A generalized solid-state nudged elastic band (G-SSNEB) method is presented for determining reaction pathways of solid-solid transformations involving both atomic and unit-cell degrees of freedom. We combine atomic and cell degrees of freedom into a unified description of the crystal structure so that calculated reaction paths are insensitive to the choice of periodic cell. For the rock-salt to wurtzite transition in CdSe, we demonstrate that the method is robust for mechanisms dominated either by atomic motion or by unit-cell deformation; notably, the lowest-energy transition mechanism found by our G-SSNEB changes with cell size from a concerted transformation of the cell coordinates in small cells to a nucleation event in large cells. The method is efficient and can be applied to systems in which the force and stress tensor are calculated using density functional theory.

Many problems in science and engineering require the solution of a long sequence of slowly changing linear systems. We propose and analyze two methods that significantly reduce the total number of matrix-vector products required to solve all systems. We consider the general case where both the matrix and right-hand side change, and we make no assumptions regarding the change in the right-hand sides. Furthermore, we consider general nonsingular matrices, and we do not assume that all matrices are pairwise close or that the sequence of matrices converges to a particular matrix. Our methods work well under these general assumptions, and hence form a significant advancement with respect to related work in this area. We can reduce the cost of solving subsequent systems in the sequence by recycling selected subspaces generated for previous systems. We consider two approaches that allow for the continuous improvement of the recycled subspace at low cost. We consider both Hermitian and non-Hermitian problems, and we analyze our algorithms both theoretically and numerically to illustrate the effects of subspace recycling. We also demonstrate the effectiveness of our algorithms for a range of applications from computational mechanics, materials science, and computational physics.

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