Friday 14 March 2025
A team of researchers has made a significant breakthrough in developing a globally convergent numerical method for solving a notoriously difficult class of problems known as coefficient inverse problems. These types of problems arise when trying to reconstruct the properties of a medium, such as its conductivity or permittivity, from measurements of how waves propagate through it.
The problem is that these measurements are inherently noisy and incomplete, making it challenging to accurately determine the underlying properties of the medium. Moreover, many coefficient inverse problems are ill-posed, meaning that small changes in the measurement data can lead to large errors in the reconstructed solution.
To overcome these challenges, the researchers developed a novel convexification method that combines Carleman estimates with a viscosity term. The key insight is that by adding a carefully chosen amount of noise to the problem, it becomes possible to globally converge to the correct solution, even in the presence of noisy measurements.
The team’s approach starts by reformulating the coefficient inverse problem as a minimization problem, where the goal is to find the medium properties that minimize the difference between measured and simulated waveforms. However, this minimization problem is typically non-convex, meaning that it has multiple local minima, making it difficult to determine the global minimum.
To overcome this issue, the researchers introduced a viscosity term into the cost functional, which regularizes the solution by adding a penalty for large gradients in the medium properties. This allows them to prove that the problem is globally convex, meaning that there is a unique global minimum.
The team then used a Carleman estimate to show that the viscosity-regularized problem has a unique solution, and that this solution converges to the true medium properties as the regularization parameter goes to zero. The Carleman estimate is a powerful tool that allows them to bound the error between the simulated and measured waveforms, which in turn ensures global convergence.
The researchers tested their method on a variety of problems, including one-dimensional and two-dimensional inverse problems for hyperbolic and parabolic equations. In each case, they were able to demonstrate global convergence to the true solution, even in the presence of noisy measurements.
The implications of this breakthrough are significant, as it opens up new possibilities for solving complex coefficient inverse problems in a wide range of fields, from medical imaging to materials science. The method is also highly flexible, allowing researchers to tailor the regularization parameter and other parameters to suit specific problem types and measurement data.
Cite this article: “Global Convergence in Coefficient Inverse Problems via Novel Convexification Method”, The Science Archive, 2025.
Coefficient Inverse Problems, Wave Propagation, Noisy Measurements, Ill-Posed Problems, Convexification Method, Carleman Estimates, Viscosity Term, Regularization Parameter, Global Convergence, Numerical Methods.
