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AMS Colloquium 1/31/14

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Written by Chris Robbins

Posted on 02 February 2014

In the field of acoustics and wave functions, problems can often get very complex and near unsolvable using methods already known and in use. Because of this, the need to develop accurate numerical approximation models in this field is constantly growing. Professor Victor Dominguez of the Universidad Publica de Navarra in Spain shared some of his most recent work in this area at Friday's installment of the AMS department's weekly colloquium series in his presentation, "Regularized Integral Equations for Acoustic Transmission Problems In Smooth Domains."

In particular, Dominguez chose to focus on problems dealing with the Helmholtz equation, a time independent partial differential equation common in the acoustics field. His goal was to develop a simpler integration equation alternative to transmission problems involving this equation. He attempted to do this by incorporating the use of admittance operators, which essentially provide a mapping of the transmission problem's boundary conditions to the surfaces and interfaces of the media through and around which waves are travelling. Finding a solution in this manner, according to Dominguez, is very straight forward and not too complex to develop, but at the cost of being very difficult to compute.

To deal with this computational issue, Dominguez set about attempting to construct an adequate model that could be used to approximate these solutions within certain domain and precision levels that would be simpler to formulate. He was able to accomplish this based upon estimates of linear combinations of Dirichlet-to-Neumann mappings (assuming they exist in this situation, according to Dominguez). Once completed, the set of newly formulated integral equations could be used to show that the acoustic fields in question can be solved in terms of layer potentials as well as showing that the densities of these field layers are in fact the solutions of the system of integral equations. The model and its incorporated admittance operators also compact the system into a perturbation of in Sobolev spaces, a vector space of functions that is often the only place to find natural solutions to partial differential equations, as well as make the system uniquely solvable.

When first beginning his research on this specific subject, Prof. Dominguez had set clear objectives of deriving new integral equations for Helmholtz acoustic problems that could be done relatively simply and were uniquely solvable. In addition, he also hoped to frame this model as a compact perturbation of the identity matrices in Sobolev spaces, which he claims are great for doing numerical and computational work with the models. Dominguez was able to conclude that he had accomplished all of these points beforehand, calling this system "a new way to regularize boundary integral equations." This model works beyond the theoretical as well, as it works well in practical problems and is competitive with other methodologies out there on this same subject. It is also great for the numerics of these acoustic situations as estimates can be done with this system of equations at lower computational times and costs. As part of his future work, Dominguez would like to extend the scope of these ideas and apply them to new problems outside of merely Helmholtz acoustics applications.