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 KolmogorovFokkerPlanck 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
ndimensions, 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
ndimensions (see [5]).
In [6] and [11], 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
multisubdomains. 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
forwardbackward 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 [10], we have
designed two parallel schemes, based on Schwarz Waveform Relaxation procedures,
for the numerical solution of the KolmogorovFokkerPlanck equation. The
KolmogorovFokkerPlanck operator is hypoelliptic and
it has properties of both hyperbolic and parabolic operators. Schwarz Waveform
Relaxation procedures decompose the spatiotemporal
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 wellposedeness 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 NavierStokes equation, with rotation, coupled to
thermodynamics and salinity diffusiontransport 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 my PhD thesis, we have studied
overlapping optimized Schwarz domain decomposition methods to parallelize the
numerical resolution of the equation in both theoretical and numerical aspects.
d. Control theory
In the note [7], I
have studied overlapping domain decomposition methods for optimal control
systems governed by wave equations.