From nanophotonics to aeroplanes, there are many applications that involve scattering in unbounded domains. Typically, one is interested in situations and geometries where there are no known analytical solutions and one has to resort to numerical algorithms to solve the problem using a computer. Such numerical algorithms should give physically meaningful solutions and hopefully obtain them with the minimal computational cost and time.

Boundary element methods (BEM) are broadly used to model such scattering problems and lead densely populated linear systems, which are usually solved by means of iterative solvers. This means that their computational efficiency relies on clever matrix approximation and compression techniques to reduce memory costs (like fast multipole methods or hierarchical matrices), and usually also on preconditioners, which reduce the number of solver iterations needed to reach an approximated solution. A good preconditioner should be easy to compute and is often labelled as optimal when its application makes the condition number remain low and almost constant regardless of how big the linear systems are.

Of course, scatterers have many different shapes and topological properties, depending on the application, and they pose different mathematical challenges (and fun). For instance, when the scatterer has corners, the solutions can have singularities and whatever method one is using to compute a solution should take this into account and be as accurate as possible.

Oxford Mathematician Carolina Urzua-Torres and her collaborators were interested in scatterers that are (bounded) infinitely thin structures, so-called screens or open surfaces in the literature, and which model, for instance, the microstrip patch antennas used in wireless communications. They have edges and they might have corners or even holes. In other words, they have a lot of particular features that differ from the standard case and that are difficult to deal with numerically. Although a lot of progress had been made in their mathematical and numerical understanding, it was still not known how to build optimal preconditioners for that case.

“A very popular and robust preconditioning approach in BEM is the so-called Calderón preconditioning that exploits the Calderón identities between the boundary integral operators (BIOs) to arrive to a well-conditioned system. Unfortunately, these identities do not hold on screens. Thus, we needed to find operators satisfying similar identities and that would give us the same nice properties. We chose to start by attempting this in the unit disk.

We had to dig into the mathematical properties of the Sobolev spaces and operators on these geometries in order to find closed-form formulas for the inverse operators of the BIOs for the Laplacian on the unit disk. We also took care that these formulas were amenable to Galerkin discretization, so we could actually use them in our code. Then we combined this new knowledge and the mathematical structure of the problems to come up with a new approach to precondition systems arising from Laplace, Helmholtz, and time-harmonic Maxwell equations on different shapes of screens.

Besides the desired optimality, this new approach has several good numerical properties. First, we can use it on many types of locally refined meshes, which is very important in resolving the singularities that the solutions have. Second, it fixes the low-frequency breakdown in the case of Maxwell equations, which is relevant for many applications. Moreover, it is ‘bandwith robust’, meaning that the preconditioner can be used for different frequencies and the numerical method will not breakdown, although it should be noted that the preconditioner’s performance will deteriorate the higher the frequency. Finally, it is easy to incorporate in existing BEM codes, which is very convenient.

Indeed, last week I met an engineer who wants to use the new preconditioner for Maxwell in industry projects and for him this implementation simplicity and bandwith robustness were of great importance. The truth is that at the moment there are no preconditioners that work for all frequencies, which is the reason why there is still a lot of research activity designing solutions for different ranges of frequency. However, given the many complexities of the problems engineers are interested in, he explained to me that they want to avoid at all costs having different things for different cases. Therefore, the fact that our approach will work in all cases, even if not great, promises a reasonable trade-off and should be good enough for the problems they are interested in. I am really looking forward to hearing about his practical experience with our preconditioner. At the end of the day, we want people to test and use our work in real-life applications.”

To find out more about the research please click the links below:
Inverses on disks
Preconditioning for Laplace (which also applies to Helmholtz) and also here
Compact-equivalent for Maxwell
Preconditioning for Maxwell - see Chapter 7 of Carolina's Thesis

By the way, for those of you who wondered, in the world of sci-fi BEM stands for bug-eyed monster, a particular feature of the genre in the 1950s. The image above comes from 'Astounding Stories, May 1931, Dark Moon' by Charles Willard Diffin.

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