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arXiv 2023年

作者： Sawicki, Jakub Berner, Rico Loos, Sarah A.M. Anvari, Mehrnaz Bader, Rolf Barfuss, Wolfram Botta, Nicola Brede, Nuria Franović, Igor Gauthier, Daniel J. Goldt, Sebastian Hajizadeh, Aida Hövel, Philipp Karin, Omer Lorenz-Spreen, Philipp Miehl, Christoph Mölter, Jan Olmi, Simona Schöll, Eckehard Seif, Alireza Tass, Peter A. Volpe, Giovanni Yanchuk, Serhiy Kurths, Jürgen Potsdam Institute for Climate Impact Research Telegrafenberg Potsdam14473 Germany Fachhochschule Nordwestschweiz FHNW Leonhardsstrasse 6 Basel4009 Switzerland Department of Physics Humboldt-Universität zu Berlin Newtonstraße 15 Berlin12489 Germany DAMTP University of Cambridge Wilberforce Road CambridgeCB3 0WA United Kingdom Fraunhofer Institute for Algorithms and Scientific Computing Schloss Birlinghoven Sankt-Augustin53757 Germany Institute of Systematic Musicology University of Hamburg Germany Transdisciplinary Research Area: Sustainable Futures University of Bonn Bonn53113 Germany University of Bonn Bonn53113 Germany Department of Computer Science and Engineering Chalmers University of Technology Göteborg412 96 Sweden Department of Computer Science University of Potsdam An der Bahn 2 Potsdam14476 Germany Scientific Computing Laboratory Center for the Study of Complex Systems Institute of Physics Belgrade University of Belgrade Pregrevica 118 Belgrade11080 Serbia The Ohio State University Department of Physics and Electrical and Computer Engineering 191 West Woodruff Ave ColumbusOH43210 United States ResCon Technologies LLC 1275 Kinnear Rd. Suite 239 ColumbusOH43212 United States Trieste Italy Research Group Comparative Neuroscience Leibniz Institute for Neurobiology Magdeburg Germany Theoretical Physics Saarland University Saarbrücken66123 Germany Center for Biophysics Saarland University Saarbrücken66123 Germany Department of Mathematics Imperial College London United Kingdom Center for Adaptive Rationality Max Planck Institute for Human Development Lentzeallee 94 Berlin14195 Germany Max Planck Institute for Brain Research Max-von-Laue-Straße 4 Frankfurt am Main60438 Germany Technical University of Munich School of Life Sciences Alte Akademie 8 Freising85354 Germany Department of Mathematics School of Computation Information and Technology Technical University of Munich Boltzmannstraße 3 Garching bei München8574

Adaptivity is a dynamical feature that is omnipresent in nature, socio-economics, and technology. For example, adaptive couplings appear in various real-world systems like the power grid, social, and neural networks, and they form the backbone of closed-loop control strategies and machine learning algorithms. In this article, we provide an interdisciplinary perspective on adaptive systems. We reflect on the notion and terminology of adaptivity in different disciplines and discuss which role adaptivity plays for various fields. We highlight common open challenges, and give perspectives on future research directions, looking to inspire interdisciplinary approaches. Copyright © 2023, The Authors. All rights reserved.

关键词： Dynamical systems

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A Full Multigrid Method for Distributed Control Problems Constrained by Stokes Equations

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高等学校计算数学学报（英文版） 2017年第3期10卷 639-655页

作者： M.M.Butt Y.Yuan Institute of Computational Mathematics and Scientific/Engineering Computing Chinese Academy of Sciences Beijing 100080 China Institute of Computational Mathematics and Scientific/Engineering Computing Chinese Academy of SciencesBeijing 100080China and Higher Education Department Government of the PunjabLahore 54000Pakistan

A full multigrid method with coarsening by a factor-of-three to distributed control problems constrained by Stokes equations is *** optimal control problem with cost functional of velocity and/or pressure tracking-type is considered with Dirichlet boundary *** optimality system that results from a Lagrange multiplier framework,form a linear system connecting the state,adjoint,and control *** investigate multigrid methods with finite difference discretization on staggered grids.A coarsening by a factor-of-three is used on staggered grids that results nested hierarchy of staggered grids and simplified the inter-grid transfer operators.A distributive-Gauss-Seidel smoothing scheme is employed to update the stateand adjoint-variables and a gradient update step is used to update the control *** experiments are presented to demonstrate the effectiveness and efficiency of the proposed multigrid framework to tracking-type optimal control problems.

