Volume 21, Issue 4

- This paper presents the multiharmonic analysis of a distributed parabolic optimal control problem in a time-periodic setting. We prove the existence and uniqueness of the solution of some weak space-time variational formulation for the parabolic time-periodic boundary value problem appearing in the constraints for the optimal control problem. Since the cost functional is quadratic, the optimal control problem is uniquely solvable as well. In order to solve the optimal control problem numerically, we state its optimality system and discretize it by the multiharmonic finite element method leading to a system of linear algebraic equations which decouples into smaller systems. We construct preconditioners for these systems which yield robust convergence rates and optimal complexity for the preconditioned minimal residual method. All systems can be solved totally in parallel. Furthermore, we present a complete analysis for the error introduced by the multiharmonic finite element discretization as well as some numerical results confirming our theoretical findings.

- We propose a new smoothing function in this paper by smoothing the symmetric perturbed Fischer-Burmeister function. Based on this new function, we present a new noninterior continuation method for solving the second-order cone programming. We adopt a variant merit function. Our algorithm needs to solve only one system of linear equations and to perform only one line search at each iteration. It can start from an arbitrary point and does not require the iteration points to be in the set of strictly feasible solutions. We prove the global and local quadratic convergence of the algorithm under suitable assumptions. Numerical results indicate that our algorithm performs well.

- Principal angles between subspaces (PABS) (also called canonical angles) serve as a classical tool in mathematics, statistics, and applications, e.g., data mining. Traditionally, PABS are introduced via their cosines. The cosines and sines of PABS are commonly defined using the singular value decomposition. We utilize the same idea for the tangents, i.e., explicitly construct matrices, such that their singular values are equal to the tangents of PABS, using several approaches: orthonormal and non-orthonormal bases for subspaces, as well as projectors. Such a construction has applications, e.g., in analysis of convergence of subspace iterations for eigenvalue problems.