会议主题：研讨偏微分方程高效算法以及多物理场问题模拟的最新研究进展。

会议时间：2021年4月16-18日。

邀请专家：（以姓氏首字母顺序排序）

会议议程

会议地点：四川师范大学数学学院学术报告厅201

会议承办单位：四川师范大学科学技术处、四川师范大学数学科学学院

联系人：李鸿亮  Email：lhl@sicnu.edu.cn

联系电话：028-84761502.

Title: MIM: A deep mixed residual method for solving high-order partial differential equations

Speaker: Jingrun Chen, Soochow University（陈景润，苏州大学）

Abstract: In recent years, a significant amount of attention has been paid to solve partial differential equations (PDEs) by deep learning. For example, deep Galerkin method uses the PDE residual in the least-squares sense as the loss function and a deep neural network (DNN) to approximate the PDE solution. In this work, we propose a deep mixed residual method (MIM) to solve PDEs with high-order derivatives. In MIM, we first rewrite a high-order PDE into a first-order system, very much in the same spirit as local discontinuous Galerkin method and mixed finite element method in classical numerical methods for PDEs. We then use the residual of first-order system in the least-squares sense as the loss function, which is in close connection with least-squares finite element method. Numerous results of MIM with different loss functions and different choice of DNNs are given. In most cases, MIM provides better approximations (not only for high-derivatives of the PDE solution but also for the PDE solution itself) than DGM with nearly the same DNN and the same execution time, sometimes by more than one order of magnitude. When different DNNs are used, in many cases, MIM provides even better approximations than MIM with only one DNN, sometimes by more than one order of magnitude. Numerical results also show some interesting connections between MIM and classical numerical methods. Therefore, we expect MIM to open up a possibly systematic way to understand and improve deep learning for solving PDEs from the perspective of classical numerical analysis.

Title：Can DG go beyond FE in efficiency?

Speaker: Ruo Li, Peking University（李若，北京大学）

Abstract: The discontinuous Galerkin method has attracted tremendous amount of attentions in the last decades since it has been applied to problems with regular solutions, the 2nd order elliptic equation for example. In spite of its well-known advantages, the efficiency of discontinuous Galerkin method for problems with very regular solutions is a weak point which has often been attacked at. In this talk, I will show that the discontinuous Galerkin method may go beyond the continuous finite element method in efficiency for elliptic problems, where is the traditional area for the finite element method to outperform. Our technique to help DG out is to construct a brand-new approximate space which will be clarified in my talk.

Title: Efficient Model Reduction Methods for PDEs with Random Inputs and Applications for Bayesian Inversion

Speaker: Qifeng Liao, （廖奇峰，上海科技大学）

Abstract: Over the past few decades there has been a rapid development in numerical methods for solving partial differential equations (PDEs) with random inputs. This explosion in interest has been driven by the need of conducting uncertainty quantification for practical problems. In particular, uncertainty quantification for problems with high-dimensional random inputs gains a lot of interest. It is known that traditional Monte Carlo methods converge slowly. New spectral methods such as polynomial chaos and collocation methods can converge quickly, but suffer from the so-called ``curse of dimensionality". Taking the sparse grid collocation method for example, when the probability space has high dimensionality, the number of points required for accurate collocation solutions can be large, and it may be costly to construct the solution. We first show that this process can be made more efficient by combining collocation with reduced basis methods, in which a greedy algorithm is used to identify a reduced problem to which the collocation method can be applied. We demonstrate with numerical experiments that this is achieved with essentially no loss of accuracy. To further resolve problems with very high-dimensional parameters, we next develop hierarchical reduced basis techniques based on an ANOVA (analysis of variance) decomposition of parameter spaces. Moreover, our reduced basis ANOVA approach can provide an efficient surrogate for high-dimensional Bayesian inverse problems. This is joint work with Howard Elman of the University of Maryland, Guang Lin of Purdue University, and Jinglai Li of the University of Birmingham.

Title: The dimensional simulation of ionization damage effect of semiconductor devices

Speaker: Hongliang Li, Sichuan Normal University（李鸿亮，四川师范大学）

Abstract: Gamma irradiation is the key environmental factor that causes the performance degradation or even a failure of satellite IC. There are two basic effects of ionizing radiation damage on semiconductor devices: total ionizing dose effect, that is, the radiation damage accumulates with the increase of radiation dose, and enhanced low dose rate sensitivity, that is, when the total radiation dose is fixed, the radiation damage increases with the decrease of dose rate. The modeling and Simulation of ionization damage effect of devices have been an important issue. In this report, we will introduce the physical mechanism of ionization damage, quantitative physical model, finite element discrete numerical scheme, and three-dimensional numerical simulation results of total ionizing dose effect of MOS transistor and ELDRS effect of bipolar transistor.

