A Robust Carleman Contraction Mapping Approach for Inverse Scattering Problems with Numerical Demonstrations

Core Concepts

This paper presents a robust approach to solve the inverse scattering problem using a Carleman contraction mapping method. The method does not require a precise initial guess and has low computational cost.

Abstract

The paper addresses the inverse scattering problem in a domain Ω, where the input data involves the waves generated by the interaction of plane waves with unknown scatterers fully occluded inside Ω. The goal is to determine the spatially distributed dielectric constant c(x) inside Ω.
The approach consists of two main stages:

Eliminate the unknown dielectric constant c from the governing equation, resulting in a system of partial differential equations (PDEs).

Develop a Carleman contraction mapping method to effectively solve this system of PDEs. This method is robust, as it does not require a precise initial guess of the true solution, and has low computational cost.

The key steps are:

Introduce a change of variable v to eliminate c from the governing equation, leading to a system of nonlinear PDEs for v.
Construct a Carleman contraction mapping by incorporating a Carleman weight function into the least-squares optimization problem for v.
Prove the convergence of the iterative scheme to the true solution v* under suitable assumptions.
Once v is found, compute the desired solution c using the original equation.

Numerical examples are presented to demonstrate the effectiveness of the proposed method.

Stats

The paper does not contain any explicit numerical data or statistics. The focus is on the theoretical development of the Carleman contraction mapping method and its application to the inverse scattering problem.

Quotes

The paper does not contain any striking quotes that support the key logics. The content is mainly technical in nature, presenting the mathematical derivation and analysis of the proposed method.

Key Insights Distilled From

A robust approach with numerical demonstrations for the inverse scattering problem using a Carleman contraction map

by Phuong M. Ng... at arxiv.org 04-08-2024

https://arxiv.org/pdf/2404.04145.pdf

Deeper Inquiries

How can the proposed Carleman contraction mapping method be extended to handle more general inverse scattering problems, such as those with multiple scatterers or non-planar incident waves

The proposed Carleman contraction mapping method can be extended to handle more general inverse scattering problems by adapting the algorithm to accommodate multiple scatterers and non-planar incident waves. For scenarios with multiple scatterers, the method can be modified to consider the interactions and effects of each scatterer on the incident waves. This would involve updating the system of equations to include the contributions from each scatterer and adjusting the optimization process to account for the additional complexity.
In the case of non-planar incident waves, the method can be adjusted to incorporate the directional components of the incident waves in multiple dimensions. This would require expanding the basis functions to capture the variations in wave direction and modifying the Carleman weight function to account for the changes in wave propagation.
Overall, the extension to handle more general inverse scattering problems would involve enhancing the algorithm's flexibility and adaptability to different scenarios, ensuring that it can effectively capture the complexities of the scattering phenomena.

What are the limitations of the current approach, and how can it be further improved to handle more challenging scenarios, such as noisy or incomplete measurement data

The current approach has limitations in handling scenarios with noisy or incomplete measurement data. To improve the method's robustness in such challenging scenarios, several enhancements can be considered:

Noise Reduction Techniques: Implementing noise reduction algorithms or filters to preprocess the measurement data before applying the Carleman contraction mapping method. This can help improve the accuracy of the reconstructed solution by reducing the impact of noise.

Regularization Methods: Incorporating regularization techniques into the optimization process to prevent overfitting and enhance the stability of the solution. Regularization can help mitigate the effects of noisy data and improve the generalization of the algorithm.

Data Completion Strategies: Developing strategies to handle incomplete measurement data, such as interpolation or extrapolation methods to fill in missing information. This can help ensure that the algorithm has sufficient data to generate an accurate reconstruction.

Sensitivity Analysis: Conducting sensitivity analysis to assess the algorithm's performance under different levels of noise and incomplete data. This can provide insights into the method's limitations and guide further improvements.

By addressing these limitations and implementing enhancements tailored to handle noisy or incomplete data, the Carleman contraction mapping method can be further improved to tackle more challenging scenarios in inverse scattering problems.

Can the Carleman contraction mapping method be applied to other types of inverse problems beyond scattering, and what are the potential applications in fields like medical imaging, nondestructive testing, or geophysical exploration

The Carleman contraction mapping method can be applied to a wide range of inverse problems beyond scattering, offering potential applications in various fields such as medical imaging, nondestructive testing, and geophysical exploration. Some potential applications include:

Medical Imaging: The method can be utilized in medical imaging techniques like MRI, CT scans, and ultrasound imaging to reconstruct internal structures and detect abnormalities with improved accuracy and resolution.

Nondestructive Testing: In the field of nondestructive testing, the method can be used to inspect and evaluate the integrity of structures, materials, and components without causing damage, enabling efficient and reliable testing processes.

Geophysical Exploration: In geophysics, the method can aid in subsurface imaging, seismic analysis, and mineral exploration by reconstructing geological structures and properties from scattered wave data, providing valuable insights for resource exploration and environmental studies.

By adapting the Carleman contraction mapping method to suit the specific requirements of these applications, it can offer advanced solutions for a diverse range of inverse problems in various scientific and industrial domains.