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Sparse Approximation of Data-Driven Polynomial Chaos Expansions: An Induced Sampling Approach

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Communications in Mathematical Research 2020年第2期36卷 128-153页

作者： Ling Guo Akil Narayan Yongle Liu Tao Zhou Department of Mathematics Shanghai Normal UniversityShanghai 200234China Department of Mathematics and Scientific Computing and Imaging InstituteUniversity of UtahSalt Lake CityUT 84112USA Department of Mathematics Southern University of Science and TechnologyShenzhen 518055China LSEC Institute of Computational Mathematics and Scientific/Engineering ComputingAcademy of Mathematics and Systems ScienceChinese Academy of SciencesBeijing 100190China.

One of the open problems in the field of forward uncertainty quantification(UQ)is the ability to form accurate assessments of uncertainty having only incomplete information about the distribution of random *** challenge is to efficiently make use of limited training data for UQ predictions of complex engineering problems,particularly with high dimensional random *** address these challenges by combining data-driven polynomial chaos expansions with a recently developed preconditioned sparse approximation approach for UQ *** first task in this two-step process is to employ the procedure developed in[1]to construct an"arbitrary"polynomial chaos expansion basis using a finite number of statistical moments of the random *** second step is a novel procedure to effect sparse approximation via l1 minimization in order to quantify the forward *** enhance the performance of the preconditioned l1 minimization problem,we sample from the so-called induced distribution,instead of using Monte Carlo(MC)sampling from the original,unknown probability *** demonstrate on test problems that induced sampling is a competitive and often better choice compared with sampling from asymptotically optimal measures(such as the equilibrium measure)when we have incomplete information about the *** demonstrate the capacity of the proposed induced sampling algorithm via sparse representation with limited data on test functions,and on a Kirchoff plating bending problem with random Young’s modulus.

关键词： Uncertainty quantification data-driven polynomial chaos expansions sparse approximation equilibrium measure induced measure

A DECOUPLING TWO-GRID METHOD FOR THE STEADY-STATE POISSON-NERNST-PLANCK EQUATIONS

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Journal of computational mathematics 2019年第4期37卷 556-578页

作者： Ying Yang Benzhuo Lu Yan Xie School of Mathematics and Computing Science Guangxi Colleges and Universities Key Laboratory of Data Analysis and Computation Guilin University of Electronic Technology Guilin 541004 China LSEC Institute of Computational Mathematics and Scientific/Engineering Computing the National Center for Mathematics and Interdisciplinary Sciences Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing 100190 China LSEC Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing 100190 China

Poisson-Nernst-Planck equations are widely used to describe the electrodiffusion of ions in a solvated biomolecular system. Two kinds of two-grid finite element algorithms are proposed to decouple the steady-state Poisson-Nernst-Planck equations by coarse grid finite element approximations. Both theoretical analysis and numerical experiments show the efficiency and effectiveness of the two-grid algorithms for solving Poisson-Nernst-Planck equations.

关键词： Poisson-Nernst-Planck equations Two-grid finite element method Decoupling method Error analysis Gummel iteration

Stellar Blend Image Classification Using computationally Efficient Gaussian Processes

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arXiv 2024年

作者： Eleh, Chinedu Zhang, Yunli Bidese, Rafael Priest, Benjamin W. Muyskens, Amanda L. Molinari, Roberto Billor, Nedret Department of Mathematics and Statistics Auburn University AuburnAL United States Department of Mechanical Engineering Auburn University AuburnAL United States Department of Biosystems Engineering Auburn University AuburnAL United States Center for Applied Scientific Computing Lawrence Livermore National Laboratory LivermoreCA United States Computational Engineering Division Lawrence Livermore National Laboratory LivermoreCA United States

Stellar blends, where two or more stars appear blended in an image, pose a significant visualization challenge in astronomy. Traditionally, distinguishing these blends from single stars has been costly and resource-intensive, involving sophisticated equipment and extensive expert analysis. This is especially problematic for analyzing the vast data volumes from surveys, such as Legacy Survey of Space and Time (LSST), Sloan Digital Sky Survey (SDSS), Dark Energy Spectroscopic Instrument (DESI), Legacy Imaging Survey and the Zwicky Transient Facility (ZTF). To address these challenges, we apply different normalizations and data embeddings on low resolution images of single stars and stellar blends, which are passed as inputs into machine learning methods and to a computationally efficient Gaussian process model (MuyGPs). MuyGPs consistently outperforms the benchmarked models, particularly on limited training data. Moreover, MuyGPs with rth root local min-max normalization achieves 83.8% accuracy. Furthermore, MuyGPs’ ability to produce confidence bands ensures that predictions with low confidence can be redirected to a specialist for efficient human-assisted labeling. © 2024, CC BY.

