| Abstract: |
| The forward and inverse problems for radiative transfer are critical in many applications, such as climate modeling, optical tomography, and remote sensing. Both problems present major challenges, particularly large memory requirements and computational expense, due to the high dimensionality of the equation and the iterative nature of solving the inverse problem.
To tackle these challenges, we investigate the hp-adaptive mesh refinement approach, which has proved effective in efficiently representing solutions where they are smooth with high-order approximations, while also providing the flexibility to resolve local features through adaptive refinements.
For the forward problem, we demonstrate that exponential convergence with respect to degrees of freedom (DOFs) can be achieved even when the solution exhibits certain levels of sharp gradients. For the inverse problem, we introduce a goal-oriented hp-adaptive mesh refinement method that can blend the two optimization processes -- one for inversion and one for mesh adaptivity -- thereby reducing computational cost and memory requirements. Numerical tests are presented to validate the theoretical predictions. |
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