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arXiv 2023年

作者： Bartuska, Arved Carlon, André Gustavo Espath, Luis Krumscheid, Sebastian Tempone, Raúl Department of Mathematics RWTH Aachen University Gebäude-1953 1.OG Pontdriesch 14-16 161 Aachen52062 Germany Thuwal23955-6900 Saudi Arabia School of Mathematical Sciences University of Nottingham NottinghamNG7 2RD United Kingdom Alexander von Humboldt Professor in Mathematics for Uncertainty Quantification RWTH Aachen University Germany Scientific Computing Center Institute for Applied and Numerical Mathematics Karlsruhe Institute of Technology Germany

Nested integration of the form (formula presented), characterized by an outer integral connected to an inner integral through a nonlinear function f, is a challenging problem in various fields, such as engineering and mathematical finance. The available numerical methods for nested integration based on Monte Carlo (MC) methods can be prohibitively expensive owing to the error propagating from the inner to the outer integral. Attempts to enhance the efficiency of these approximations using the quasi-MC (QMC) or randomized QMC (rQMC) method have focused on either the inner or outer integral approximation. This work introduces a novel nested rQMC method that simultaneously addresses the approximation of the inner and outer integrals. The method leverages the unique nested integral structure to offer a more efficient approximation mechanism. Incorporating Owen’s scrambling techniques, we address integrands exhibiting infinite variation in the Hardy–Krause sense, enabling theoretically sound error estimates. As the primary contribution, we derive asymptotic error bounds for the bias and variance of our estimator, along with the regularity conditions under which these bounds can be attained. Moreover, we derive a truncation scheme for applications in the context of expected information gain estimation and indicate how to use importance sampling to remedy the measure concentration arising in the inner integral. We verify the estimator quality through numerical experiments by comparing the computational efficiency of the nested rQMC method against standard nested MC estimation for two case studies: one in thermomechanics and the other in pharmacokinetics. These examples highlight the computational savings and enhanced applicability of the proposed approach. Copyright © 2023, The Authors. All rights reserved.

关键词： Importance sampling

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Benchmarks for single-phase flow in fractured porous media

arXiv

引用

arXiv 2017年

作者： Flemisch, Bernd Berre, Inga Boon, Wietse Fumagalli, Alessio Schwenck, Nicolas Scotti, Anna Stefansson, Ivar Tatomir, Alexandru Department of Hydromechanics and Modelling of Hydrosystems University of Stuttgart Pfaffenwaldring 61 Stuttgart70569 Germany Department of Mathematics University of Bergen Allégaten 41 Bergen5007 Norway Laboratory for Modeling and Scientific Computing MOX Politecnico di Milano p.za Leonardo da Vinci 32 Milano20133 Italy Department of Applied Geology Geosciences Center University of Göttingen Goldschmidtstrasse 3 Göttingen37077 Germany

This paper presents several test cases intended to be benchmarks for numerical schemes for single-phase fluid flow in fractured porous media. A number of solution strategies are compared, including a vertex and a cell-centered finite volume method, a non-conforming embedded discrete fracture model, a primal and a dual extended finite element formulation, and a mortar discrete fracture model. The proposed benchmarks test the schemes by increasing the difficulties in terms of network geometry, e.g. intersecting fractures, and physical parameters, e.g. low and high fracture-matrix permeability ratio as well as heterogeneous fracture permeabilities. For each problem, the results presented by the participants are the number of unknowns, the approximation errors in the porous matrix and in the fractures with respect to a reference solution, and the sparsity and condition number of the discretized linear system. All data and meshes used in this study are publicly available for further comparisons. Copyright © 2017, The Authors. All rights reserved.

关键词： Number theory

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MULTILEVEL RANDOMIZED QUASI-MONTE CARLO ESTIMATOR FOR NESTED INTEGRATION

arXiv

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arXiv 2024年

作者： Bartuska, Arved Carlon, André Gustavo Espath, Luis Krumscheid, Sebastian Tempone, Raúl Department of Mathematics RWTH Aachen University Gebäude-1953 1.OG Pontdriesch 14-16 161 Aachen52062 Germany Thuwal23955-6900 Saudi Arabia School of Mathematical Sciences University of Nottingham NottinghamNG7 2RD United Kingdom RWTH Aachen University Germany Scientific Computing Center Institute for Applied and Numerical Mathematics Karlsruhe Institute of Technology Germany

