Communications in Mathematical Sciences

Volume 12 (2014)

Number 6

Blended finite element method and its convergence for three-dimensional image reconstruction using $L^2$-gradient flow

Pages: 989 – 1015

DOI: http://dx.doi.org/10.4310/CMS.2014.v12.n6.a1

Authors

Guoliang Xu (LSEC, ICMSEC, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, China)

Chong Chen (LSEC, ICMSEC, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, China)

Abstract

The gradient-flow-based explicit and semi-implicit finite element methods proposed in our earlier papers have been applied to solve various variational models for image reconstruction in cryo-electron microscopy and X-ray computed tomography, respectively. In this paper, we develop a gradient-flow-based blended finite element method for solving the variational model of three-dimensional image reconstruction. The method can be regarded as a linear combination of the explicit and semi-implicit schemes, which combines the advantages of both. The computational cost of each iteration of the method is less than that of the semi-implicit situation. In addition, the convergence rate of the method is faster because the temporal step-size can be larger than that of the explicit scheme. Furthermore, the convergence analysis for the blended finite element method is presented. The numerical results also show that the method is more efficient than the explicit and semi-implicit methods.

Keywords

three-dimensional image reconstruction, blended finite element method, convergence analysis, variational model, cryo-electron tomography, X-ray computed tomography

2010 Mathematics Subject Classification

65K05, 65R32, 92C55

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