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A role of random slow manifolds in detecting stochastic bifurcation

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arXiv 2018年

作者： He, Ziying Cai, Rui Duan, Jinqiao Liu, Xianming Center for Mathematical Sciences Huazhong University of Sciences and Technology Wuhan430074 China School of Mathematics and Statistics Huazhong University of Sciences and Technology Wuhan430074 China Hubei Key Laboratory for Engineering Modeling and Scientific Computing Huazhong University of Sciences and Technology Wuhan430074 China Department of Applied Mathematics Illinois Institute of Technology ChicagoIL60616 United States

We consider the relation for the stochastic equilibrium states between the reduced system on a random slow manifold and the original system. This provides a theoretical basis for the reduction about sophisticated detailed models by the random slow manifold without significant damage to the overall qualitative properties. Based on this result, we reveal a role of random slow manifolds in detecting stochastic bifurcation by an example. The example exhibits a stochastic bifurcation phenomenon and possesses a random slow manifold that carries the stochastic bifurcation information of the system. Specifically, the lower dimensional reduced system on the random slow manifold retains the stochastic bifurcation phenomenon from the original *** Codes 60H10, 58J65 Copyright © 2018, The Authors. All rights reserved.

关键词： Perturbation techniques

Phase-field-based lattice Boltzmann method for the transport of insoluble surfactant in two-phase flows

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arXiv 2024年

作者： Zhan, Chengjie Liang, Hong Chai, Zhenhua Shi, Baochang 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 Department of Physics Hangzhou Dianzi University Hangzhou310018 China

In this work, we present a general second-order phase-field model for the transport of insoluble surfactant in incompressible two-phase flows. In this model, the second-order local Allen-Cahn equation is applied for interface capturing, a general form of the simple scalar transport equation [S. S. Jain, J. Comput. Phys. 515, 113277 (2024)] is adopted for interface-confined surfactant, and the consistent and conservative Navier-Stokes equations with the Marangoni force is used for fluid flows. To solve this model, we further developed a mesoscopic lattice Boltzmann (LB) method, in which the LB model for surfactant transport equation is proposed under the general LB framework for the convection-diffusion type equation, and it can correctly recover the governing equation for surfactant transport. The accuracy of the present LB method is tested by several benchmark problems, and the numerical results show it has a good performance for the transport of the insoluble surfactant in two-phase flows. © 2024, CC BY-NC-ND.

关键词： Navier Stokes equations

Slow manifolds for stochastic systems with non-Gaussian stable Lévy noise

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arXiv 2017年

作者： Yuan, Shenglan Hu, Jianyu Liu, Xianming Duan, Jinqiao Center for Mathematical Sciences Huazhong University of Sciences and Technology Wuhan430074 China School of Mathematics and Statistics Huazhong University of Sciences and Technology Wuhan430074 China Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Sciences and Technology Wuhan430074 China Department of Applied Mathematics Illinois Institute of Technology ChicagoIL60616 United States

This work is concerned with the dynamics of a class of slow-fast stochastic dynamical systems with non-Gaussian stable Lévy noise with a scale parameter. Slow manifolds with exponentially tracking property are constructed, eliminating the fast variables to reduce the dimension of these coupled dynamical systems. It is shown that as the scale parameter tends to zero, the slow manifolds converge to critical manifolds in distribution, which helps understand long time dynamics. The approximation of slow manifolds with error estimate in distribution are also considered. Copyright © 2017, The Authors. All rights reserved.

关键词： Differential equations

Coarse-Grid Selection Using Simulated Annealing

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arXiv 2021年

作者： Zaman, Tareq U. MacLachlan, Scott P. Olson, Luke N. West, Matt Scientific Computing Program Memorial University of Newfoundland NL Canada Department of Mathematics and Statistics Memorial University of Newfoundland NL Canada Department of Computer Science University of Illinois at Urbana-Champaign IL United States Department of Mechanical Science and Engineering University of Illinois at Urbana-Champaign IL United States

Multilevel techniques are efficient approaches for solving the large linear systems that arise from discretized partial differential equations and other problems. While geometric multigrid requires detailed knowledge about the underlying problem and its discretization, algebraic multigrid aims to be less intrusive, requiring less knowledge about the origin of the linear system. A key step in algebraic multigrid is the choice of the coarse/fine partitioning, aiming to balance the convergence of the iteration with its cost. In work by MacLachlan and Saad [1], a constrained combinatorial optimization problem is used to define the "best" coarse grid within the setting of a two-level reduction-based algebraic multigrid method and is shown to be NP-complete. Here, we develop a new coarsening algorithm based on simulated annealing to approximate solutions to this problem, which yields improved results over the greedy algorithm developed previously. We present numerical results for test problems on both structured and unstructured meshes, demonstrating the ability to exploit knowledge about the underlying grid structure if it is available. © 2021, CC BY.

