/tsci180824220y.

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In this paper, we consider the numerical solution of the time-fractional non-linear Klein-Gordon equation. We propose a spectral collocation method in both temporal and spatial discretizations with a spectral expansion of Jacobi interpolation polynomial for this equation. A rigorous error analysis is provided for the spectral methods to show both the errors of approximate solutions and the errors of approximate derivatives of the solutions decaying exponentially in infinity-norm and weighted L2-norm. Numerical tests are carried out to confirm the theoretical results.

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18

Zeng, Xueying, Lixin Shen, Yuesheng Xu, and Jian Lu. "Matrix completion via minimizing an approximate rank." Analysis and Applications 17, no. 05 (2019): 689–713. http://dx.doi.org/10.1142/s0219530519400025.

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The low rank matrix completion problem which aims to recover a matrix from that having missing entries has received much attention in many fields such as image processing and machine learning. The rank of a matrix may be measured by the [Formula: see text] norm of the vector of its singular values. Due to the nonconvexity and discontinuity of the [Formula: see text] norm, solving the low rank matrix completion problem which is clearly NP hard suffers from computational challenges. In this paper, we propose a constrained matrix completion model in which a novel nonconvex continuous rank surroga

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19

Siahlooei, Esmaeil, and Seyed Abolfazl Shahzadeh Fazeli. "Two Iterative Methods for Solving Linear Interval Systems." Applied Computational Intelligence and Soft Computing 2018 (October 8, 2018): 1–13. http://dx.doi.org/10.1155/2018/2797038.

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Conjugate gradient is an iterative method that solves a linear system Ax=b, where A is a positive definite matrix. We present this new iterative method for solving linear interval systems Ãx̃=b̃, where Ã is a diagonally dominant interval matrix, as defined in this paper. Our method is based on conjugate gradient algorithm in the context view of interval numbers. Numerical experiments show that the new interval modified conjugate gradient method minimizes the norm of the difference of Ãx̃ and b̃ at every step while the norm is sufficiently small. In addition, we present another iterative met

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20

Jacobsen, Andrew, Matthew Schlegel, Cameron Linke, Thomas Degris, Adam White, and Martha White. "Meta-Descent for Online, Continual Prediction." Proceedings of the AAAI Conference on Artificial Intelligence 33 (July 17, 2019): 3943–50. http://dx.doi.org/10.1609/aaai.v33i01.33013943.

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This paper investigates different vector step-size adaptation approaches for non-stationary online, continual prediction problems. Vanilla stochastic gradient descent can be considerably improved by scaling the update with a vector of appropriately chosen step-sizes. Many methods, including AdaGrad, RMSProp, and AMSGrad, keep statistics about the learning process to approximate a second order update—a vector approximation of the inverse Hessian. Another family of approaches use meta-gradient descent to adapt the stepsize parameters to minimize prediction error. These metadescent strategies are

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21

Bonesky, Thomas, Kamil S. Kazimierski, Peter Maass, Frank Schöpfer, and Thomas Schuster. "Minimization of Tikhonov Functionals in Banach Spaces." Abstract and Applied Analysis 2008 (2008): 1–19. http://dx.doi.org/10.1155/2008/192679.

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Tikhonov functionals are known to be well suited for obtaining regularized solutions of linear operator equations. We analyze two iterative methods for finding the minimizer of norm-based Tikhonov functionals in Banach spaces. One is the steepest descent method, whereby the iterations are directly carried out in the underlying space, and the other one performs iterations in the dual space. We prove strong convergence of both methods.

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22

Sandilya, Ruchi, and Sarvesh Kumar. "Convergence Analysis of Discontinuous Finite Volume Methods for Elliptic Optimal Control Problems." International Journal of Computational Methods 13, no. 02 (2016): 1640012. http://dx.doi.org/10.1142/s0219876216400120.

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In this paper, we discuss the convergence analysis of discontinuous finite volume methods applied to distribute the optimal control problems governed by a class of second-order linear elliptic equations. In order to approximate the control, two different methodologies are adopted: one is the method of variational discretization and second is piecewise constant discretization technique. For variational discretization method, optimal order of convergence in the [Formula: see text]-norm for state, adjoint state and control variables is derived. Moreover, optimal order of convergence in discrete [

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23

Shi, Xiulian. "Spectral Collocation Methods for Fractional Integro-Differential Equations with Weakly Singular Kernels." Journal of Mathematics 2022 (October 25, 2022): 1–9. http://dx.doi.org/10.1155/2022/3767559.

