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Lijian Jiang

Lijian Jiang appears in the imported research catalog. Authorship, coauthor and topic links are available while profile ownership is still unclaimed.

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math.NANumerical AnalysisMachine Learningmath.DSComputation

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Published work

13 published item(s)

preprint2026arXiv

Coupling-Informed Transport Maps for Bayesian Filtering in Nonlinear Dynamical Systems

A likelihood-free transport filtering method is proposed based on the couplings between state and observation variables. By exploiting a block-triangular structure in the transport map, the analysis step of filtering is reformulated as the minimization of the maximum mean discrepancy (MMD) between the true joint measure and its transport-based approximation. To circumvent the non-convexity in the MMD optimization, we introduce a training-free transport filter method via gradient flows, which leads to an analytic computation for the transport map that implies the steepest descent direction of the MMD. The proposed approach accurately approximates non-Gaussian filtering posteriors and avoids particle collapse. We provide a convergence analysis for the expectation of the MMD between the approximated posterior and the truth posterior. Finally, we extend the method to high-dimensional problems through domain localization. Numerical examples demonstrate the superior performance of our approach over conventional filtering methods in nonlinear, non-Gaussian scenarios.

0 citations0 reviewsNo code linkedOpen access
preprint2026arXiv

Reinforcement learning based data assimilation for unknown state model

Data assimilation (DA) has increasingly emerged as a critical tool for state estimation across a wide range of applications. It is significantly challenging when the governing equations of the underlying dynamics are unknown. To this end, various machine learning approaches have been employed to construct a surrogate state transition model in a supervised learning framework, which relies on pre-computed training datasets. However, it is often infeasible to obtain noise-free ground-truth state sequences in practice. To address this challenge, we propose a novel method that integrates reinforcement learning with ensemble-based Bayesian filtering methods, enabling the learning of surrogate state transition model for unknown dynamics directly from noisy observations, without using true state trajectories. Specifically, we treat the process for computing maximum likelihood estimation of surrogate model parameters as a sequential decision-making problem, which can be formulated as a discrete-time Markov decision process (MDP). Under this formulation, learning the surrogate transition model is equivalent to finding an optimal policy of the MDP, which can be effectively addressed using reinforcement learning techniques. Once the model is trained offline, state estimation can be performed in the online stage using filtering methods based on the learned dynamics. The proposed framework accommodates a wide range of observation scenarios, including nonlinear and partially observed measurement models. A few numerical examples demonstrate that the proposed method achieves superior accuracy and robustness in high-dimensional settings.

0 citations0 reviewsNo code linkedOpen access
preprint2024arXiv

Regularized coupling multiscale method for thermomechanical coupled problems

The coupling effects in multiphysics processes are often neglected in designing multiscale methods. The coupling may be described by a non-positive definite operator, which in turn brings significant challenges in multiscale simulations. In the paper, we develop a regularized coupling multiscale method based on the generalized multiscale finite element method (GMsFEM) to solve coupled thermomechanical problems, and it is referred to as the coupling generalized multiscale finite element method (CGMsFEM). The method consists of defining the coupling multiscale basis functions through local regularized coupling spectral problems in each coarse-grid block, which can be implemented by a novel design of two relaxation parameters. Compared to the standard GMsFEM, the proposed method can not only accurately capture the multiscale coupling correlation effects of multiphysics problems but also greatly improve computational efficiency with fewer multiscale basis functions. In addition, the convergence analysis is also established, and the optimal error estimates are derived, where the upper bound of errors is independent of the magnitude of the relaxation coefficient. Several numerical examples for periodic, random microstructure, and random material coefficients are presented to validate the theoretical analysis. The numerical results show that the CGMsFEM shows better robustness and efficiency than uncoupled GMsFEM.

