Doubly Inertial Coupled Dynamical System for Linear Inverse Problems
- 1Institute of Biomedical Sciences, Tokushima University, 3-18-15 Kuramoto, Tokushima 770-8509, Japan
- 2Faculty of Science, Tanta University, El-Giesh St., Tanta 31527, Egypt
Abstract
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.
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BibTeX
@article{kasai2026doubly,
title = {{Doubly Inertial Coupled Dynamical System for Linear Inverse Problems}},
author = {Ryosuke Kasai and Tetsuya Yoshinaga and Omar M. Abou Al-Ola},
journal = {Mathematics},
year = {2026},
month = oct,
doi = {10.3390/math14203713},
url = {https://www.mdpi.com/2227-7390/14/20/3713},
note = {CC BY 4.0},
}
RIS
TY - JOUR
TI - Doubly Inertial Coupled Dynamical System for Linear Inverse Problems
AU - Ryosuke Kasai
AU - Tetsuya Yoshinaga
AU - Omar M. Abou Al-Ola
PY - 2026
DA - 2026/10/13
JO - Mathematics
DO - 10.3390/math14203713
UR - https://www.mdpi.com/2227-7390/14/20/3713
AB - 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.
SN - 2227-7390
ER -