Title: Robust Finite Elements for Strain Gradient Elasticity

Speaker：Pingbing Ming（明平兵）

Abstract: I will give an overview of the robust finite elements for the strain elasticity model. We firstly prove an H² Korn's inequality and the corresponding broken H² Korn's inequality, the former yields the well-posedness of the strain elasticity model, while the latter serves as the guideline for designing the robust finite elements, the robustness is understood in the sense that the rate of the convergence is uniform with respect to the small material parameter. A family of finite elements are proposed and analyzed, numerical results are also reported to confirm the accuracy and efficiency of the elements.

Title：Steady and unsteady flexural shell models and its numerical computation

Speaker：Xiaoqin Shen（沈晓芹，西安理工大学）

Abstract：The theory of elastic shells is one of the important branches of the theory of elasticity. In this talk, we discuss steady and unsteady flexural shell models. For steady flexural shell model, we propose a conforming finite element method coupling penalty method. For unsteady flexural shell model, we construct space-time full discretization schemes. The corresponding analyses of existence, uniqueness, stability, convergence and priori error estimates are given. Finally, we provide numerical experiments with several kinds of shells to demonstrate the efficiency of models and the stability and convergence of numerical schemes.

Title：High order time-domain numerical methods for magnetohydrodynamic equations

Speaker：Liwei Xu（徐立伟，电子科技大学）

Abstract：Magnetohydrodynamic (MHD) equations are of great importance in sciences and engineering. In this talk, we present some high order time-domain numerical schemes, mainly based on decouple BDF or Crank-Nicolson schemes, for several kinds of magnetohydrodynamic equations, including incompressible MHD, resistive MHD,and two-phase MHD equations etc. Related theoretical and numerical results will be given to illustrate the accuracy of numerical schemes.

Title：Discontinuous Galerkin method for a distributed optimal control problem of time fractional diffusion equation

Speaker: Xiaoping Xie, Sichuan University（谢小平，四川大学）

Abstract: This talk is devoted to the numerical analysis of a control constrained distributed optimal control problem subject to a time fractional diffusion equation with non-smooth initial data. The solutions of the state and co-state are decomposed into singular and regular parts, and some growth estimates are obtained for the singular parts. Following the variational discretization concept, a full discretization is applied to the state and co-state equations by using conforming linear finite element method in space and piecewise constant discontinuous Galerkin method in time. Error estimates are derived by employing the growth estimates. In particular, graded temporal grids are adopted to obtain the first-order temporal accuracy. Finally, numerical experiments are provided to verify the theoretical results.

Title: Model reduction based on the Onsager variational principle

Speaker: Xianmin Xu, （许现民，中国科学院数学与系统科学研究院）

Abstract: The Onsager variational principle is a fundamental law for irreversible processes in nonequilibrium statistical physics. It has been used to model many complicated phenomena in soft matter. In this talk, we will show it can be used as a powerful approximation tool for many complicated two-phase flow problems. In particular, it can be used to derive some efficient numerical method for wetting problems on inhomogeneous surfaces. The variational principle also provides a theoretical tool for the moving finite element method for gradient flow systems. All collaborators will be acknowledged during the talk.

Title: An inverse eigenvalue problem for modified pseudo-Jacobi matrices

Speaker: Wei-Ru Xu, Sichuan Normal University（徐伟孺，四川师范大学）

Abstract: In the non-Hermitian quantum mechanics, there exists the construction problem of pseudo-Jacobi matrices, which is derived from the discretization and truncation of Schrodinger equation, the construction of Hamiltonian system of an indefinite Toda lattice and the symmetry reduction of the Wess-Zumino-Novikov-Witten model in quantum field theory, etc. In mathematics, this problem is referred to as pseudo-Jacobi matrix inverse eigenvalue problem, which concerns the reconstruction of a specified pseudo-Jacobi matrix from the prescribed spectral data. In this talk, we introduce an inverse eigenvalue problem for matrices that are obtained from pseudo-Jacobi matrices by only modifying the (1,r)-th and (r,1)-th entries,                                               . Necessary and sufficient conditions under which the problem is solvable are derived. Uniqueness results are presented, and a numerical algorithm to reconstruct the matrices from the given spectral data is proposed using the modified unsymmetric Lanczos algorithm. Illustrative examples are provided.

Title：Numerical Methods for Thermodynamically Consistent Model Reduction

Speaker: Haijun Yu （于海军，中国科学院数学与系统科学研究院）

Abstract：Mathematical modeling of complex dissipation systems is a challenging task. Starting from the first principle or molecular dynamics, one can formally write down some very accurate mathematical models with few assumptions on molecules, but those models are usually not computable due to huge number of freedoms or high dimensions. In this talk, we briefly introduce two approaches to build computable low-dimensional models from the underlying high-dimensional model or simulated trajectory data. The thermodynamical properties are kept in both approaches. We will use the Navier-Stokes equations and Fokker-Planck equations for Newtonian and non-Newtonian fluids as two examples to demonstrate the methods.

编辑：数学科学学院