Kernel Ridge Regression with Constraint of Helmholtz Equation for Sound Field Interpolation

Kernel ridge regression with the constraint of the Helmholtz equation for three-dimensional sound field interpolation is proposed. Interpolation of a sound field is formulated as an optimization problem whose objective function to be minimized is a weighted sum of the squared residuals and the squared norm of the interpolating sound field. We show that this problem corresponds to kernel ridge regression and that the reproducing kernel for this problem is given by the zeroth-order spherical Bessel function of the first kind. In this kernel, the physical constraint on the sound fields, i.e., the Helmholtz equation, is taken into consideration, which is a major advantage over other commonly used kernels such as the Gaussian kernel. This advantage is also confirmed by numerical simulations.