Zhu Wang

Hongfei Fu, Hong Wang, Zhu Wang

In this paper, a reduced-order model (ROM) based on the proper orthogonal decomposition and the discrete empirical interpolation method is proposed for efficiently simulating time-fractional partial differential equations (TFPDEs). Both linear and nonlinear equations are considered. We demonstrate the effectiveness of the ROM by several numerical examples, in which the ROM achieves the same accuracy of the full-order model (FOM) over a long-term simulation while greatly reducing the computational cost. The proposed ROM is then regarded as a surrogate of FOM and is applied to an inverse problem for identifying the order of the time-fractional derivative of the TFPDE model. Based on the Levenberg--Marquardt regularization iterative method with the Armijo rule, we develop a ROM-based algorithm for solving the inverse problem. For cases in which the observation data is either uncontaminated or contaminated by random noise, the proposed approach is able to achieve accurate parameter estimation efficiently.

Traian Iliescu, Zhu Wang

We develop a variational multiscale proper orthogonal decomposition reduced-order model for turbulent incompressible Navier-Stokes equations. The error analysis of the full discretization of the model is presented. All error contributions are considered: the spatial discretization error (due to the finite element discretization), the temporal discretization error (due to the backward Euler method), and the proper orthogonal decomposition truncation error. Numerical tests for a three-dimensional turbulent flow past a cylinder at Reynolds number Re=1000 show the improved physical accuracy of the new model over the standard Galerkin and mixing-length proper orthogonal decomposition reduced-order models. The high computational efficiency of the new model is also showcased. Finally, the theoretical error estimates are confirmed by numerical simulations of a two-dimensional Navier-Stokes problem.

Traian Iliescu, Zhu Wang

This paper presents a theoretical and numerical investigation of the following practical question: Should the time difference quotients of the snapshots be used to generate the proper orthogonal decomposition basis functions? The answer to this question is important, since some published numerical studies use the time difference quotients, whereas other numerical studies do not. The criterion used in this paper to answer this question is the rate of convergence of the error of the reduced order model with respect to the number of proper orthogonal decomposition basis functions. Two cases are considered: the no_DQ case, in which the snapshot difference quotients are not used, and the DQ case, in which the snapshot difference quotients are used. The error estimates suggest that the convergence rates in the $C^0(L^2)$-norm and in the $C^0(H^1)$-norm are optimal for the DQ case, but suboptimal for the no_DQ case. The convergence rates in the $L^2(H^1)$-norm are optimal for both the DQ case and the no_DQ case. Numerical tests are conducted on the heat equation and on the Burgers equation. The numerical results support the conclusions drawn from the theoretical error estimates. Overall, the theoretical and numerical results strongly suggest that, in order to achieve optimal pointwise in time rates of convergence with respect to the number of proper orthogonal decomposition basis functions, one should use the snapshot difference quotients.

Xuping Xie, David Wells, Zhu Wang et al.

Standard ROMs generally yield spurious numerical oscillations in the simulation of convection-dominated flows. Regularized ROMs use explicit ROM spatial filtering to decrease these spurious numerical oscillations. The Leray ROM is a recently introduced regularized ROM that utilizes explicit ROM spatial filtering of the convective term in the Navier-Stokes equations. This paper presents the numerical analysis of the finite element discretization of the Leray ROM. Error estimates for the ROM differential filter, which is the explicit ROM spatial filter used in the Leray ROM, are proved. These ROM filtering error estimates are then used to prove error estimates for the Leray ROM. Finally, both the ROM filtering error estimates and the Leray ROM error estimates are numerically investigated in the simulation of the two-dimensional Navier-Stokes equations with an analytic solution.

Thi-Thao-Phuong Hoang, Lili Ju, Zhu Wang

The paper is concerned with overlapping domain decomposition and exponential time differencing for the diffusion equation discretized in space by cell-centered finite differences. Two localized exponential time differencing methods are proposed to solve the fully discrete problem: the first method is based on Schwarz iteration applied at each time step and involves solving stationary problems in the subdomains at each iteration, while the second method is based on the Schwarz waveform relaxation algorithm in which time-dependent subdomain problems are solved at each iteration. The convergence of the associated iterative solutions to the corresponding fully discrete multidomain solution and to the exact semi-discrete solution is rigorously proved. Numerical experiments are carried out to confirm theoretical results and to compare the performance of the two methods.

