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1. Dispersion analysis of finite difference and discontinuous Galerkin schemes for Maxwell’s equations in linear Lorentz media

   Jiang, Yan; Sakkaplangkul, Puttha; Bokil, Vrushali A; Cheng, Yingda; Li, Fengyan (May 2019, Journal of computational physics)

   In this paper, we consider Maxwell’s equations in linear dispersive media described by a single-pole Lorentz model for electronic polarization. We study two classes of commonly used spatial discretizations: finite difference methods (FD) with arbitrary even order accuracy in space and high spatial order discontinuous Galerkin (DG) finite element methods. Both types of spatial discretizations are coupled with second order semi-implicit leap-frog and implicit trapezoidal temporal schemes. By performing detailed dispersion analysis for the semi-discrete and fully discrete schemes, we obtain rigorous quantification of the dispersion error for Lorentz dispersive dielectrics. In particular, comparisons of dispersion error can be made taking into account the model parameters, and mesh sizes in the design of the two types of schemes. This work is a continuation of our previous research on energy-stable numerical schemes for nonlinear dispersive optical media [6,7]. The results for the numerical dispersion analysis of the reduced linear model, considered in the present paper, can guide us in the optimal choice of discretization parameters for the more complicated and nonlinear models. The numerical dispersion analysis of the fully discrete FD and DG schemes, for the dispersive Maxwell model considered in this paper, clearly indicate the dependence of the numerical dispersion errors on spatial and temporal discretizations, their order of accuracy, mesh discretization parameters and model parameters. The results obtained here cannot be arrived at by considering discretizations of Maxwell’s equations in free space. In particular, our results contrast the advantages and disadvantages of using high order FD or DG schemes and leap-frog or trapezoidal time integrators over different frequency ranges using a variety of measures 

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2. BOME! Bilevel Optimization Made Easy: A Simple First-Order Approach

   Bo Liu; Mao Ye; Stephen Wright; Peter Stone; Qiang Liu (January 2022, Advances in neural information processing systems)

   Bilevel optimization (BO) is useful for solving a variety of important machine learning problems including but not limited to hyperparameter optimization, meta- learning, continual learning, and reinforcement learning. Conventional BO methods need to differentiate through the low-level optimization process with implicit dif- ferentiation, which requires expensive calculations related to the Hessian matrix. There has been a recent quest for first-order methods for BO, but the methods pro- posed to date tend to be complicated and impractical for large-scale deep learning applications. In this work, we propose a simple first-order BO algorithm that de- pends only on first-order gradient information, requires no implicit differentiation, and is practical and efficient for large-scale non-convex functions in deep learning. We provide a non-asymptotic convergence analysis of the proposed method to stationary points for non-convex objectives and present empirical results that show its superior practical performance. 

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3. Conformal symplectic and relativistic optimization

   https://doi.org/10.1088/1742-5468/abcaee  

   França, Guilherme; Sulam, Jeremias; Robinson, Daniel P; Vidal, René (December 2020, Journal of Statistical Mechanics: Theory and Experiment)

   Abstract Arguably, the two most popular accelerated or momentum-based optimization methods in machine learning are Nesterov’s accelerated gradient and Polyaks’s heavy ball, both corresponding to different discretizations of a particular second order differential equation with friction. Such connections with continuous-time dynamical systems have been instrumental in demystifying acceleration phenomena in optimization. Here we study structure-preserving discretizations for a certain class of dissipative (conformal) Hamiltonian systems, allowing us to analyse the symplectic structure of both Nesterov and heavy ball, besides providing several new insights into these methods. Moreover, we propose a new algorithm based on a dissipative relativistic system that normalizes the momentum and may result in more stable/faster optimization. Importantly, such a method generalizes both Nesterov and heavy ball, each being recovered as distinct limiting cases, and has potential advantages at no additional cost. 

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4. Reduced Order Models for the Quasi-Geostrophic Equations: A Brief Survey

   https://doi.org/10.3390/fluids6010016  

   Mou, Changhong; Wang, Zhu; Wells, David R.; Xie, Xuping; Iliescu, Traian (January 2021, Fluids)

   null (Ed.)

   Reduced order models (ROMs) are computational models whose dimension is significantly lower than those obtained through classical numerical discretizations (e.g., finite element, finite difference, finite volume, or spectral methods). Thus, ROMs have been used to accelerate numerical simulations of many query problems, e.g., uncertainty quantification, control, and shape optimization. Projection-based ROMs have been particularly successful in the numerical simulation of fluid flows. In this brief survey, we summarize some recent ROM developments for the quasi-geostrophic equations (QGE) (also known as the barotropic vorticity equations), which are a simplified model for geophysical flows in which rotation plays a central role, such as wind-driven ocean circulation in mid-latitude ocean basins. Since the QGE represent a practical compromise between efficient numerical simulations of ocean flows and accurate representations of large scale ocean dynamics, these equations have often been used in the testing of new numerical methods for ocean flows. ROMs have also been tested on the QGE for various settings in order to understand their potential in efficient numerical simulations of ocean flows. In this paper, we survey the ROMs developed for the QGE in order to understand their potential in efficient numerical simulations of more complex ocean flows: We explain how classical numerical methods for the QGE are used to generate the ROM basis functions, we outline the main steps in the construction of projection-based ROMs (with a particular focus on the under-resolved regime, when the closure problem needs to be addressed), we illustrate the ROMs in the numerical simulation of the QGE for various settings, and we present several potential future research avenues in the ROM exploration of the QGE and more complex models of geophysical flows. 

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5. A direct method for generating quadratic curvilinear tetrahedral meshes using an advancing front approach

   https://doi.org/10.5281/zenodo.5559211  

   Mohammadi, Fariba; Shontz, Suzanne M. (October 2021, Proc. of the 29th International Meshing Roundtable)

   Computational modeling and simulation of real-world problems, e.g., various applications in the automotive, aerospace, and biomedical industries, often involve geometric objects which are bounded by curved surfaces. The geometric modeling of such objects can be performed via high-order meshes. Such a mesh, when paired with a high-order partial differential equation (PDE) solver, can realize more accurate solution results with a decreased number of mesh elements (in comparison to a low-order mesh). There are several types of high-order mesh generation approaches, such as direct methods, a posteriori methods, and isogeometric analysis (IGA)-based spline modeling approaches. In this paper, we propose a direct, high-order, curvilinear tetrahedral mesh generation method using an advancing front technique. After generating the mesh, we apply mesh optimization to improve the quality and to take advantage of the degrees of freedom available in the initially straight-sided quadratic elements. Our method aims to generate high-quality tetrahedral mesh elements from various types of boundary representations including the cases where no computer-aided design files are available. Such a method is essential, for example, for generating meshes for various biomedical models where the boundary representation is obtained from medical images instead of CAD files. We present several numerical examples of second-order tetrahedral meshes generated using our method based on input triangular surface meshes. 

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