0 citations0 reviewsNo code linkedOpen access
preprint2023arXiv

Data-driven reduced-order modeling for nonautonomous dynamical systems in multiscale media

In this article, we present data-driven reduced-order modeling for nonautonomous dynamical systems in multiscale media using Koopman operators. Different from the case of autonomous dynamical systems, the Koopman operator family of nonautonomous dynamical systems significantly depend on a time pair. In order to effectively estimate the time-dependent Koopman operators, a moving time window is used to decompose the snapshot data, and the extended dynamic mode decomposition method is applied to computing the Koopman operators in each local temporal domain. Many physical properties in multiscale media often vary in very different scales. In order to capture multiscale information well, the dimension of collected data may be high. To accurately construct the models of dynamical systems in multiscale media, we use high spatial dimension of observation data. It is challenging to compute the Koopman operators using the very high dimensional data. Thus, the strategy of reduced-order modeling is proposed to treat the difficulty. The proposed reduced-order modeling includes two stages: offline stage and online stage. In offline stage, a block-wise low rank decomposition is used to reduce the spatial dimension of initial snapshot data. For the nonautonomous dynamical systems, real-time observation data may be required to update the Koopman operators. The online reduced-order modeling is proposed to correct the offline reduced-order modeling. Three methods are developed for the online reduced-order modeling: fully online, semi-online and adaptive online. The adaptive online method automatically selects the fully online or semi-online and can achieve a good trade-off between modeling accuracy and efficiency.

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preprint2023arXiv

Perron-Frobenius operator filter for stochastic dynamical systems

The filtering problems are derived from a sequential minimization of a quadratic function representing a compromise between model and data. In this paper, we use the Perron-Frobenius operator in stochastic process to develop a Perron-Frobenius operator filter. The proposed method belongs to Bayesian filtering and works for non-Gaussian distributions for nonlinear stochastic dynamical systems. The recursion of the filtering can be characterized by the composition of Perron-Frobenius operator and likelihood operator. This gives a significant connection between the Perron-Frobenius operator and Bayesian filtering. We numerically fulfil the recursion through approximating the Perron-Frobenius operator by Ulam's method. In this way, the posterior measure is represented by a convex combination of the indicator functions in Ulam's method. To get a low rank approximation for the Perron-Frobenius operator filter, we take a spectral decomposition for the posterior measure by using the eigenfunctions of the discretized Perron-Frobenius operator. A convergence analysis is carried out and shows that the Perron-Frobenius operator filter achieves a higher convergence rate than the particle filter, which uses Dirac measures for the posterior. The proposed method is explored for the data assimilation of the stochastic dynamical systems. A few numerical examples are presented to illustrate the advantage of the Perron-Frobenius operator filter over particle filter and extend Kalman filter.

0 citations0 reviewsNo code linkedOpen access
preprint2022arXiv

Online multiscale model reduction for nonlinear stochastic PDEs with multiplicative noise

In this paper, an online multiscale model reduction method is presented for stochastic partial differential equations (SPDEs) with multiplicative noise, where the diffusion coefficient is spatially multiscale and the noise perturbation nonlinearly depends on the diffusion dynamics. It is necessary to efficiently compute all possible trajectories of the stochastic dynamics for quantifying model's uncertainty and statistic moments. The multiscale diffusion and nonlinearity may cause the computation intractable. To overcome the multiscale difficulty, a constraint energy minimizing generalized multiscale finite element method (CEM-GMsFEM) is used to localize the computation and obtain an effective coarse model. However, the nonlinear terms are still defined on a fine scale space after the Galerkin projection of CEM-GMsFEM is applied to the nonlinear SPDEs. This significantly impacts on the simulation efficiency by CEM-GMsFEM. To this end, a stochastic online discrete empirical interpolation method (DEIM) is proposed to treat the stochastic nonlinearity. The stochastic online DEIM incorporates offline snapshots and online snapshots. The offline snapshots consist of the nonlinear terms at the approximate mean of the stochastic dynamics and are used to construct an offline reduced model. The online snapshots contain some information of the current new trajectory and are used to correct the offline reduced model in an increment manner. The stochastic online DEIM substantially reduces the dimension of the nonlinear dynamics and enhances the prediction accuracy for the reduced model. Thus, the online multiscale model reduction is constructed by using CEM-GMsFEM and the stochastic online DEIM. A priori error analysis is carried out for the nonlinear SPDEs. We present a few numerical examples with diffusion in heterogeneous porous media and show the effectiveness of the proposed model reduction.