Nested integration problems arise in various scientific and engineering applications, including Bayesian experimental design, financial risk assessment, and uncertainty quantification. These nested integrals take the form R f (R g(y, x) dx) dy, for nonlinear f, making them computationally challenging, particularly in high-dimensional settings. Although widely used for single integrals, traditional Monte Carlo (MC) methods can be inefficient when encountering complexities of nested integration. This work introduces a novel multilevel estimator, combining deterministic and randomized quasi-MC (rQMC) methods to handle nested integration problems efficiently. In this context, the inner number of samples and the discretization accuracy of the inner integrand evaluation constitute the level. We provide a comprehensive theoretical analysis of the estimator, deriving error bounds demonstrating significant reductions in bias and variance compared with standard methods. The proposed estimator is particularly effective in scenarios where the integrand is evaluated approximately, as it adapts to different levels of resolution without compromising precision. We verify the performance of our method via numerical experiments, focusing on estimating the expected information gain of experiments. When applied to Gaussian noise in the experiment, a truncation scheme ensures finite error bounds at the same computational complexity as in the bounded noise case up to multiplicative logarithmic terms. The results reveal that the proposed multilevel rQMC estimator outperforms existing MC and rQMC approaches, offering a substantial reduction in computational costs and offering a powerful tool for practitioners dealing with complex, nested integration problems across various *** Codes 62F15, 65C05, 65D30, 65D32 © 2024, CC0.

关键词： Risk assessment

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Scalable Multiscale-Spectral Gfem with an Application to Composite Aero-Structures

SSRN

引用

SSRN 2023年

作者： Bénézech, Jean Seelinger, Linus Bastian, Peter Butler, Richard Dodwell, Timothy Ma, Chupeng Scheichl, Robert Centre for Integrated Materials Processes and Structures University of Bath BathBA2 7AY United Kingdom Institute for Applied Mathematics Interdisciplinary Center for Scientific Computing Heidelberg University Heidelberg Germany Institute of Data Science and AI University of Exeter UK & digiLab Exeter Exeter United Kingdom Department of Mathematical Sciences University of Bath Bath United Kingdom Institute of Scientific Research Great Bay University Dongguan China

In this paper, the first large-scale application of multiscale-spectral generalized finite element methods (MS-GFEM) to composite aero-structures is presented. The crucial novelty lies in the introduction of A-harmonicity in the local approximation spaces, which in contrast to [Babuska, Lipton,Multiscale Model. Simul.9,[Babuska, Lipton, Multiscale Model. Simul. 9, 2011] is enforced more efficiently via a constraint in the local eigenproblems. This significant modification leads to excellent approximation properties, which turn out to be essential to capture accurately material strains and stresses with a low dimensional approximation space, hence maximising model order reduction. The implementation of the framework in the DUNE software package, as well as a detailed description of all components of the method are presented and exemplified on a composite laminated beam under compressive loading. The excellent parallel scalability of the method, as well as its superior performance compared to the related, previously introduced GenEO method are demonstrated on two realistic application cases, including a C-shaped wing spar with complex geometry. Further, by allowing low-cost approximate solves for closely related models or geometries this efficient, novel technology provides the basis for future applications in optimisation or uncertainty quantification on challenging problems in composite aero-structures. © 2023, The Authors. All rights reserved.

关键词： Finite element method

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Laser interferometric gravitational wave detection in space and structure formation in the early universe

引用

Chinese Astronomy and Astrophysics 2015年第4期39卷 411-446页

作者： Gong, Xue-Fei Xu, Sheng-Nian Yuan, Ye-Fei Bai, Shan Bian, Xing Cao, Zhou-Jian Chen, Ge-Rui Dong, Peng Gao, Tian-Shu Gao, Wei Huang, Shuang-Lin Luo, Zi-Ren Shao, Ming-Xue Sun, Bao-San Tang, Wen-Lin Yu, Pin Xu, Peng Zang, Yun-Long Zhang, Hai-Peng Lau, Yun-Kau Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing 100190 China University of Science and Technology of China Department of Astronomy Hefei 230026 China Theoretisch-Physikalisches Institut Friedrich-Schiller-Universität Jena Jena D-07743 Germany University of Chinese Academy of Sciences Beijing 100049 China State Key Laboratory of Scientific and Engineering Computing Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing 100190 China State Key Laboratory of Theoretical Physics Institute of Theoretical Physics Chinese Academy of Sciences Beijing 100190 China Beijing University of Technology College of Applied Sciences Beijing 100124 China Morningside Center of Mathematics Chinese Academy of Sciences Beijing 100190 China Capital Normal University Beijing 100089 China Henan University Department of Mathematics Kaifeng 475001 China Institute of Mechanics Chinese Academy of Sciences Beijing 100190 China Henan Polytechnic University Jiaozuo Henan 454000 China Huazhong University of Science and Technology School of Physics Wuhan 430074 China Tshinghua University Mathematical Sciences Center Beijing 100083 China Institute of Theoretical Physics Chinese Academy of Sciences Beijing 100190 China

