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Time parallelization for hyperbolic and parabolic problems

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arXiv 2025年

作者： Gander, Martin J. Wu, Shu-Lin Zhou, Tao Department of Mathematics University of Geneva CP64 Geneva 41211 Switzerland School of Mathematics and Statistics Northeast Normal University Changchun130024 China Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and Systems Science Chinese Academy of Sciences China

Time parallelization, also known as PinT (Parallel-in-Time) is a new research direction for the development of algorithms used for solving very large scale evolution problems on highly parallel computing architectures. Despite the fact that interesting theoretical work on PinT appeared as early 1964, it was not until 2004, when processor clock speeds reached their physical limit, that research in PinT took off. A distinctive characteristic of parallelization in time is that information flow only goes forward in time, meaning that time evolution processes seem necessarily to be sequential. Nevertheless, many algorithms have been developed over the last two decades to do PinT computations, and they are often grouped into four basic classes according to how the techniques work and are used: shooting-type methods;waveform relaxation methods based on domain decomposition;multigrid methods in space-time;and direct time parallel methods. However, over the past few years, it has been recognized that highly successful PinT algorithms for parabolic problems struggle when applied to hyperbolic problems. We focus in this survey therefore on this important aspect, by first providing a summary of the fundamental differences between parabolic and hyperbolic problems for time parallelization. We then group PinT algorithms into two basic groups: the first group contains four effective PinT techniques for hyperbolic problems, namely Schwarz Waveform Relaxation with its relation to Tent Pitching;Parallel Integral Deferred Correction;ParaExp;and ParaDiag. While the methods in the first group also work well for parabolic problems, we then present PinT methods especially designed for parabolic problems in the second group: Parareal: the Parallel Full Approximation Scheme in Space-Time;Multigrid Reduction in Time;and Space-Time Multigrid. We complement our analysis with numerical illustrations using four time-dependent PDEs: the heat equation;the advection-diffusion equation;Burgers’ equa

关键词： Wave equations

ANALYSIS ACCELERATED MIRROR DESCENT VIA HIGH-RESOLUTION ODES

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arXiv 2023年

作者： Yuan, Ya-Xiang Zhang, Yi Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing100190 China University of Chinese Academy of Sciences Beijing100049 China

Mirror descent plays a crucial role in constrained optimization and acceleration schemes, along with its corresponding low-resolution ordinary differential equations (ODEs) framework have been proposed. However, the low-resolution ODEs are unable to distinguish between Polyak's heavy-ball method and Nesterov's accelerated gradient method. This problem also arises with accelerated mirror descent. To address this issue, we derive high-resolution ODEs for accelerated mirror descent and propose a general Lyapunov function framework to analyze its convergence rate in both continuous and discrete time. Furthermore, we demonstrate that accelerated mirror descent can minimize the squared gradient norm at an inverse cubic rate. © 2023, CC BY.

关键词： Lyapunov functions

Can Efficient Fourier-Transform Techniques Favorably Impact on Broadband computational Electromagnetism?

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arXiv 2024年

作者： Anderson, Thomas G. Lyon, Mark Yin, Tao Bruno, Oscar P. Department of Computational Applied Mathematics and Operations Research Rice University Houston77098 United States Department of Mathematics & Statistics University of New Hampshire Durham03824 United States LSEC Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing100190 China Department of Computing and Mathematical Sciences California Institute of Technology Pasadena91125 United States

In view of recently demonstrated joint use of novel Fourier-transform techniques and effective high-accuracy frequency domain solvers related to the Method of Moments, it is argued that a set of trans-formative innovations could be developed for the effective, accurate and efficient simulation of problems of wave propagation and scattering of broadband, time-dependent wavefields. This contribution aims to convey the character of these methods and to highlight their applicability in computational modeling of electromagnetic configurations across various fields of science and *** Codes 65R10, 65R20 Copyright © 2024, The Authors. All rights reserved.

