A new class of accelerated regularization methods, with application to bioluminescence tomography

@article{Gong2020ANC,
  title={A new class of accelerated regularization methods, with application to bioluminescence tomography},
  author={Rongfang Gong and Bernd Hofmann and Yehui Zhang},
  journal={Inverse Problems},
  year={2020},
  volume={36}
}
In this paper we propose a new class of iterative regularization methods for solving ill-posed linear operator equations. The prototype of these iterative regularization methods is in the form of second order evolution equation with a linear vanishing damping term, which can be viewed not only as an extension of the asymptotical regularization, but also as a continuous analog of the Nesterov’s acceleration scheme. New iterative regularization methods are derived from this continuous model in… 
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References

SHOWING 1-10 OF 37 REFERENCES
A New Coupled Complex Boundary Method for Bioluminescence Tomography
In this paper, we introduce and study a new method for solving inverse source problems, through a working model that arises in bioluminescence tomography (BLT). In the BLT problem, one constructs
A dynamical regularization algorithm for solving inverse source problems of elliptic partial differential equations
This study considers the inverse source problem for elliptic partial differential equations with both Dirichlet and Neumann boundary data. The unknown source term is to be determined by additional
On the second-order asymptotical regularization of linear ill-posed inverse problems
ABSTRACT In this paper, we establish an initial theory regarding the second-order asymptotical regularization (SOAR) method for the stable approximate solution of ill-posed linear operator equations
A coupled complex boundary expanding compacts method for inverse source problems
Abstract In this paper, we consider an inverse source problem for elliptic partial differential equations with both Dirichlet and Neumann boundary conditions. The unknown source term is to be
A novel coupled complex boundary method for inverse source problems
In this paper, we consider an inverse source problem for elliptic partial differential equations with Dirichlet and Neumann boundary data. The unknown source term is to be determined by additional
A novel coupled complex boundary method for solving inverse source problems
In this paper, we consider an inverse source problem for elliptic partial differential equations with Dirichlet and Neumann boundary data. The unknown source term is to be determined from additional
Solving an Inverse Problem from Bioluminescence Tomography by Minimizing an Energy-like Functional
A new method was proposed for the inverse source problem of the molecular imaging by solving a certain minimization problem though Tikhonov regularization method.The unique existence of the solution
RTE-based bioluminescence tomography: A theoretical study
Molecular imaging has become a rapidly developing area in biomedical imaging. Bioluminescence tomography (BLT) is an emerging and promising molecular imaging technology. Light propagation within
On the asymptotical regularization of nonlinear ill-posed problems
We investigate the method of asymptotical regularization for solving nonlinear ill-posed problems F(x)=y, where, instead of y, noisy data ydelta epsilon Y with //y-ydelta //<or= delta are given and
...
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