关键词： Dark energy

A Scalable Real-Time Data Assimilation Framework for Predicting Turbulent Atmosphere Dynamics

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arXiv 2024年

作者： Yin, Junqi Liang, Siming Liu, Siyan Bao, Feng Chipilski, Hristo G. Lu, Dan Zhang, Guannan National Center for Computational Sciences Oak Ridge National Laboratory Oak RidgeTN37831 United States Department of Mathematics Florida State University TallahasseeFL32306 United States Computational Sciences and Engineering Division Oak Ridge National Laboratory Oak RidgeTN37831 United States Department of Scientific Computing Florida State University TallahasseeFL32306 United States Computer Science and Mathematics Division Oak Ridge National Laboratory Oak RidgeTN37831 United States

The weather and climate domains are undergoing a significant transformation thanks to advances in AI-based foundation models such as FourCastNet, GraphCast, ClimaX and Pangu-Weather. While these models show considerable potential, they are not ready yet for operational use in weather forecasting or climate prediction. This is due to the lack of a data assimilation method as part of their workflow to enable the assimilation of incoming Earth system observations in real time. This limitation affects their effectiveness in predicting complex atmospheric phenomena such as tropical cyclones and atmospheric rivers. To overcome these obstacles, we introduce a generic real-time data assimilation framework and demonstrate its end-to-end performance on the Frontier supercomputer. This framework comprises two primary modules: an ensemble score filter (EnSF), which significantly outperforms the state-of-the-art data assimilation method, namely, the Local Ensemble Transform Kalman Filter (LETKF);and a vision transformer-based surrogate capable of real-time adaptation through the integration of observational data. The ViT surrogate can represent either physics-based models or AI-based foundation models. We demonstrate both the strong and weak scaling of our framework up to 1024 GPUs on the Exascale supercomputer, Frontier. Our results not only illustrate the framework's exceptional scalability on high-performance computing systems, but also demonstrate the importance of supercomputers in real-time data assimilation for weather and climate predictions. Even though the proposed framework is tested only on a benchmark surface quasigeostrophic (SQG) turbulence system, it has the potential to be combined with existing AI-based foundation models, making it suitable for future operational implementations. Copyright © 2024, The Authors. All rights reserved.

关键词： Supercomputers

The effect of smooth parametrizations on nonconvex optimization landscapes

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arXiv 2022年

作者： Levin, Eitan Kileel, Joe Boumal, Nicolas Dept. of Computing and Mathematical Sciences California Institute of Technology United States Dept. of Mathematics Oden Institute for Computational Engineering and Sciences University of Texas at Austin United States Switzerland

We develop new tools to study landscapes in nonconvex optimization. Given one optimization problem, we pair it with another by smoothly parametrizing the domain. This is either for practical purposes (e.g., to use smooth optimization algorithms with good guarantees) or for theoretical purposes (e.g., to reveal that the landscape satisfies a strict saddle property). In both cases, the central question is: how do the landscapes of the two problems relate? More precisely: how do desirable points such as local minima and critical points in one problem relate to those in the other problem? A key finding in this paper is that these relations are often determined by the parametrization itself, and are almost entirely independent of the cost function. Accordingly, we introduce a general framework to study parametrizations by their effect on landscapes. The framework enables us to obtain new guarantees for an array of problems, some of which were previously treated on a case-by-case basis in the literature. Applications include: optimizing low-rank matrices and tensors through factorizations;solving semidefinite programs via the Burer-Monteiro approach;training neural networks by optimizing their weights and biases;and quotienting out *** Codes 53Z50, 49J53, 90C26, 90C46 Copyright © 2022, The Authors. All rights reserved.

关键词： Optimization

A Partial Exact Penalty Function Approach for Constrained Optimization

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arXiv 2023年

作者： Xiao, Nachuan Liu, Xin Toh, Kim-Chuan The Institute of Operations Research and Analytics National University of Singapore Singapore State Key Laboratory of Scientific and Engineering Computing Academy of Mathematics and Systems Science Chinese Academy of Sciences University of Chinese Academy of Sciences China Department of Mathematics Institute of Operations Research and Analytics National University of Singapore Singapore119076 Singapore

In this paper, we focus on a class of constrained nonlinear optimization problems (NLP), where some of its equality constraints define a closed embedded submanifold M in Rn. Although NLP can be solved directly by various existing approaches for constrained optimization in Euclidean space, these approaches usually fail to recognize the manifold structure of M. To achieve better efficiency by utilizing the manifold structure of M in directly applying these existing optimization approaches, we propose a partial penalty function approach for NLP. In our proposed penalty function approach, we transform NLP into the corresponding constraint dissolving problem (CDP) in the Euclidean space, where the constraints that define M are eliminated through exact penalization. We establish the relationships on the constraint qualifications between NLP and CDP, and prove that NLP and CDP have the same stationary points and KKT points in a neighborhood of the feasible region under mild conditions. Therefore, various existing optimization approaches developed for constrained optimization in the Euclidean space can be directly applied to solve NLP through CDP. Preliminary numerical experiments demonstrate that by dissolving the constraints that define M, CDP gains superior computational efficiency when compared to directly applying existing optimization approaches to solve NLP, especially in high dimensional scenarios. Copyright © 2023, The Authors. All rights reserved.