关键词： Frequency domain analysis

Symplectic Discretization Approach for Developing New Proximal Point Algorithm

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arXiv 2023年

The rapid advancements in high-dimensional statistics and machine learning have increased the use of first-order methods. Many of these methods can be regarded as instances of the proximal point algorithm. Given the importance of the proximal point algorithm, there has been growing interest in developing its accelerated variants. However, some existing accelerated proximal point algorithms exhibit oscillatory behavior, which can impede their numerical convergence rate. In this paper, we first introduce an ODE system and demonstrate its o(1/t2) convergence rate and weak convergence property. Next, we apply the Symplectic Euler Method to discretize the ODE and obtain a new accelerated proximal point algorithm, which we call the Symplectic Proximal Point Algorithm. The reason for using the Symplectic Euler Method is its ability to preserve the geometric structure of the ODEs. Theoretically, we demonstrate that the Symplectic Proximal Point Algorithm achieves an o(1/k2) convergence rate and that the sequences generated by our method converge weakly to the solution set. Practically, our numerical experiments illustrate that the Symplectic Proximal Point Algorithm significantly reduces oscillatory behavior, leading to improved long-time behavior and faster numerical convergence *** Codes 47J20, 47H05, 65K15, 65Y20, 90C25, 90C52 © 2023, CC BY.

关键词： Lyapunov functions

Low-regret shape optimization in the presence of missing Dirichlet data

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arXiv 2024年

作者： Kunisch, Karl Simon, John Sebastian H. Johann Radon Institute for Computational and Applied Mathematics Austrian Academy of Sciences Altenberger Strasse 69 Linz4040 Austria Institute of Mathematics and Scientific Computing University of Graz Heinrichstrasse 36 GrazA-8010 Austria

A shape optimization problem subject to an elliptic equation in the presence of missing data on the Dirichlet boundary condition is considered. It is formulated by optimizing the deformation field that varies the spatial domain where the Poisson equation is posed. To take into consideration the missing boundary data the problem is formulated as a no-regret problem and approximated by low-regret problems. This approach allows to obtain deformation fields which are robust against the missing information. The formulation of the regret problems was achieved by employing the Fenchel transform. Convergence of the solutions of the low-regret to the no-regret problems is analysed, the gradient of the cost is characterized and a first order numerical method is proposed. Numerical examples illustrate the robustness of the low-regret deformation fields with respect to missing data. This is likely the first time that a numerical investigation is reported on for the level of effectiveness of the low-regret approach in the presence of missing data in an optimal control problem. © 2024, CC BY.

关键词： Shape optimization

ANALYSIS AND APPROXIMATION TO PARABOLIC OPTIMAL CONTROL PROBLEMS WITH MEASURE-VALUED CONTROLS IN TIME

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arXiv 2024年

作者： Gong, Wei Liang, Dongdong The State Key Laboratory of Scientific and Engineering Computing Institute of Computational Mathematics National Center for Mathematics and Interdisciplinary Sciences Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing100190 China The Hong Kong Polytechnic University Shenzhen Research Institute Shenzhen518057 China

In this paper, we investigate an optimal control problem governed by parabolic equations with measure-valued controls over time. We establish the well-posedness of the optimal control problem and derive the first-order optimality condition using Clarke’s subgradients, revealing a sparsity structure in time for the optimal control. Consequently, these optimal control problems represent a generalization of impulse control for evolution equations. To discretize the optimal control problem, we employ the space-time finite element method. Here, the state equation is approximated using piecewise linear and continuous finite elements in space, alongside a Petrov-Galerkin method utilizing piecewise constant trial functions and piecewise linear and continuous test functions in time. The control variable is discretized using the variational discretization concept. For error estimation, we initially derive a priori error estimates and stabilities for the finite element discretizations of the state and adjoint equations. Subsequently, we establish weak-* convergence for the control under the norm M(I¯c;L2(ω)), with a convergence order of O(h 21 + τ 41 ) for the *** Codes 65M06, 68M99, 49M40 © 2024, CC BY-SA.

