###### Domain Decomposition Methods – Parallel Computing

The theory of domain decomposition methods:

Applications of domain decomposition methods:

Stochastic differential equations and financial maths

Kinetic equations

Computational Fluid dynamics and primitive equation

Control theory

 A numerical test of domain decomposition methods for the heat equation in 2 dimension (see [1] for more simulations and details). A numerical test of domain decomposition methods for the Kolmogorov-Fokker-Planck equation (see [11] for more simulations and details).

1.  The theory of domain decomposition methods:

Domain decomposition methods are procedures to parallelize and solve partial differential equations numerically, where each iteration involves the solutions of the original equations on smaller subdomains. It was originally proposed by H. A. Schwarz in 1870 and was then used and extended by P. L. Lions as parallel algorithms in solving partial differential equations. Together with the development of domain decomposition methods, a theory of convergence for the methods is really needed and many efforts have been made in this direction; however, the problem still remains open for both optimized and classical methods with general nonlinearities, domains and dimensions. In my work, I have developed a new machinery to study the convergence of domain decomposition methods.

The tool was originated in [9] where I have studied the classical domain decomposition algorithms for a semilinear heat equation in n-dimensions, where the nonlinear term leads to explosion of solutions in finite time. I then prove that there exists a common existence time for all iterations.

I have also succeeded extending this technique to prove a convergence in the continuous sense of optimized Schwarz methods for linear parabolic equations in n-dimensions (see [5]).

In [6] and [10], I have completely developed the techniques and presented convergence proofs of overlapping classical and optimized Schwarz methods for elliptic and parabolic semilinear equations, in the general forms, for general multi-subdomains. The techniques seem to be very powerful and have been applied successfully to various problems: kinetic equations, stochastic differential equations, primitive equation and the problem of controlling the wave equation.

I have also proved that Schwarz methods converge when we increase the overlapping size (see [8]).

In the work [1] (195 pages), I have computed the optimized parameters of optimized Schwarz methods with Robin and Ventcell transmission condition and checked them numerically.

2.  Applications of domain decomposition methods:

I have used domain decomposition methods to parallelize the numerical resolution of several problems in various research areas.

a.  Stochastic differential equations and financial maths

The theory of forward-backward stochastic differential equations (FBSDEs) has been a very active field of research since the first work of Pardoux and Peng and Antonelli came out in the early 1990s. These equations appear in a large number of application fields in stochastic control and financial mathematics. Motivated by the idea of imposing paralleling computing on solving stochastic differential equations, I have introduced a new domain decomposition scheme to solve FBSDEs parallely. I have reconstructed the four step scheme by Ma, Protter and Yong with some different conditions and then associate it with the idea of domain decomposition methods. To my knowledge, this is the first domain decomposition scheme for stochastic differential equations (see [4]).

b.  Kinetic equations

In [11], we have designed two parallel schemes, based on Schwarz Waveform Relaxation procedures, for the numerical solution of the Kolmogorov-Fokker-Planck equation. The Kolmogorov-Fokker-Planck operator is hypoelliptic and it has properties of both hyperbolic and parabolic operators. Schwarz Waveform Relaxation procedures decompose the spatio-temporal computational domain into subdomains and solve (in parallel) subproblems, that are coupled through suitable conditions at the interfaces to recover the solution of the global problem. We consider coupling conditions of both Dirichlet and Robin types. We have proved well-posedeness of the schemes subproblems and convergence for the proposed algorithms. We have corroborated our findings with some numerical tests. The theoretical and numerical results in this paper show that the equation is more parabolic than hyperbolic, in the regime of domain decomposition.

c.  Computational fluid dynamics and primitive equation

Describing large scale dynamics of oceans and atmosphere, the primitive equation is derived from the Navier-Stokes equation, with rotation, coupled to thermodynamics and salinity diffusion-transport equations, which account for the buoyancy forces and stratification effects under the Boussinesq approximation. While studying the equation numerically, we face the problem of global or regional simulations of the ocean evolution. Due to the large size of global simulations and the interaction between global and regional models, we need efficient parallel algorithms. In [12], we have studied overlapping optimized Schwarz domain decomposition methods to parallelize the numerical resolution of the equation in both theoretical and numerical aspects. This is a part of my PhD thesis.

d.  Control theory

In the note [7], I have studied overlapping domain decomposition methods for optimal control systems governed by wave equations.