The laser interferometric gravitational wave detection in space is an important and realistic approach to studying some key problems of astronomy and cosmology, such as the stellar evolution and galactic formation in the early universe, the co-evolution of black holes and galaxies, etc. After performing two stages of the Advance Research Project of Space Science commissioned by the Chinese Academy of Sciences, and by weighing the feasibility of technology and the foresight of science, we have selected the gravitational wave sources including the high-redshift intermediate-and-large mass binary black hole mergers, and the intermediate-mass black holes in clusters and other dense dynamic environments, as the main scientific objectives, and given a preliminary design for the project of Chinese space gravitational wave detection in the frequency range from millihertz to hertz. Based on this, here we introduce briefly the science of the space gravitational wave detection, as a new measure of astronomical observations, as well as the scientific objectives and potential detectivity of the design for the Chinese space gravitational wave detection mission.

关键词： Binary black hole coalescence Gravitational wave intermediate mass black hole ΛCDM hierarchical structure formation picture

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gLaSDI: Parametric Physics-informed Greedy Latent Space Dynamics Identification

arXiv

引用

arXiv 2022年

作者： He, Xiaolong Choi, Youngsoo Fries, William D. Belof, Jonathan L. Chen, Jiun-Shyan Department of Structural Engineering University of California La Jolla San DiegoCA92093 United States Center for Applied Scientific Computing Lawrence Livermore National Laboratory LivermoreCA94550 United States Applied Mathematics School of Mathematical Sciences University of Arizona TucsonAZ85721 United States Materials Science Division Physical and Life Science Directorate Lawrence Livermore National Laboratory LivermoreCA94550 United States

A parametric adaptive physics-informed greedy Latent Space Dynamics Identification (gLaSDI) method is proposed for accurate, efficient, and robust data-driven reduced-order modeling of high-dimensional nonlinear dynamical systems. In the proposed gLaSDI framework, an autoencoder discovers intrinsic nonlinear latent representations of high-dimensional data, while dynamics identification (DI) models capture local latent-space dynamics. An interactive training algorithm is adopted for the autoencoder and local DI models, which enables identification of simple latent-space dynamics and enhances accuracy and efficiency of data-driven reduced-order modeling. To maximize and accelerate the exploration of the parameter space for the optimal model performance, an adaptive greedy sampling algorithm integrated with a physics-informed residual-based error indicator and random-subset evaluation is introduced to search for the optimal training samples on the fly. Further, to exploit local latent-space dynamics captured by the local DI models for an improved modeling accuracy with a minimum number of local DI models in the parameter space, a k-nearest neighbor convex interpolation scheme is employed. The effectiveness of the proposed framework is demonstrated by modeling various nonlinear dynamical problems, including Burgers equations, nonlinear heat conduction, and radial advection. The proposed adaptive greedy sampling outperforms the conventional predefined uniform sampling in terms of accuracy. Compared with the high-fidelity models, gLaSDI achieves 17 to 2,658× speed-up with 1 to 5% relative errors. Copyright © 2022, The Authors. All rights reserved.

关键词： Nonlinear dynamical systems

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Certified data-driven physics-informed greedy auto-encoder simulator

arXiv

引用

arXiv 2022年

A parametric adaptive greedy Latent Space Dynamics Identification (gLaSDI) framework is developed for accurate, efficient, and certified data-driven physics-informed greedy auto-encoder simulators of high-dimensional nonlinear dynamical systems. In the proposed framework, an auto-encoder and dynamics identification models are trained interactively to discover intrinsic and simple latent-space dynamics. To effectively explore the parameter space for optimal model performance, an adaptive greedy sampling algorithm integrated with a physics-informed error indicator is introduced to search for optimal training samples on the fly, outperforming the conventional predefined uniform sampling. Further, an efficient k-nearest neighbor convex interpolation scheme is employed to exploit local latent-space dynamics for improved predictability. Numerical results demonstrate that the proposed method achieves 121 to 2,658 × speed-up with 1 to 5% relative errors for radial advection and 2D Burgers dynamical problems. Copyright © 2022, The Authors. All rights reserved.

