arXiv subjects

Changhong Mou

Publications and source records attributed to Changhong Mou.

31 records · Page 2Linked to original sources

arXiv2022

A Two-Level Galerkin Reduced Order Model for the Steady Navier-Stokes Equations

We propose, analyze, and investigate numerically a novel two-level Galerkin reduced order model (2L-ROM) for the efficient and accurate numerical simulation of the steady Navier-Stokes equations. In the first step of the 2L-ROM, a relatively low-dimensional nonlinear system is solved. In the second step, the Navier-Stokes equations are linearized around the solution found in the first step, and a higher-dimensional system for the linearized problem is solved. We prove an error bound for the new 2L-ROM and compare it to the standard one level ROM (1L-ROM) in the numerical simulation of the steady Burgers equation. The 2L-ROM significantly decreases (by a factor of $2$ and even $3$) the 1L-ROM computational cost, without compromising its numerical accuracy.

math.NA↗

arXiv2022

An Energy-Based Lengthscale for Reduced Order Models of Turbulent Flows

In this paper, we propose a novel reduced order model (ROM) lengthscale that is constructed by using energy distribution arguments. The new energy-based ROM lengthscale is fundamentally different from the current ROM lengthscales, which are built by using dimensional arguments. To assess the novel, energy-based ROM lengthscale, we compare it with a standard, dimensionality-based ROM lengthscale in two fundamentally different types of models: (i) the mixing-length ROM (ML-ROM), which is a ROM closure model; and (ii) the evolve-filter-relax ROM (EFR-ROM), which is a regularized ROM. We test the four combinations (i.e., ML-ROM and EFR-ROM equipped with the energy-based and dimensionality-based lengthscales) in the numerical simulation of the turbulent channel flow at $Re_τ = 395$. The numerical investigation yields the following conclusions: (i) The new energy-based ROM lengthscale is significantly (almost two orders of magnitude) larger than the standard dimensionality-based ROM lengthscale. As a result, the energy-based lengthscale yields more stable ML-ROMs and EFR-ROMs than the dimensionality-based lengthscale. (ii) The energy-based lengthscale displays the correct asymptotic behavior with respect to the ROM dimension, whereas the dimensionality-based lengthscale does not. (iii) The energy-based lengthscale yields ML-ROMs and (when significant filtering is effected) EFR-ROMs whose parameters are less sensitive (i.e., more robust) than the parameters of the ML-ROMs and EFR-ROMs based on the dimensionality-based lengthscale. The novel energy-based lengthscale could enable the development of better scale-aware ROM strategies for flow-specific applications and is expected to have long term applications in nuclear reactor thermal-hydraulics.

math.NA↗

arXiv2022

Verifiability of the Data-Driven Variational Multiscale Reduced Order Model

In this paper, we focus on the mathematical foundations of reduced order model (ROM) closures. First, we extend the verifiability concept from large eddy simulation to the ROM setting. Specifically, we call a ROM closure model verifiable if a small ROM closure model error (i.e., a small difference between the true ROM closure and the modeled ROM closure) implies a small ROM error. Second, we prove that a data-driven ROM closure (i.e., the data-driven variational multiscale ROM) is verifiable. Finally, we investigate the verifiability of the data-driven variational multiscale ROM in the numerical simulation of the one-dimensional Burgers equation and a two-dimensional flow past a circular cylinder at Reynolds numbers $Re=100$ and $Re=1000$.

math.NA↗

arXiv2022

Stochastic Data-Driven Variational Multiscale Reduced Order Models

Trajectory-wise data-driven reduced order models (ROMs) tend to be sensitive to training data, and thus lack robustness. We propose to construct a robust stochastic ROM closure (S-ROM) from data consisting of multiple trajectories from random initial conditions. The S-ROM is a low-dimensional time series model for the coefficients of the dominating proper orthogonal decomposition (POD) modes inferred from data. Thus, it achieves reduction both space and time, leading to simulations orders of magnitude faster than the full order model. We show that both the estimated POD modes and parameters in the S-ROM converge when the number of trajectories increases. Thus, the S-ROM is robust when the training data size increases. We demonstrate the S-ROM on a 1D Burgers equation with a viscosity $ν= 0.002$ and with random initial conditions. The numerical results verify the convergence. Furthermore, the S-ROM makes accurate trajectory-wise predictions from new initial conditions and with a prediction time far beyond the training range, and it quantifies the spread of uncertainties due to the unresolved scales.

math.NA↗

arXiv2022

An Efficient Data-Driven Multiscale Stochastic Reduced Order Modeling Framework for Complex Systems

