The Flexible Preconditioned Conjugate Gradient Method
In numerically challenging applications, sophisticated preconditioners are used, which may lead to variable preconditioning, changing between iterations. Even if the preconditioner is symmetric positive-definite on every iteration, the fact that it may change makes the arguments above invalid, and in practical tests leads to a significant slow down of the convergence of the algorithm presented above. Using the Polak–Ribière formula
instead of the Fletcher–Reeves formula
may dramatically improve the convergence in this case. This version of the preconditioned conjugate gradient method can be called flexible, as it allows for variable preconditioning. The implementation of the flexible version requires storing an extra vector. For a fixed preconditioner, so both formulas for are equivalent in exact arithmetic, i.e., without the round-off error.
The mathematical explanation of the better convergence behavior of the method with the Polak–Ribière formula is that the method is locally optimal in this case, in particular, it converges not slower than the locally optimal steepest descent method.
Read more about this topic: Conjugate Gradient Method
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