We present a straightforward and verified method of deciding whether the point x ∈ R, n > 1, such that ∇f(x) = 0, is the local minimizer, maximizer or just a saddle point of a real-valued function f . The method scales linearly with dimensionality of the problem and never produces false results.

Abstract The preconditioned iterative solution of large-scale saddle-point systems is great importance in numerous application areas, many them involving partial differential equations. Robustness with respect to certain problem parameters often a concern, and it can be addressed by identifying proper scalings preconditioner building blocks. In this paper, we consider new perspective finding ef...

We propose a class of regularized Hermitian and skew-Hermitian splitting methods for the solution of large, sparse linear systems in saddle-point form. These methods can be used as stationary iterative solvers or as preconditioners for Krylov subspace methods. We establish unconditional convergence of the stationary iterations and we examine the spectral properties of the corresponding precondi...

We examine block-diagonal preconditioners and efficient variants of indefinite preconditioners for block two-by-two generalized saddle-point problems. We consider the general, nonsymmetric, nonsingular case. In particular, the (1,2) block need not equal the transposed (2,1) block. Our preconditioners arise from computationally efficient splittings of the (1,1) block. We provide analyses for the...

The problem of sampling low lying, first-order saddle points on a high dimensional surface is discussed and a method presented for improving the sampling efficiency. The discussion is in the context of an energy surface for a system of atoms and thermally activated transitions in solids treated within the harmonic approximation to transition state theory. Given a local minimum as an initial sta...

Recently, some attractive primal-dual algorithms have been proposed for solving a saddle-point problem, with particular applications in the area of total variation (TV) image restoration. This paper focuses on the convergence analysis of existing primal-dual algorithms and shows that the involved parameters of those primal-dual algorithms (including the step sizes) can be significantly enlarged...

We study H(div) preconditioning for the saddle-point systems that arise in a stochastic Galerkin mixed formulation of the steady-state diffusion problem with random data. The key ingredient is a multigrid V-cycle for an H(div) operator with random weight function acting on a certain tensor product space of random fields with finite variance. We build on the ArnoldFalk-Winther multigrid algorith...