Suitable reduced order models (ROMs) are computationally efficient tools in characterizing key dynamical and statistical features of nature. In this paper, a systematic multiscale stochastic ROM framework is developed for complex systems with strong chaotic or turbulent behavior. The new ROMs are fundamentally different from the traditional Galerkin ROM (G-ROM) or those deterministic ROMs that aim at minimizing the path-wise errors and applying mainly to laminar systems. Here, the new ROM focuses on recovering the large-scale dynamics to the maximum extent while it also exploits cheap but effective conditional linear functions as the closure terms to capture the statistical features of the medium-scale variables and its feedback to the large scales. In addition, physics constraints are incorporated into the new ROM. One unique feature of the resulting ROM is that it facilitates an efficient and accurate scheme for nonlinear data assimilation, the solution of which is provided by closed analytic formulae. Such an analytic solvable data assimilation solution significantly accelerates the computational efficiency and allows the new ROM to avoid many potential numerical and sampling issues in recovering the unobserved states from partial observations. The overall model calibration is efficient and systematic via explicit mathematical formulae. The new ROM framework is applied to complex nonlinear systems, in which the intrinsic turbulent behavior is either triggered by external random forcing or deterministic nonlinearity. It is shown that the new ROM significantly outperforms the G-ROM in both scenarios in terms of reproducing the dynamical and statistical features as well as recovering unobserved states via the associated efficient data assimilation scheme.

physics.comp-ph↗

arXiv2022

Reduced Order Model Closures: A Brief Tutorial

In this paper, we present a brief tutorial on reduced order model (ROM) closures. First, we carefully motivate the need for ROM closure modeling in under-resolved simulations. Then, we construct step by step the ROM closure model by extending the classical Galerkin framework to the spaces of resolved and unresolved scales. Finally, we develop the data-driven variational multiscale ROM closure and then we test it in fluid flow simulations. Our tutorial on ROM closures is structured as a sequence of questions and answers, and is aimed at first year graduate students and advanced undergraduate students. Our goal is not to explain the "how," but the "why." That is, we carefully explain the principles used to develop ROM closures, without focusing on particular approaches. Furthermore, we try to keep the technical details to a minimum and describe the general ideas in broad terms while citing appropriate references for details.

math.NA↗

arXiv2021

A Numerical Investigation of the Lengthscale in the Mixing-Length Reduced Order Model of the Turbulent Channel Flow

In this paper, we propose a novel reduced order model (ROM) lengthscale definition that is based on energy distribution arguments. This novel ROM lengthscale is fundamentally different from the current ROM lengthscales, which are generally based on dimensional arguments. As a first step in the assessment of the new, energy based ROM lengthscale, we compare it with a standard, dimensional based ROM lengthscale in the mixing-length ROM (ML-ROM) simulation of the turbulent channel flow at $Re_τ = 395$. The numerical investigation shows that the energy based ROM lengthscale yields a significantly more stable ML-ROM than the dimensional based ROM lengthscale. The new energy based lengthscale definition could allow the development of scale-aware reduced order modeling strategies that are better suited for flow-specific applications.

physics.flu-dyn↗

arXiv2020

Reduced Order Models for the Quasi-Geostrophic Equations: A Brief Survey

Reduced order models (ROMs) are computational models whose dimension is significantly lower than those obtained through classical numerical discretizations (e.g., finite element, finite difference, finite volume, or spectral methods). Thus, ROMs have been used to accelerate numerical simulations of many query problems, e.g., uncertainty quantification, control, and shape optimization. Projection-based ROMs have been particularly successful in the numerical simulation of fluid flows. In this brief survey, we summarize some recent ROM developments for the quasi-geostrophic equations (QGE) (also known as the barotropic vorticity equations), which are a simplified model for geophysical flows in which rotation plays a central role, such as wind-driven ocean circulation in mid-latitude ocean basins. Since the QGE represent a practical compromise between efficient numerical simulations of ocean flows and accurate representations of large scale ocean dynamics, these equations have often been used in the testing of new numerical methods for ocean flows. ROMs have also been tested on the QGE for various settings in order to understand their potential in efficient numerical simulations of ocean flows. In this paper, we survey the ROMs developed for the QGE in order to understand their potential in efficient numerical simulations of more complex ocean flows: We explain how classical numerical methods for the QGE are used to generate the ROM basis functions, we outline the main steps in the construction of projection-based ROMs (with a particular focus on the under-resolved regime, when the closure problem needs to be addressed), we illustrate the ROMs in the numerical simulation of the QGE for various settings, and we present several potential future research avenues in the ROM exploration of the QGE and more complex models of geophysical flows.

