We present a dynamical-system approach to linear inverse problems based on a newly introduced discrepancy measure called the mutual power divergence (MPD). The MPD evaluates the mutual consistency between the measurement and the prediction and serves as a Lyapunov function for constructing a continuous-time dynamical system associated with the inverse problem. The equilibrium points of the system coincide with solutions of the linear model, and their stability is established using Lyapunov theory and LaSalle’s invariance principle. By discretizing the continuous-time dynamics with a multiplicative Euler scheme, we obtain an iterative reconstruction algorithm with multiplicative updates. The resulting iteration constitutes a two-parameter extension of the simultaneous multiplicative algebraic reconstruction technique. Numerical experiments on tomographic reconstruction problems demonstrate that the proposed algorithm achieves improved reconstruction accuracy over existing multiplicative methods.
Kasai et al. (Mon,) studied this question.