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**数值方法与理论前沿研讨会**

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主办：四川大学数学学院

Hosted by：School of Mathematics，Sichuan University

会议日程安排

Conference Schedule

2018年6月8日至2018年6月11日，四川大学，成都

8^{th} June to 11^{th} June，2018

Sichuan University，Chengdu，China

（最后附有会议手册，欢迎一键下载）

**一．住宿安排及交通指引**

**住宿/会议地点**： 瑞熙城市酒店

**酒店地址：**** **** **** **成都市武侯区科华中路139号

**酒店电话：** （+028）8521 9888

交通指引：

1. **双流国际机场**

至瑞熙城市酒店

滴滴或出租车：约20分钟43元。

2. **火车东站**

至瑞熙城市酒店

滴滴或出租车：约15分钟24元。

**二．联系人**

王 皓：wangh@scu.edu.cn, 13882264928

唐庆粦：qinglin_tang@scu.edu.cn, 15210219290

三． **会议日程**

**6月8日（周五）**

14:00–22:00 会议注册， 瑞熙城市酒店

**6月9日（周六） **

8:00 - 8:45 签到以及开幕式：谢小平

**主持人： 谢小平**

8:45 - 9:20 明平兵 (中国科学院 计算数学与科学工程计算研究所)

A hybrid numerical method for composite materials with local defects

9:20 - 9:55 张 辉 (北京师范大学)

A positivity-preserving, energy stable and convergent numerical scheme for the Cahn-Hilliard equation

with a Flory-Huggins-deGennes energy

9:55 - 10:15 茶歇和照相

**主持人： 唐庆粦**

10:15 - 10:50 胡光辉 (澳门大学)

An efficient and high order finite volume framework for Euler equations

10:50 - 11:25 倪国喜 (北京应用物理与计算数学研究所)

A conservative numerical method for compressible reactive fluids

11:25 - 12:00 王 柱 (University of South Carolina)

Efficient algorithms for simulating ensembles of parameterized flow problems

12:00 - 14:00 午 餐

**主持人： 刘长丽**

14:00 - 14:35 黄宏财 (I-Shou University)

Study of Effective Condition Number

14:35 - 15:10 王汉权 (云南财经大学)

Numerical studies on coherent pulse progression of mid-infrared quantum- cascade lasers under

group-velocity dispersion and self-phase modulation

15:10 - 15:30 茶 歇

**主持人： 王皓**

15:30 - 16:05 陈黄鑫 (厦门大学)

HDG methods for the Maxwell equations problems

16:05 - 16:40 王 东 (University of Utah)

A generalized MBO diffusion generated method and its applications

16:40 - 17:25 魏 柯 复旦大学

Nonconvex optimization for spectral compressed sensing

18:00 - 晚 宴

**6****月10****日（周日）**

**主持人： 张世全**

8:45 - 9:20 李文彬 (哈尔滨工业大学深圳校区)

Level-set based structured solution for geophysical imaging

9:20 - 9:55 吴 昊 (清华大学)

Some recent results for waveform based earthquake location

## 9:55 - 10:30** **张晨松 (中国科学院数学与系统科学研究院)

## **Numerical simulation of discrete fracture networks in petroleum reservoir simulation **

10:30 - 10:50 茶 歇

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**主持人： 马强**

10:50 - 11:25 蒋 维 (武汉大学)

Modeling and theoretical analysis for solid-state dewetting problems

11:25-12:00 贺巧琳 (四川大学)

Numerical study of phase transition in VdW fluid and asymptotic stability of solutions of compressible

Navier-Stokes-Allen-Cahn system

12:00 - 14:00 午 餐

14:00 - 18:00 自由讨论

**6月11日（周一）** 自由讨论/离会

四． **报告题目和摘要**

**报告人：**陈黄鑫 (厦门大学)

**题目：**HDG methods for the Maxwell equations

**摘要****：**In this talk we will introduce a new HDG method for the steady state Maxwell equations based on a mixed curl-curl formulation. We use a non-trivial subspace of polynomials of degree k+1 to approximate the numerical tangential trace of the electric field on the faces. If the dual operator of the Maxwell equation has adequate regularity, the order of convergence of L2-error for the electric field is k+2. From the point of view of degrees of freedom of the globally coupled unknown: numerical trace, this HDG method achieves superconvergence for the electric field without post-processing. When we consider the Maxwell equations with low regularity of electric field, another HDG method and its a priori and a posteriori error estimates will be discussed. Some numerical results will be shown to demonstrate the efficiency of the HDG methods for the Maxwell equations.

