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We present a hp-inverse model to estimate a smooth, non-negative sourcefunction from a limited number of observations for a two-dimensional linearsource inversion problem. A standard least-square inverse model is formulatedby using a set of Gaussian radial basis functions (GRBF) on a rectangular meshsystem with a uniform grid space. Here, the choice of the mesh system ismodeled as a random variable and the generalized polynomial chaos (gPC)expansion is used to represent the random mesh system. It is shown that theconvolution of gPC and GRBF provides hierarchical basis functions for thelinear source inverse model with the $hp$-refinement capability. We propose amixed l_1 and l_2 regularization to exploit the hierarchical nature of thebasis functions to find a sparse solution. The $hp$-inverse model has anadvantage over the standard least-square inverse model when the number of datais limited. It is shown that the hp-inverse model provides a good estimate ofthe source function even when the number of unknown parameters ($m$) is muchlarger the number of data ($n$), e.g., m/n > 40.

Development of hp-inverse model by using generalized polynomial chaos