Matroidal approaches to rough sets via closure operators

AbstractThis paper studies rough sets from the operator-oriented view by matroidal approaches. We firstly investigate some kinds of closure operators and conclude that the Pawlak upper approximation operator is just a topological and matroidal closure operator. Then we characterize the Pawlak upper approximation operator in terms of the closure operator in Pawlak matroids, which are first defined in this paper, and are generalized to fundamental matroids when partitions are generalized to coverings. A new covering-based rough set model is then proposed based on fundamental matroids and properties of this model are studied. Lastly, we refer to the abstract approximation space, whose original definition is modified to get a one-to-one correspondence between closure systems (operators) and concrete models of abstract approximation spaces. We finally examine the relations of four kinds of abstract approximation spaces, which correspond exactly to the relations of closure systems

Conjugate gradient-boundary element solution for distributed elliptic optimal control problems

AbstractAn optimality system of equations for the optimal control problem governed by Helmholtz-type equations is derived. By the associated first-order necessary optimality condition, we obtain the conjugate gradient method (CGM) in the continuous case. Introducing the sequence of higher-order fundamental solutions, we propose an iterative algorithm based on the conjugate gradient-boundary element method using the multiple reciprocity method (CGM+MRBEM) for solving the discrete control input. This algorithm has an advantage over that of the existing literatures because the main attribute (the reduced dimensionality) of the boundary element method is fully utilized. Finally, the local error estimates for this scheme are obtained, and a test problem is given to illustrate the efficiency of the proposed method

Generalized set-valued variational inclusions in q-uniformly smooth Banach spaces

AbstractIn this paper, a class of generalized set-valued variational inclusions in Banach spaces are introduced and studied, which include many variational inclusions studied by others in recent years. By using some new and innovative techniques, several existence theorems for the generalized set-valued variational inclusions in q-uniformly smooth Banach spaces are established, and some perturbed iterative algorithms for solving this kind of set-valued variational inclusions are suggested and analyzed. Our results improve and generalize many known algorithms and results