**报告人：**贺巧琳(四川大学)

**题目：**Numerical study of phase transition in VdW fluid and asymptotic stability of solutions of compressible Navier-Stokes-Allen-Cahn system

**摘要****：**In this talk, we use a relaxation scheme for conservation laws to study liquid-vapor phase transition modeled by the van der Waals equation, which introduces a small parameter ε and a new variable. In the case of diffuse interface model for two immiscible fluids, in order to understand the motion of the interfaces between immiscible fluids, we use a relaxation scheme coupled with convex splitting method to solve a compressible Navier-Stokes-Allen-Cahn system. Numerical results are given.

**报告人：**胡光辉 (澳门大学)

**题目：**An efficient and high order finite volume framework for Euler equations

**摘要****：**In this talk, I will give a brief introduction of our recent work on high order finite volume methods for Euler equations. The framework consists of a Newton iteration for the linearisation of Euler equations, and a geometrical multigrid method for solving the linearised system. A non-oscillatory k-exact reconstruction method is proposed for the quality solution reconstruction, and NURBS are used to handle the curve representation. To enhance the efficiency, an h-adaptive module is introduced in the algorithm in which the Hierarchy Geometry Tree is used to manage the mesh data, and an adjoint-based analysis is used to generate the error indicator. Numerical experiments successfully show the effectiveness of the proposed algorithm.

**报告人：**黄宏财 (I-Shou University)

**题目：**Study of effective condition number

**摘要****：**For solving the linear algebraic equations **A****=b** with the symmetric and positive definite matrix **A**, the traditional condition number in the 2-norm is defined by Cond=, where and are the maximal and the minimal eigenvalues of the matrix **A**, respectively. The condition number is used to provide the bounds of the relative errors from the perturbation of both **A** and **b**. Such a Cond can only be reached by the worst situation of all rounding errors and all** b**. For the given b the true relative errors may be smaller, or even much smaller than the Cond, which is called the effective condition number in Chan and Foulser (1988) and Christiansen and Hansen (1994). The condition number originated from Wilkinson (1963), and it has been using for stability analysis. In this study, the new effective condition number Cond_eff=**b** , is proposed to explore the stability of numerical partial differential equations. Since the effective condition number is smaller, and even much smaller than the condition number, the effective condition number may provide a sharp estimation of stability analysis. In fact, the effective condition number was first studied in Rice (1981), but is not noticeable in the community of linear algebra. However, the effective condition number is remarkably advantageous over the traditional condition number for numerical partial differential equations. The materials of this talk are taken from the monograph [Z.C. Li, H.T. Huang, Y. Wei, A.H.-D Cheng, Effective condition number for Numerical Partial Differential Equations (Second Edition), Science Press, Beijing, 2015.]

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**报告人：**蒋维 (武汉大学)

**题目：**Modeling and theoretical analysis for solid-state dewetting problems

**摘要****：**In this talk, I will talk about our work about modeling solid-state dewetting problems. Taking the 2D case for example, I will explain the main idea behind these approaches; some extensions to the 3D case will be also presented. If time permits, I will talk about our recent work about using Onsager’s variational principle to derive reduced models with applications to solid-state dewetting.

**报告人：**李文彬 (哈尔滨工业大学深圳校区)

**题目：**Level-set based structured solution for geophysical imaging

**摘要****：**We study the inverse problems arising from geophysical imaging. A level-set-based parametric approach is proposed to find structured solutions with piecewise continuous structure and interface structure. We have studied traveltime tomography, inversion of potential field data, and joint inversion of heterogeneous data; a systematic methodology has been developed to handle these problems in an Eulerian framework. The structural parameterization improves resolutions of interfaces in the imaging, and alleviates ambiguities in the interpretations of geophysical data. The proposed method has been successfully applied to the inversion of field data from realistic mineral explorations.

