Nested iterations for symmetric eigenproblems

The evaluation of the leftmost eigenspectrum of large sparse symmetric matrices is of great interest in many physical and engineering applications. Recently an efficient iterative method called DACG (deflation-accelerated conjugate gradient) based on the conjugate gradient optimization of successive deflated Rayleigh quotients, was developed. In the present paper the authors study the possibility of increasing the efficiency of this scheme by means of a nested iteration (NI) technique that acts on nested finite element grids and calculates an improved initial guess to be used by the DACG procedure. The new NI-DACG method is tested on two sample problems of large size, which make use of nested regular and irregular finite element grids. The results obtained in the calculation of the 40 leftmost eigenpairs for both problems show that the computational efficiency of DACG is increased by a factor of five for the regular mesh and two for the irregular one. They also show the limitation of simple (low-order) interpolation schemes for the intergrid transfer in improving the initial eigenvector estimates, and call for more research in this area