On the limited memory BFGS method for large scale optimization
Mathematical Programming: Series A and B
The use of the L-curve in the regularization of discrete ill-posed problems
SIAM Journal on Scientific Computing
Perspectives in Flow Control and Optimization
Perspectives in Flow Control and Optimization
Induction heating processes optimization a general optimal control approach
Journal of Computational Physics
Energy minimization using Sobolev gradients: application to phase separation and ordering
Journal of Computational Physics
A computational framework for the regularization of adjoint analysis in multiscale PDE systems
Journal of Computational Physics
Frequency Domain Optical Tomography Based on the Equation of Radiative Transfer
SIAM Journal on Scientific Computing
Adjoint-based optimization of PDE systems with alternative gradients
Journal of Computational Physics
Journal of Computational Physics
Journal of Computational Physics
Fundamentals of Computerized Tomography: Image Reconstruction from Projections
Fundamentals of Computerized Tomography: Image Reconstruction from Projections
Journal of Computational Physics
Journal of Computational and Applied Mathematics
Application of Sobolev gradient method to Poisson-Boltzmann system
Journal of Computational Physics
A New Sobolev Gradient Method for Direct Minimization of the Gross-Pitaevskii Energy with Rotation
SIAM Journal on Scientific Computing
Nonlinear least squares and Sobolev gradients
Applied Numerical Mathematics
Hi-index | 31.45 |
Optical tomography is mathematically treated as a non-linear inverse problem where the optical properties of the probed medium are recovered through the minimization of the errors between the experimental measurements and their predictions with a numerical model at the locations of the detectors. According to the ill-posed behavior of the inverse problem, some regularization tools must be performed and the Tikhonov penalization type is the most commonly used in optical tomography applications. This paper introduces an optimized approach for optical tomography reconstruction with the finite element method. An integral form of the cost function is used to take into account the surfaces of the detectors and make the reconstruction compatible with all finite element formulations, continuous and discontinuous. Through a gradient-based algorithm where the adjoint method is used to compute the gradient of the cost function, an alternative inner product is employed for preconditioning the reconstruction algorithm. Moreover, appropriate re-parameterization of the optical properties is performed. These regularization strategies are compared with the classical Tikhonov penalization one. It is shown that both the re-parameterization and the use of the Sobolev cost function gradient are efficient for solving such an ill-posed inverse problem.