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In this paper, we propose and analyze a spectral approximation for the numerical solutions of fractional integro-differential equations with weakly kernels. First, the original equations are transformed into an equivalent weakly singular Volterra integral equation, which possesses nonsmooth solutions. To eliminate the singularity of the solution, we introduce some suitable smoothing transformations, and then use Jacobi spectral collocation method to approximate the resulting equation. Later, the spectral accuracy of the proposed method is investigated in the infinity norm and weighted L 2 norm

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24

Carpentier, Jean, and Sebastien Blandin. "Approximate Gradient Descent Convergence Dynamics for Adaptive Control on Heterogeneous Networks." Proceedings of the International Conference on Automated Planning and Scheduling 29 (May 25, 2021): 68–76. http://dx.doi.org/10.1609/icaps.v29i1.3461.

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Adaptive control is a classical control method for complex cyber-physical systems, including transportation networks. In this work, we analyze the convergence properties of such methods on exemplar graphs, both theoretically and numerically. We first illustrate a limitation of the standard backpressure algorithm for scheduling optimization, and prove that a re-scaling of the model state can lead to an improvement in the overall system optimality by a factor of at most O(k) depending on the network parameters, where k characterizes the network heterogeneity. We exhaustively describe the associa

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25

Innocenti, Luca, Leonardo Banchi, Sougato Bose, Alessandro Ferraro, and Mauro Paternostro. "Approximate supervised learning of quantum gates via ancillary qubits." International Journal of Quantum Information 16, no. 08 (2018): 1840004. http://dx.doi.org/10.1142/s021974991840004x.

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We present strategies for the training of a qubit network aimed at the ancilla-assisted synthesis of multi-qubit gates based on a set of restricted resources. By assuming the availability of only time-independent single and two-qubit interactions, we introduce and describe a supervised learning strategy implemented through momentum-stochastic gradient descent with automatic differentiation methods. We demonstrate the effectiveness of the scheme by discussing the implementation of nontrivial three qubit operations, including a QFT and a half-adder gate.

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26

Han, Xu, Jiasong Wu, Lu Wang, Yang Chen, Lotfi Senhadji, and Huazhong Shu. "Linear Total Variation Approximate Regularized Nuclear Norm Optimization for Matrix Completion." Abstract and Applied Analysis 2014 (2014): 1–8. http://dx.doi.org/10.1155/2014/765782.

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Matrix completion that estimates missing values in visual data is an important topic in computer vision. Most of the recent studies focused on the low rank matrix approximation via the nuclear norm. However, the visual data, such as images, is rich in texture which may not be well approximated by low rank constraint. In this paper, we propose a novel matrix completion method, which combines the nuclear norm with the local geometric regularizer to solve the problem of matrix completion for redundant texture images. And in this paper we mainly consider one of the most commonly graph regularized

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27

Buong, Nguyen, Nguyen Anh, and Khuat Binh. "Steepest-descent Ishikawa iterative methods for a class of variational inequalities in Banach spaces." Filomat 34, no. 5 (2020): 1557–69. http://dx.doi.org/10.2298/fil2005557b.

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In this paper, for finding a fixed point of a nonexpansive mapping in either uniformly smooth or reflexive and strictly convex Banach spaces with a uniformly G?teaux differentiable norm, we present a new explicit iterative method, based on a combination of the steepest-descent method with the Ishikawa iterative one. We also show its several particular cases one of which is the composite Halpern iterative method in literature. The explicit iterative method is also extended to the case of infinite family of nonexpansive mappings. Numerical experiments are given for illustration.

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28

Yu, Hong, Tongjun Sun, and Na Li. "The Time DiscontinuousH1-Galerkin Mixed Finite Element Method for Linear Sobolev Equations." Discrete Dynamics in Nature and Society 2015 (2015): 1–10. http://dx.doi.org/10.1155/2015/618258.

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We combine theH1-Galerkin mixed finite element method with the time discontinuous Galerkin method to approximate linear Sobolev equations. The advantages of these two methods are fully utilized. The approximate schemes are established to get the approximate solutions by a piecewise polynomial of degree at mostq-1with the time variable. The existence and uniqueness of the solutions are proved, and the optimalH1-norm error estimates are derived. We get high accuracy for both the space and time variables.

