| Incomplete factorization preconditioners. Incomplete factorization
preconditioners were the first widely used technique for solving challenging systems
in industrial applications such as semiconductor simulation, oil-reservoir simulations,
computational fluid dynamics simulations, etc. To make this technique more robust
and effective, many issues remain to be solved. |
| Approximate inverse preconditioners. This relatively new
technique motivated by the advent of high performance computers. Many promising results
have been reported in recent years. The potential of this technique for many challenging
applications (such as structural analysis, CFD…) are emerging. |
| Domain decomposition preconditioners. The domain decomposition
approach is another example of a technique which was promoted by new computer technology.
The coarse grain parallel computer is a perfect environment for this mathematical
approach. A large problem can be dissected into several small problems and solved
on different processors. The use of domain decomposition to construct an effective
parallel preconditioner is a popular choice for many large scale applications. Many
theoretical and practical problems of this approach are undertaken by many top researchers
in the world. |
| Multi-level preconditoners. Multi-level methods have demonstrated
their superiority in efficiency as a solver for large problems. However, the robustness
and/or the sophisticated implementation of this technique inhibits the use of this
technique. |
| Finite element preconditioners. This preconditioner is derived
from finite element methods and is used often in stress analysis and some commercial
software. |
| Computational Fluid Dynamics applications (CFD). CFD applications
have a long history of using preconditioning techniquesin their simulation process.
However, the need to solve much larger and ill-conditioned sparse matrices arising
in the new generation of the aircrafts design process makes effective preconditioning
techniques more crucial. |
| Stress analysis applications. Due to the difficulties inherited
inthis area, many state-of-the-art preconditioning techniques such as hierarchical
basis preconditioners, approximate inverse preconditioners, element by element preconditioners,
etc. have been tried on these applications. The huge size of the matrices arising
in these applications also makes the search for an effective parallel solution technique
a very active research area. |
| Computational finance applications. Solution of multi-asset
option pricing problems gives rise to a multi-dimensional time dependent PDE. The
resulting matrix problems are not M-matrices and are often difficult to solve. |
| Preconditioning differential-convolution operators
arising in image processing. In regularized least squares approaches to image reconstruction,
one often has to precondition operators consisting of a sum of a differential operator
arising from the regularization and a convolution operator arising from the blurring.
These operators are particularly challenging to precondition due to the ill-conditioning
of both types of operators as well as their different spectral properties. Ideas
from fast transform preconditioners, operator-splitting, and multilevel methods have
been developed. |
| Multiphase subsurface flow applications. Due to the complicated
structure of the subsurface and several multi-phase components in one complex system,
the resulting matrices in this application are usually very ill-conditioned. The
significant impact of these models on the economy and environment makes this application
very important. |
| Petroleum industry applications. There is a long history
of using preconditioning to solve their very challenging problems. Their experiences
and findings are very useful to the research community in this area. |
| Semiconductor device applications. The small size, sophisticated
structure of the semiconductor devices, and the stiff changes of the physics cause
the resulting matrices to be very ill-conditioned. The requirements for three dimensional
models including quantum effects even further complicate the solution process. |
