Over the last two decades, classical Schwarz methods have been extended to systems of hyperbolic partial differential equations, using characteristic transmission conditions, and it has been observed that the classical Schwarz method can be convergent even without overlap in certain cases. This is in strong contrast to the behavior of classical Schwarz methods applied to elliptic problems, for ...

We discuss some of the geometric properties, such as foliated Schwarz symmetry, monotonicity along axial and affine-radial directions, first eigenfunctions a Zaremba problem for Laplace operator on annular domains. Together with shape calculus, these fine properties help us to prove that eigenvalue is strictly decreasing inner ball moves towards boundary outer ball.

During the last two decades many domain decomposition algorithms have been constructed and lot of techniques have been developed to prove the convergence of the algorithms at the continuous level. Among the techniques used to prove the convergence of classical Schwarz algorithms, the first technique is the maximum principle used by Schwarz. Adopting this technique M. Gander and H. Zhao proved a...

We propose an iterative algorithm for high-dimensional sparse grid regression with penalty. The algorithm is of additive Schwarz type and is thus parallel. We show that the convergence of the algorithm is better than additive Schwarz and examples demonstrate that convergence is between that of additive Schwarz and multiplicative Schwarz procedures. Similarly, the method shows improved performan...

We propose a modiied approach to the abstract convergence theory of the additive and multiplicative Schwarz iterative methods associated with a subspace splitting for solving symmetric positive deenit linear problems. In particular, we obtain a satisfactory bound for the convergence rate of the appropriately scaled multiplicative Schwarz method directly in terms of the condition number of the c...