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29

Karakida, Ryo, and Kazuki Osawa. "Understanding approximate Fisher information for fast convergence of natural gradient descent in wide neural networks*." Journal of Statistical Mechanics: Theory and Experiment 2021, no. 12 (2021): 124010. http://dx.doi.org/10.1088/1742-5468/ac3ae3.

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Abstract Natural gradient descent (NGD) helps to accelerate the convergence of gradient descent dynamics, but it requires approximations in large-scale deep neural networks because of its high computational cost. Empirical studies have confirmed that some NGD methods with approximate Fisher information converge sufficiently fast in practice. Nevertheless, it remains unclear from the theoretical perspective why and under what conditions such heuristic approximations work well. In this work, we reveal that, under specific conditions, NGD with approximate Fisher information achieves the same fast

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30

KIM, MI-YOUNG. "DISCONTINUOUS GALERKIN METHODS FOR THE LOTKA–MCKENDRICK EQUATION WITH FINITE LIFE-SPAN." Mathematical Models and Methods in Applied Sciences 16, no. 02 (2006): 161–76. http://dx.doi.org/10.1142/s0218202506001108.

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We consider a model of population dynamics whose mortality function is unbounded and the solution is not regular near the maximum age. A continuous-time discontinuous Galerkin method to approximate the solution is described and analyzed. Our results show that the scheme is convergent, in L∞(L2) norm, at the rate of r + 1/2 away from the maximum age and that it is convergent at the rate of l - 1/(2q) + α/2 in L2(L2) norm, near the maximum age, if u ∈ L2(Wl,2q), where q ≥ 1, 1/2 ≤ l ≤ r + 1, r is the degree of the polynomial of the approximation space, and α is the growth rate of the mortality f

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31

Bertrand, Fleurianne, Zhiqiang Cai, and Eun Young Park. "Least-Squares Methods for Elasticity and Stokes Equations with Weakly Imposed Symmetry." Computational Methods in Applied Mathematics 19, no. 3 (2019): 415–30. http://dx.doi.org/10.1515/cmam-2018-0255.

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AbstractThis paper develops and analyzes two least-squares methods for the numerical solution of linear elasticity and Stokes equations in both two and three dimensions. Both approaches use the{L^{2}}norm to define least-squares functionals. One is based on the stress-displacement/velocity-rotation/vorticity-pressure (SDRP/SVVP) formulation, and the other is based on the stress-displacement/velocity-rotation/vorticity (SDR/SVV) formulation. The introduction of the rotation/vorticity variable enables us to weakly enforce the symmetry of the stress. It is shown that the homogeneous least-squares

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32

Zhu, Jun, Changwei Chen, Shoubao Su, and Zinan Chang. "Compressive Sensing of Multichannel EEG Signals via lq Norm and Schatten-p Norm Regularization." Mathematical Problems in Engineering 2016 (2016): 1–7. http://dx.doi.org/10.1155/2016/2189563.

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In Wireless Body Area Networks (WBAN) the energy consumption is dominated by sensing and communication. Recently, a simultaneous cosparsity and low-rank (SCLR) optimization model has shown the state-of-the-art performance in compressive sensing (CS) recovery of multichannel EEG signals. How to solve the resulting regularization problem, involving l0 norm and rank function which is known as an NP-hard problem, is critical to the recovery results. SCLR takes use of l1 norm and nuclear norm as a convex surrogate function for l0 norm and rank function. However, l1 norm and nuclear norm cannot well

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Zhou, Jingcheng, Wei Wei, Ruizhi Zhang, and Zhiming Zheng. "Damped Newton Stochastic Gradient Descent Method for Neural Networks Training." Mathematics 9, no. 13 (2021): 1533. http://dx.doi.org/10.3390/math9131533.

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First-order methods such as stochastic gradient descent (SGD) have recently become popular optimization methods to train deep neural networks (DNNs) for good generalization; however, they need a long training time. Second-order methods which can lower the training time are scarcely used on account of their overpriced computing cost to obtain the second-order information. Thus, many works have approximated the Hessian matrix to cut the cost of computing while the approximate Hessian matrix has large deviation. In this paper, we explore the convexity of the Hessian matrix of partial parameters a

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Zhang, Bin, Liuliu Wang, Shuang Li, Futai Xie, and Lideng Wei. "Airborne Single-Pass Multi-Baseline InSAR Layover Separation Method Based on Multi-Look Compressive Sensing." Applied Sciences 12, no. 24 (2022): 12658. http://dx.doi.org/10.3390/app122412658.

