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A Stochastic Galerkin Lattice Boltzmann Method for Incompressible Fluid Flows with Uncertainties

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SSRN

SSRN 2024年

作者： Zhong, Mingliang Xiao, Tianbai Krause, Mathias J. Frank, Martin Simonis, Stephan Scientific Computing Center Karlsruhe Institute of Technology Eggenstein-Leopoldshafen76344 Germany State Key Laboratory of High Temperature Gas Dynamics Centre for Interdisciplinary Research in Fluids Institute of Mechanics Chinese Academy of Sciences Beijing100190 China School of Engineering Science University of Chinese Academy of Sciences Beijing100049 China Institute for Applied and Numerical Mathematics Karlsruhe Institute of Technology Karlsruhe76131 Germany Institute of Mechanical Process Engineering and Mechanics Karlsruhe Institute of Technology Karlsruhe76131 Germany Lattice Boltzmann Research Group Karlsruhe Institute of Technology Karlsruhe76131 Germany

Efficient modeling and simulation of uncertainties in computational fluid dynamics (CFD) remains a crucial challenge. In this paper, we present the first stochastic Galerkin (SG) lattice Boltzmann method (LBM) built upon the generalized polynomial chaos (gPC). The proposed method offers an efficient and accurate approach that depicts the propagation of uncertainties in stochastic incompressible flows. Formal analysis shows that the SG LBM preserves the correct Chapman-Enskog asymptotics and recovers the corresponding macroscopic fluid equations. Numerical experiments, including the Taylor-Green vortex flow, lid-driven cavity flow, and isentropic vortex convection, are presented to validate the solution algorithm. The results demonstrate that the SG LBM achieves the expected spectral convergence and the computational cost is significantly reduced compared to the sampling-based non-intrusive approaches, e.g., the routinely used Monte Carlo method. We obtain a speedup factor of 5.72 compared to Monte Carlo sampling in a randomized two-dimensional Taylor-Green vortex flow test case. By leveraging the accuracy and flexibility of LBM and the efficiency of gPC-based SG, the proposed SG LBM provides a powerful framework for uncertainty quantification in CFD practice. © 2024, The Authors. All rights reserved.

关键词： computational fluid dynamics

Achievable Rate Maximization Pattern Design for Reconfigurable MIMO Antenna Array

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arXiv 2022年

作者： Wang, Haonan Li, Ang Liu, Ya-Feng Qin, Qibo Song, Lingyang Li, Yonghui The School of Information and Communications Engineering Faculty of Electronic and Information Engineering Xi'an Jiaotong University Shaanxi Xi'An710049 China The State Key Laboratory of Integrated Services Networks Xidian University Shaanxi Xi'An China The 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 Wireless Network RAN Research Department Shanghai Huawei Technologies Co. Ltd. Shanghai201206 China The School of Electrical Engineering and Computer Science Peking University Beijing100871 China The School of Electrical and Information Engineering The University of Sydney SydneyNSW2006 Australia

Reconfigurable multiple-input multiple-output can provide performance gains over traditional MIMO by reshaping the channels, i.e., introducing more channel realizations. In this paper, we focus on the achievable rate maximization pattern design for reconfigurable MIMO systems. Firstly, we introduce the matrix representation of pattern reconfigurable MIMO (PR-MIMO), based on which a pattern design problem is formulated. To further reveal the effect of the radiation pattern on the wireless channel, we consider pattern design for both the single-pattern case where the optimized radiation pattern is the same for all the antenna elements, and the multi-pattern case where different antenna elements can adopt different radiation patterns. For the single-pattern case, we show that the pattern design is equivalent to a redistribution of gains among all scattering paths, and an eigenvalue optimization based solution is obtained. For the multi-pattern case, we propose a sequential optimization framework with manifold optimization and eigenvalue decomposition to obtain near-optimal solutions. Numerical results validate the superiority of PR-MIMO systems over traditional MIMO in terms of achievable rate, and also show the effectiveness of the proposed solutions. Copyright © 2022, The Authors. All rights reserved.

关键词： MIMO systems

***: A Comprehensive Genomic Visualisation Database for Malaysian Species

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***: A Comprehensive Genomic Visualisation Database for Mala...

