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1. FFT-OT: A Fast Algorithm for Optimal Transportation

   Lei, Na; Gu, Xianfeng (July 2021, Proceedings of International Conference on Computer Vision (ICCV))

   null (Ed.)

   An optimal transportation map finds the most economical way to transport one probability measure to the other. It has been applied in a broad range of applications in vision, deep learning and medical images. By Brenier theory, computing the optimal transport map is equivalent to solving a Monge-Amp\`ere equation. Due to the highly non-linear nature, the computation of optimal transportation maps in large scale is very challenging. This work proposes a simple but powerful method, the FFT-OT algorithm, to tackle this difficulty based on three key ideas. First, solving Monge-Amp\`ere equation is converted to a fixed point problem; Second, the obliqueness property of optimal transportation maps are reformulated as Neumann boundary conditions on rectangular domains; Third, FFT is applied in each iteration to solve a Poisson equation in order to improve the efficiency. Experiments on surfaces captured from 3D scanning and reconstructed from medical imaging are conducted, and compared with other existing methods. Our experimental results show that the proposed FFT-OT algorithm is simple, general and scalable with high efficiency and accuracy. 

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2. Efficient Approximation of Optimal TransportationMap by Pogorelov Map

   An, Dongsheng; Lei, Na; Chen, Wei; Luo, Zhongxuan; Zhao, Tong; Si, Hang; Gu, Xianfeng (October 2021, Proceedings of International Meshing Roundtable)

   null (Ed.)

   Optimal transportation (OT) finds the most economical way to transport one measure to another and plays an important role in geometric modeling and processing. Based on the Brenier theorem, the OT problem is equivalent to the Alexandrov problem, which is the dual to the Pogorelov problem. Although solving the Alexandrov/Pogorelov problem are both equivalent to solving the Monge-Amp\`{e}re equation, the former requires second type boundary condition and the latter requires much simpler Dirichlet boundary condition. Hence, we propose to use the Pogorelov map to approximate the OT map. The Pogorelov problem can be solved by a convex geometric optimization framework, in which we need to ensure the searching inside the admissible space. In this work, we prove the discrete Alexandrov maximum principle, which gives an apriori estimate of the searching. Our experimental results demonstrate that the Pogorelov map does approximate the OT map well with much more efficient computation. 

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3. AdaptiveMesh RefinementMethod for Optimal Control Using Nonsmoothness Detection andMesh Size Reduction

   https://doi.org/DOI: 10.1016/j.franklin.2015.05.028  

   Liu, F. (January 2015, Journal of the Franklin Institute)

   An adaptive mesh refinement method for solving optimal control problems is developed. The method employs orthogonal collocation at Legendre–Gauss–Radau points, and adjusts both the mesh size and the degree of the approximating polynomials in the refinement process. A previously derived convergence rate is used to guide the refinement process. The method brackets discontinuities and improves solution accuracy by checking for large increases in higher-order derivatives of the state. In regions between discontinuities, where the solution is smooth, the error in the approximation is reduced by increasing the degree of the approximating polynomial. On mesh intervals where the error tolerance has been met, mesh density may be reduced either by merging adjacent mesh intervals or lowering the degree of the approximating polynomial. Finally, the method is demonstrated on two examples from the open literature and its performance is compared against a previously developed adaptive method. 

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4. An Error Estimation and Mesh Refinement Method Applied to Optimal Libration Point Orbit Transfers

   Haman, G V; Rao, A V (July 2024, American Control Conference)

   An adaptive mesh refinement method for nu- merically solving optimal control problems is described. The method employs collocation at the Legendre-Gauss-Radau points. Within each mesh interval, a relative error estimate is derived based on the difference between the Lagrange polynomial approximation of the state and an adaptive forward- backward explicit integration of the state dynamics. Accuracy in the method is achieved by adjusting the number of mesh intervals and degree of the approximating polynomial in each mesh interval. The method is demonstrated on time-optimal transfers from an L1 halo orbit to an L2 halo orbit in the Earth- Moon system, and performance is compared against previously developed mesh refinement methods. 

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5. Inverse radiative transfer with goal-oriented hp-adaptive mesh refinement: adaptive-mesh inversion

   https://doi.org/10.1088/1361-6420/acf785  

   Du, Shukai; Stechmann, Samuel N. (September 2023, Inverse Problems)

   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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