## Section: New Results

### Nonlinear solvers and time decomposition methods

#### Autoadaptative limited memory Broyden method

Participants : Frédéric Guyomarch, Mohammed Ziani.

In the context of Mohammed Ziani's thesis we have started to study the convergence of non linear solvers on classical physical cases. We start with the well-known Newton's algorithm and use a few rules to have a more robust convergence (Armijo's criteria) and also to have faster convergence (forcing terms). Then we investigate Broyden's method. This method tries to "learn" the Jacobian matrix along the non linear iterations. But it also requires to store all the so called Broyden's directions in memory thus a classical heuristic of limited memory is often used. The goal is to shrink the memory needed for the Broyden's directions but this new parameter is quite hard to tune. Therefore, we propose a new method, called autoadaptative limited memory Broyden's method, which doesn't have the drawback of an extra parameter. The method starts with only one Broyden's direction and increases the size of its space when it detects a poor convergence. This need of extra storage is detected when the biggest singular value of the Broyden's vector we discarded goes above the nonlinear residual [39] , [18] .

#### Newton method and Proper Orthogonal Decomposition

Participant : Damien Tromeur-Dervout.

The acceleration of Newton methods by proper orthogonal decomposition of the time iterate solutions of a non stationary non linear problems have been proposed with the lid driven cavity problems in stream function biharmonic formulation [52] . The developed methodology satisfies the constraints of metacomputer/grid architecture with a client-server approach [16] . First results with modest acceleration on the IPARS code of M. Wheeler of the University of Texas at Austin can be found in [62] .

#### Reactive transport

Participants : Caroline de Dieuleveult, Jocelyne Erhel.

This work was done in the context of the MOMAS GdR ( 8.1.2 ), in collaboration with M. Kern (Estime Inria team at Rocquencourt), and in the context of the Andra contract ( 8.1.1 ), in collaboration with A. Dimier (Andra).

Reactive transport models are complex nonlinear PDEs, coupling the transport engine with the reaction operator. We consider here chemical reactions at equilibrium. We have compared the different solutions in the literature and we have proposed to use a PDAE (Partial Differential Algebraic Equations) framework, a method of lines and a DAE solver. In contrary to other approaches, we solve neither the nonlinear chemistry equations nor the transport equations, but the linearized chemistry equations, coupled with the transport equations. We have developed a prototype in Matlab, on a 1D domain, which shows the efficiency of the method [26] (paper in preparation). We have also analyzed the softwares MT3D and TRACES for transport equations and the software PHREEQC for chemistry equations, in order to use components of these libraries for our global method [59] .

#### Time and space domain decompositions for systems of PDE

Participant : Damien Tromeur-Dervout.

Three approaches have been developed concerning the solving of big ODEs/ADEs systems and time domain decomposition. The first one consists to enhance domain decomposition methods by introducing a definition of a coarse grid based on the relative tolerance of the time integrator. This allows to solve stiff problems as shown in [10] . The second approach, consists in an automatic preconditioned Schur domain decomposition to solve the linearized jacobian problem with no a priori knowledge on the structure of the Jacobian matrix [37] . The third approach combines a time domain decomposition with the spectral deferred correction in order to obtain a two level parallelism with pipelining the iterations to increase the accuracy on the solution [36] .

#### Parallel-in-time integration

Participant : Noha Makhoul.

The rescaling method has been designed for solving differential equations related to evolution problems by generating, through a change of variables, a sequence of time slices such that the time variable and the solution are restored to zero in the beginning of each slice, and the rescaled solution is controlled by a uniform criterion for ending slices.

This method has the advantage of leading to the solving of similar rescaled models and has been very efficient for solving evolution problems whose solution was explosive in finite time.

The sequential implementation of the rescaling method has shown the existence of a relation between the initial values of the successive time slices (ratio phenomenon). Approximating this relation allows the prediction of the initial values in order to start on a parallel-in-time integration through a prediction-correction scheme, with the double advantage of similarity between the rescaled models on the successive slices, on each of which there is absence of any singularity, allowing (due to the uniform criterion for ending slices) the solution to be sufficiently regular .

RaPTI Algorithm (Ratio-based Parallel Time Integration) starts with running the rescaling method sequentially on a few slices and computing the ratios of the successive initial values, in order to determine the way they are related and predict the following initial values necessary for the starting of the prediction-correction scheme.

Unlike the other parallel-in-time algorithms, RaPTI does not involve any sequential computation (except for the first slices) and generates time slices whose sizes vary with the behavior of the solution insuring a similarity between the slices.

This algorithm has first been tested on the reaction-diffusion equation, showing a fast convergence [44] , [43] .

#### Adaptation of the rescaling method to oscillatory evolution problems

Participant : Noha Makhoul.

The rescaling method has then been adapted to oscillatory evolution problems such those following from the population dynamics, namely the logistic predator-prey model of Lotka-Volterra (with 2 or 3 species) [65] . Such models lead to trajectories presenting an oscillating behavior which will approach a constant. Their ``orbits'' in a phase-portrait environment shows inwards spirals toward a stable equilibrium point.

After the sequential adaptation of the rescaling method (that showed again a ratio phenomenon), RaPTI algorithm has successfully been tested leading to a fast convergence.

Then, after a survey over the mathematical models for infectious diseases, the rescaling method has been applied to the endemic classical SIR model (whose behavior is very similar to the one of the Lotka-Volterra model) [73] . At present, we are investigating the age-structured SIR epidemic model with vertical transmission [74] , [69] .