关键词： Optimal control systems

TRACKING OPTIMAL FEEDBACK CONTROL UNDER UNCERTAIN PARAMETERS

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arXiv 2024年

作者： Guth, Philipp A. Kunisch, Karl Rodrigues, Sérgio S. Johann Radon Institute for Computational and Applied Mathematics ÖAW Altenbergerstrasse 69 Linz4040 Austria Institute of Mathematics and Scientific Computing Karl-Franzens University of Graz Heinrich-strasse 36 Graz8010 Austria

Optimal control problems of tracking type for a class of linear systems with uncertain parameters in the dynamics are investigated. An affine tracking feedback control input is obtained by considering the minimization of an energy-like functional depending on a finite ensemble of training/sample parameters. It is computed from the nonnegative definite solution of an associated differential Riccati equation. Simulations are presented showing the tracking performance of the computed input for trained as well as untrained *** Codes 34F05, 49J15, 49J20, 49N10, 93B52, 93C15, 93C20 Copyright © 2024, The Authors. All rights reserved.

关键词： Feedback control

Improved Turbo Message Passing for Compressive Robust Principal Component Analysis: Algorithm Design and Asymptotic Analysis

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arXiv 2024年

作者： He, Zhuohang Ma, Junjie Yuan, Xiaojun The National Key Laboratory of Wireless Communications University of Electronic Science and Technology of China Chengdu611731 China The Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing100864 China

Compressive Robust Principal Component Analysis (CRPCA) naturally arises in various applications as a means to recover a low-rank matrix low-rank matrix L and a sparse matrix S from compressive measurements. In this paper, we approach the problem from a Bayesian inference perspective. We establish a probabilistic model for the problem and develop an improved turbo message passing (ITMP) algorithm based on the sum-product rule and the appropriate approximations. Additionally, we establish a state evolution framework to characterize the asymptotic behavior of the ITMP algorithm in the large-system limit. By analyzing the established state evolution, we further propose sufficient conditions for the global convergence of our algorithm. Our numerical results validate the theoretical results, demonstrating that the proposed asymptotic framework accurately characterize the dynamical behavior of the ITMP algorithm, and the phase transition curve specified by the sufficient condition agrees well with numerical simulations. © 2024, CC BY.

关键词： Asymptotic analysis

Helmholtz decomposition based windowed Green function methods for elastic scattering problems on a half-space

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arXiv 2023年

作者： Yin, Tao Zhang, Lu Zheng, Weiying Zhu, Xiaopeng 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 Zhejiang University Hangzhou310027 China LSEC NCMIS Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and System Sciences Chinese Academy of Sciences Beijing100190 China School of Mathematical Science University of Chinese Academy of Sciences Beijing100049 China

This paper proposes a new Helmholtz decomposition based windowed Green function (HD-WGF) method for solving the time-harmonic elastic scattering problems on a half-space with Dirichlet boundary conditions in both 2D and 3D. The Helmholtz decomposition is applied to separate the pressure and shear waves, which satisfy the Helmholtz and Helmholtz/Maxwell equations, respectively, and the corresponding boundary integral equations of type (I + T) = f, that couple these two waves on the unbounded surface, are derived based on the free-space fundamental solution of Helmholtz equation. This approach avoids the treatment of the complex elastic displacement tensor and traction operator that involved in the classical integral equation method for elastic problems. Then a smooth "slow-rise" windowing function is introduced to truncate the boundary integral equations and a "correction" strategy is proposed to ensure the uniformly fast convergence for all incident angles of plane incidence. Numerical experiments for both two and three dimensional problems are presented to demonstrate the accuracy and efficiency of the proposed method. Copyright © 2023, The Authors. All rights reserved.

关键词： Boundary integral equations