A stochastic galerkin method for the direct and inverse random source problems of the helmholtz equation

This paper investigates a novel approach for solving both the direct and inverse random source problems of the one-dimensional Helmholtz equation with additive white noise, based on the generalized polynomial chaos (gPC) approximation. The direct problem is to determine the wave field that is emitted from a random source, while the inverse problem is to use the boundary measurements of the wave field at various frequencies to reconstruct the mean and variance of the source. The stochastic Helmholtz equation is reformulated in such a way that the random source is represented by a collection of mutually independent random variables. The stochastic Galerkin method is employed to transform the model equation into a two-point boundary value problem for the gPC expansion coefficients. The explicit connection between the sine or cosine transform of the mean and variance of the random source and the analytical solutions for the gPC coefficients is established. The advantage of these analytical solutions is that the gPC coefficients are zero for basis polynomials of degree higher than one, which implies that the total number of the gPC basis functions increases proportionally to the dimension, and indicates that the stochastic Galerkin method has the potential to be used in practical applications involving random variables of higher dimensions. By taking the inverse sine or cosine transform of the data, the inverse problem can be solved, and the statistical information of the random source such as the mean and variance can be obtained straightforwardly as the gPC basis functions are orthogonal. Numerical experiments are conducted to demonstrate the efficiency of the proposed method.