2022 IEEE International Conference on computing, ICOCO 2022

作者： Chan, Vei Siang Mohamed, Farhan Bin Salleh, Faezah Mohd Iglesias, Andres Wagiran, Alina Binti Selamat, Ali Bin Universiti Teknologi Malaysia School of Computing Faculty of Engineering Johor Bahru Malaysia Universiti Teknologi Malaysia Media and Game Innovation Center of Excellence Johor Bahru Malaysia Universiti Teknologi Malaysia Dept. of Biosciences Faculty of Science Johor Bahru Malaysia University of Cantabria Dept. of Applied Mathematics and Computational Sciences Santander Spain University of Technology Malaysia Malaysia-Japan International Institute of Technology Kuala Lumpur Malaysia

ISBN: (纸本)9781665489966

Malaysia is one of the most megadiverse regions in the world. Therefore, it is important to record these rich biodiversity data to safeguard and improve national conservation efforts. However, the lack of nationwide genomic sequencing hampers a more sophisticated species identification, data management, and analytics activities. Thus, this project aims to design and implement ***, a comprehensive one-stop online DNA reference database with integrated DNA visualisation and digital references to allow the users to search, retrieve and analyse the collected DNA sequencing data interactively. This paper documented the system architecture and implementation methods to create ***. The results of *** can display taxonomy information of biodiversity in the web's user interfaces and visualise the complex set of DNA data to allow new insights. Lastly, the authors identified several future works to improve *** including improve user interaction, add visualisation methods to support more bioinformatics tasks, and conduct formal user evaluation. © 2022 IEEE.

关键词： Taxonomies

Automated computation of topological derivatives with application to nonlinear elasticity and reaction–diffusion problems

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Computer Methods in Applied Mechanics and engineering 2022年 398卷

作者： Gangl, P. Sturm, K. Institute of Applied Mathematics Graz University of Technology Steyrergasse 30/III Graz8010 Austria Friedrich Alexander University Erlangen-Nürnberg Cauerstraße 11 Erlangen91058 Germany Johann Radon Institute for Computational and Applied Mathematics Austrian Academy of Sciences Altenberger Straße 69 Linz4040 Austria Institute of Analysis and Scientific Computing TU Wien Wiedner Hauptstr. 8-10 Vienna1040 Austria

While topological derivatives have proven useful in applications of topology optimization and inverse problems, their mathematically rigorous derivation remains an ongoing research topic, in particular in the context of nonlinear partial differential equation (PDE) constraints. We present a systematic yet formal approach for the numerical computation of topological derivatives of a large class of PDE-constrained topology optimization problems with respect to arbitrary inclusion shapes. Scalar and vector-valued as well as linear and nonlinear elliptic PDE constraints are considered in two and three space dimensions including a nonlinear elasticity model and nonlinear reaction–diffusion problems. The systematic procedure follows a Lagrangian approach for computing topological derivatives. For problems where the exact formula is known, the numerically computed values show good coincidence. Moreover, by inserting the computed values into the topological asymptotic expansion, we verify that the obtained values satisfy the expected behavior also for other, previously unknown problems, indicating the correctness of the procedure. © 2022 The Author(s)

关键词： Inverse problems

Improving the Noise Estimation of Latent Neural Stochastic Differential Equations

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arXiv 2024年

作者： Heck, Linus Gelbrecht, M. Schaub, M. Boers, N. Institute for Computing and Information Sciences Radboud University Nijmegen Netherlands Earth System Modelling School of Engineering and Design TU Munich Germany Computational Network Science Dept of Computer Science RWTH Aachen University Germany Department of Computer Science RWTH Aachen University Germany Complexity Science Potsdam Institute for Climate Impact Research Potsdam Germany Department of Mathematics and Statistics University of Exeter Exeter United Kingdom

Latent neural stochastic differential equations (SDEs) have recently emerged as a promising approach for learning generative models from stochastic time series data. However, they systematically underestimate the noise level inherent in such data, limiting their ability to capture stochastic dynamics accurately. We investigate this underestimation in detail and propose a straightforward solution: by including an explicit additional noise regularization in the loss function, we are able to learn a model that accurately captures the diffusion component of the data. We demonstrate our results on a conceptual model system that highlights the improved latent neural SDE’s capability to model stochastic bistable dynamics. © 2024, CC BY-NC-ND.

关键词： Stochastic models

An Approximation Theory Framework for Measure-Transport Sampling Algorithms

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arXiv 2023年

作者： Baptista, Ricardo Hosseini, Bamdad Kovachki, Nikola B. Marzouk, Youssef Sagiv, Amir Computing and Mathematical Sciences Caltech PasadenaCA United States Applied Mathematics University of Washington SeattleWA United States NVIDIA Corporation Santa ClaraCA United States Center for Computational Science and Engineering MIT CambridgeMA United States Technion - Israel Institute of Technology Haifa Israel

This article presents a general approximation-theoretic framework to analyze measure transport algorithms for probabilistic modeling. A primary motivating application for such algorithms is sampling-a central task in statistical inference and generative modeling. We provide a priori error estimates in the continuum limit, i.e., when the measures (or their densities) are given, but when the transport map is discretized or approximated using a finite-dimensional function space. Our analysis relies on the regularity theory of transport maps and on classical approximation theory for high-dimensional functions. A third element of our analysis, which is of independent interest, is the development of new stability estimates that relate the distance between two maps to the distance (or divergence) between the pushforward measures they define. We present a series of applications of our framework, where quantitative convergence rates are obtained for practical problems using Wasserstein metrics, maximum mean discrepancy, and Kullback-Leibler divergence. Specialized rates for approximations of the popular triangular Knöthe-Rosenblatt maps are obtained, followed by numerical experiments that demonstrate and extend our theory. Copyright © 2023, The Authors. All rights reserved.

