  This paper presents a weakly intrusive strategy for computing a low-rank
approximation of the solution of a system of nonlinear parameter-dependent
equations. The proposed strategy relies on a Newton-like iterative solver which
only requires evaluations of the residual of the parameter-dependent equation
and of a preconditioner (such as the differential of the residual) for
instances of the parameters independently. The algorithm provides an
approximation of the set of solutions associated with a possibly large number
of instances of the parameters, with a computational complexity which can be
orders of magnitude lower than when using the same Newton-like solver for all
instances of the parameters. The reduction of complexity requires efficient
strategies for obtaining low-rank approximations of the residual, of the
preconditioner, and of the increment at each iteration of the algorithm. For
the approximation of the residual and the preconditioner, weakly intrusive
variants of the empirical interpolation method are introduced, which require
evaluations of entries of the residual and the preconditioner. Then, an
approximation of the increment is obtained by using a greedy algorithm for
low-rank approximation, and a low-rank approximation of the iterate is finally
obtained by using a truncated singular value decomposition. When the
preconditioner is the differential of the residual, the proposed algorithm is
interpreted as an inexact Newton solver for which a detailed convergence
analysis is provided. Numerical examples illustrate the efficiency of the
method.
