Table of Content

04 February 2023, Volume 41 Issue 1

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REQUIRED NUMBER OF ITERATIONS FOR SPARSE SIGNAL RECOVERY VIA ORTHOGONAL LEAST SQUARES

Haifeng Li, Jing Zhang, Jinming Wen, Dongfang Li

2023, 41(1):  1-17.  DOI: 10.4208/jcm.2104-m2020-0093

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In countless applications, we need to reconstruct a K-sparse signal x ∈ Rⁿ from noisy measurements y=Φx+v, where Φ∈ R^m×n is a sensing matrix and v ∈ R^m is a noise vector. Orthogonal least squares (OLS), which selects at each step the column that results in the most significant decrease in the residual power, is one of the most popular sparse recovery algorithms. In this paper, we investigate the number of iterations required for recovering x with the OLS algorithm. We show that OLS provides a stable reconstruction of all K-sparse signals x in [2.8K] iterations provided that Φ satisfies the restricted isometry property (RIP). Our result provides a better recovery bound and fewer number of required iterations than those proposed by Foucart in 2013.

RECONSTRUCTION OF SPARSE POLYNOMIALS VIA QUASI-ORTHOGONAL MATCHING PURSUIT METHOD

Renzhong Feng, Aitong Huang, Ming-Jun Lai, Zhaiming Shen

2023, 41(1):  18-38.  DOI: 10.4208/jcm.2104-m2020-0250

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In this paper, we propose a Quasi-Orthogonal Matching Pursuit (QOMP) algorithm for constructing a sparse approximation of functions in terms of expansion by orthonormal polynomials. For the two kinds of sampled data, data with noises and without noises, we apply the mutual coherence of measurement matrix to establish the convergence of the QOMP algorithm which can reconstruct s-sparse Legendre polynomials, Chebyshev polynomials and trigonometric polynomials in s step iterations. The results are also extended to general bounded orthogonal system including tensor product of these three univariate orthogonal polynomials. Finally, numerical experiments will be presented to verify the e ectiveness of the QOMP method.

RECONSTRUCTED DISCONTINUOUS APPROXIMATION TO STOKES EQUATION IN A SEQUENTIAL LEAST SQUARES FORMULATION

Ruo Li, Fanyi Yang

2023, 41(1):  39-71.  DOI: 10.4208/jcm.2104-m2020-0231

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We propose a new least squares nite element method to solve the Stokes problem with two sequential steps. The approximation spaces are constructed by the patch reconstruction with one unknown per element. For the rst step, we reconstruct an approximation space consisting of piecewise curl-free polynomials with zero trace. By this space, we minimize a least squares functional to obtain the numerical approximations to the gradient of the velocity and the pressure. In the second step, we minimize another least squares functional to give the solution to the velocity in the reconstructed piecewise divergence-free space. We derive error estimates for all unknowns under both L² norms and energy norms. Numerical results in two dimensions and three dimensions verify the convergence rates and demonstrate the great exibility of our method.

SECOND ORDER UNCONDITIONALLY STABLE AND CONVERGENT LINEARIZED SCHEME FOR A FLUID-FLUID INTERACTION MODEL

Wei Li, Pengzhan Huang, Yinnian He

2023, 41(1):  72-93.  DOI: 10.4208/jcm.2104-m2020-0265

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In this paper, a fully discrete finite element scheme with second-order temporal accuracy is proposed for a fluid-fluid interaction model, which consists of two Navier-Stokes equations coupled by a linear interface condition. The proposed fully discrete scheme is a combination of a mixed finite element approximation for spatial discretization, the secondorder backward differentiation formula for temporal discretization, the second-order Gear's extrapolation approach for the interface terms and extrapolated treatments in linearization for the nonlinear terms. Moreover, the unconditional stability is established by rigorous analysis and error estimate for the fully discrete scheme is also derived. Finally, some numerical experiments are carried out to verify the theoretical results and illustrate the accuracy and efficiency of the proposed scheme.

