This paper presents a collocation method with an iterative linear system solver to compute periodic solutions of a system of autonomous delay differential equations (DDEs). We show that the linearized collocation system is equivalent to a discretization of the linearized periodic boundary value problem (BVP). This linear BVP is solved using the Newton-Picard single shooting method ([Int. J. Bif...

Elliptic problems with oscillating coefficients can be approximated up to arbitrary accuracy by using sufficiently fine meshes, i.e., by resolving the fine scale. Well-known multiscale finite elements [5, 9] can be regarded as direct numerical homogenization methods in the sense that they provide approximations of the corresponding (unfeasibly) large linear systems by much smaller systems while...

Direct methods for solving sparse systems of linear equations are fast and robust, but can consume an impractical amount of memory, particularly for large three-dimensional problems. Preconditioned iterative solvers have the potential to solve very large systems with a fraction of the memory used by direct methods. The diversity of preconditioners makes it difficult to analyze them in a unified...

Based on Banach's principle, we formally obtain possible choices for an error vector in the direct inversion in the iterative subspace (DIIS) method. These choices not only include all previously proposed error vectors, but also a new type of error vector which is computationally efficient and applicable to much wider range of problems. The error vector analysis also reveals a strong connection...

Article history: Received 31 January 2011 Received in revised form 17 August 2011 Accepted 6 November 2011 Available online 16 November 2011

The Modified Direct Method (MDM) is an iterative mesh smoothing method for smoothing planar and surface meshes, which is developed from the non-iterative smoothing method originated by Balendran [1]. When smooth planar meshes, the performance of the MDM is effectively identical to that of Laplacian smoothing, for triangular and quadrilateral meshes; however, the MDM outperforms Laplacian smooth...

We extend the application of the direct inversion in the iterative subspace ~DIIS! technique to the ridge method for finding transition states ~TS!. The latter is not a quasi-Newton-type algorithm, which is the only class of geometry optimization methods that has been combined with DIIS. With this new combination, we obtain a factor of two speedup due to DIIS, similar to the DIIS-related speedu...

We present an algorithm that is a new combination of the direct inversion in the iterative subspace ~DIIS! and the generalized valence bond ~GVB! methods. The proposed algorithm is based on applying the DIIS directly to the orbitals updated via the GVB scheme as opposed to the conventional approach of applying DIIS to a series of composite Fock matrices ~CFMs!. The new method results in GVB con...

We propose and study a block-iterative projections method for solving linear equations and/or inequalities. The method allows diagonal component-wise relaxation in conjunction with orthogonal projections onto the individual hyperplanes of the system, and is thus called diagonally-relaxed orthogonal projections (DROP). Diagonal relaxation has proven useful in accelerating the initial convergence...

The main general reference for this and the next two lectures is a recent book [6] and its preliminary version [5] which is freely available on the web. 1 Why iterative methods? As we already know, there are two major classes of methods to solve linear systems: direct methods (like LU factorization, Cholesky factorization, etc.) and iterative methods. We underline the following properties of th...