Qiuqi Li

Lijian Jiang, Qiuqi Li

In this paper, we propose a model's sparse representation based on reduced mixed generalized multiscale finite element (GMsFE) basis methods for elliptic PDEs with random inputs. Mixed generalized multiscale finite element method (GMsFEM) is one of the accurate and efficient approaches to solve multiscale problem in a coarse grid with local mass conservation. When the inputs of the PDEs are parameterized by the random variables, the GMsFE basis functions usually depend on the random parameters. This leads to a large number degree of freedoms for the mixed GMsFEM and substantially impacts on the computation efficiency. In order to overcome the difficulty, we develop reduced mixed GMsFE basis methods such that the multiscale basis functions are independent of the random parameters and span a low-dimensional space. To this end, a greedy algorithm is used to find a set of optimal samples from a training set scattered in the parameter space. Reduced mixed GMsFE basis functions are constructed based on the optimal samples using two optimal sampling strategies: basis-oriented cross-validation and proper orthogonal decomposition. Although the dimension of the space spanned by the reduced mixed GMsFE basis functions is much smaller than the dimension of the original full order model, the online computation still depends on the number of coarse degree of freedoms. To significantly improve the online computation, we integrate the reduced mixed GMsFE basis methods with sparse tensor approximation and obtain a sparse representation for the model's outputs. The sparse representation is very efficient for evaluating the model's outputs for many instances of parameters. To illustrate the efficacy of the proposed methods, we present a few numerical examples for multsicale problems with random inputs.

Lingling Ma, Qiuqi Li, Lijian Jiang

In this paper, a local-global model reduction method is presented to solve stochastic optimal control problems governed by partial differential equations (PDEs). If the optimal control problems involve uncertainty, we need to use a few random variables to parameterize the uncertainty. The stochastic optimal control problems require solving coupled optimality system for a large number of samples in the stochastic space to quantify the statistics of the system response and explore the uncertainty quantification. Thus the computation is prohibitively expensive. To overcome the difficulty, model reduction is necessary to significantly reduce the computation complexity. We exploit the advantages from both reduced basis method and Generalized Multiscale Finite Element Method (GMsFEM) and develop the local-global model reduction method for stochastic optimal control problems with PDE constraints. This local-global model reduction can achieve much more computation efficiency than using only local model reduction approach and only global model reduction approach. We recast the stochastic optimal problems in the framework of saddle-point problems and analyze the existence and uniqueness of the optimal solutions of the reduced model. In the local-global approach, most of computation steps are independent of each other. This is very desirable for scientific computation. Moreover, the online computation for each random sample is very fast via the proposed model reduction method. This allows us to compute the optimality system for a large number of samples. To demonstrate the performance of the local-global model reduction method, a few numerical examples are provided for different stochastic optimal control problems.

Qiuqi Li, Lijian Jiang

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.

Qiuqi Li, Chang Liu, Yifei Yang

Dynamic mode decomposition (DMD) is a widely used data-driven algorithm for predicting the future states of dynamical systems. However, its standard formulation often struggles with poor long-term predictive accuracy. To address this limitation, we propose a localized DMD (LDMD) framework that improves prediction performance by integrating DMD's strong linear forecasting capabilities with time-domain segmentation techniques. In this framework, the temporal domain is segmented into multiple subintervals, within which snapshot matrices are constructed and localized predictions are performed. We first present the localized DMD method with predefined segmentation, and then explore an adaptive segmentation strategy to further enhance computational efficiency and prediction robustness. Furthermore, we conduct an error analysis that provides the upper bound of the local and global truncation error for the proposed framework. The effectiveness of LDMD is demonstrated on four benchmark problems-Burgers', Allen-Cahn, nonlinear Schrodinger, and Maxwell's equations. Numerical results show that LDMD significantly enhances long-term predictive accuracy while preserving high computational efficiency.

Qiuqi Li, Yiting Liu, Jin Zhao et al.

Parametric partial differential equations (PDEs) are fundamental for modeling a wide range of physical and engineering systems influenced by uncertain or varying parameters. Traditional neural network-based solvers, such as Physics-Informed Neural Networks (PINNs) and Deep Galerkin Methods, often face challenges in generalization and long-time prediction efficiency due to their dependence on full space-time approximations. To address these issues, we propose a novel and scalable framework that significantly enhances the Neural Galerkin Method (NGM) by incorporating the Meta-Auto-Decoder (MAD) paradigm. Our approach leverages space-time decoupling to enable more stable and efficient time integration, while meta-learning-driven adaptation allows rapid generalization to unseen parameter configurations with minimal retraining. Furthermore, randomized sparse updates effectively reduce computational costs without compromising accuracy. Together, these advancements enable our method to achieve physically consistent, long-horizon predictions for complex parameterized evolution equations with significantly lower computational overhead. Numerical experiments on benchmark problems demonstrate that our methods performs comparatively well in terms of accuracy, robustness, and adaptability.