Krylov Subspace Methods for Regularized Models in Acoustic Temperature Reconstruction From Simulated and Real Measurements

In a microwave heating system, there is usually an anomaly like thermal runaway or local overheating due to the propagation characteristics of electromagnetic waves during the heating process. The anomalous situation can be coped with by analyzing the temperature information obtained from acoustic travel time tomography. The aim of the research is to investigate the possibility of applying the acoustic approach to microwave heating. When applying acoustic tomography, an ill-conditioned least-squares problem is formulated, and the problem is hard to solve for super bad condition numbers of the coefficient matrices. In computational mathematics, researchers make a lot of effort in developing Krylov subspace methods to solve ill-conditioned problems. Some existing methods show slow convergence for practical applications, which motivates us to improve one Krylov subspace method based on real cases. Our method can be treated as an application of heavy ball minimal residual (HBMR) method to regularized least-squares problems. Based on the data collected from a measurement platform that simulates the internal circumstance of the heating system, our method is compared with certain classical Krylov subspace methods from aspects of convergence rate, reconstruction accuracy, and anti-interference capacity. Numerical experiments demonstrate a significant improvement in convergence rate on our method for regularized ill-conditioned problems. In addition, we display how to select an optimal kernel function. Details of choosing the optimal parameter are shown as well.