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Due to the small number of baselines (2–3), the traditional L1 norm compressive sensing method for layover solution in InSAR has poor separation ability and height estimation stability and a long operation time. This paper, based on the idea of multi-look, adopts a multi-look compressive sensing method and a multi-look compressive sensing method based on separable approximate sparse reconstruction. The layover separation method based on multi-look compressive sensing adopts the surrounding pixels around the current point as independent observations together with this point to increase the obse

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SHISHKIN, GRIGORII I., and PETR N. VABISHCHEVICH. "PARALLEL DOMAIN DECOMPOSITION METHODS WITH THE OVERLAPPING OF SUBDOMAINS FOR PARABOLIC PROBLEMS." Mathematical Models and Methods in Applied Sciences 06, no. 08 (1996): 1169–85. http://dx.doi.org/10.1142/s0218202596000493.

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For a model of two-dimensional boundary value problem for a second-order parabolic equation, finite difference schemes on the base of a domain decomposition method, oriented on modern parallel computers, is constructed. In the used finite difference schemes iterations at time levels are not applied; some subdomains overlap. We study two classes of schemes characterized by synchronous and asynchronous implementations. It is shown that, under refining grids, the approximate solutions do converge to the exact one in the uniform grid norm.

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Yadav, Sangita, and Amiya K. Pani. "Superconvergent discontinuous Galerkin methods for nonlinear parabolic initial and boundary value problems." Journal of Numerical Mathematics 27, no. 3 (2019): 183–202. http://dx.doi.org/10.1515/jnma-2018-0035.

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Abstract In this article, we discuss error estimates for nonlinear parabolic problems using discontinuous Galerkin methods which include HDG method in the spatial direction while keeping time variable continuous. When piecewise polynomials of degree k ⩾ 1 are used to approximate both the potential as well as the flux, it is shown that the error estimate for the semi-discrete flux in L∞(0, T; L2)-norm is of order k + 1. With the help of a suitable post-processing of the semi-discrete potential, it is proved that the resulting post-processed potential converges with order of convergence $\begin{

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Wang, Yi Yan, and Shi Yun Wu. "Adaptive Image Denoising Approach Based on Generalized Lp Norm Variational Model." Applied Mechanics and Materials 556-562 (May 2014): 4851–55. http://dx.doi.org/10.4028/www.scientific.net/amm.556-562.4851.

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The well-known methods based on gradient dependent regularizers such as total variation (TV) model often suffer the staircase effect and the loss of edge details. In order to overcome such drawbacks, an adaptive variational approach is proposed. First, we introduced a Gaussian smoothed image as the variable of the Lp norm, and then we employed the difference curvature instead of gradient as new edge indicator, which can effectively distinguish between ramps and edges. In the proposed model, the regularization term and fidelity term are both adaptive. At object edges, the regularization term is

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Cao, Yongcan, and Huixin Zhan. "Efficient Multi-objective Reinforcement Learning via Multiple-gradient Descent with Iteratively Discovered Weight-Vector Sets." Journal of Artificial Intelligence Research 70 (January 20, 2021): 319–49. http://dx.doi.org/10.1613/jair.1.12270.

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   Solving multi-objective optimization problems is important in various applications where users are interested in obtaining optimal policies subject to multiple (yet often conflicting) objectives. A typical approach to obtain the optimal policies is to first construct a loss function based on the scalarization of individual objectives and then derive optimal policies that minimize the scalarized loss function. Albeit simple and efficient, the typical approach provides no insights/mechanisms on the optimization of multiple objectives due to the lack of ability to quantify th

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Jurisic Bellotti, Maja, and Mladen Vucic. "Sparse FIR Filter Design Based on Signomial Programming." Elektronika ir Elektrotechnika 26, no. 1 (2020): 40–45. http://dx.doi.org/10.5755/j01.eie.26.1.23560.

