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

NONLINEAR SYSTEMS AND OPTIMIZATION
7.19 Inexact range-space Krylov solvers for linear systems arising
from inverse problems.
S. Gratton : INPT-IRIT, UNIVERSITY OF TOULOUSE AND ENSEEIHT, France ; J. Tshimanga : INPT-
ENSEEIHT AND IRIT, France ; Ph. L. Toint : FUNDP UNIVERSITY OF NAMUR, Belgium
The object of our work in [ALG26] is twofold. Firstly, range-space variants of standard Krylov iterative
solvers are introduced for unsymmetric and symmetric linear systems. These are characterized by possibly
much lower storage and computational costs than their full-space counterparts, which is crucial in data
assimilation applications and other inverse problems. Secondly, it is shown that the computational cost may
be further reduced by using inexact matrix-vector products : formal error bounds are derived on the size
of the residuals obtained under two different accuracy models, and it is shown why a model controlling
forward error on the product result is often preferable to one controlling backward error on the operator.
Numerical examples finally illustrate the developed concepts and methods.
7.20 An observation-space formulation of variational assimilation
using a restricted preconditioned conjugate gradient algorithm.
S. Gratton : INPT-IRIT, UNIVERSITY OF TOULOUSE AND ENSEEIHT, France ; J. Tshimanga : INPT-
ENSEEIHT AND IRIT, France
In [ALG53], we consider parameters estimation problems involving a set of m physical observations,
where an unknown vector of n parameters is defined as the solution of a nonlinear least-squares problem.
We assume that the problem is regularized by a quadratic penalty term. When solution techniques based
on successive linearization are considered, as in the incremental four-dimensional variational (4D-Var)
techniques for data assimilation, a sequence of linear systems with particular structure has to be solved.
We exhibit a subspace of dimension m that contains the solution of these linear systems, and derive a
variant of the conjugate gradient algorithm that is more efficient in terms of memory and computational
costs than its standard form, when m is smaller than n. The new algorithm, which we call the Restricted
Preconditioned Conjugate Gradient (RPCG), can be viewed as an alternative to the so-called Physical-space
Statistical Analysis System (PSAS) algorithm, which is another approach to solve the linear problem. In
addition, we show that the non-monotone and somehow chaotic behavior of PSAS algorithm when viewed
in the model space, experimentally reported by some authors, can be fully suppressed in RPCG.
Moreover, since preconditioning and reorthogonalization of residuals vectors are often used in practice to
accelerate convergence in high dimension data assimilation, we show how to reformulate these techniques
within subspaces of dimension m in RPCG. Numerical experiments are reported, on an idealized data
assimilation system based on the heat equation, that clearly show the effectiveness of our algorithm for
large scale problems.
7.21 The exact condition number of the truncated singular value
solution of a linear ill-posed problem.
S. Gratton : INPT-IRIT, UNIVERSITY OF TOULOUSE AND ENSEEIHT, France ; J. Tshimanga : INPT-
ENSEEIHT AND IRIT, France
The main result in [ALG54] is the investigation of an explicit expression of the condition number of the
truncated least squares solution of Ax = b. The result is derived using the notion of the Fréchet derivative
together with the product norm ‖[αA,βb]‖ F , with α,β > 0, for the data space and the 2-norm for the
30 Jan. 2010 – Dec. 2011