Adaptive Preconditioned Variational Methods For Solving Self-Adjoint Partial Differential Equations

The implementation of the Preconditioned Conjugate Gradient method for the solution of large linear systems arising from the discretization of differential operators, requires the pre-determination of only one iteration parameter. The numerical determination of the optimal value of this constant parameter, involves the spectral bounds of some matrices and can be obtained in O ( N 2 ) sine function evaluations, where 1/ N is the discretization mesh size. It is shown that this parameter can be chosen in a stable manner in O (1) operations per iteration, if it is allowed to vary with the iteration index from information derived from the gradient parameters.

Keywords:

• Steepest Descent
• Conjugate Gradient
• Preconditioning
• Adaptive Methods