This work proposes a hyper-reduction method for nonlinear parametric dynamical systems characterized by gradient fields such as Hamiltonian systems and gradient flows. The gradient structure is associated with conservation of invariants or with dissipation and hence plays a crucial role in the description of the physical properties of the system. Traditional hyper-reduction of nonlinear gradient fields yields efficient approximations that, however, lack the gradient structure. We focus on Hamiltonian gradients and we propose to first decompose the nonlinear part of the Hamiltonian, mapped into a suitable reduced space, into the sum of d terms, each characterized by a sparse dependence on the system state. Then, the hyper-reduced approximation is obtained via discrete empirical interpolation (DEIM) of the Jacobian of the derived d-valued nonlinear function. The resulting hyper-reduced model retains the gradient structure and its computationally complexity is independent of the size of the full model. Moreover, a priori error estimates show that the hyper-reduced model converges to the reduced model and the Hamiltonian is asymptotically preserved. Whenever the nonlinear Hamiltonian gradient is not globally reducible, i.e. its evolution requires high-dimensional DEIM approximation spaces, an adaptive strategy is performed. This consists in updating the hyper-reduced Hamiltonian via a low-rank correction of the DEIM basis. Numerical tests demonstrate the applicability of the proposed approach to general nonlinear operators and runtime speedups compared to the full and the reduced models.
Preserving the positivity of the deformation gradient determinant in intergrid interpolation by combining RBFs and SVD: application to cardiac electromechanics
Some remarks on filtered polynomial interpolation at Chebyshev nodes
本文研究了四类Chebyshev节点上的滤波de la Vallée Poussin（VP）插值。这种近似模型是有趣的应用，因为它结合了经典的拉格朗日多项式逼近（插值和多项式保持）与过滤近似（一致有界的勒贝格常数和减少的吉布斯现象）的优点。在这里，我们重点介绍一些在滤波VP插值应用中有用的附加功能。特别地，我们分析了VP插值多项式的导数所提供的同时逼近。此外，我们还证明了VP逼近算子在Sobolev和Hölder-Zygmund空间中的一致有界性，其中几个积分-微分模型是唯一稳定可解的.
A unified framework for original energy dissipation of equilibrium-preserving exponential Runge--Kutta methods for $L^2$ gradient flows
School of Science and Engineering, The Chinese University of Hong Kong, Shenzhen（香港中文大学深圳校区科学与工程学院）；Department of Applied Mathematics, The Hong Kong Polytechnic University（香港理工大学应用数学系）
Superconvergence analysis of linear FEM based on the polynomial preserving recovery and Richardson extrapolation for Helmholtz equation with high wave number
Beijing Computational Science Research Center（北京计算科学研究中心）；Department of Mathematics, Nanjing University（南京大学数学系）；Department of Mathematics, Wayne State University（韦恩州立大学数学系）
A structure-preserving discretisation of SO(3)-rotation fields for finite Cosserat micropolar elasticity
University of Luxembourg（卢森堡大学）；University of Exeter（埃克塞特大学）；Technische Universität Berlin（柏林工业大学）；Karlsruhe Institute of Technology（卡尔斯鲁厄理工学院）；Technische Universität Dortmund（多特蒙德工业大学）
Asymptotic-preserving and energy-conserving methods for a hyperbolic approximation of the BBM equation
Johannes Gutenberg University Mainz（美因茨约翰内斯·古腾堡大学）；Indian Institute of Technology Kanpur（坎普尔印度理工学院）；King Abdullah University of Science and Technology (KAUST)（阿卜杜拉国王科技大学）；Hasselt University（哈瑟尔特大学）
