In this paper, we consider the sparse distributed control problem constrained by a random elliptic equation, which we reformulate as a nonsmooth stochastic optimization problem in Hilbert space. By incorporating the advantages of the stochastic approximation approach and the alternating direction method of multipliers (ADMM), we propose a stochastic ADMM algorithm. This method decouples the stochasticity arising from the random equation constraint from the nonsmoothness of the control objective, allowing them to be tackled separately within the iterations. We introduce stochastic gradients and develop a proximal linearization technique for the stochastic subproblem, allowing each subproblem to admit a closed-form solution. The convergence and a high-probability bound of the proposed method are analyzed for the model problem. Numerical results demonstrate the effectiveness and efficiency of our method.

The rate of convergence of the augmented Lagrangian method for solving nonlinear programming is studied under the Jacobian uniqueness conditions. It is demonstrated that, for a given multiplier vector $(\mu, \lambda)$, the rate of convergence of the augmented Lagrangian method is linear with respect to $\| (\mu, \lambda) - (\mu^{*}, \lambda^{*}) \|$ and the ratio constant is proportional to $1/c$ when the ratio $\| (\mu, \lambda)-(\mu^{*}, \lambda^{*}) \| /c$ is small enough, where $c$ is the penalty parameter that exceeds a threshold $c^{*} > 0$ and $(\mu^{*}, \lambda^{*})$ is the multiplier corresponding to a local minimum point. Importantly, the ratio constant of the Q-linear convergence of the sequence of multiplier vectors is estimated by the second-order derivative of the value function of the nonlinear optimization problem. This characterization gives an explicit expression for the rate constant of the Q-linear convergence of the sequence of multiplier vectors.

In this paper, we consider a class of three-composite nonconvex optimization problems, in which the nonsmooth function is further composed with a linear operator. This problem has many applications such as sparse signal recovery, image processing and machine learning. Based on the conjugate duality theory, we present an accelerated preconditioned primal-dual gradient algorithm for this problem. Compared with the existing algorithms, our algorithm only needs to calculate the proximal mapping of the conjugate function $h^∗$ which is always convex and lower semicontinuous and it does not need to calculate the proximal mapping of nonconvex functions. This may significantly reduce the computation load. We prove that the sequence generated by the proposed algorithm globally converges to a critical point when the function satisfies the Kurdyka- Lojasiewicz property. We also obtain the convergence rate of the proposed algorithm. Finally, numerical results on sparse signal recovery and image processing illustrate the efficiency and competitiveness of the proposed algorithm.

In this work, a subdiffusion equation with constant time delay $\tau$ is considered. First, the regularity of the solution to the considered problem is investigated, finding that its first-order time derivative exhibits singularity at $t = 0^+$ and its second-order time derivative shows singularity at both $t = 0^+$ and $\tau^+$, while the solution can be decomposed into its singular and regular components. Then, we derive a fully discrete finite element scheme to solve the considered problem based on the standard Galerkin finite element method in space and the Grünwald-Letnikov type approximation in time. The analysis shows that the developed numerical scheme is stable. In order to discuss the error estimate, a new discrete Grönwall inequality is established. Under the above decomposition of the solution, we obtain a local error estimate in time for the developed numerical scheme. Finally, some numerical tests are provided to support our theoretical analysis.

In this paper, we design novel high order probabilistic numerical algorithms for forward backward stochastic differential equations. Moreover, we derive the error estimates and prove the high order convergence rates of the proposed schemes. Because the proposed scheme involves conditional expectations, an estimator based on the multilevel Monte Carlo method is applied to approximate the conditional expectations. Furthermore, we theoretically demonstrate that the computational complexity of our numerical method is proportional to the square of prescribed accuracy. Numerical experiments are given to illustrate the theoretical results.

This paper is devoted to identifying the source term and initial value simultaneously in a time-fractional Black-Scholes equation, which is an ill-posed problem. The inverse problem is transformed into a system of operator equations, and under certain source conditions, conditional stability is established. We propose a regularization method with two differential operators to solve the problem, error estimates by rules of a priori and a posteriori regularization parameter selection are derived, respectively. Numerical experiments are presented to validate the effectiveness of the proposed regularization method.

Superconvergent behavior for nonlinear Kirchhoff-type with damping is researched by a structure-preserving nonconforming finite element method (FEM). A new implicit energy dissipation scheme is developed and the numerical solution is bounded in energy norm. The existence of the numerical solution is obtained with the help of the Brouwer fixed-point theorem and then the uniqueness is gained. Superconvergence characteristics is revealed by the properties of the nonconforming FE and a special splitting technique. Numerical tests confirm the correctness of the theoretical research results.

In this paper, we analyze the virtual element method for the symmetric second order elliptic eigenvalue problem with variable coefficients in two and three dimensions, which reduces the number of degrees of freedom of the standard virtual element method. We attempt to prove the interpolation theory and stability analysis for the serendipity nodal virtual element space, which provides new stabilization terms for the virtual element schemes. Then we prove the spectral approximation and the optimal a priori error estimates. Moreover, we construct a fully computable residual-type a posteriori error estimator applied to the adaptive serendipity virtual element method and prove its upper and lower bounds with respect to the approximation error. Finally, we show numerical examples to verify the theoretical results and show the comparison between standard and serendipity virtual element methods.

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.

This paper develops a low order weak Galerkin (WG) finite element method for the steady thermally coupled incompressible magnetohydrodynamics flow. In the interior of elements, the WG scheme uses piecewise linear polynomials for the approximations of the velocity, the magnetic field and the temperature, and piecewise constants for the approximations of the pressure and the magnetic pseudo-pressure; and on the interfaces of elements, the scheme uses piecewise constants for the numerical traces of velocity and the temperature, and piecewise linear polynomials for the numerical traces of the magnetic fields, the pressure and the magnetic pseudo-pressure. This WG method is shown to yield globally divergence-free approximations of the velocity and magnetic fields. Existence and uniqueness results as well as optimal a priori error estimates for the discrete scheme are obtained. A convergent linearized iterative algorithm is presented. Numerical experiments are provided to verify the theoretical analysis.