关键词： Approximation algorithms

On Minimization of Upper Bound for the Convergence Rate of the QHSS Iteration Method

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Communications on Applied mathematics and Computation 2019年第2期1卷 263-282页

作者： Wen-Ting Wu State Key Laboratory of Scientific/Engineering Computing Institute of Computational Mathematics and Scientific/Engineering ComputingAcademy of Mathematics and Systems ScienceChinese Academy of Sciences. P.O. Box 2719. Beijing 100190China School of Mathematical Sciences University of Chinese Academy of SciencesBeijing 100049China

For an upper bound of the spectral radius of the QHSS (quasi Hermitian and skew-Hermitian splitting) iteration matrix which can also bound the contraction factor of the QHSS iteration method,we give its minimum point under the conditions which guarantee that the upper bound is strictly less than one. This provides a good choice of the involved iteration parameters,so that the convergence rate of the QHSS iteration method can be significantly improved.

关键词： System of linear equations Non-Hermitian matrix QHSS iteration method Convergence rate

Quantum simulation of the Fokker-Planck equation via Schrödingerization

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arXiv 2024年

作者： Jin, Shi Liu, Nana Yu, Yue School of Mathematical Sciences Institute of Natural Sciences MOE-LSC Shanghai Jiao Tong University Shanghai200240 China University of Michigan Shanghai Jiao Tong University Joint Institute Shanghai200240 China School of Mathematics and Computational Science Hunan Key Laboratory for Computation and Simulation in Science and Engineering Key Laboratory of Intelligent Computing and Information Processing of Ministry of Education Xiangtan University Hunan Xiangtan411105 China

This paper studies a quantum simulation technique for solving the Fokker-Planck equation. Traditional semi-discretization methods often fail to preserve the underlying Hamiltonian dynamics and may even modify the Hamiltonian structure, particularly when incorporating boundary conditions. We address this challenge by employing the Schrödingerization method - it converts any linear partial and ordinary differential equation with non-Hermitian dynamics into systems of Schrödinger-type equations. It does so via the so-called warped phase transformation that maps the equation into one higher dimension. We explore the application in two distinct forms of the Fokker-Planck equation. For the conservation form, we show that the semi-discretization-based Schrödingerization is preferable, especially when dealing with non-periodic boundary conditions. Additionally, we analyze the Schrödingerization approach for unstable systems that possess positive eigenvalues in the real part of the coefficient matrix or differential operator. Our analysis reveals that the direct use of Schrödingerization has the same effect as a stabilization procedure. For the heat equation form, we propose a quantum simulation procedure based on the time-splitting technique, and give explicitly its corresponding quantum circuit. We discuss the relationship between operator splitting in the Schrödingerization method and its application directly to the original problem, illustrating how the Schrödingerization method accurately reproduces the time-splitting solutions at each step. Furthermore, we explore finite difference discretizations of the heat equation form using shift operators. Utilizing Fourier bases, we diagonalize the shift operators, enabling efficient simulation in the frequency space. Providing additional guidance on implementing the diagonal unitary operators, we conduct a comparative analysis between diagonalizations in the Bell and the Fourier bases, and show that the former generally exhibit

关键词： Hamiltonians

A New Global Divergence Free and Pressure-Robust HDG Method for Tangential Boundary Control of Stokes Equations

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arXiv 2022年

作者： Chen, Gang Gong, Wei Mateos, Mariano Singler, John R. Zhang, Yangwen School of Mathematics Sichuan University Chengdu China NCMIS LSEC Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing China Dpto. de Matemáticas Universidad de Oviedo Campus de Gijón Spain Department of Mathematics and Statistics Missouri University of Science and Technology RollaMO United States Department of Mathematical Science Carnegie Mellon University PittsburghPA United States

In [ESAIM: M2AN, 54(2020), 2229-2264], we proposed an HDG method to approximate the solution of a tangential boundary control problem for the Stokes equations and obtained an optimal convergence rate for the optimal control that reflects its global regularity. However, the error estimates depend on the pressure, and the velocity is not divergence free. The importance of pressure-robust numerical methods for fluids was addressed by John et al. [SIAM Review, 59(2017), 492-544]. In this work, we devise a new HDG method to approximate the solution of the Stokes tangential boundary control problem;the HDG method is also of independent interest for solving the Stokes equations. This scheme yields a H(div) conforming, globally divergence free, and pressure-robust solution. To the best of our knowledge, this is the first time such a numerical scheme has been obtained for an optimal boundary control problem for the Stokes equations. We also provide numerical experiments to show the performance of the new HDG method and the advantage over the non pressure-robust *** Codes 35Q35, 49M25, 65N30 Copyright © 2022, The Authors. All rights reserved.

关键词： Numerical methods