| Abstract: |
| The inverse problem in radiative transfer is critical for many applications, such as optical tomography and remote sensing. However, solving it numerically presents significant challenges, including high memory requirements and computational expense due to the problem`s high dimensionality and the iterative nature of the solution process.
To address these challenges, we propose a goal-oriented hp-adaptive mesh refinement method for solving the inverse radiative transfer problem. A novel aspect of this approach is that it simultaneously combines two optimization processes -- one for inversion and one for mesh adaptivity. By leveraging the connection between duality-based mesh adaptivity and adjoint-based inversion techniques, we introduce a goal-oriented error estimator that is computationally inexpensive and can efficiently guide mesh refinement to solve the inverse problem numerically.
For discretizing both the forward and adjoint problems, we employ discontinuous Galerkin spectral element methods. Using the goal-oriented error estimator, we then design an hp-adaptive algorithm to refine the meshes. Numerical experiments demonstrate that this method accelerates convergence and reduces memory usage, highlighting the efficiency of the goal-oriented mesh adaptive approach. |
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