**报告人：**倪国喜 (北京应用物理与计算数学研究所)

**题目：**A conservative numerical method for compressible reactive fluids

**摘要****：**In this report, we present a conservative interface method for reactive fluids problems, the standard finite volume scheme on Cartesian grids is modified by considering computational cells being cut by interface. the discretized governing equations are updated conservatively , the method treats the topological changes naturally by combining interface description and geometric operations with a level set technique. Extensive tests in 1D are carried out, and 2D examples suggest that the present scheme is able to handle CJ problems in a straightforward way with good robustness and accuracy.

**报告人：**明平兵 (中国科学院 计算数学与科学工程计算研究所)

**题目：**A hybrid numerical method for composite materials with local defects

**摘要****：**We present a new hybrid numerical method for multiscale partial differential equations, which simultaneously captures the global macroscopic information and resolves the local microscopic events over regions of relatively small size. This is particularly suitable for composite materials with local defects. The method couples concurrently the microscopic coefficients in the region of interest with the homogenized coefficients elsewhere. The cost of the method is comparable to the heterogeneous multiscale method, while being able to recover microscopic information of the solution. The convergence of the method is proved for problems with bounded and measurable coefficients, while the rate of convergence is established for problems with rapidly oscillating periodic or almost-periodic coefficients. Numerical results are reported to show the efficiency and accuracy of the proposed method.This is a join work with Yufang Huang and Jianfeng Lu.

**报告人：**王柱 (University of South Carolina)

**题目：**Efficient algorithms for simulating ensembles of parameterized flow problems.

**摘要****：**Many computational fluid dynamics applications require multiple simulations of a flow under different input conditions. In this talk, we consider such settings for which one needs to perform a sequence of simulations based on the Navier-Stokes equations, each having different initial condition data, boundary condition data, forcing functions, and/or coefficients such as the viscosity. For such settings, we propose ensemble methods to accelerate the solutions. The main idea is to manipulate the time-stepping scheme so that all the problems could share a common coefficient matrix, then, instead of solving a sequence of linear systems with one right-hand-side vector, the method needs to solve one linear system with multiple right-hand-sides. The computational efficiency is then improved by using block iterative algorithms. Rigorous analyses are given proving the conditional stability and establishing error estimates for the proposed algorithms. Numerical experiments are presented to illustrate the analyses.

**报告人：**王东 (University of Utah)

**题目：**An generalized MBO diffusion generated method and its applications

**摘要****：**A variety of tasks in inverse problems and data analysis can be formulated as the variational problem of minimizing the Dirichlet energy of a function that takes values in a certain submanifold and possibly satisfies additional constraints. These additional constraints may be used to enforce fidelity to data or other structural constraints arising in the particular problem considered. I’ll present a general framework for minimizing such a functional. I’ll give examples of how this method can be used for constructing smooth orthogonal matrix valued functions, finding Dirichlet partitions, finding equilibrium state of foam bubbles, and manifold-valued image processing. For the first problem, I’ll prove the stability of the method by introducing an appropriate Lyapunov function, generalizing a result of Esedoglu and Otto to matrix-valued functions. I’ll also state a convergence result for the method. This is based on some joint works with Braxton Osting.

**报告人：**王汉权 (云南财经大学)

**题目：**Numerical studies on coherent pulse progression of mid-infrared quantum-cascade lasers under group-velocity dispersion and self-phase modulation

**摘要****：**The effect of group-velocity dispersion (GVD) and self-phase modulation (SPM) on the coherent pulse progression in mid-infrared quantum-cascade lasers (QCLs) is investigated. The theoretical model is built based on the Maxwell–Bloch formulism accounting for the couplings among the electric field, the polarization, and the population inversion. The pulse evolution in time-spatial domains is simulated by the finite difference method with prior nondimensionalization, which is necessary for convergent solution. It is found that the anomalous GVD, which receives less attention in the study of QCL dynamics, can significantly narrow the spectrum splitting between side modes. The SPM can broaden the linewidth of the spectral modes. Their combined effects can lead the possibility of forming solitons.

