Advances in Inverse Problems and Numerical Methods

The field of inverse problems and numerical methods is witnessing significant developments, with a focus on innovative techniques for solving complex equations and improving the accuracy of solutions. Researchers are exploring new approaches, such as the use of Carleman convexification and phase retrieval via the Wentzel-Kramers-Brillouin approximation, to address challenging inverse problems. Additionally, semi-analytical methods and perturbation solutions are being developed to tackle nonlinear boundary value problems. The Lightning Method, a recent development in numerical solutions, is being applied to solve linear PDEs with spectral accuracy. Data-driven self-supervised learning is also being used to detect singularities in solutions of partial differential equations. Noteworthy papers include: The 'Inverse scattering without phase' paper, which presents a robust numerical framework for reconstructing spatially varying dielectric constants from phaseless backscattering measurements. The 'Data-Driven Self-Supervised Learning for the Discovery of Solution Singularity for Partial Differential Equations' paper, which proposes a self-supervised learning framework for estimating the location of singularities in PDE solutions.

Sources

Inverse scattering without phase: Carleman convexification and phase retrieval via the Wentzel--Kramers--Brillouin approximation

Semi Analytical Solution of a Nonlinear Oblique Boundary Value Problem

The lightning method for the heat equation

Data-Driven Self-Supervised Learning for the Discovery of Solution Singularity for Partial Differential Equations

An inverse-free fixed-time stable dynamical system and its forward-Euler discretization for solving generalized absolute value equations

On the Intensity-based Inversion Method for Quantitative Quasi-Static Elastography

A modified Levenberg-Marquardt method for estimating the elastic material parameters of polymer waveguides using residuals between autocorrelated frequency responses

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