physics.flu-dyn↗

arXiv2020

Data-Driven Variational Multiscale Reduced Order Models

We propose a new data-driven reduced order model (ROM) framework that centers around the hierarchical structure of the variational multiscale (VMS) methodology and utilizes data to increase the ROM accuracy at a modest computational cost. The VMS methodology is a natural fit for the hierarchical structure of the ROM basis: In the first step, we use the ROM projection to separate the scales into three categories: (i) resolved large scales, (ii) resolved small scales, and (iii) unresolved scales. In the second step, we explicitly identify the VMS-ROM closure terms, i.e., the terms representing the interactions among the three types of scales. In the third step, we use available data to model the VMS-ROM closure terms. Thus, instead of phenomenological models used in VMS for standard numerical discretizations (e.g., eddy viscosity models), we utilize available data to construct new structural VMS-ROM closure models. Specifically, we build ROM operators (vectors, matrices, and tensors) that are closest to the true ROM closure terms evaluated with the available data. We test the new data-driven VMS-ROM in the numerical simulation of four test cases: (i) the 1D Burgers equation with viscosity coefficient $ν= 10^{-3}$; (ii) a 2D flow past a circular cylinder at Reynolds numbers $Re=100$, $Re=500$, and $Re=1000$; (iii) the quasi-geostrophic equations at Reynolds number $Re=450$ and Rossby number $Ro=0.0036$; and (iv) a 2D flow over a backward facing step at Reynolds number $Re=1000$. The numerical results show that the data-driven VMS-ROM is significantly more accurate than standard ROMs.

math.NA↗

arXiv2020

A Multifidelity Ensemble Kalman Filter with Reduced Order Control Variates

This work develops a new multifidelity ensemble Kalman filter (MFEnKF) algorithm based on linear control variate framework. The approach allows for rigorous multifidelity extensions of the EnKF, where the uncertainty in coarser fidelities in the hierarchy of models represent control variates for the uncertainty in finer fidelities. Small ensembles of high fidelity model runs are complemented by larger ensembles of cheaper, lower fidelity runs, to obtain much improved analyses at only small additional computational costs. We investigate the use of reduced order models as coarse fidelity control variates in the MFEnKF, and provide analyses to quantify the improvements over the traditional ensemble Kalman filters. We apply these ideas to perform data assimilation with a quasi-geostrophic test problem, using direct numerical simulation and a corresponding POD-Galerkin reduced order model. Numerical results show that the two-fidelity MFEnKF provides better analyses than existing EnKF algorithms at comparable or reduced computational costs.

math.NA↗

arXiv2020

Lagrangian Data-Driven Reduced Order Modeling of Finite Time Lyapunov Exponents

There are two main strategies for improving the projection-based reduced order model (ROM) accuracy: (i) improving the ROM, i.e., adding new terms to the standard ROM; and (ii) improving the ROM basis, i.e., constructing ROM bases that yield more accurate ROMs. In this paper, we use the latter. We propose new Lagrangian inner products that we use together with Eulerian and Lagrangian data to construct new Lagrangian ROMs. We show that the new Lagrangian ROMs are orders of magnitude more accurate than the standard Eulerian ROMs, i.e., ROMs that use standard Eulerian inner product and data to construct the ROM basis. Specifically, for the quasi-geostrophic equations, we show that the new Lagrangian ROMs are more accurate than the standard Eulerian ROMs in approximating not only Lagrangian fields (e.g., the finite time Lyapunov exponent (FTLE)), but also Eulerian fields (e.g., the streamfunction). We emphasize that the new Lagrangian ROMs do not employ any closure modeling to model the effect of discarded modes (which is standard procedure for low-dimensional ROMs of complex nonlinear systems). Thus, the dramatic increase in the new Lagrangian ROMs' accuracy is entirely due to the novel Lagrangian inner products used to build the Lagrangian ROM basis.

physics.flu-dyn↗

arXiv2019

Data-Driven Correction Reduced Order Models for the Quasi-Geostrophic Equations: A Numerical Investigation

This paper investigates the recently introduced data-driven correction reduced order model (DDC-ROM) in the numerical simulation of the quasi-geostrophic equations. The DDC-ROM uses available data to model the correction term that is generally used to represent the missing information in low-dimensional ROMs. Physical constraints are added to the DDC-ROM to create the constrained data-driven correction reduced order model (CDDC-ROM) in order to further improve its accuracy and stability. Finally, the DDC-ROM is tested on time intervals that are longer than the time interval over which it was trained. The numerical investigation shows that, for low-dimensional ROMs, both the DDC-ROM and CDDC-ROM perform better than the standard Galerkin ROM (G-ROM) and the CDDC-ROM provides the best results.

physics.flu-dyn↗

arXiv2018

Commutation Error in Reduced Order Modeling of Fluid Flows

For reduced order models (ROMs) of fluid flows, we investigate theoretically and computationally whether differentiation and ROM spatial filtering commute, i.e., whether the commutation error (CE) is nonzero. We study the CE for the Laplacian and two ROM filters: the ROM projection and the ROM differential filter. Furthermore, when the CE is nonzero, we investigate whether it has any significant effect on ROMs that are constructed by using spatial filtering. As numerical tests, we use the Burgers equation with viscosities $ν=10^{-1}$ and $ν=10^{-3}$ and a 2D flow past a circular cylinder at Reynolds numbers $Re=1$ and $Re=100$. Our investigation shows that: (i) the CE exists, and (ii) the CE has a significant effect on ROM development for low Reynolds numbers, but not so much for higher Reynolds numbers.

math.NA↗