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1. 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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2. Residual-based a posteriori error estimates of mixed methods for a three-field Biot’s consolidation model

   https://doi.org/10.1093/imanum/draa074  

   Li, Yuwen; Zikatanov, Ludmil T (October 2020, IMA Journal of Numerical Analysis)

   null (Ed.)

   Abstract We present residual-based a posteriori error estimates of mixed finite element methods for the three-field formulation of Biot’s consolidation model. The error estimator are upper and lower bounds of the space-time discretization error up to data oscillation. As a by-product, we also obtain a new a posteriori error estimate of mixed finite element methods for the heat equation. 

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3. A posteriori error estimates of Darcy flows with Robin-type jump interface conditions

   Lee, Jeonghun J (November 2024, Computers and mathematics with applications)

   In this work we develop an a posteriori error estimator for mixed finite element methods of Darcy flow problems with Robin-type jump interface conditions. We construct an energy-norm type a posteriori error estimator using the Stenberg post-processing. The reliability of the estimator is proved using an interface-adapted Helmholtz-type decomposition and an interface-adapted Scott-Zhang type interpolation operator. A local efficiency and the reliability of post-processed pressure are also proved. Numerical results illustrating adaptivity algorithms using our estimator are included. 

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4. An anisotropic hp-adaptation framework for ultraweak discontinuous Petrov-Galerkin formulations

   https://doi.org/10.1016/j.camwa.2024.05.025  

   Chakraborty, Ankit; Henneking, Stefan; Demkowicz, Leszek (August 2024, Computers & Mathematics with Applications)

   In this article, we present a three-dimensional anisotropic hp-mesh refinement strategy for ultraweak discontinuous Petrov-Galerkin (DPG) formulations with optimal test functions. The refinement strategy utilizes the built-in residual-based error estimator accompanying the DPG discretization. The refinement strategy is a two-step process: (a) use the built-in error estimator to mark and isotropically hp-refine elements of the (coarse) mesh to generate a finer mesh; (b) use the reference solution on the finer mesh to compute optimal ℎ-and 𝑝-refinements of the selected elements in the coarse mesh. The process is repeated with coarse and fine mesh being generated in every adaptation cycle, until a prescribed error tolerance is achieved. We demonstrate the performance of the proposed refinement strategy using several numerical examples on hexahedral meshes. 

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5. Probabilistic error estimation for non-intrusive reduced models learned from data of systems governed by linear parabolic partial differential equations

   https://doi.org/10.1051/m2an/2021010  

   Uy, Wayne Isaac; Peherstorfer, Benjamin (May 2021, ESAIM: Mathematical Modelling and Numerical Analysis)

   null (Ed.)

   This work derives a residual-based a posteriori error estimator for reduced models learned with non-intrusive model reduction from data of high-dimensional systems governed by linear parabolic partial differential equations with control inputs. It is shown that quantities that are necessary for the error estimator can be either obtained exactly as the solutions of least-squares problems in a non-intrusive way from data such as initial conditions, control inputs, and high-dimensional solution trajectories or bounded in a probabilistic sense. The computational procedure follows an offline/online decomposition. In the offline (training) phase, the high-dimensional system is judiciously solved in a black-box fashion to generate data and to set up the error estimator. In the online phase, the estimator is used to bound the error of the reduced-model predictions for new initial conditions and new control inputs without recourse to the high-dimensional system. Numerical results demonstrate the workflow of the proposed approach from data to reduced models to certified predictions. 

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