The use of inner preconditioned conjugate gradient iterations in large sparse nonlinear optimization problems c I. E. Kaporin Computing Center of Russian Academy of Sciences, Vavilova 40, Moscow 119991, Russia E-mail:
[email protected],
[email protected]
1
Introduction
In our presentation, we consider the problem of the (sub)optimum termination of inner linear Preconditioned Conjugate Gradient (PCG) iterations used within Inexact Newton nonlinear solvers [2, 4, 3]. In general, we show that for certain solution error measure , where is the number of nonlinear (outer) iteration, one has
(1)
where is a certain characterization of the problem nonlinearity ("!# when the problem is linear), and $ &% '( is a special error measure for the solution of some inner linear equation using iterations of the PCG method (this linear error is zero whenever % !) ). We present a simple practical rule for choosing which does not assume the knowledge of and makes it possible to avoid making redundant PCG iterations within each nonlinear step. The general outline is as follows. Let the nonlinear convergence is measured by +*
-, +./
0
,213 54
Then (1) yields the following sufficient condition for the nonlinear iterations to be converged: * 6 7 89 .
6 ?>A@CBED 6 GF 8 : 8 V@5BED 6 WF 8 : 8 SRTK ,
6 U>V@5B D 6 F XZY\ []
HKJLI 8 _Q 8 8 `a , 4 .W^ 8 ^?* 6
Hence, the stopping criterion can now be formulated in terms of choosing the inner iteration numbers 8 providing for a reasonably small value of each ratio b
c! HKJdI _Qe 8 4
Since the quantity 8 can be readily evaluated at any -th inner iteration, b cf.[3], one can stop at d !f 8 where the first local minimum of 8 is attained.
2
Nonlinear Least Squares Problem
Let g be a differentiable mapping g h5ikj\lmiknE/+oqpsrt/
and let the nonlinear least squares problem uv
B
