A Crank-Nicolson finite volume scheme for the modeling of the Riesz space
distributed-order diffusion equation is proposed. The corresponding linear system has a
symmetric positive definite Toeplitz matrix. It can be efficiently stored in $\mathcal{O}$($NK$) memory. Moreover, for the finite volume scheme, a fast version of conjugate gradient (FCG)
method is developed. Compared with the Gaussian elimination method, the computational complexity is reduced from $\mathcal{O}$($MN$^{3} + $NK$) to $\mathcal{O}$($l$_{$A$}$MN$log$N$ + $NK$), where $l$_{$A$} is
the average number of iterations at a time level. Further reduction of the computational
cost is achieved due to use of a circulant preconditioner. The preconditioned fast finite
volume method is combined with the Levenberg-Marquardt method to identify the free
parameters of a distribution function. Numerical experiments show the efficiency of the
method.