Using the Diffuse Approximation for Optimizing the Location of Anti-Sound Sources

Diffuse approximation is a new tool for reconstructing sampled fields p , usable whatever the dimension of the space considered. At a given point x, it finds an estimation of the Taylor development of the field p , by using a weighted least squares method which favours the sample points nearest to x. Diffuse approximation thus provides an approximation to p and also to the partial derivatives concerned in the Taylor expansion. Viewed by the authors as a replacement for finite element methods, diffuse approximation is used here as an estimation tool for a function and its gradients in a process optimizing source positions in an active absorption system. Starting with experimental data, which are fields radiated by a source situated at a certain number of given points in a continuous set of possible positions of anti-noise sources, diffuse approximation allows prevision of fields radiated by a source at any point in the set, with field gradients relative to the co-ordinates of the source. From this information, it is possible to optimize the position of anti-noise sources in a region of the configuration space of the source system where the problem is convex. Unfortunately, such regions are small, and one also has to deal with the ordinary difficulties of non-convex optimization. The results obtained are convincing, because of both the precision of the estimates and the simplicity of the implementation. On the other hand, the problem of optimizing anti-noise source positions being essentially non-convex, the proposed solution provides a tool which is efficient for a small number of sources, but the application of which is limited when applied to the general problem.