The field of inverse problems and numerical methods is rapidly evolving, with a focus on developing efficient and accurate algorithms for solving complex problems. Recent research has centered on improving iterative inversion schemes, such as the Born series and the reduced inverse Born series, to avoid local minima and linearization. Additionally, there is a growing interest in using Prony series approximations for analyzing relaxation data in glassy states.

Another significant direction is the development of regularization methods, including projected iterated Tikhonov regularization, to solve ill-posed problems in low precision. The Lebesgue constant for uniform approximation of differential forms has also been explored, providing insights into the estimation of approximation errors.

Frequency-weighted extended balanced truncation methods have been proposed for discrete and continuous-time linear time-invariant plants, offering a-priori error bounds and numerical validation. Computing equilibrium points of electrostatic potentials has also been investigated, with algorithms based on piecewise approximation of the potential function by Taylor series.

Noteworthy papers in this area include: The paper on a superfast direct solver for type-III inverse nonuniform discrete Fourier transform, which proposes a novel decomposition method enabling quasi-linear complexity. The paper on greedy techniques for inverse problems, which introduces a novel greedy framework for optimal selection of indirect measurements in operator codomains. The paper on a unified low-rank ADI framework, which shows that low-rank ADI methods for Lyapunov, Sylvester, and Riccati equations can be shared, substantially increasing computational efficiency.