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The goal of sparse FIR filter design is to minimize the number of nonzero filter coefficients, while keeping its frequency response within specified boundaries. Such a design can be formally expressed via minimization of l0-norm of filter’s impulse response. Unfortunately, the corresponding minimization problem has combinatorial complexity. Therefore, many design methods are developed, which solve the problem approximately, or which solve the approximate problem exactly. In this paper, we propose an approach, which is based on the approximation of the l0-norm by an lp-norm with 0 < p < 1

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Ficarella, Elisa, Luciano Lamberti, and Sadik Ozgur Degertekin. "Mechanical Identification of Materials and Structures with Optical Methods and Metaheuristic Optimization." Materials 12, no. 13 (2019): 2133. http://dx.doi.org/10.3390/ma12132133.

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This study presents a hybrid framework for mechanical identification of materials and structures. The inverse problem is solved by combining experimental measurements performed by optical methods and non-linear optimization using metaheuristic algorithms. In particular, we develop three advanced formulations of Simulated Annealing (SA), Harmony Search (HS) and Big Bang-Big Crunch (BBBC) including enhanced approximate line search and computationally cheap gradient evaluation strategies. The rationale behind the new algorithms—denoted as Hybrid Fast Simulated Annealing (HFSA), Hybrid Fast Harmon

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Si, Weijian, Xinggen Qu, Yilin Jiang, and Tao Chen. "Multiple Sparse Measurement Gradient Reconstruction Algorithm for DOA Estimation in Compressed Sensing." Mathematical Problems in Engineering 2015 (2015): 1–6. http://dx.doi.org/10.1155/2015/152570.

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A novel direction of arrival (DOA) estimation method in compressed sensing (CS) is proposed, in which the DOA estimation problem is cast as the joint sparse reconstruction from multiple measurement vectors (MMV). The proposed method is derived through transforming quadratically constrained linear programming (QCLP) into unconstrained convex optimization which overcomes the drawback thatl1-norm is nondifferentiable when sparse sources are reconstructed by minimizingl1-norm. The convergence rate and estimation performance of the proposed method can be significantly improved, since the steepest d

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Xiong, Huan. "DPCD: Discrete Principal Coordinate Descent for Binary Variable Problems." Proceedings of the AAAI Conference on Artificial Intelligence 36, no. 9 (2022): 10391–98. http://dx.doi.org/10.1609/aaai.v36i9.21281.

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Binary optimization, a representative subclass of discrete optimization, plays an important role in mathematical optimization and has various applications in computer vision and machine learning. Generally speaking, binary optimization problems are NP-hard and difficult to solve due to the binary constraints, especially when the number of variables is very large. Existing methods often suffer from high computational costs or large accumulated quantization errors, or are only designed for specific tasks. In this paper, we propose an efficient algorithm, named Discrete Principal Coordinate Desce

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Wunderlich, Jonathan, and Michael Plum. "Computer-assisted Existence Proofs for One-dimensional Schrödinger-Poisson Systems." Acta Cybernetica 24, no. 3 (2020): 373–91. http://dx.doi.org/10.14232/actacyb.24.3.2020.6.

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Motivated by the three-dimensional time-dependent Schrödinger-Poisson system we prove the existence of non-trivial solutions of the one-dimensional stationary Schrödinger-Poisson system using computer-assisted methods. Starting from a numerical approximate solution, we compute a bound for its defect, and a norm bound for the inverse of the linearization at the approximate solution. For the latter, eigenvalue bounds play a crucial role, especially for the eigenvalues "close to" zero. Therefor, we use the Rayleigh-Ritz method and a corollary of the Temple-Lehmann Theorem to get enclosures

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Sun, Min, Jing Liu, and Yaru Wang. "Two Improved Conjugate Gradient Methods with Application in Compressive Sensing and Motion Control." Mathematical Problems in Engineering 2020 (May 5, 2020): 1–11. http://dx.doi.org/10.1155/2020/9175496.

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To solve the monotone equations with convex constraints, a novel multiparameterized conjugate gradient method (MPCGM) is designed and analyzed. This kind of conjugate gradient method is derivative-free and can be viewed as a modified version of the famous Fletcher–Reeves (FR) conjugate gradient method. Under approximate conditions, we show that the proposed method has global convergence property. Furthermore, we generalize the MPCGM to solve unconstrained optimization problem and offer another novel conjugate gradient method (NCGM), which satisfies the sufficient descent property without any l

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Zuo, Ming, Shuguo Xie, Xian Zhang, and Meiling Yang. "DOA Estimation Based on Weighted l1-norm Sparse Representation for Low SNR Scenarios." Sensors 21, no. 13 (2021): 4614. http://dx.doi.org/10.3390/s21134614.