Jun Liu, Zhu Wang

In this paper, we analyze the convergence of several discretize-then-optimize algorithms, based on either a second-order or a fourth-order finite difference discretization, for solving elliptic PDE-constrained optimization or optimal control problems. To ensure the convergence of a discretize-then-optimize algorithm, one well-accepted criterion is to choose or redesign the discretization scheme such that the resultant discretize-then-optimize algorithm commutes with the corresponding optimize-then-discretize algorithm. In other words, both types of algorithms would give rise to exactly the same discrete optimality system. However, such an approach is not trivial. In this work, by investigating a simple distributed elliptic optimal control problem, we first show that enforcing such a stringent condition of commutative property is only sufficient but not necessary for achieving the desired convergence. We then propose to add some suitable $H_1$ semi-norm penalty/regularization terms to recover the lost convergence due to the inconsistency caused by the loss of commutativity. Numerical experiments are carried out to verify our theoretical analysis and also validate the effectiveness of our proposed regularization techniques.

Zhu Wang

For nonlinear reduced-order models, especially for those with non-polynomial nonlinearities, the computational complexity still depends on the dimension of the original dynamical system. As a result, the reduced-order model loses its computational efficiency, which, however, is its the most significant advantage. Nonlinear dimensional reduction methods, such as the discrete empirical interpolation method, have been widely used to evaluate the nonlinear terms at a low cost. But when the finite element method is utilized for the spatial discretization, nonlinear snapshot generation requires inner products to be fulfilled, which costs lots of off-line time. Numerical integrations are also needed over elements sharing the selected interpolation points during the simulation, which keeps on-line time high. In this paper, we extend the finite element method with interpolated coefficients to nonlinear reduced-order models. It approximates the nonlinear function in the reduced-order model by its finite element interpolation, which makes coefficient matrices of the nonlinear terms pre-computable and, thus, leads to great savings in the computational efforts. Due to the separation of spatial and temporal variables in the finite element interpolation, the discrete empirical interpolation method can be directly applied on the nonlinear functions. Therefore, the main computational hurdles when applying the discrete empirical interpolation method in the finite element context are conquered. We also establish a rigorous asymptotic error estimation, which shows that the proposed approach achieves the same accuracy as that of the standard finite element method under certain smoothness assumptions of the nonlinear functions. Several numerical tests are presented to validate the proposed method and verify the theoretical results.

Thi-Thao-Phuong Hoang, Lili Ju, Wei Leng et al.

In this paper we present and analyze a general framework for constructing high order explicit local time stepping (LTS) methods for hyperbolic conservation laws. In particular, we consider the model problem discretized by Runge-Kutta discontinuous Galerkin (RKDG) methods and design LTS algorithms based on strong stability preserving Runge-Kutta (SSP-RK) schemes, that allow spatially variable time step sizes to be used for time integrations in different regions. The proposed algorithms are of predictor-corrector type, in which the interface information along the time direction is first predicted based on the SSP-RK approximations and Taylor expansions, and then the fluxes over the region of interface are corrected to conserve mass exactly at each time step. Following the proposed framework, we detail the corresponding LTS schemes with accuracy up to the fourth order, and prove their conservation property and nonlinear stability for the scalar conservation laws. Numerical experiments are also presented to demonstrate excellent performance of the proposed LTS algorithms.

Yan Luo, Zhu Wang

A first-order, Monte Carlo ensemble method has been recently introduced for solving parabolic equations with random coefficients in [26], which is a natural synthesis of the ensemble-based, Monte Carlo sampling algorithm and the ensemble-based, first-order time stepping scheme. With the introduction of an ensemble average of the diffusion function, this algorithm leads to a single discrete system with multiple right-hand sides for a group of realizations, which could be solved more efficiently than a sequence of linear systems. In this paper, we pursue in the same direction and develop a new multilevel Monte Carlo ensemble method for solving random parabolic partial differential equations. Comparing with the approach in [26], this method possesses a high-order accuracy in time and further reduces the computational cost by using the multilevel Monte Carlo method. Rigorous numerical analysis shows the method achieves the optimal rate of convergence. Several numerical experiments are presented to illustrate the theoretical results.

Xiangru Li, Zhu Wang, Si Zeng et al.

