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

7 Nonlinear Systems and Optimization
7.1 A Contribution to the Conditioning of the Total Least-Squares
Problem.
M. Baboulin : UNIVERSITY OF PARIS-SUD AND INRIA, France ; S. Gratton : INPT-IRIT, UNIVERSITY
OF TOULOUSE AND ENSEEIHT, France
In [ALG10], we derive closed formulas for the condition number of a linear function of the total leastsquares
solution. Given an overdetermined linear system Ax = b, we show that this condition number can be
computed using the singular values and the right singular vectors of [A,b] and A. We also provide an upper
bound that requires the computation of the largest and the smallest singular value of [A,b] and the smallest
singular value of A. In numerical examples, we compare these values and the resulting forward error
bounds with the error estimates given by Van Huffel and Vandewalle [The Total Least Squares Problem :
Computational Aspects and Analysis, Frontiers Appl. Math. 9, SIAM, Philadelphia, 1991], and we show
the limitation of the first order approach.
7.2 A retrospective trust-region method for unconstrained
optimization.
F. Bastin : UNIVERSITY OF MONTRÉAL, Canada ; V. Malmedy : FUNDP UNIVERSITY OF NAMUR,
Belgium ; M. Mouffe : CERFACS, France ; Ph. L. Toint : FUNDP UNIVERSITY OF NAMUR, Belgium ;
D. Tomanos : FUNDP UNIVERSITY OF NAMUR, Belgium
In [ALG11], we introduce a new trust-region method for unconstrained optimization where the radius
update is computed using the model information at the current iterate rather than at the preceding one.
The update is then performed according to how well the current model retrospectively predicts the value of
the objective function at last iterate. Global convergence to first- and second-order critical points is proved
under classical assumptions and preliminary numerical experiments on CUTEr problems indicate that the
new method is very competitive.
7.3 Quasi-Newton methods for solving a two-dimensional acoustic
waveform inversion.
M. Biari : ENSEEIHT AND CERFACS, France
We are interested in solving acoustic waveform inversion problem with the help of Quasi-Newton methods
with application to geophysics. The theory is reviewed and we focus on the efficient solution of the adjoint
problem with iterative methods. Two-grid methods are investigated as preconditioner of Krylov subspace
methods, closely following strategies developed for the direct problem in X. Pinel’s PhD thesis [ALG66].
Two-dimensional academic applications are considered. The master project has been supervised by S.
Gratton and X. Vasseur.
24 Jan. 2010 – Dec. 2011