• Abstract

In this paper, we introduce a nonlinear finite volume scheme preserving discrete strong extremum principle (DSEP) for diffusion equations on tetrahedral meshes. In the construction of our nonlinear scheme, the key is to reformulate a discrete normal flux with local extremum principle structure, which is based on a modification of a second order linear scheme. In the construction of existing cell-centered finite volume schemes that maintain the discrete maximum principle, it is required to represent auxiliary unknowns as convex combinations of primary unknowns, which results in strong constraints on the smoothness of the mesh and diffusion coefficient. By contrast, our new scheme avoids this kind of constraints. Moreover, we will prove that there holds the DSEP for any solution of our scheme and there exists at least one solution preserving DSEP for our scheme. Furthermore, a modified Picard iteration with the Anderson acceleration (mP-AA) for solving the nonlinear scheme is proposed, and the nonlinear convergence of the modified Picard iteration is also proved. Finally, numerical examples are presented to show that the new scheme preserves DSEP and obtains second order accuracy, as well as the mP-AA method is effective.

• General Summary

• Abstract

In this paper, our focus is on examining the robustness of the central scheme in two dimensions. Although stability analyses are available in the literature for the scheme’s solution of scalar conservation laws, the associated Courant-Friedrichs-Lewy (CFL) number is often notably small, occasionally degenerating to zero. This challenge is traced back to the initial data reconstruction. The interface value limiter used in the reconstruction proves insufficient to maintain the invariant region of the updated solutions. To overcome this limitation, we introduce the vertex value limiter, resulting in a more suitable CFL number that is half of the one-dimensional value. We present a unified analysis of stability applicable to both types of limiters. This enhanced stability condition enables the utilization of larger time steps, offering improved resolution to the solution and ensuring faster simulations. Our analysis extends to general conservation laws, encompassing scalar problems and nonlinear systems. We support our findings with numerical examples, validating our claims and showcasing the robustness of the enhanced scheme.

• General Summary

• Abstract

Many inverse problems that appear in applications can be modeled as an operator equation. In practice, most of these problems are ill-posed, and computing solutions to such problems in an efficient manner is challenging and has been of greatest interest among researchers in the recent past. While many approaches are developed within infinite-dimensional Hilbert space settings, practical applications often require solutions in finite-dimensional spaces, and we need to discretize the problem. In this manuscript, we study a novel discretization scheme along with a class of regularization techniques for solving linear ill-posed problems and obtain the optimal order error estimates under an a priori parameter choice strategy. We illustrate the computational efficacy of the proposed scheme through numerical examples, and the results demonstrate that the proposed scheme is more economical due to the amount of discrete information needed to solve the problem is significantly lower than the traditional finite-dimensional approach.

• General Summary

• Abstract

In this paper, a three-term conjugate gradient projection method that employs both inertial and relaxed techniques is proposed to find approximate solutions for unconstrained nonlinear pseudo-monotone equations. The search direction generated at each iteration possesses the sufficient descent and trust region properties independent of the line search technique used. The global convergence of the proposed method is shown without the Lipschitz continuity of the underlying mapping. Moreover, the asymptotic and non-asymptotic convergence rates in terms of iteration complexity are established under the local Lipschitz continuity assumption. To our knowledge, this is the first time in the literature that an iteration-complexity analysis has been conducted for inertial-relaxed gradient-type projection methods. Numerical experiments on large-scale benchmark test problems are conducted to demonstrate the effectiveness and efficiency of the proposed algorithm. Furthermore, the applicability and practicality of the proposed method are also verified by applying it to solve sparse signal restoration problems.

• General Summary

• Abstract

This paper investigates an interface-coupled fractional subdiffusion model, featuring two subdiffusion equations in adjacent domains connected by an interface allowing bidirectional energy transfer. The fractional derivative, accounting for long-term medium effects, introduces challenges in theoretical analysis and computational efficiency. We propose a partitioned time-stepping algorithm using higher-order extrapolations on the interface term to decouple the system with improved temporal accuracy, combined with finite element spatial approximations. Rigorous theoretical analysis demonstrates unconditional stability and optimal $L^2$ norm error estimates, supported by several numerical experiments.

• General Summary

• Abstract

In this paper, we propose an accelerated stochastic variance reduction gradient method with a trust-region-like framework, referred as the NMSVRG-TR method. Based on NMSVRG, we incorporate a Katyusha-like acceleration step into the stochastic trust region scheme, which improves the convergence rate of the SVRG methods. Under appropriate assumptions, the linear convergence of the algorithm is provided for strongly convex objective functions. Numerical experiment results show that our algorithm is generally superior to some existing stochastic gradient methods.

• General Summary

• 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.

• General Summary

• Abstract

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.

• General Summary

• Abstract

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.

• General Summary

• Abstract

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.

• General Summary

• Abstract

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.

• General Summary

• Abstract

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.

• General Summary

• Abstract

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.

• General Summary

• Abstract

Stochastic gradient descent (SGD) methods have gained widespread popularity for solving large-scale optimization problems. However, the inherent variance in SGD often leads to slow convergence rates. We introduce a family of unbiased stochastic gradient estimators that encompasses existing estimators from the literature and identify a gradient estimator that not only maintains unbiasedness but also achieves minimal variance. Compared with the existing estimator used in SGD algorithms, the proposed estimator demonstrates a significant reduction in variance. By utilizing this stochastic gradient estimator to approximate the full gradient, we propose two mini-batch stochastic conjugate gradient algorithms with minimal variance. Under the assumptions of strong convexity and smoothness on the objective function, we prove that the two algorithms achieve linear convergence rates. Numerical experiments validate the effectiveness of the proposed gradient estimator in reducing variance and demonstrate that the two stochastic conjugate gradient algorithms exhibit accelerated convergence rates and enhanced stability.

• General Summary