• Abstract

In this paper, the novel optimization model for solving tensor completion with noise is proposed, its objective function is a convex combination of the minimum nuclear norm and maximum nuclear norm. The necessary condition and sufficient condition of the stationary point and optimal solution are discussed. Based on the proximal gradient algorithm and feasible direction method, we design the new algorithm for solving the proposed nonconvex and nonsmooth optimization problem and prove that the sub-sequence generated by the new algorithm converges to the stationary point. Finally, experimental results on the random sample completions and images show that the proposed optimization and algorithm are superior to the compared algorithms in CPU time or precision.

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In this paper, we propose an adaptive algorithm for L1-fidelity color image restoration by using saturation-value total variation. The main contribution of this paper is to employ the generalized cross validation method efficiently and automatically to estimate the regularization parameter in a saturation-value total variation plus L1-fidelity color image restoration model. We consider Poisson noise and mixed noise in this paper, and the experimental results show that the visual quality and the SSIM/PSNR/SAM values of the restored images by using the proposed algorithm are competitive with other tested existing methods, which makes the proposed algorithm to be comparable both quantitatively and qualitatively.

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In practical applications, the Allen-Cahn (AC) equation is commonly used to model microstructure evolutions, including alloy solidification, crystal growth, fingerprint image restoration, and image segmentation. However, when we discretize the AC equation with a conventional finite difference scheme, the directional bias in error terms introduces anisotropy into the numerical results, affecting interface dynamics. To address this issue, we use two- and three-dimensional isotropic finite difference schemes to solve the AC equation. Stability of the proposed algorithm is verified by deriving the time step constraints in both 2D and 3D domains. To demonstrate the sharp estimation of the stability constraints, we conducted several numerical experiments and found the maximum principle is guaranteed under the analyzed time-step constraint.

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A fractional-order mathematical model of lung cancer is used to describe the dynamics of tumor growth and the interactions between cancer cells and immune cells. To obtain approximate solutions and better understand the behavior of the state functions, a pseudo-operational collocation scheme employing shifted Jacobi polynomials as basis functions is introduced. Initially, the existence and uniqueness of solutions to the model are established using the Leray-Schauder fixed-point theorem. Error bounds for the residual functions are estimated within a Jacobi-weighted $L^2$-space. To enhance the accuracy and reliability of the results, two distinct strategies are implemented: sensitivity analysis and feedback control. The feedback control of the proposed pseudo-operational spectral method is performed using the method of Lagrange multipliers, marking its first application in this context. Spectral solutions are derived by applying the pseudo-operational scheme to both the original model and the model with control functions. Improved performance and outputs are anticipated following the application of the feedback control strategy. Finally, comprehensive biological interpretations of the results are provided, offering insights into the practical implications of the model.

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This paper is devoted to the study of a sub-diffusion equation involving a Clarke subdifferential boundary condition. It describes transport of particles governed by the anomalous diffusion in media with boundary semipermeability. The weak formulation of the model problem results in a time fractional parabolic hemivariational inequality. We first construct an abstract hemivariational evolutionary inclusion and prove its unique solvability using a time-discretization approximation, known as the Rothe method. In addition, a numerical approach based on a finite difference scheme in time and finite dimensional approximation in space is proposed and analyzed for the abstract problem. These results are then applied to establish the convergence of the numerical solution of the model problem. Under appropriate regularity assumptions, an optimal order error estimate for the linear finite element method is derived. Some numerical examples are provided to support the theoretical results.

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Consider the inverse acoustic scattering of time-harmonic point sources by a locally perturbed interface with bounded obstacles embedded in the lower half-space. A Newton-type iterative method is proposed to simultaneously reconstruct the locally rough interface and embedded obstacles by taking partial near-field measurements in the upper half-space. The method relies on a differentiability analysis of the scattering problem with respect to the locally rough interface and the embedded obstacle, which is established by introducing a kind of new shape derivatives and reducing the original model to an equivalent system of integral equations defined in a bounded domain. With a slight modification, the inversion method can be easily generalized to reconstruct local perturbations of a global rough interface. Finally, numerical results are presented to illustrate the effectiveness of the inversion method with the multi-frequency data.

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Multiple solutions are common in various non-convex problems arising from industrial and scientific computing. Nonetheless, understanding the nontrivial solutions’ qualitative properties seems limited, partially due to the lack of efficient and reliable numerical methods. In this paper, we design a dedicated numerical method to explore these nontrivial solutions further. We first design an improved adaptive orthogonal basis deflation method by combining the adaptive orthogonal basis method with a bisection-deflation algorithm. We then apply the proposed new method to study the impact of domain changes on multiple solutions of certain nonlinear elliptic equations. When the domain varies from a circular disk to an elliptical disk, the corresponding functional value changes dramatically for some particular solutions, which indicates that these nontrivial solutions in the circular domain may become unstable in the elliptical domain. Moreover, several theoretical results on multiple solutions in the existing literature are verified. For the nonlinear sine-Gordon equation with parameter $λ,$ nontrivial solutions are found for $λ > λ_2,$ here $λ_2$ is the second eigenvalue of the corresponding linear eigenvalue problem. For the singularly perturbed Ginzburg-Landau equation, highly concentrated solutions are numerically verified, suggesting that their convergent limit is a delta function when the perturbation parameter goes to zero.