关键词： PDE-constrained optimization stokes equations multigrid staggered grids finite difference

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Gradient flow based discretized kohn-sham density functional theory

arXiv

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arXiv 2019年

作者： Dai, Xiaoying Wang, Qiao Zhou, Aihui LSEC Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing100190 China School of Mathematical Sciences University of Chinese Academy of Sciences Beijing100049 China

In this paper, we propose and analyze a gradient ow based Kohn-Sham density functional theory. First, we prove that the critical point of the gradient ow based model can be a local minimizer of the Kohn-Sham total energy. Then we apply a midpoint scheme to carry out the temporal discretization. It is shown that the critical point of the Kohn-Sham energy can be well-approximated by the scheme. In particular, based on the midpoint scheme, we design an orthogonality preserving iteration scheme to minimize the Kohn-Sham energy and show that the orthogonality preserving iteration scheme produces approximations that are orthogonal and conver- gent to a local minimizer under reasonable assumptions. Finally, we report numerical experiments that support our theory. Copyright © 2019, The Authors. All rights reserved.

关键词： Dynamical systems

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Automatic Variationally Stable Analysis for Finite Element Computations: Transient Convection-Diffusion Problems

arXiv

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arXiv 2020年

作者： Valseth, Eirik Behnoudfar, Pouria Dawson, Clint Romkes, Albert Oden Institute for Computational Engineering and Sciences University of Texas at Austin AustinTX78712 United States The Department of Data Science The Norwegian University of Life Science Drøbakveien 31 Ås1433 Norway Department of Scientific Computing and Numerical Analysis Simula Research Laboratory Kristian Augusts gate 23 Oslo0164 Norway PerthWA6152 Australia Department of Mechanical Engineering South Dakota School of Mines & Technology Rapid CitySD57701 United States

We present an application of stable finite element (FE) approximations of convection-diffusion initial boundary value problems (IBVPs) using a weighted least squares FE method, the automatic variationally stable finite element (AVS-FE) method [1]. The transient convection-diffusion problem leads to issues in classical FE methods as the differential operator can be considered a singular perturbation in both space and time. The stability property of the AVS-FE method, allows us significant flexibility in the construction of FE approximations in both space and time. Thus, in this paper, we take two distinct approaches to the FE discretization of the convection-diffusion problem: i) considering a space-time approach in which the temporal discretization is established using finite elements, and ii) a method of lines approach in which we employ the AVS-FE method in space whereas the temporal domain is discretized using the generalized-α method. We also consider another space-time technique in which the temporal direction is partitioned, thereby leading to finite space-time"slices" in an attempt to reduce the computational cost of the space-time discretizations. We present numerical verifications for these approaches, including numerical asymptotic convergence studies highlighting optimal convergence properties. Furthermore, in the spirit of the discontinuous Petrov-Galerkin (DPG) method by Demkowicz and Gopalakrishnan [2–6], the AVS-FE method also leads to readily available a posteriori error estimates through a Riesz representer of the residual of the AVS-FE approximations. Hence, the norm of the resulting local restrictions of these estimates serve as error indicators in both space and time for which we present multiple numerical verifications in mesh adaptive *** Codes 65M60 65M12 65M20 65M50 Copyright © 2020, The Authors. All rights reserved.

关键词： Finite element method

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An energy stable and maximum bound preserving scheme with variable time steps for time fractional Allen-Cahn equation

arXiv

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arXiv 2020年

作者： Liao, Hong-Lin Tang, Tao Zhou, Tao Department of Mathematics Nanjing University of Aeronautics and Astronautics Nanjing211106 China Division of Science and Technology BNU-HKBU United International College ZhuhaiGuangdong Province China Department of Mathematics International Center for Mathematics Southern University of Science and Technology ShenzhenGuangdong China LSEC Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing100190 China

