Adaptive finite element methods for optimization problems for second order linear elliptic partial differential equations subject to pointwise constraints on the $\mathcal{ℓ}^2$-norm of the gradient of the state are considered. In a weak duality setting, i.e. without assuming a constraint qualification such as the existence of a Slater point, residual based a posteriori error estimators are derived. To overcome the lack in constraint qualification on the continuous level, the weak Fenchel dual is utilized. Several numerical tests illustrate the performance of the proposed error estimators.

The main aim of this paper is to study the nonconforming linear triangular Crouzeix-Raviart type finite element approximation of planar linear elasticity problem with the pure displacement boundary value on anisotropic general triangular meshes satisfying the maximal angle condition and coordinate system condition. The optimal order error estimates of energy norm and $L^2$-norm are obtained, which are independent of lamé parameter $λ$. Numerical results are given to demonstrate the validity of our theoretical analysis.

In this paper, we establish a convergence result of the cyclic reduction (CR) algorithm for a class of weakly overdamped quadratic matrix polynomials without assumption that the partial multiplicities of the $n$th largest eigenvalue are all equal to 2. Our result can be regarded as a complement of that by Guo, Higham and Tisseur [SIAM J. Matrix Anal. Appl., 30 (2009), pp. 1593-1613]. The numerical example indicates that the convergence behavior of the CR algorithm is largely dictated by our theory.

This paper is concerned with the reconstruction of the radiation wave field in the exterior of a bounded two- or three-dimensional domain from the knowledge of Cauchy data on a part of the boundary of the aforementioned domain. It is described by the Cauchy problem for the Helmholtz equation. By using the Dirichlet-to-Neumann map, this problem is transformed into an operator equation with compact operator. We rigorously justify the asymptotic behaviors of singular values of the compact operator. Then a projection method with regularization is applied to solve the operator equation, and the convergence of the regularization method is discussed. Finally, several numerical examples are presented to illustrate the approach. The results demonstrate that the algorithm is effective.

The total variation (TV) minimization problem is widely studied in image restoration. Although many alternative methods have been proposed for its solution, the Newton method remains not usable for the primal formulation due to no convergence. A previous study by Chan, Zhou and Chan [15] considered a regularization parameter continuation idea to increase the domain of convergence of the Newton method with some success but no robust parameter selection schemes. In this paper, we consider a homotopy method for the same primal TV formulation and propose to use curve tracking to select the regularization parameter adaptively. It turns out that this idea helps to improve substantially the previous work in efficiently solving the TV Euler-Lagrange equation. The same idea is also considered for the two other methods as well as the deblurring problem, again with improvements obtained. Numerical experiments show that our new methods are robust and fast for image restoration, even for images with large noisy-to-signal ratio.

In this paper, we consider a class of the stochastic linear complementarity problems (SLCPs) with finitely many elements. A feasible semismooth damped Gauss-Newton algorithm for the SLCP is proposed. The global and local quadratic convergence of the proposed algorithm are obtained under suitable conditions. Some numerical results are reported in this paper, which confirm the good theoretical properties of the proposed algorithm.