关键词： Constrained optimization

A Consistent and Conservative Axisymmetric Lattice Boltzmann Method for Multiphase Electro-Hydrodynamic Flows

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SSRN

SSRN 2024年

作者： Liu, Xi Chai, Zhenhua Shi, Baochang Yuan, Xiaolei School of Mathematics and Statistics Huazhong University of Science and Technology Wuhan430074 China Institute of Interdisciplinary Research for Mathematics and Applied Science Huazhong University of Science and Technology Wuhan430074 China Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and Technology Wuhan430074 China College of Mathematics and Information Science Hebei University Baoding071002 China

In this paper, a consistent and conservative mathematical model is first developed for multiphase electro-hydrodynamic (EHD) flows, which has some distinct features in the volume conservation, the consistency of reduction, and the consistency of mass-momentum transport. Then a two-relaxation-time axisymmetric lattice Boltzmann (LB) method is proposed for multiphase EHD flows, and the present LB method consists of three LB models, which are used to solve the axisymmetric reduction-consistent and conservative Allen-Cahn equation for the multiphase field, the axisymmetric Poisson equation for the electric potential, and the axisymmetric consistent and conservative Navier-Stokes equations for the flow field. Owing to the reduction-consistent property, the present LB method can be used to handle a set of M (1≤M≤N) EHD fluids in N-phase system. To test the capacity of the LB method, the deformations of a single two-phase droplet and a ternary-phase compound droplet under a uniform electric field are considered. It is found that different from a single droplet with two deformation modes, the compound droplet has four deformation modes, and additionally, the effects of the electric strength, the conductivity ratio and the permittivity ratio on the compound droplet are also investigated. Finally, the compound droplet in the quaternary-phase system is also explored, and there are eight deformation modes that have not been reported in the available literature. © 2024, The Authors. All rights reserved.

关键词： Navier Stokes equations

Discrete energy analysis of the third-order variable-step BDF time-stepping for diffusion equations

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arXiv 2022年

作者： Liao, Hong-Lin Tang, Tao Zhou, Tao School of Mathematics Nanjing University of Aeronautics and Astronautics Nanjing211106 China MIIT Nanjing211106 China Division of Science and Technology BNU-HKBU United International College Guangdong Province Zhuhai China Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing100190 China

This is one of our series works on discrete energy analysis of the variable-step BDF schemes. In this part, we present stability and convergence analysis of the third-order BDF (BDF3) schemes with variable steps for linear diffusion equations, see e.g. [SIAM J. Numer. Anal., 58:2294-2314] and [Math. Comp., 90: 1207-1226] for our previous works on the BDF2 scheme. To this aim, we first build up a discrete gradient structure of the variable-step BDF3 formula under the condition that the adjacent step ratios are less than 1.4877, by which we can establish a discrete energy dissipation law. Mesh-robust stability and convergence analysis in the L2 norm are then obtained. Here the mesh robustness means that the solution errors are well controlled by the maximum time-step size but independent of the adjacent time-step ratios. We also present numerical tests to support our theoretical *** Codes 65M06, 65M12 © 2022, CC BY-NC-SA.

关键词： Partial differential equations

Phase-field based lattice Boltzmann method for containerless freezing

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arXiv 2024年

作者： Huang, Jiangxu Wang, Lei Chai, Zhenhua Shi, Baochang School of Mathematics and Statistics Huazhong University of Science and Technology Wuhan430074 China School of Mathematics and Physics China University of Geosciences Wuhan430074 China Institute of Interdisciplinary Research for Mathematics and Applied Science Huazhong University of Science and Technology Wuhan430074 China Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and Technology Wuhan430074 China

In this paper, a phase-field based lattice Boltzmann (LB) method is proposed to simulate solid-liquid phase change phenomena in multiphase systems. The method couples the thermal properties of the solidification front with the dynamics of the droplet interface, which enables the description of the complex interfacial changes during solid-liquid phase change process. The volume expansion or shrinkage of the droplet caused by the change of density during phase change is represented by adding a mass source term to the continuum equation. We first test the developed LB method by the three-phase Stefan problem and the droplet solidification on a cold surface, and the numerical results are in good agreement with the analytical and experimental solutions. Then the LB method is used to study the solidification problem with bubbles. The results show that the model can accurately capture the effect of bubbles on the solidification process, which is in good agreement with the previous work. Finally, a parametric study is carried out to examine the influences of some physical parameters on the sessile droplet solidification, and it is found that the time of droplet solidification increases with the increase of droplet volume and contact angle. Copyright © 2024, The Authors. All rights reserved.

关键词： Freezing