AMS Colloquium: “Fast Solvers” and partial differential equations

As applications of partial differential equations have advanced in recent times, so too have the size and complexity of the equations involved. This has led to a growing demand for solving techniques that can be implemented to quickly and efficiently find solutions and solution sets to these equations. It is no surprise, then, that this growing demand has led to a growing field of finding these “fast solvers”, and some recent work in this field was the topic of this week’s installment of the AMS department’s colloquia series.

Presented by Dr. Gunnar Martinsson of the University of Colorado at Boulder, some light was shed onto the development and implementation of these methods, particularly the “fast direct” technique. What separates the fast direct method from past alternatives is that it relies upon numerical approximation rather than truly solving the systems. While most former techniques employed the use of Gaussian elimination in solving systems of PDEs, a direct solver such as this requires a defined tolerance level, and then iteratively reaches a value within the predefined tolerance level. This estimation can then assume the same (or at least a very similar) value as what the true solution would have been if found by other methods.

The advantages that the fast direct method has over alternatives, according to Dr. Martinsson, are that they can be used to compute a final solution much more quickly, even applications that are in need of large numbers of iterations and computations. Also, in addition to their basic uses in their standard forms, these direct solvers can be modified and manipulated to construct models for other applications, such as spectral decomposition problems.

How these solvers are actually constructed comes from a combination of several steps. First off, the domain of the problem situation and the necessary tolerance levels for the approximated solution must be defined. Next, local solution operators are found from the domain through a brute force method, systematically going through and narrowing down the field of all possible operators to only those deemed relevant (by predetermined assumptions and criteria). This brute force solver returns the operators in the form of Dirichlet-to-Neumann operators, which aid in simplifying the later computations and reduce the computational costs in terms of time. Due to these lowered computational cost, the hope is that these fast direct solvers will be able to be implemented further in various other applications.