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preprint2016arXiv

A dimension reduction method with applications for coefficient inversion of diffusion equations

In this paper, we present a dimension reduction method to reduce the dimension of parameter space and state space and efficiently solve inverse problems. To this end, proper orthogonal decomposition (POD) and radial basis function (RBF) are combined to represent the solution of forward model with a form of variable separation. This POD-RBF method can be used to efficiently evaluate the model's output. A gradient regularization method is presented to solve the inverse problem with fast convergence. A generalized cross validation method is suggested to select the regularization parameter and differential step size for the gradient computation. Because the regularization method needs many model's evaluations. This is desirable for POD-RBF method. Thus, the POD-RBF method is integrated with the gradient regularization method to provide an efficient approach to solve inverse problems. We focus on the coefficient inversion of diffusion equations using the proposed approach. Based on different types of measurement data and different basis functions for coefficients, we present a few numerical examples for the coefficient inversion. The numerical results show that accurate reconstruction for the coefficient can be achieved efficiently.

0 citations0 reviewsNo code linkedOpen access
preprint2016arXiv

A novel variable-separation method based on sparse representation for stochastic partial differential equations

In this paper, we propose a novel variable-separation (NVS) method for generic multivariate functions. The idea of NVS is extended to to obtain the solution in tensor product structure for stochastic partial differential equations (SPDEs). Compared with many widely used variation-separation methods, NVS shares their merits but has less computation complexity and better efficiency. NVS can be used to get the separated representation of the solution for SPDE in a systematic enrichment manner. No iteration is performed at each enrichment step. This is a significant improvement compared with proper generalized decomposition. Because the stochastic functions of the separated representations obtained by NVS depend on the previous terms, this impacts on the computation efficiency and brings great challenge for numerical simulation for the problems in high stochastic dimensional spaces. In order to overcome the difficulty, we propose an improved least angle regression algorithm (ILARS) and a hierarchical sparse low rank tensor approximation (HSLRTA) method based on sparse regularization. For ILARS, we explicitly give the selection of the optimal regularization parameters at each step based on least angle regression algorithm (LARS) for lasso problems such that ILARS is much more efficient. HSLRTA hierarchically decomposes a high dimensional problem into some low dimensional problems and brings an accurate approximation for the solution to SPDEs in high dimensional stochastic spaces using limited computer resource. A few numerical examples are presented to illustrate the efficacy of the proposed methods.

0 citations0 reviewsNo code linkedOpen access
preprint2016arXiv

Multiscale model reduction method for Bayesian inverse problems of subsurface flow

This work presents a model reduction approach to the inverse problem in the application of subsurface flows. For the Bayesian inverse problem, the forward model needs to be repeatedly computed for a large number of samples to get a stationary chain. This requires large computational efforts. To significantly improve the computation efficiency, we use generalized multiscale finite element method and least-squares stochastic collocation method to construct a reduced computational model. To avoid the difficulty of choosing regularization parameter, hyperparameters are introduced to build a hierarchical model. We use truncated Karhunen-Loeve expansion (KLE) to reduce the dimension of the parameter spaces and decrease the mixed time of Markov chains. The techniques of hyperparameter and KLE are incorporated into the model reduction method. The reduced model is constructed offline. Then it is computed very efficiently in the online sampling stage. This strategy can significantly accelerate the evaluation of the Markov chain and the resultant posterior distribution converges fast. We analyze the convergence for the approximation between the posterior distribution by the reduced model and the reference posterior distribution by the full-order model. A few numerical examples in subsurface flows are carried out to demonstrate the performance of the presented model reduction method with application of the Bayesian inverse problem.