In this work, we propose a Crank-Nicolson-type scheme with variable steps for the time fractional Allen-Cahn equation. The proposed scheme is shown to be unconditionally stable (in a variational energy sense), and is maximum bound preserving. Interestingly, the discrete energy stability result obtained in this paper can recover the classical energy dissipation law when the fractional order α → 1. That is, our scheme can asymptotically preserve the energy dissipation law in the α → 1 limit. This seems to be the first work on variable time-stepping scheme that can preserve both the energy stability and the maximum bound principle. Our Crank-Nicolson scheme is build upon a reformulated problem associated with the Riemann-Liouville derivative. As a by product, we build up a reversible transformation between the L1-type formula of the Riemann-Liouville derivative and a new L1-type formula of the Caputo derivative, with the help of a class of discrete orthogonal convolution kernels. This is the first time such a discrete transformation is established between two discrete fractional derivatives. We finally present several numerical examples with an adaptive time-stepping strategy to show the effectiveness of the proposed *** Codes 35Q99, 65M06, 65M12, 74A50 © 2020, CC BY-NC-SA.

关键词： Energy dissipation

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Adaptive step size strategies for orthogonality constrained line search methods

arXiv

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arXiv 2019年

作者： Dai, Xiaoying Zhang, Liwei Zhou, Aihui LSEC Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing100190 China School of Mathematical Sciences University of Chinese Academy of Sciences Beijing100049 China

In this paper, we propose an adaptive step size strategy for a class of line search methods for orthogonality constrained minimization problems, which avoids the classic backtracking procedure. We prove the convergence of the line search methods equipped with our adaptive step size strategy under some mild *** then apply the adaptive algorithm to electronic structure calculations, which show that our strategy is efficient and recommended. Copyright © 2019, The Authors. All rights reserved.

关键词： Adaptive algorithms

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Modeling subgrid-scale force and divergence of heat flux of compressible isotropic turbulence by artificial neural network

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Physical Review Fluids 2019年第10期4卷 104605-104605页

作者： Chenyue Xie Ke Li Chao Ma Jianchun Wang Shenzhen Key Laboratory of Complex Aerospace Flows Center for Complex Flows and Soft Matter Research Department of Mechanics and Aerospace Engineering Southern University of Science and Technology Shenzhen 518055 People's Republic of China Institute of Computational Mathematics and Scientific Engineering Computing Chinese Academy of Sciences Beijing 100190 People's Republic of China The Program in Applied and Computational Mathematics Princeton University Princeton New Jersey 08544 USA

In this paper, the subgrid-scale (SGS) force and the divergence of SGS heat flux of compressible isotropic turbulence are modeled directly by an artificial neural network (ANN), which serves as a data-driven SGS modeling tool for large-eddy simulations (LESs). The unclosed SGS force and divergence of SGS heat flux are modeled based on the local stencil geometry with Galilean invariance. The input features include the first-order and second-order derivatives of filtered velocity and temperature, filtered density, and its first-order derivative. It is shown that the proposed ANN-F7 model shows an advantage over the gradient model in the a priori test. Specifically, the ANN-F7 model gives larger correlation coefficients and smaller relative errors than the gradient model. In an a posteriori analysis, the ANN-F7 model performs better than the dynamic Smagorinsky model (DSM) and dynamic mixed model (DMM) in the prediction of the statistical properties of flow fields at the Taylor microscale Reynolds number Reλ ranging from 180 to 250. The DSM and DMM models lead to the typical tilted spectral distribution of velocity, where low wave numbers are too energy rich, while those near the cutoff are damped too strongly. In contrast, it is shown that the velocity spectrum predicted by the ANN-F7 model almost overlaps with the filtered direct numerical simulation data. Besides, the ANN-F7 model reconstructs the probability density functions of SGS force and divergence of SGS heat flux much better than the DSM and DMM models. An artificial neural network with reasonable physical input features can deepen our understanding of turbulence modeling.

关键词： Compressible flows Turbulence Artificial neural networks

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On the complexity of an augmented lagrangian method for nonconvex optimization

arXiv

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arXiv 2019年

作者： Grapiglia, Geovani N. Yuan, Ya-Xiang Departamento de Matemática Universidade Federal do Paraná Centro Politécnico Cx. postal 19.081 Paraná Curitiba81531-980 Brazil State Key Laboratory of Scientific/Engineering Computing Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and Systems Science Chinese Academy of Sciences Zhong Guan Cun Donglu 55 Beijing100190 China