The accuracy of the estimated stellar atmospheric parameter decreases evidently with the decreasing of spectral signal-to-noise ratio (SNR) and there are a huge amount of this kind observations, especially in case of SNR$<$30. Therefore, it is helpful to improve the parameter estimation performance for these spectra and this work studied the ($T_\texttt{eff}, \log~g$, [Fe/H]) estimation problem for LAMOST DR8 low-resolution spectra with 20$\leq$SNR$<$30. We proposed a data-driven method based on machine learning techniques. Firstly, this scheme detected stellar atmospheric parameter-sensitive features from spectra by the Least Absolute Shrinkage and Selection Operator (LASSO), rejected ineffective data components and irrelevant data. Secondly, a Multi-layer Perceptron (MLP) method was used to estimate stellar atmospheric parameters from the LASSO features. Finally, the performance of the LASSO-MLP was evaluated by computing and analyzing the consistency between its estimation and the reference from the APOGEE (Apache Point Observatory Galactic Evolution Experiment) high-resolution spectra. Experiments show that the Mean Absolute Errors (MAE) of $T_\texttt{eff}, \log~g$, [Fe/H] are reduced from the LASP (137.6 K, 0.195 dex, 0.091 dex) to LASSO-MLP (84.32 K, 0.137 dex, 0.063 dex), which indicate evident improvements on stellar atmospheric parameter estimation. In addition, this work estimated the stellar atmospheric parameters for 1,162,760 low-resolution spectra with 20$\leq$SNR$<$30 from LAMOST DR8 using LASSO-MLP, and released the estimation catalog, learned model, experimental code, trained model, training data and test data for scientific exploration and algorithm study.

Yuankai Teng, Zhu Wang, Lili Ju et al.

Due to the curse of dimensionality and the limitation on training data, approximating high-dimensional functions is a very challenging task even for powerful deep neural networks. Inspired by the Nonlinear Level set Learning (NLL) method that uses the reversible residual network (RevNet), in this paper we propose a new method of Dimension Reduction via Learning Level Sets (DRiLLS) for function approximation. Our method contains two major components: one is the pseudo-reversible neural network (PRNN) module that effectively transforms high-dimensional input variables to low-dimensional active variables, and the other is the synthesized regression module for approximating function values based on the transformed data in the low-dimensional space. The PRNN not only relaxes the invertibility constraint of the nonlinear transformation present in the NLL method due to the use of RevNet, but also adaptively weights the influence of each sample and controls the sensitivity of the function to the learned active variables. The synthesized regression uses Euclidean distance in the input space to select neighboring samples, whose projections on the space of active variables are used to perform local least-squares polynomial fitting. This helps to resolve numerical oscillation issues present in traditional local and global regressions. Extensive experimental results demonstrate that our DRiLLS method outperforms both the NLL and Active Subspace methods, especially when the target function possesses critical points in the interior of its input domain.

Anthony Gruber, Max Gunzburger, Lili Ju et al.

The popularity of deep convolutional autoencoders (CAEs) has engendered new and effective reduced-order models (ROMs) for the simulation of large-scale dynamical systems. Despite this, it is still unknown whether deep CAEs provide superior performance over established linear techniques or other network-based methods in all modeling scenarios. To elucidate this, the effect of autoencoder architecture on its associated ROM is studied through the comparison of deep CAEs against two alternatives: a simple fully connected autoencoder, and a novel graph convolutional autoencoder. Through benchmark experiments, it is shown that the superior autoencoder architecture for a given ROM application is highly dependent on the size of the latent space and the structure of the snapshot data, with the proposed architecture demonstrating benefits on data with irregular connectivity when the latent space is sufficiently large.

Yuankai Teng, Xiaoping Zhang, Zhu Wang et al.

Partial differential equations are often used to model various physical phenomena, such as heat diffusion, wave propagation, fluid dynamics, elasticity, electrodynamics and image processing, and many analytic approaches or traditional numerical methods have been developed and widely used for their solutions. Inspired by rapidly growing impact of deep learning on scientific and engineering research, in this paper we propose a novel neural network, GF-Net, for learning the Green's functions of linear reaction-diffusion equations in an unsupervised fashion. The proposed method overcomes the challenges for finding the Green's functions of the equations on arbitrary domains by utilizing physics-informed approach and the symmetry of the Green's function. As a consequence, it particularly leads to an efficient way for solving the target equations under different boundary conditions and sources. We also demonstrate the effectiveness of the proposed approach by experiments in square, annular and L-shape domains.