0 citations0 reviewsNo code linkedOpen access
preprint2013arXiv

A hybrid HDMR for mixed multiscale finite element method with application for flows in random porous media

Stochastic modeling has become a popular approach to quantify uncertainty in flows through heterogeneous porous media. The uncertainty in heterogeneous structure properties is often parameterized by a high-dimensional random variable. This leads to a deterministic problem in a high-dimensional parameter space and the numerical computation becomes very challengeable as the dimension of the parameter space increases. To efficiently tackle the high-dimensionality, we propose a hybrid high dimensional model representation (HDMR) technique, through which the high-dimensional stochastic model is decomposed into a moderate-dimensional stochastic model in a most active random space and a few one-dimensional stochastic models. The derived low-dimensional stochastic models are solved by incorporating sparse grid stochastic collocation method into the proposed hybrid HDMR. The porous media properties such as permeability are often heterogeneous. To treat the heterogeneity, we use a mixed multiscale finite element method (MMsFEM) to simulate each of derived stochastic models. To capture the non-local spatial features of the porous media and the important effects of random variables, we can hierarchically incorporate the global information individually from each of random parameters. This significantly enhances the accuracy of the multiscale simulation. The synergy of the hybrid HDMR and the MMsFEM reduces the stochastic model of flows in both stochastic space and physical space, and significantly decreases the computation complexity. We carefully analyze the proposed HDMR technique and the derived stochastic MMsFEM. A few numerical experiments are carried out for two-phase flows in random porous media and support the efficiency and accuracy of the MMsFEM based on the hybrid HDMR.

0 citations0 reviewsNo code linkedOpen access
preprint2012arXiv

A resourceful splitting technique with applications to deterministic and stochastic multiscale finite element methods

In this paper we use a splitting technique to develop new multiscale basis functions for the multiscale finite element method (MsFEM). The multiscale basis functions are iteratively generated using a Green's kernel. The Green's kernel is based on the first differential operator of the splitting. The proposed MsFEM is applied to deterministic elliptic equations and stochastic elliptic equations, and we show that the proposed MsFEM can considerably reduce the dimension of the random parameter space for stochastic problems. By combining the method with sparse grid collocation methods, the need for a prohibitive number of deterministic solves is alleviated. We rigorously analyze the convergence of the proposed method for both the deterministic and stochastic elliptic equations. Computational complexity discussions are also offered to supplement the convergence analysis. A number of numerical results are presented to confirm the theoretical findings.

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preprint2012arXiv

Expanded mixed multiscale finite element methods and their applications for flows in porous media

We develop a family of expanded mixed Multiscale Finite Element Methods (MsFEMs) and their hybridizations for second-order elliptic equations. This formulation expands the standard mixed Multiscale Finite Element formulation in the sense that four unknowns (hybrid formulation) are solved simultaneously: pressure, gradient of pressure, velocity and Lagrange multipliers. We use multiscale basis functions for the both velocity and gradient of pressure. In the expanded mixed MsFEM framework, we consider both cases of separable-scale and non-separable spatial scales. We specifically analyze the methods in three categories: periodic separable scales, $G$- convergence separable scales, and continuum scales. When there is no scale separation, using some global information can improve accuracy for the expanded mixed MsFEMs. We present rigorous convergence analysis for expanded mixed MsFEMs. The analysis includes both conforming and nonconforming expanded mixed MsFEM. Numerical results are presented for various multiscale models and flows in porous media with shales to illustrate the efficiency of the expanded mixed MsFEMs.

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preprint2011arXiv

Mixed Multiscale Finite Volume Methods for Elliptic Problems in Two-phase Flow Simulations

We develop a framework for constructing mixed multiscale finite volume methods for elliptic equations with multiple scales arising from flows in porous media. Some of the methods developed using the framework are already known \cite{jennylt03}; others are new. New insight is gained for the known methods and extra flexibility is provided by the new methods. We give as an example a mixed MsFV on uniform mesh in 2-D. This method uses novel multiscale velocity basis functions that are suited for using global information, which is often needed to improve the accuracy of the multiscale simulations in the case of continuum scales with strong non-local features. The method efficiently captures the small effects on a coarse grid. We analyze the new mixed MsFV and apply it to solve two-phase flow equations in heterogeneous porous media. Numerical examples demonstrate the accuracy and efficiency of the proposed method for modeling the flows in porous media with non-separable and separable scales.

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