This paper is concerned with a new approach for preconditioning large sparse least squares problems. Based on the idea of the approximate inverse preconditioner, which was originally developed for square matrices, we construct a generalized approximate inverse (GAINV) $M$ which approximately minimizes $\|I-MA\|_{\mathrm{F}}$ or $\|I-AM\|_{\mathrm{F}}$. Then, we also discuss the theoretical issues such as the equivalence between the original least squares problem and the preconditioned problem. Finally, numerical experiments on problems from Matrix Market collection and random matrices show that although the preconditioning is expensive, it pays off in certain cases.

We propose four conservative schemes for the regularized long-wave (RLW) equation. The RLW equation has three invariants: mass, momentum, and energy. Our schemes are designed by using the discrete variational derivative method to inherit appropriate conservation properties from the equation. Two of our schemes conserve mass and momentum, while the other two schemes conserve mass and energy. With one of our schemes, we prove the numerical solution stability, the existence of the solutions, and the convergence of the solutions. Through some numerical computation examples, we demonstrate the efficiency and robustness of our schemes.

In this paper the sinc-Galerkin method, as well as the sinc-collocation method, based on the double exponential transformation (DE transformation) for singularly perturbed boundary value problems of second order ordinary differential equation is considered. A large merit of the present method exists in that we can apply the standard sinc method with only a small care for perturbation parameter. Through several numerical experiments we confirmed higher efficiency of the present method than that of other methods, e.g., sinc method based on the single exponential (SE) transformation, as the number of sampling points increases.

An explicit scheme for time-dependent convection-diffusion problems is presented. It is shown that convenient bounds for the time step value ensure $L^{\infty}$ stability, in both space and time, for piecewise linear finite element discretizations in any space dimension. Convergence results in the same sense are also demonstrated under certain conditions. Numerical results certify the good performance of the scheme.

In this aricle, the author considers mathematical formulation and numerical solutions of distributed and Neumann boundary optimal control problems associated with the stationary Bénard problem. The solution of the optimal control problem is obtained by controlling of the source term of the equations and/or Neumann boundary conditions. Then the author considers the approximation, by finite element methods, of the optimality system and derive optimal error estimates. The convergence of a simple gradient method is proved and some numerical results are given.