Development of a spectral source inverse model by using generalized polynomial chaos

We present a spectral inverse model to estimate a smooth source function from a limited number of observations for an advection-diffusion problem. A standard least-square inverse model is formulated by using a set of Gaussian radial basis functions (GRBF) on a rectangular mesh system. Here, the choice of the collocation points is modeled as a random variable and the generalized polynomial chaos (gPC) expansion is used to represent the random mesh system. It is shown that the convolution of gPC and GRBF provides hierarchical basis functions for a spectral source estimation. We propose a mixed l1 and l2 regularization to exploit the hierarchical nature of the basis polynomials to find a sparse solution. The spectral inverse model has an advantage over the standard least-square inverse model when the number of data is limited. It is shown that the spectral inverse model provides a good estimate of the source function even when the number of unknown parameters (m) is much larger the number of data (n), e.g., m/n > 50.