Partial Differential Equations - Modelling and ... - ResearchGate

154 G. Bencteux et al.
and set
Z k+1
1 = ˜Z k 1 + α ∗ ˜Zk 2 √1+(α∗
) 2 ,
Zk+1 2 = −α∗ ˜Zk 1 + ˜Z 2
k √
1+(α∗ ) . (12)
2
This algorithm operates at two levels: a fine level where two problems of
dimension N b are solved (rather than one problem of dimension 2N b ); a coarse
level where a problem of dimension 2 is solved.
The local step monotonically reduces the objective function; however, it
may not converge to the global optimum. The technical problem is that the
Lagrange multipliers associated with the constraint 〈Z 1 ,Z 2 〉 = 0 may converge
to different values in the two subproblems associated with the local step. The
global step again reduces the value of the objective function since ˜Z 1
k and
˜Z 2 k are feasible in the global step. The combined algorithm (local step +
global step), therefore, makes the objective function monotonically decrease.
The simple case H 1 = H 2 is interesting to consider. First, if the algorithm
is initialized with Z2
0 = 0 in the first line of (9), it is easily seen that the
local step is sufficient to converge to the global minimizer, in one single step.
Second, it has been proved in [Bar05] that for a more general initial guess and
under some assumption on the eigenvalues of the matrix H 1 , this algorithm
globally converges to an optimal solution of (8). Ongoing work [BCHL] aims
at generalizing the above proof when the additional assumption on eigenvalues
is omitted. The analysis of the convergence in the case H 1 ≠ H 2 is a longer
term goal.
3 The Multilevel Domain Decomposition (MDD)
Algorithm
We define, for all p-tuple (C i ) 1≤i≤p ,
) p∑ ( )
E
((C i ) 1≤i≤p = Tr H i C i Ci
t , (13)
i=1
and set by convention
U 0 = U p =0. (14)
It has been shown in [BCHL07] that updating the block sizes m i along the
iterations is crucial to make the domain decomposition algorithm converge
toward a good approximation of the solution to (5). It is, however, observed
in practice that after a few iterations, the block sizes have converged (they do
not vary in the course of the following iterations). This is why, for the sake of
clarity, we have chosen to present here a simplified version of the algorithm
where block sizes are held constant along the iterations. For a description of
the complete algorithm with variable block sizes, we refer to [BCHL07].
At iteration k, we have at hand a set of matrices (Ci k) 1≤i≤p such that
Ci k ∈M n,mi (R), [Ci k]t Ci k = I mi ,[Ci k]t TCi+1 k = 0. We now explain how to
compute the new iterate (C k+1
i ) 1≤i≤p .