关键词： Nonlinear dynamical systems

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Scalable multiscale-spectral GFEM with an application to composite aero-structures

arXiv

引用

arXiv 2022年

作者： Bénézech, Jean Seelinger, Linus Bastian, Peter Butler, Richard Dodwell, Timothy Ma, Chupeng Scheichl, Robert Centre for Integrated Materials Processes and Structures University of Bath BathBA2 7AY United Kingdom Institute for Applied Mathematics Interdisciplinary Center for Scientific Computing Heidelberg University Heidelberg Germany Institute of Data Science and AI University of Exeter Exeter United Kingdom DigiLab Exeter United Kingdom Department of Mathematical Sciences University of Bath Bath United Kingdom Institute of Scientific Research Great Bay University Dongguan China

In this paper, the first large-scale application of multiscale-spectral generalized finite element methods (MS-GFEM) to composite aero-structures is presented. The crucial novelty lies in the introduction of A-harmonicity in the local approximation spaces, which in contrast to [Babuska, Lipton, Multiscale Model. Simul. 9, 2011] is enforced more efficiently via a constraint in the local eigenproblems. This significant modification leads to excellent approximation properties, which turn out to be essential to capture accurately material strains and stresses with a low dimensional approximation space, hence maximising model order reduction. The implementation of the framework in the Distributed and Unified Numerics Environment (DUNE) software package, as well as a detailed description of all components of the method are presented and exemplified on a composite laminated beam under compressive loading. The excellent parallel scalability of the method, as well as its superior performance compared to the related, previously introduced GenEO method are demonstrated on two realistic application cases, including a C-shaped wing spar with complex geometry. Further, by allowing low-cost approximate solves for closely related models or geometries this efficient, novel technology provides the basis for future applications in optimisation or uncertainty quantification on challenging problems in composite aero-structures. Copyright © 2022, The Authors. All rights reserved.

关键词： Finite element method

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Potential numerical techniques and challenges for atmospheric modeling

Potential numerical techniques and challenges for atmospheri...

引用

作者： Li, J. Steppeler, J. Fang, F. Pain, C.C. Zhu, J. Peng, X. Dong, L. Li, Y. Tao, L. Leng, W. Wang, Y. Zheng, J. International Center for Climate and Environment Sciences Institute of Atmospheric Physics Chinese Academy of Sciences Beijing China Hamburg Germany Applied Modelling and Computation Group Department of Earth Science and Engineering Imperial College London London United Kingdom International Center for Climate and Environment Sciences Institute of Atmospheric Physics Chinese Academy of Sciences Department of Atmospheric Sciences University of Chinese Academy of Sciences Beijing China State Key Laboratory of Severe Weather Chinese Academy of Meteorological Sciences Beijing China State Key Laboratory of Numerical Modeling for Atmospheric Sciences and Geophysical Fluid Dynamics Institute of Atmospheric Physics Chinese Academy of Sciences Beijing China State Key Laboratory of Scientific and Engineering Computing Chinese Academy of Sciences Beijing China School of Mathematics and Statistics Henan University Kaifeng China Center for Excellence in Regional Atmospheric Environment Institute of Urban Environment Chinese Academy of Sciences Xiamen China

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Call for participation: Verification benchmarks for single-phase flow in three-dimensional fractured porous media

arXiv

引用

arXiv 2018年

作者： Berre, Inga Boon, Wietse Flemisch, Bernd Fumagalli, Alessio Gläser, Dennis Keilegavlen, Eirik Scotti, Anna Stefansson, Ivar Tatomir, Alexandru Department of Mathematics University of Bergen Allégaten 41 Bergen5007 Norway Department of Hydromechanics and Modelling of Hydrosystems University of Stuttgart Pfaffenwaldring 61 Stuttgart70569 Germany Laboratory for Modeling and Scientific Computing MOX Politecnico di Milano p.za Leonardo da Vinci 32 Milano20133 Italy Department of Applied Geology Geosciences Center University of Göttingen Goldschmidtstrasse 3 Göttingen37077 Germany Christian Michelsen Research Bergen Norway

This call for participation proposes four benchmark tests to verify and compare numerical schemes to solve single-phase flow in fractured porous media. With this, the two-dimensional suite of benchmark tests presented by Flemisch et al.1 is extended to include three-dimensional problems. Moreover, transport simulations are included as a means to compare discretization methods for flow. With this publication, we invite researchers to contribute to the study by providing results to the test cases based on their applied discretization methods. Copyright © 2018, The Authors. All rights reserved.

关键词： Discrete event simulation

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