In this paper, we present a dimension reduction method to reduce the dimension of parameter space and state space and efficiently solve inverse problems. To this end, proper orthogonal decomposition (POD) and radial basis function (RBF) are combined to represent the solution of forward model with a form of variable separation. This POD-RBF method can be used to efficiently evaluate the models output. A gradient regularization method is presented to solve the inverse problem with fast convergence. A generalized cross validation method is suggested to select the regularization parameter and differential step size for the gradient computation. Because the regularization method needs many models evaluations. This is desirable for POD-RBF method. Thus, the POD-RBF method is integrated with the gradient regularization method to provide an efficient approach to solve inverse problems. We focus on the coefficient inversion of diffusion equations using the proposed approach. Based on different types of measurement data and different basis functions for coefficients, we present a few numerical examples for the coefficient inversion. The numerical results show that accurate reconstruction for the coefficient can be achieved efficiently.
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