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The numerical analysis of stochastic time-fractional equations exhibits a significantly low-order convergence rate since the limited regularity of model caused by the nonlocal operator and the presence of noise. In this work, we consider stochastic time-fractional equations driven by integrated white noise, where $^CD^α_t ψ(x, t),$ $0 < α < 2$ and $I^γ_t\dot{W} (x, t),$ $0 < γ < 1.$ We first establish the regularity of the mild solution. Then superlinear convergence rate

with sufficiently small ε term in the exponent is established based on the modified two-step backward difference formula methods. Here $d$ represents the spatial dimension, $ψ^n$ denotes the approximate solution at the $n$-th time step, and $\mathbb{E}$ is the expectation operator. Numerical experiments are performed to verify the theoretical results. To the best of our knowledge, this is the first topic on the superlinear convergence analysis for the stochastic time-fractional equations with integrated white noise.

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This paper investigates two spectral volume (SV) methods applied to 2D linear hyperbolic conservation laws on rectangular meshes. These methods utilize upwind fluxes and define control volumes using Gauss-Legendre (LSV) and right-Radau (RRSV) points within mesh elements. Within the framework of Petrov-Galerkin method, a unified proof is established to show that the proposed LSV and RRSV schemes are energy stable and have optimal error estimates in the $L^2$ norm. Additionally, we demonstrate superconvergence properties of the SV method at specific points and analyze the error in cell averages under appropriate initial and boundary discretizations. As a result, we show that the RRSV method coincides with the standard upwind discontinuous Galerkin method for hyperbolic problems with constant coefficients. Numerical experiments are conducted to validate all theoretical findings.

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In this work, we investigate a thermoviscoelastic system governed by the Guyer-Krumhansl model. The system is free from the paradox of infinite heat propagation speed and, furthermore, is more suitable for modeling complex problems involving heterogeneous materials on a macroscale and at room temperature. Firstly, we establish the well-posedness of the system using the theory of semigroups of linear operators, and then we prove uniform exponential decay with respect to a given physical parameter using the multiplier method. Subsequently, we discretize the system and propose a monotone and consistent numerical scheme using finite differences. The convergence of the numerical solution is proven by the Lax equivalence theorem. Finally, we present numerical experiments using MATLAB to demonstrate the accuracy and efficiency of the scheme that reproduce the theoretical results.

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In this paper, we design a novel explicit second order scheme with one step for forward backward stochastic differential equations, and the Crank-Nicolson scheme is a specific case of our proposed framework. We first establish a rigorous stability result, and then we derive precise error estimates. Moreover, we confirm that the proposed novel scheme is second order convergent. The theoretical results for the proposed methods are supported by numerical experiments.

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In this work, a nonlinear subdiffusion initial-boundary value problem with a variable exponent is considered, whose solution behaves a weak singularity at initial time. By adding a corrected term to the nonuniform L1 scheme, a novel scheme is investigated to approximate the time-fractional Caputo derivative with a variable exponent. This scheme allows us to use a smaller grading parameter $r$ to obtain a similar level of accuracy as that of the L1 method. Combining the proposed scheme with the finite element method in space and the Newton linearization for the nonlinear term, a fully discrete scheme is constructed. To obtain the unconditional optimal error estimate, the temporal-spatial splitting technique is adopted to derive the boundedness of the computed solution $U^n_h$ in $L^∞$-norm. With the help of this bound and the discrete fractional Gronwall inequality, the optimal error analysis without certain temporal restrictions dependent on the spatial mesh size is derived. Furthermore, by using a simple postprocessing technique of the computed solution, the convergence order in the spatial direction is improved. Finally, numerical experiments are presented to verify the theoretical findings.

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Graph energy is a spectrum-based graph invariant that has been studied extensively in network sciences. However, as far as we are aware, most of the existing works try to establish theoretical bounds, and there are few efficient algorithms for computing energy of extremely large-scale graph. To fill-in this gap, we first propose a randomized algorithm for evaluating energy of large-scale graphs, under the assumption that the adjacency matrix is approximately low (numerical) rank. However, the number of sampling used in this algorithm is difficult to determine in advance, and the graph energy is often underestimated. In order to improve the quality of the evaluation, we then propose a non-restarted randomized algorithm that updates the columns of the search basis incrementally. The error analysis and the convergence of the algorithm are established. However, the non-restarted algorithm may suffer from heavy overhead as the iteration proceeds. So as to release the overhead of the non-restarted algorithm, we finally propose a restarted randomized algorithm for evaluating energy of extremely large-scale graphs. The rationality of the restarted algorithm is given. Extensive numerical experiments are performed on extremely large-scale real-world and synthetic graphs, to show the effectiveness of our strategies and efficiency of the proposed algorithms.

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Seeking the external equitable partitions (EEPs) of networks under unknown structures is an emerging problem in network analysis. The special structure of EEPs has found widespread applications in many fields such as cluster synchronization and consensus dynamics. While most literature focuses on utilizing the special structural properties of EEPs for network studies, there has been little work on the extraction of EEPs or their connection with graph signals. In this paper, we address the interesting connection between low pass graph signals and EEPs, which, as far as we know, is the first time. We provide a method BE-EEPs for extracting EEPs from low pass graph signals and propose an optimization model, which is essentially a problem involving nonnegative orthogonality matrix decomposition. We derive theoretical error bounds for the performance of our proposed method under certain assumptions and apply three algorithms to solve the resulting model, including the K-means algorithm, the practical exact penalty method and the iterative Lagrangian approach. Numerical experiments verify the effectiveness of the proposed method. Under strong low pass graph signals, the iterative Lagrangian and K-means perform equally well, outperforming the exact penalty method. However, under complex weak low pass signals, all three perform equally well.

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