TABLE OF CONTENTS The purpose of the LATIN method is to build an algorithm to compute corrections of an approximate state thanks to stages, both global and linear, defined over the entire time interval. The residuals we want to define, must characterise what could be the mean value over ]0, t f] of the corrections by using the Finite Element subspaces 
and T. But, to avoid expensive computations we do not want to compute these corrections by using the Finite Element subspaces. In order to define the residuals, we present the LATIN algorithm as if we used the Finite Element subspaces. Hence we modify the formulation of the linear correction stage to employ the reduced-order model for the computation of the corrections.
The Finite Element problem presented in section 2 is both global and non-linear. But, we chose a formulation such that the global equations are linear and the nonlinear equations are local equations. The LATIN method is based on three principles : P1, P2 and P3 [LAD 85] [LAD 96]:
P1 : in order to split the difficulties (global and non-linear equations), two groups of equations are created from the reference problem : a group of local equations and a group of linear equations;
P2 : the algorithm is an iterative procedure that provides a solution for each group of equations at each iteration, these solutions are defined over the structure and over the entire time...