关键词： Simulated annealing

Construction and Validation of an Obstetric Risk Classification Engine to Support Intensive Care Unit (ICU) Admission Using Machine Learning

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SSRN

SSRN 2022年

作者： Soares, Fabiano Miguel Rose, Lívia Ohana da Rocha Oak Cecatti, José Guilherme Luz, Adriana Gomes Awe, Oluwafunmilola Deborah Esmi, Estevão Laureano de Carvalho Pacagnella, Rodolfo Department of Obstetrics and Gynecology Faculty of Medical Sciences State University of Campinas SP Campinas Brazil Department of Applied and Computational Mathematics Institute of Mathematics Statistics and Scientific Computing State University of Campinas SP Campinas Brazil Campus Universitário R. Alexander Flemming 101. Barão Geraldo Campinas SP. 13083-872 Brazil

Introduction: Maternal mortality remains a global problem, despite the efforts of many international organizations such as the World Health Organization to reduce it. In recent times, scientific evidence useful in the reduction of maternal mortality has become well established, which include implementation of bundles in emergency, and provision of Intensive Care Units (ICU) for obstetric complications. This is a major factor in the management of pregnant women, however, there is a need for optimization of the Intensive Care Unit (ICU) when the need arises. We aimed to develop a clinical support tool to guide decisions on which obstetric population required ICU ***: This study describes the development and validation of an obstetric risk classification engine among patients with maternal morbidity We used a supervised machine learning technique in a multicentric database of pregnant population including 9,550 women with Severe Maternal Morbidity. A decision tree model and two regression tree models were tested (Random Forest, Gradient boosting Machine -GBM, and Extreme Gradient Boosting -XGBoost) and after application, the accuracy, sensitivity and specificity were compared and the differences between the predictions were analyzed using Wilcoxon ***: The XGBoost algorithm showed the highest efficiency with an accuracy of 85%, sensitivity of 42%, specificity of 97%, AUC on the ROC curve of 86.7% and the estimated prevalence of ICU use (11.6%, in contrast to the actual prevalence of ICU, 21.52%), indicating the need to optimize the use of intensive ***: The Risk engine developed showed positive results, hence, can serve as a reinforcement factor in the management of women with complications so that those who need intensive care can be correctly identified, thereby optimizing the use of intensive beds. © 2022, The Authors. All rights reserved.

关键词： Intensive care units

Multilevel hierarchical decomposition of finite element white noise with application to multilevel Markov chain Monte Carlo

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arXiv 2020年

作者： Fairbanks, Hillary R. Villa, Umberto Vassilevski, Panayot S. Center for Applied Scientific Computing Lawrence Livermore National Laboratory LivermoreCA United States Electrical & Systems Engineering Department Washington University in St. Louis St. LouisMO United States Fariborz Maseeh Department of Mathematics and Statistics Portland State University PortlandOR United States

In this work we develop a new hierarchical multilevel approach to generate Gaussian random field realizations in an algorithmically scalable manner that is well-suited to incorporate into multilevel Markov chain Monte Carlo (MCMC) algorithms. This approach builds off of other partial differential equation (PDE) approaches for generating Gaussian random field realizations;in particular, a single field realization may be formed by solving a reaction-diffusion PDE with a spatial white noise source function as the righthand side. While these approaches have been explored to accelerate forward uncertainty quantification tasks, e.g. multilevel Monte Carlo, the previous constructions are not directly applicable to multilevel MCMC frameworks which build fine scale random fields in a hierarchical fashion from coarse scale random fields. Our new hierarchical multilevel method relies on a hierarchical decomposition of the white noise source function in L2 which allows us to form Gaussian random field realizations across multiple levels of discretization in a way that fits into multilevel MCMC algorithmic frameworks. After presenting our main theoretical results and numerical scaling results to showcase the utility of this new hierarchical PDE method for generating Gaussian random field realizations, this method is tested on a four-level MCMC algorithm to explore its feasibility. Copyright © 2020, The Authors. All rights reserved.

