Research lines Probabilistic Domain Decomposition
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| Probabilistic Domain Decomposition |
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Development of new efficient algorithms
for High Performance Supercomputers based on probabilistic domain decomposition methods
The probabilistically induced domain decomposition method (PDD) was proposed in 2005 initially for numerically solving boundary-value elliptic problems. From then the PDD method has been successfully extended for solving a wide range of problems described through partial differential equations (PDEs).
The class of equations for which method has been applied include linear elliptic and parabolic equations, general semilinear parabolic equations, linear purely advection-dominated equations, the non-linear Vlasov-Poison system of equations governing plasma physics, and the Telegraph equation. Applications include all kind of diffusion and advection problems, finance (Black-Scholes and similar equations), plasma physics, and electrical transmission lines. The result is a domain decomposition technique based on a probabilistic method that is suited for massively parallel computers, enjoying full scalability and fault tolerance.
One of the most relevant numerical methods for solving PDEs is that of Domain
Decomposition (DD), which is well suited to handle multiscale and
multiphysics. In fact, accomplishing a domain decomposition is
considered one of the most successful strategies to solve PDEs problems
exploiting parallel architectures, since it is based on splitting the
original domain into as many subdomains as the available processors.
Then, a local solver is used, independently, on each subdomain. The
overall solution is finally reconstructed combining all such partial
results. However, due to the global nature of the typical PDEs problems,
realizing the aforementioned splitting requires interfacial
communication, across the internal (artificial) boundaries generated by
the splitting procedure. In practice, such a interfacial
communication entails an communication overhead, the heavier the more
processors are being used. Consequently, it may be possible that
no advantage could be obtained using more and more processors, hence
scalability itself is doubtful.
In the classical domain decomposition splitting the domain into any number of subdomains
to be solved in parallel induces an interfacial communication. A new type of domain decomposition based on probabilistic methods The PDD method is a numerical algorithm for solving PDEs, capable of exploiting the best features of two well known strategies: domain decomposition method, and the Monte Carlo method. A domain decomposition approach has been used to split the given space-time domain into as many subdomains as available processors. The solutions on the interfaces separating the subdomains, being unknown, are computed by interpolating on the nodal points where the solution is obtained probabilistically. Diagram showing the probabilistic domain decomposition (PDD) at work.
It is shown schematically how a 2D bounday value problem for a parabolic partial differential equation can be solved in practice using a PDD method. The probabilistic computation for nonlinear problems consists of evaluating averages on suitably-generated branching stochastic processes, which play a role similar to that of random paths in linear problems. In contrast to the classical deterministic method, the probabilistic approach allows to compute the solution at single points internal to the domain, without the need for first generating a computational grid and solving the full problem. This fact is of paramount importance because, once the solution on the interfaces has been computed, the tasks of evaluating the solutions inside each subdomains turn out to be totally independent of one another, and thus can be assigned to an arbitrary number of processors without any intercommunication overhead. The probabilistic domain decomposition (PDD) allows to split the domain into independent subdomains
once the solution at the interfaces was previously computed. Selected publications
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