Three multi-level mixed finite element methods for the steady Boussinesq equations are analyzed and discussed in this paper. The nonlinear and multi-variables coupled problem on a coarse mesh with the mesh size $h_0$ is solved firstly, and then, a series of decoupled and linear subproblems with the Stokes, Oseen and Newton iterations are solved on the successive and refined grids with the mesh sizes $h_j$, $j$ = 1, 2, . . . , $J$. The computational scales are reduced and the computational costs are saved. Furthermore, the uniform stability and convergence results in both $L^2$ - and $H^1$ -norms of are derived under some uniqueness conditions by using the mathematical induction and constructing the dual problems. Theoretical results show that the multi-level methods have the same order of numerical solutions in the $H^1$ -norm as the one level method with the mesh sizes $h_j$ = $h^2_j$−1, $j$ = 1, 2, . . . , $J$. Finally, some numerical results are provided to investigate and compare the effectiveness of the multi-level mixed finite element methods.

Inspired by Polyak’s heavy-ball method, this paper proposes an adaptive deterministic block coordinate descent method with momentum (mADBCD) for efficiently solving large-scale linear least-squares problems. The proposed method introduces a novel column selection criterion based on the Euclidean norm of the residual vector of the normal equation. In contrast to classical block coordinate descent methods, mADBCD does not require a fixed pre-partitioning of the column indices of the coefficient matrix and avoids the expensive computation of Moore Penrose pseudoinverses of submatrices at each iteration. The method adaptively updates the block index set at each step, thereby improving both flexibility and scalability. When the coefficient matrix is of full column rank, we prove that mADBCD converges linearly to the unique solution of the least-squares problem. Numerical experiments are conducted to show that mADBCD outperforms several recent block coordinate descent methods in terms of iteration count and CPU time. In particular, when solving extremely sparse least-squares problems, mADBCD is the first block coordinate descent method reported to achieve CPU time nearly comparable to that of the classical least squares QR (LSQR) method [Paige and Saunders, ACM Trans. Math. Softw., 8 (1982)].

Recovery of block sparse signals with partially-known block support information is of particular importance in compressed sensing. A uniform sufficient condition guaranteeing stable recovery of non-strictly block $k$-sparse signals is established via the weighted $ℓ_{2,p}−\alpha ℓ_{2,q}$ nonconvex minimization method, and the reconstruction error is precisely bounded in terms of the residual of block-sparsity and the measurement error. Furthermore, a series of contrastive numerical experiments reveal that exploiting the approximate block-sparsity characteristic and the nonuniform prior block support estimate substantially promotes the performance of reconstruction for block-structural signals.

In this paper, we propose the parareal algorithms for stochastic Maxwell equations with the damping term driven by additive noise. The proposed algorithms proceed as two-level temporal parallelizable integrators with the stochastic exponential integrator as the coarse $\mathscr{G}$-propagator and both the exact solution integrator and the stochastic exponential integrator as the fine $\mathscr{F}$-propagator. The mean-square convergence order of the proposed algorithms consistently increases to $k,$ regardless of whether the exact solution integrator or the stochastic exponential integrator is chosen as the fine $\mathscr{F}$-propagator. Several numerical experiments are illustrated in order to verify our theoretical findings for different choices of the iteration number k and the damping coefficient $σ.$

An explicit numerical method is developed for a class of non-autonomous time-changed stochastic differential equations, whose coefficients obey Hölder’s continuity in terms of the time variables and are allowed to grow super-linearly in terms of the state variables. The strong convergence of the method in the finite time interval is proved and the convergence rate is obtained. Numerical simulations are provided.

Image restoration based on total variation has been widely studied owing to its edgepreservation properties. In this study, we consider the total variation infimal convolution (TV-IC) image restoration model for eliminating mixed Poisson-Gaussian noise. Based on the alternating direction method of multipliers (ADMM), we propose a complete splitting proximal bilinear constraint ADMM algorithm to solve the TV-IC model. We prove the convergence of the proposed algorithm under mild conditions. In contrast with other algorithms used for solving the TV-IC model, the proposed algorithm does not involve any inner iterations, and each subproblem has a closed-form solution. Finally, numerical experimental results demonstrate the efficiency and effectiveness of the proposed algorithm.

We present a decoupled, linearly implicit numerical scheme with energy stability and mass conservation for solving the coupled Cahn-Hilliard system.The time-discretization is done by leap-frog method with the scalar auxiliary variable (SAV) approach.It only needs to solve three linear equations at each time step, where each unknown variable can be solved independently.It is shown that the semi-discrete scheme has second-order accuracy in the temporal direction.Such convergence results are proved by a rigorous analysis of the boundedness of the numerical solution and the error estimates at different time-level.Numerical examples are presented to further confirm the validity of the methods.

This paper presents three regularized models for the logarithmic Klein-Gordon equation. By using a modified Crank-Nicolson method in time and the Galerkin finite element method (FEM) in space, a fully implicit energy-conservative numerical scheme is constructed for the local energy regularized model that is regarded as the best one among the three regularized models. Then, the cut-off function technique and the time-space error splitting technique are innovatively combined to rigorously analyze the unconditionally optimal and high-accuracy convergence results of the numerical scheme without any coupling condition between the temporal step size and the spatial mesh width. The theoretical framework is uniform for the other two regularized models. Finally, numerical experiments are provided to verify our theoretical results. The analytical techniques in this work are not limited in the FEM, and can be directly extended into other numerical methods. More importantly, this work closes the gap for the unconditional error/stability analysis of the numerical methods for the logarithmic systems in higher dimensional spaces.

In this paper, we establish the oracle inequalities of highly corrupted linear observations b = Ax0 + f0 + e ∈ R m .Here the vector x0 ∈ R n with n ≫ m is a (approximately) sparse signal and f0 ∈ R m is a sparse error vector with nonzero entries that can be possible infinitely large, e ∼ N (0, σ 2 Im) represents the Gaussian random noise vector.We extend the oracle inequality x -x0 2 2 i min{|x0(i)| 2 , σ 2 } for Dantzig selector and Lasso models in [E.