CERFACS CERFACS Scientific Activity Report Jan. 2010 – Dec. 2011

NONLINEAR SYSTEMS AND OPTIMIZATION
parameters is identified on these problems, yielding a satisfactory compromise between reliability and
efficiency. The resulting default algorithm is then compared with alternative optimization techniques such
as mesh refinement and direct solution of the fine-level problem. It is also shown that its behaviour is similar
to that of multigrid algorithms for linear systems.
7.13 Stopping rules and backward error analysis for boundconstrained
optimization.
S. Gratton : INPT-IRIT, UNIVERSITY OF TOULOUSE AND ENSEEIHT, France ;
M. Mouffe : CERFACS, France ; Ph. L. Toint : FUNDP UNIVERSITY OF NAMUR, Belgium
In [ALG23], termination criteria for the iterative solution of bound-constrained optimization problems are
examined in the light of backward error analysis. It is shown that the problem of determining a suitable
perturbation on the problem’s data corresponding to the definition of the backward error is analytically
solvable under mild assumptions. Moreover, a link between existing termination criteria and this solution is
clarified, indicating that some standard measures of criticality may be interpreted in the sense of backward
error analysis. The backward error problem is finally considered from the multicriteria optimization point
of view and some numerical illustration is provided.
7.14 On a class of limited memory preconditioners for large scale
linear systems with multiple right-hand sides.
S. Gratton : INPT-IRIT, UNIVERSITY OF TOULOUSE AND ENSEEIHT, France ; A. Sartenaer : FUNDP
UNIVERSITY OF NAMUR, Belgium ; J. Tshimanga : INPT-ENSEEIHT AND IRIT, France
This work studies a class of limited memory preconditioners (LMPs) for solving linear (positive-definite)
systems of equations with multiple right-hand sides. We propose a class of (LMPs), whose construction
requires a small number of linearly independent vectors. After exploring the theoretical properties of the
preconditioners, we focus on three particular members : spectral-LMP, quasi-Newton-LMP, and Ritz-LMP.
We show that the first two are well known, while the third is new. Numerical tests indicate that the Ritz-
LMP is efficient on a real-life nonlinear optimization problem arising in a data assimilation system for
oceanography. For more information, see [ALG25].
7.15 Approximate invariant subspaces and quasi-Newton
optimization methods.
S. Gratton : INPT-IRIT, France ; Ph. L. Toint : FUNDP UNIVERSITY OF NAMUR, Belgium
In [ALG21], new approximate secant equations are shown to result from the knowledge of (problem
dependent) invariant subspace information, which in turn suggests improvements in quasi-Newton
methods for unconstrained minimization. A new limited-memory Broyden-Fletcher-Goldfarb-Shanno using
approximate secant equations is then derived and its encouraging behaviour illustrated on a small collection
of multilevel optimization examples. The smoothing properties of this algorithm are considered next, and
automatic generation of approximate eigenvalue information demonstrated. The use of this information for
improving algorithmic performance is finally investigated on the same multilevel examples.
28 Jan. 2010 – Dec. 2011