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Iterative sinc-convolution method for solving planar D-bar equation with application to EIT.

Mahdi Abbasi1, Ahmad-Reza Naghsh-Nilchi

  • 1Department of Computer Engineering, Engineering Faculty, University of Isfahan, Iran.

International Journal for Numerical Methods in Biomedical Engineering
|August 8, 2014
PubMed
Summary

A new sinc-convolution algorithm efficiently solves 2D D-bar integral equations, overcoming limitations of previous methods. This numerical approach enhances conductivity imaging and inverse scattering solutions with faster computation and better convergence.

Keywords:
D-bar equationEITinverse scatteringsinc-convolution

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Area of Science:

  • Computational Mathematics
  • Applied Physics
  • Electrical Engineering

Background:

  • Numerical solutions for D-bar integral equations are crucial for inverse scattering problems.
  • Existing methods like product integrals and multigrid have drawbacks such as high computational complexity and slow convergence.

Purpose of the Study:

  • To introduce a novel and efficient sinc-convolution algorithm for solving the two-dimensional D-bar integral equation.
  • To address the singularity problem in D-bar equations, which has been inadequately tackled previously.

Main Methods:

  • The sinc-convolution method replaces multidimensional convolution-form integrals with algebraic equations using collocation.
  • Separation of variables is employed to avoid formulating large full matrices, significantly reducing computational complexity.

Main Results:

  • The proposed sinc-convolution algorithm demonstrates drastically reduced computational complexity compared to existing methods.
  • The method achieves exponential convergence with a rate of O(e-cN).
  • Simulations for an electrical impedance tomography problem confirm the algorithm's efficiency.

Conclusions:

  • The sinc-convolution algorithm offers an efficient and effective solution for the 2D D-bar integral equation.
  • This method overcomes the computational and convergence limitations of prior approaches.
  • It provides a robust tool for applications like conductivity imaging and inverse scattering.