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An adaptive mesh refinement method for numerically solving optimal control problems is developed using Legendre-Gauss-Radau direct collocation. In regions of the solution where the desired accuracy tolerance has not been met, the mesh is refined by either increasing the degree of the approximating polynomial in a mesh interval or dividing a mesh interval into subintervals. In regions of the solution where the desired accuracy tolerance has been met, the mesh size may be reduced by either merging adjacent mesh intervals or decreasing the degree of the approximating polynomial in a mesh interval. Coupled with the mesh refinement method described in this paper is a newly developed relative error estimate that is based on the differences between solutions obtained from the collocation method and those obtained by solving initial-value and terminal-value problems in each mesh interval using an interpolated control obtained from the collocation method. Because the error estimate is based on explicit simulation, the solution obtained via collocation is in close agreement with the solution obtained via explicit simulation using the control on the final mesh, which ensures that the control is an accurate approximation of the true optimal control. The method is demonstrated on three examples from the open literature, and the results obtained show an improvement in final mesh size when compared against previously developed mesh refinement methods.
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Inverse radiative transfer with goal-oriented hp-adaptive mesh refinement: adaptive-mesh inversion
Abstract The inverse problem for radiative transfer is important in many applications, such as optical tomography and remote sensing. Major challenges include large memory requirements and computational expense, which arise from high-dimensionality and the need for iterations in solving the inverse problem. Here, to alleviate these issues, we propose adaptive-mesh inversion: a goal-orientedhp-adaptive mesh refinement method for solving inverse radiative transfer problems. One novel aspect here is that the two optimizations (one for inversion, and one for mesh adaptivity) are treated simultaneously and blended together. By exploiting the connection between duality-based mesh adaptivity and adjoint-based inversion techniques, we propose a goal-oriented error estimator, which is cheap to compute, and can efficiently guide the mesh-refinement to numerically solve the inverse problem. We use discontinuous Galerkin spectral element methods to discretize the forward and the adjoint problems. Then, based on the goal-oriented error estimator, we propose anhp-adaptive algorithm to refine the meshes. Numerical experiments are presented at the end and show convergence speed-up and reduced memory occupation by the goal-oriented mesh adaptive method.
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- Award ID(s):
- 2324368
- PAR ID:
- 10463635
- Publisher / Repository:
- IOP Publishing
- Date Published:
- Journal Name:
- Inverse Problems
- Volume:
- 39
- Issue:
- 11
- ISSN:
- 0266-5611
- Page Range / eLocation ID:
- Article No. 115002
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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