Optimization of Acoustic Source Strength in the Problems of Active Noise Control

We consider a problem of eliminating the unwanted time-harmonic noise on a predetermined region of interest. The desired objective is achieved by active means, i.e., by introducing additional sources of sound called control sources, which generate the appropriate annihilating acoustic signal (antisound). A general solution for the control sources has been obtained previously in both continuous and discrete formulation of the problem. In the current paper, we focus on optimizing the overall absolute acoustic source strength of the control sources. Mathematically, this amounts to the minimization of multivariable complex-valued functions in the sense of L1 with conical constraints, which are only “marginally” convex. The corresponding numerical optimization problem appears very challenging even for the most sophisticated state-of-the-art methodologies, and even when the dimension of the grid is small and the waves are long. Our central result is that the global L1-optimal solution can, in fact, be obtained without solving the numerical optimization problem. This solution is given by a special layer of monopole sources on the perimeter of the protected region. We provide a rigorous proof of global L1 minimality for both continuous and discrete optimization problems in the one-dimensional case. We also provide numerical evidence that corroborates our result in the two-dimensional case, when the protected domain is a cylinder. Even though we cannot fully justify it, we believe that the same result holds in the general case, i.e., for multidimensional settings and domains of arbitrary shape. We formulate this notion as a conjecture at the end of the paper.