In this paper we study the worst-case complexity of an inexact Augmented Lagrangian method for nonconvex constrained problems. Assuming that the penalty parameters are bounded, we prove a complexity bound of O(| log(ǫ)|) outer iterations for the referred algorithm to generate an ǫ-approximate KKT point, for ǫ ∈ (0, 1). When the penalty parameters are unbounded, we prove an outer iteration complexity bound of O (ǫ−2/(α−1)), where α > 1 controls the rate of increase of the penalty parameters. For linearly constrained problems, these bounds yield to evaluation complexity bounds of O(| log(ǫ)|2ǫ−2) and O ( ǫ− ( 2(2+ α−1 α) +2 )), respectively, when appropriate first-order methods (p = 1) are used to approximately solve the unconstrained subproblems at each iteration. In the case of problems having only linear equality constraints, the latter bounds are improved to O(| log(ǫ)|2ǫ−(p+1)/p) and O ( ǫ− ( α−4 1 + p+1 p )), respectively, when appropriate p-order methods (p ≥ 2) are used as inner solvers. Copyright © 2019, The Authors. All rights reserved.

关键词： Nonlinear programming

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High-order partitioned spectral deferred correction solvers for multiphysics problems

arXiv

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arXiv 2019年

作者： Huang, Daniel Z. Pazner, Will Persson, Per-Olof Zahr, Matthew J. Institute for Computational and Mathematical Engineering Stanford University United States Center for Applied Scientific Computing Lawrence Livermore National Laboratory United States Department of Mathematics University of California Berkeley United States Department of Aerospace and Mechanical Engineering University of Notre Dame United States

We present an arbitrarily high-order, conditionally stable, partitioned spectral deferred correction (SDC) method for solving multiphysics problems using a sequence of pre-existing single-physics solvers. This method extends the work in [1, 2], which used implicit-explicit Runge-Kutta methods (IMEX) to build high-order, partitioned multiphysics solvers. We consider a generic multiphysics problem modeled as a system of coupled ordinary differential equations (ODEs), coupled through coupling terms that can depend on the state of each subsystem;therefore the method applies to both a semi-discretized system of partial differential equations (PDEs) or problems naturally modeled as coupled systems of ODEs. The sufficient conditions to build arbitrarily high-order partitioned SDC schemes are derived. Based on these conditions, various of partitioned SDC schemes are designed. The stability of the first-order partitioned SDC scheme is analyzed in detail on a coupled, linear model problem. We show that the scheme is conditionally stable, and under conditions on the coupling strength, the scheme can be unconditionally stable. We demonstrate the performance of the proposed partitioned solvers on several classes of multiphysics problems including a simple linear system of ODEs, advection-diffusion-reaction systems, and fluid-structure interaction problems with both incompressible and compressible flows, where we verify the design order of the SDC schemes and study various stability properties. We also directly compare the accuracy, stability, and cost of the proposed partitioned SDC solver with the partitioned IMEX method in [1, 2] on this suite of test problems. The results suggest that the high-order partitioned SDC solvers are more robust than the partitioned IMEX solvers for the numerical examples considered in this work, while the IMEX methods require fewer implicit solves. Copyright © 2019, The Authors. All rights reserved.

关键词： Stability

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DRIFT-PRESERVING NUMERICAL INTEGRATORS FOR STOCHASTIC HAMILTONIAN SYSTEMS

arXiv

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arXiv 2019年

作者： Chen, Chuchu Cohen, David D'Ambrosio, Raffaele Lang, Annika State Key Laboratory of Scientific and Engineering Computing Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing100190 China Department of Mathematics and Mathematical Statistics Umeå University Umeå90187 Sweden Department of Engineering and Computer Science and Mathematics University of L'Aquila AQ L'Aquila Italy Department of Mathematical Sciences Chalmers University of Technology University of Gothenburg Göteborg41296 Sweden

The paper deals with numerical discretizations of separable nonlinear Hamiltonian systems with additive noise. For such problems, the expected value of the total energy, along the exact solution, drifts linearly with time. We present and analyze a time integrator having the same property for all times. Furthermore, strong and weak convergence of the numerical scheme along with efficient multilevel Monte Carlo estimators are studied. Finally, extensive numerical experiments illustrate the performance of the proposed numerical *** Codes 65C20. 65C30. 60H10. 60H35 Copyright © 2019, The Authors. All rights reserved.

关键词： Additive noise

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