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In this paper, a weighted l1-norm is proposed in a l1-norm-based singular value decomposition (L1-SVD) algorithm, which can suppress spurious peaks and improve accuracy of direction of arrival (DOA) estimation for the low signal-to-noise (SNR) scenarios. The weighted matrix is determined by optimizing the orthogonality of subspace, and the weighted l1-norm is used as the minimum objective function to increase the signal sparsity. Thereby, the weighted matrix makes the l1-norm approximate the original l0-norm. Simulated results of orthogonal frequency division multiplexing (OFDM) signal demonst

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He, Guodong, Maozhong Song, Shanshan Zhang, Huiping Qin, and Xiaojuan Xie. "GPS Sparse Multipath Signal Estimation Based on Compressive Sensing." Wireless Communications and Mobile Computing 2021 (May 11, 2021): 1–9. http://dx.doi.org/10.1155/2021/5583429.

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A GPS sparse multipath signal estimation method based on compressive sensing is proposed. A new 0 norm approximation function is designed, and the parameter of the approximate function is gradually reduced to realize the approximation of 0 norm. The sparse signal is reconstructed by a modified Newton method. The reconstruction performance of the proposed algorithm is better than several commonly reconstruction algorithms at different sparse numbers and noise intensities. The GPS sparse multipath signal model is established, and the sparse multipath signal is estimated by the proposed reconstru

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Li, Ning, Qing-Wen Wang, and Jing Jiang. "An Efficient Algorithm for the Reflexive Solution of the Quaternion Matrix EquationAXB+CXHD=F." Journal of Applied Mathematics 2013 (2013): 1–14. http://dx.doi.org/10.1155/2013/217540.

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We propose an iterative algorithm for solving the reflexive solution of the quaternion matrix equationAXB+CXHD=F. When the matrix equation is consistent over reflexive matrixX, a reflexive solution can be obtained within finite iteration steps in the absence of roundoff errors. By the proposed iterative algorithm, the least Frobenius norm reflexive solution of the matrix equation can be derived when an appropriate initial iterative matrix is chosen. Furthermore, the optimal approximate reflexive solution to a given reflexive matrixX0can be derived by finding the least Frobenius norm reflexive

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Sun, Zhenzhen, and Yuanlong Yu. "Robust multi-class feature selection via l2,0-norm regularization minimization." Intelligent Data Analysis 26, no. 1 (2022): 57–73. http://dx.doi.org/10.3233/ida-205724.

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Feature selection is an important data preprocessing in data mining and machine learning, that can reduce the number of features without deteriorating model’s performance. Recently, sparse regression has received considerable attention in feature selection task due to its good performance. However, because the l2,0-norm regularization term is non-convex, this problem is hard to solve, and most of the existing methods relaxed it by l2,1-norm. Unlike the existing methods, this paper proposes a novel method to solve the l2,0-norm regularized least squares problem directly based on iterative hard

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Jain, Nishant, Brian Coyle, Elham Kashefi, and Niraj Kumar. "Graph neural network initialisation of quantum approximate optimisation." Quantum 6 (November 17, 2022): 861. http://dx.doi.org/10.22331/q-2022-11-17-861.

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Approximate combinatorial optimisation has emerged as one of the most promising application areas for quantum computers, particularly those in the near term. In this work, we focus on the quantum approximate optimisation algorithm (QAOA) for solving the MaxCut problem. Specifically, we address two problems in the QAOA, how to initialise the algorithm, and how to subsequently train the parameters to find an optimal solution. For the former, we propose graph neural networks (GNNs) as a warm-starting technique for QAOA. We demonstrate that merging GNNs with QAOA can outperform both approaches ind

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Masegosa, Andrés R., Rafael Cabañas, Helge Langseth, Thomas D. Nielsen, and Antonio Salmerón. "Probabilistic Models with Deep Neural Networks." Entropy 23, no. 1 (2021): 117. http://dx.doi.org/10.3390/e23010117.

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Recent advances in statistical inference have significantly expanded the toolbox of probabilistic modeling. Historically, probabilistic modeling has been constrained to very restricted model classes, where exact or approximate probabilistic inference is feasible. However, developments in variational inference, a general form of approximate probabilistic inference that originated in statistical physics, have enabled probabilistic modeling to overcome these limitations: (i) Approximate probabilistic inference is now possible over a broad class of probabilistic models containing a large number of

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