UNCONDITIONAL SUPERCONVERGENT ANALYSIS OF QUASI-WILSON ELEMENT FOR BENJAMIN-BONA-MAHONEY EQUATION

Xiangyu Shi, Linzhang Lu

2023, 41(1):  94-106.  DOI: 10.4208/jcm.2104-m2020-0233

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This article aims to study the unconditional superconvergent behavior of nonconforming quadrilateral quasi-Wilson element for nonlinear Benjamin Bona Mahoney (BBM) equation. For the generalized rectangular meshes including rectangular mesh, deformed rectangular mesh and piecewise deformed rectangular mesh, by use of the special character of this element, that is, the conforming part(bilinear element) has high accuracy estimates on the generalized rectangular meshes and the consistency error can reach order O(h²), one order higher than its interpolation error, the superconvergent estimates with respect to mesh size h are obtained in the broken H¹-norm for the semi-/fully-discrete schemes. A striking ingredient is that the restrictions between mesh size h and time step τ required in the previous works are removed. Finally, some numerical results are provided to con rm the theoretical analysis.

ENERGY AND QUADRATIC INVARIANTS PRESERVING METHODS FOR HAMILTONIAN SYSTEMS WITH HOLONOMIC CONSTRAINTS

Lei Li, Dongling Wang

2023, 41(1):  107-132.  DOI: 10.4208/jcm.2106-m2020-0205

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We introduce a new class of parametrized structure-preserving partitioned RungeKutta (α-PRK) methods for Hamiltonian systems with holonomic constraints. The methods are symplectic for any xed scalar parameter α, and are reduced to the usual symplectic PRK methods like Shake-Rattle method or PRK schemes based on Lobatto IIIA-IIIB pairs when α=0. We provide a new variational formulation for symplectic PRK schemes and use it to prove that the α-PRK methods can preserve the quadratic invariants for Hamiltonian systems subject to holonomic constraints. Meanwhile, for any given consistent initial values (p0, q0) and small step size h > 0, it is proved that there exists α^*=(h, p0, q0) such that the Hamiltonian energy can also be exactly preserved at each step. Based on this, we propose some energy and quadratic invariants preserving α-PRK methods. These α-PRK methods are shown to have the same convergence rate as the usual PRK methods and perform very well in various numerical experiments.

A SSLE-TYPE ALGORITHM OF QUASI-STRONGLY SUB-FEASIBLE DIRECTIONS FOR INEQUALITY CONSTRAINED MINIMAX PROBLEMS

Jinbao Jian, Guodong Ma, Yufeng Zhang

2023, 41(1):  133-152.  DOI: 10.4208/jcm.2106-m2020-0059

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In this paper, we discuss the nonlinear minimax problems with inequality constraints. Based on the stationary conditions of the discussed problems, we propose a sequential systems of linear equations (SSLE)-type algorithm of quasi-strongly sub-feasible directions with an arbitrary initial iteration point. By means of the new working set, we develop a new technique for constructing the sub-matrix in the lower right corner of the coe cient matrix of the system of linear equations (SLE). At each iteration, two systems of linear equations (SLEs) with the same uniformly nonsingular coe cient matrix are solved. Under mild conditions, the proposed algorithm possesses global and strong convergence. Finally, some preliminary numerical experiments are reported.

A HYBRID VISCOSITY APPROXIMATION METHOD FOR A COMMON SOLUTION OF A GENERAL SYSTEM OF VARIATIONAL INEQUALITIES, AN EQUILIBRIUM PROBLEM, AND FIXED POINT PROBLEMS

Maryam Yazdi, Saeed Hashemi Sababe

2023, 41(1):  153-172.  DOI: 10.4208/jcm.2106-m2020-0209

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In this paper, we introduce a new iterative method based on the hybrid viscosity approximation method for finding a common element of the set of solutions of a general system of variational inequalities, an equilibrium problem, and the set of common fixed points of a countable family of nonexpansive mappings in a Hilbert space. We prove a strong convergence theorem of the proposed iterative scheme under some suitable conditions on the parameters. Furthermore, we apply our main result for W-mappings. Finally, we give two numerical results to show the consistency and accuracy of the scheme.