Doubly Inertial Coupled Dynamical System for Linear Inverse Problems

We propose a doubly inertial coupled dynamical system (DICDS) for linear inverse problems, extending an existing first-order coupled dynamical system by introducing second-order dynamics for both the unknown variable x and an auxiliary variable y representing the measurement process. Rather than designing an iterative algorithm directly, we formulate the problem as a continuous-time dynamical system whose equilibrium coincides with the solution set of the underlying model, and establish its asymptotic stability via a spectral analysis yielding a necessary and sufficient condition and a Lyapunov-function argument yielding a sufficient condition that clarifies the dissipative structure of the dynamics. This stability theory covers both underdetermined systems, where the equilibrium forms a manifold, and the complementary case with at least as many measurements as unknowns, where it reduces to a unique point whose x-component is the least-squares solution. A direct Euler discretization of the DICDS is numerically unstable; instead, we formulate a separate quasi-inertial continuous-time system that shares the same equilibrium structure and whose Euler discretization yields a two-variable iterative algorithm with momentum-type updates. Numerical experiments on tomographic reconstruction show that the proposed algorithm achieves more effective noise suppression and structural preservation than the corresponding non-inertial coupled method, with the coupling and inertial mechanisms playing distinct roles depending on the measurement-noise level.