Conceptos Básicos
Reconstruction of unbounded perturbations in 3D using a Zernike basis.
Resumen
The article discusses the linearised Calderón problem in three dimensions, focusing on exact direct reconstruction of any L3 perturbation from linearised data. The method utilizes a 3D Zernike basis for efficient implementation and requires only a subset of boundary measurements for reconstruction compared to a full L2 basis. The content is structured as follows:
- Introduction to the conductivity problem in a unit ball.
- Neumann-to-Dirichlet map and its properties.
- Linearised Calderón problem for reconstructing η from knowledge of Fη.
- Detailed explanation of the 3D Zernike basis functions and their orthonormality.
- Theorem 1.1 providing a formula for reconstructing η from linearised data Fη.
- A numerical example demonstrating the ill-posedness of the problem.
- Regularisation techniques and numerical simulations for accurate measurements.
- Discussion on stability, numerical computations, and practical applications.
The appendix extends the linearised problem to bounded smooth domains in higher dimensions, emphasizing Fréchet differentiability with respect to complex-valued perturbations.
Estadísticas
Recently an algorithm was given for exact direct reconstruction of any L2 perturbation from linearised data in two-dimensional Calderón problem.
The method uses a 3D Zernike basis to obtain exact direct reconstruction of any L3 perturbation from linearised data.