**报告人：**魏轲 (复旦大学)

**题目：**Nonconvex optimization for spectral compressed sensing

**摘要：**Spectrally sparse data arise in many areas of science and engineering, for instance magnetic resonance imaging, fluorescence microscopy and radar imaging. Spectral compressed sensing is about reconstructing spectrally sparse data from incomplete information. Two different classes of nonconvex optimization algorithms are introduced to tackle this problem. They are developed by exploiting low rank structure within the data in two different ways: one is based on the embedded manifold of low rank matrices and the other is based on the factorization model of low rank matrices. Theoretical recovery guarantees will be presented for the proposed algorithms under certain random models, showing that the sampling complexity is essentially proportional to the intrinsic dimension of the problems rather the ambient dimension. Empirical observations demonstrate the efficacy of the algorithms.

**报告人：**吴昊 (清华大学)

**题目：**Some recent results for waveform based earthquake location

**摘要：**The waveform based earthquake location is essentially a PDE-constraint optimization problem. In this talk, we will present some newly developed techniques. (i) We convert the original optimization problem into the problem of finding the zero point of the auxiliary functions. But the computational cost is significantly less than that of the iterative methods. (ii) We apply the famous Wasserstein metric to locate the earthquake. The convexity of the misfit function with respect to the earthquake hypocenter and the origin time can be observed. Even for large data noise, these methods could locate the earthquake with reasonable accuracy. These approaches provide fast and accurate methods to locate the earthquakes, which may be useful for the earthquake real-time locating and the earthquake relocation.

**报告人：**张晨松 **(**中国科学院数学与系统科学研究院**)**

**题目：**Numerical Simulation of Discrete Fracture Networks in Petroleum Reservoir Simulation

**摘要：**A multi-scale hybrid-mixed method is applied to the numerical approximation of two-dimensional fluid flow in porous media with fractures. The two-dimensional fluid flow in the reservoir and the one-dimensional flow in the discrete fractures are approximated using mixed finite elements. The coupling of the two-dimensional flow with the one-dimensional fracture flow is enforced using the pressure of the one-dimensional flow as a Lagrange multiplier to express the conservation of fluid transfer between the fracture flow and the divergence of the one-dimensional fracture flux. A zero-dimensional pressure (point element) is used to express conservation of mass where fractures intersect. The issuing simulation is then reduced using the MHM method leading to surprisingly accurate results with a very reduced number of global equations. A general system was developed where fracture geometries and conductivities are specified in an input file and meshes are generated using the public domain software GMSH. Several test cases illustrate the effectiveness of the proposed approach comparing the multi-scale results with direct simulations.

**报告人：**张辉 (北京师范大学)

**题目：**A positivity-preserving, energy stable and convergent numerical scheme for the Cahn-Hilliard equation with a Flory-Huggins-deGennes energy

**摘要：**This talk is focused on the bound estimate and convergence analysis of an unconditionally energy stable scheme for the MMC-TDGL equation, a Cahn-Hilliard equation with a Flory-Huggins-deGennes energy. The numerical scheme, a finite difference algorithm based on a convex splitting technique of the energy functional, was proposed in [Sci. China Math. 59(2016),1815]. We provide a theoretical justification of the unique solvability for the proposed numerical scheme, in which a well-known difficulty associated with the singular nature of the logarithmic energy potential has to be handled. Meanwhile, a careful analysis reveals that, such a singular nature prevents the numerical solution of the phase variable reach the limit singular values, so that the positivity preserving property could be proved at a theoretical level. In particular, the natural structure of the deGennes diffusive coefficients also ensures the desired positivity-preserving property. In turn, the unconditional energy stability becomes an outcome of the unique solvability and the convex-concave decomposition for the energy functional. Moreover, an optimal rate convergence analysis is presented. In addition, a rewritten form of the surface diffusion term has facilitated the convergence analysis, in which we have made use of the special structure of concentration-dependent deGennes type coefficients.

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**组织委员会（按姓氏拼音排序）****：**

冯民富 贺巧琳 胡 兵 刘长丽 马 强 唐庆粦 王 皓 谢小平 张世全