Dynamical systems gradient method for solving ill-conditioned linear algebraic systems
This work provides an alternative numerical method for solving ill-conditioned linear systems, but it is incremental as it relies on spectral decomposition, which is a known bottleneck.
The paper proposes a Dynamical Systems Method (DSM) for solving ill-conditioned linear algebraic systems, with justified stopping rules and an algorithm based on spectral decomposition. Numerical results show it can serve as an alternative to Variational Regularization when spectral decomposition is feasible.
A version of the Dynamical Systems Method (DSM) for solving ill-conditioned linear algebraic systems is studied in this paper. An {\it a priori} and {\it a posteriori} stopping rules are justified. An algorithm for computing the solution using a spectral decomposition of the left-hand side matrix is proposed. Numerical results show that when a spectral decompositon of the left-hand side matrix is available or not computationally expensive to obtain the new method can be considered as an alternative to the Variational Regularization.