A Projection Strategy for Improving the Preconditioner in the LOBPCG

The computational methods for solving the generalized eigenvalue problems of real symmetric matrices are crucial in fields such as structural dynamics analysis. As the scale of the problems to be solved increases, higher efficiency in solving eigenvalue problems is demanded. The LOBPCG (locally optimal block preconditioned conjugate gradient) method is a promising iterative algorithm suitable for solving large-scale eigenvalue problems, capable of quickly solving multiple extreme eigenpairs. In the LOBPCG, the preconditioner can be executed by calling the truncated PCG to approximately solve the ‘inner’ linear system. However, the convergence rate of the LOBPCG is highly sensitive to the quality of its preconditioner. Only when paired with an appropriate preconditioner, the LOBPCG is notably efficient in minimizing the iterations needed for convergence. This paper proposed a projection strategy which can enhance the quality of the preconditioner, thus improving the overall efficiency and stability of the LOBPCG. The projection strategy first utilizes intermediate vectors from the PCG iterations to construct search subspaces and constraint subspaces for oblique projection, and then executes the oblique projection in truncated PCG when solving inner linear system. This oblique projection technique can find a more accurate approximate solution which minimizes the 2-norm residuals in the search subspace without significantly increasing computational cost, thereby improving the quality of the preconditioner, thus accelerating convergence of the LOBPCG.Numerical experiments show that the projection strategy can improve the LOBPCG algorithm significantly in terms of efficiency and stability.