A two-stage method for reconstruction of parameters in diffusion equations

@article{Bai2022ATM,
  title={A two-stage method for reconstruction of parameters in diffusion equations},
  author={Xuesong Bai and Elena Cherkaev and Dong Wang},
  journal={ArXiv},
  year={2022},
  volume={abs/2206.12750}
}
. Parameter reconstruction for diffusion equations has a wide range of applications. In this paper, we proposed a two-stage scheme to efficiently solve conductivity reconstruction problems for steady-state diffusion equations with solution data measured inside the domain. The first stage is based on total variation regularization of the log diffusivity and the split Bregman iteration method. In the second stage, we apply the K-means clustering for the reconstruction of “blocky” conductivity functions… 

Figures from this paper

References

SHOWING 1-10 OF 33 REFERENCES

Reconstructing material properties by deconvolution of full-field measurement images: The conductivity case

This study concerns the reconstruction of material parameters from full-field measurements. In this context the typical available data is a set of digital images that is seldom handled as such when

An Iterative Regularization Method for Total Variation-Based Image Restoration

We introduce a new iterative regularization procedure for inverse problems based on the use of Bregman distances, with particular focus on problems arising in image processing. We are motivated by

An inverse problem formulation for parameter estimation of a reaction–diffusion model of low grade gliomas

TLDR
The goal is to estimate the spatial distribution of tumor concentration, as well as the magnitude of anisotropic tumor diffusion, using a constrained optimization formulation with a reaction–diffusion model that results in a system of nonlinear partial differential equations.

Reconstruction of Coefficients in Scalar Second‐Order Elliptic Equations from Knowledge of Their Solutions

This paper concerns the reconstruction of possibly complex‐valued coefficients in a second‐order scalar elliptic equation that is posed on a bounded domain from knowledge of several solutions of that

Mixed Finite Elements for Elliptic Problems with Tensor Coefficients as Cell-Centered Finite Differences

TLDR
An expanded mixed finite element approximation of second-order elliptic problems containing a tensor coefficient is presented, and it is shown that rates of convergence are retained for the finite difference method.

The Split Bregman Method for L1-Regularized Problems

TLDR
This paper proposes a “split Bregman” method, which can solve a very broad class of L1-regularized problems, and applies this technique to the Rudin-Osher-Fatemi functional for image denoising and to a compressed sensing problem that arises in magnetic resonance imaging.

A convergence proof of the split Bregman method for regularized least-squares problems

TLDR
This paper conducts a convergence rate analysis of the ADMM algorithm using two splits for image restoration problems with quadratic data-fitting term and regularization term, and can show that the two-split ADMM algorithms can be faster than the SB method if the AL penalty parameter of theSB method is suboptimal.

Computational Methods for Inverse Problems

In verse problems arise in a number of important practical applications, ranging from biomedical imaging to seismic prospecting. This book provides the reader with a basic understanding of both the