关键词： Gaussian distribution

Deterministic and Bayesian Characterization of Quantum computing Devices

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arXiv 2023年

作者： Peng, Zhichao Appelö, Daniel Petersson, N. Anders Motamed, Mohammad Garcia, Fortino Cho, Yujin Department of Mathematics Michigan State University United States Department of Mathematics Virginia Tech United States Center for Applied Scientific Computing Lawrence Livermore National Laboratory United States Department of Mathematics and Statistics University of New Mexico United States Courant Institute of Mathematical Sciences New York University United States Quantum Coherent Device Physics Lawrence Livermore National Laboratory United States

Motivated by the noisy and fluctuating behavior of current quantum computing devices, this paper presents a data-driven characterization approach for estimating transition frequencies and decay times in a Lindbladian dynamical model of a superconducting quantum device. The data includes parity events in the transition frequency between the first and second excited states. A simple but effective mathematical model, based upon averaging solutions of two Lindbladian models, is demonstrated to accurately capture the experimental observations. A deterministic point estimate of the device parameters is first performed to minimize the misfit between data and Lindbladian simulations. These estimates are used to make an informed choice of prior distributions for the subsequent Bayesian inference. An additive Gaussian noise model is developed for the likelihood function, which includes two hyper-parameters to capture the noise structure of the data. The outcome of the Bayesian inference are posterior probability distributions of the transition frequencies, which for example can be utilized to design risk neutral optimal control pulses. The applicability of our approach is demonstrated on experimental data from the Quantum Device and Integration Testbed (QuDIT) at Lawrence Livermore National Laboratory, using a tantalum-based superconducting transmon *** Codes 62F15, 81P68 © 2023, CC BY-SA.

关键词： Probability distributions

A Phase-Field-Based Multiple-Relaxation-Time Lattice Boltzmann Method for Incompressible Multiphase Flows with Density and Viscosity Contrasts

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SSRN

SSRN 2022年

作者： Yuan, Xiaolei Wu, Yao Zhang, Chunhua Chai, Zhenhua Shi, Baochang College of Mathematics and Information Science Hebei University Baoding071002 China Institute of Interdisciplinary Research for Mathematics and Applied Science School of Mathematics and Statistics 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 Department of Mechanics and Aerospace Engineering Southern University of Science and Technology Guangdong Shenzhen518055 China

In this work, we present a unified framework of phase-field-based multiple-relaxation-time lattice Boltzmann (MRT-LB) method for incompressible multiphase flows with density and viscosity contrasts where a block-lower-triangular relaxation matrix is introduced. The present framework includes the classic MRT-LB model and central-moments-based LB model (CLBM, a particular version of the MRT-LB model). The governing equations of incompressible multiphase flows can be reproduced precisely through the direct Taylor expansion method at the second-order of expansion parameters. In this work, the pressure $p$ is re-estimated, and we prove that the present pressure expression is suitable for classic MRT model, CLBM and various lattice structures. In addition, theoretical analysis and modeling are both carried out in the velocity space. As a result, the present model has a similar structure to the traditional Bhatnagar-Gross-Krook (BGK) model and is very easy to analyze and implement. We then test the present model (mainly CLBM) by five classic physical problems. The results show that the present model is found to be accurate and has better numerical stability over the BGK model, and has the ability to simulate multiphase flow problems with large density ratio ($1000$). © 2022, The Authors. All rights reserved.

关键词： Multiphase flow

HOMOGENIZATION OF THE DISTRIBUTION-DEPENDENT STOCHASTIC ABSTRACT FLUID MODELS

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arXiv 2023年

作者： Chen, Junlong Qiu, Zhaoyang Tang, Yanbin School of Sciences Great Bay University Great Bay Institute for Advanced Study Dongguan523000 China School of Mathematics University of Science and Technology of China Anhui Hefei230026 China School of Applied Mathematics Nanjing University of Finance and Economics Nanjing210046 China School of Mathematics and Statistics Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and Technology Hubei Wuhan430074 China

In this paper, we study the homogenization of the distribution-dependent stochastic abstract fluid models by combining the two−scale convergence and martingale representative approach. A general framework of the homogenization research is established for stochastic abstract fluid models which are the type of nonlinear partial differential equations including the (distribution-dependent) stochastic Navier-Stokes equations, stochastic magneto-hydrodynamic equations, stochastic Boussinesq equations, stochastic micropolar equations, stochastic Allen-Cahn equations. Copyright © 2023, The Authors. All rights reserved.

关键词： Navier Stokes equations