科学计算方法与理论前沿进展研讨会

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主办：天津大学 应用数学中心

2020年12月17- 20日天津

一．住宿安排及交通指引

住宿地点：天津泰达国际会馆

酒店地址：天津市南开区复康路7号增2号（南开大学南门对面）

酒店电话：(022) 5869 5555

交通指引：

1. 天津滨海国际机场

地 铁：滨海国际机场地铁站乘地铁2号线（曹庄方向）9站到天津站地铁站下

天津站地铁站转乘地铁3号线（南站方向）6站到天塔地铁站下车

天塔地铁站C口出，步行约1公里到达天津泰达国际会馆

出租车：约36分钟72元（约29公里）

2. 天津站

地 铁：天津站地铁站乘地铁3号线（南站方向）6站到天塔地铁站下车

天塔地铁站C口出，步行约1公里到达天津泰达国际会馆

出租车：约23分钟32元（7.4公里）

3. 天津西站

地 铁：西站地铁站上车乘坐地铁6号线（梅林路方向）8站到红旗南路地铁站下

红旗南路地铁站转乘地铁3号线（小淀方向2站到天塔地铁站下车

天塔地铁站C口出，步行约1公里到达天津泰达国际会馆

出租车：约25分钟37元（10公里）

4. 天津南站

地 铁：南站地铁站上车乘坐地铁3号线（小淀方向）9站到天塔地铁站下车

天塔地铁站C口出，步行约1公里到达天津泰达国际会馆

出租车：约27分钟53元（16公里）

二．联系人

张 勇：sunny5zhang@163.com 13811258554

谢满庭：mtxie@tju.edu.cn 13212299867

唐庆粦：qinglin_tang@163.com 15210219290

会议秘书：张少波 16622898661

三． 会议日程

四、报告题目和摘要(按姓氏拼音排序)

Normalized gradient flow for computing ground states of Bose--Einstein condensates

蔡勇勇（北京师范大学）

The normalized gradient flow, i.e. the gradient flow with discrete normalization (GFDN) introduced in [W. Bao and Q. Du,SIAM J. Sci. Comput., 25 (2004), pp.1674--1697] or the imaginary time evolution method is one of the most popular techniques for computing the ground states of Bose--Einstein condensates (BECs). In this talk, we revisit the time discretizations for the GFDN and its generalization to the multi-component BECs. Several widely used time discretizations are demonstrated not accurate for computing the ground state solution in the general case, especially for the multi-component BECs with two or more constraints even for the most accepted linearized backward Euler schemes. More precisely, these schemes usually converge to a solution with an error depending on the time step size. To accurately and efficiently compute the ground state solution of BECs, we propose the gradient flow with Lagrange multiplier (GFLM) method which can be viewed as the modified GFDN by introducing the explicit Lagrange multiplier terms or an approximation of the continuous normalized gradient flow (CNGF).

A Fourier collocation method for Schrödinger-Poisson system with perfectly matched layer

成蓉华（云南财经大学）

In this talk，we present Fourier collocation method and apply PML technique to carry out numerical simulation for SP system. When the PML technique is applied, the original Schrödinger-type equation becomes a Schrödinger-type equation with variable coefficients. Fourier collocation method has an advantage in dealing with differential equations with variable coefficients and periodic boundary conditions. So we implement the method to solve (SP) system with PML. We carry out numerical simulation for the SP system by employing splitting method in time and Fourier collocation method in space, respectively. Numerical results show that the Fourier collocation method coupled with PML technique can absorb well the outgoing waves governed by the Schrödinger equation when the wave goes out of the physical computational boundary.

三维并行自适应有限元编程框架PHG及其在集成电路EDA中应用

崔涛（中国科学院数学与系统科学研究院）

PHG是中国科学院科学与工程计算国家重点实验室研制的一个并行自适应有限元编程框架。EDA工具是电子设计自动化（Electronic Design Automation）的简称，利用EDA工具，工程师将芯片的电路设计、性能分析、设计出IC版图的整个过程交由计算机自动处理完成，是集成电路产业的支撑工具。器件仿真、光学校正、成品率分析和优化、互连线寄生参数提取等EDA工具都涉及大量场路耦合问题。本报告将介绍PHG的基本功能及应用情况。重点介绍我们最新提出的高阶界面积分算法及其程序接口，以及集成电路紧凑模型的相关算法研究工作。

Radiation condition for time-harmonic acoustic scattering in half spaces

胡广辉（南开大学）

Wave scattering in a layered medium and in a half plane has numerousapplications in scientific and engineering areas. It is well known thatthe Sommerfeld outgoing radiation condition does not hold for time-harmonicscattering by unbounded surfaces such as grating structures and roughsurfaces. For scattering problems in a half space, the outgoing radiationcondition also depends on the form of incident waves. In this talk, wefirst review the existing radiation conditions for periodic surfaces, roughsurfaces and locally perturbed flat surfaces. Next we will present newradiation conditions for non-locally perturbed unbounded surfaces and provea Sommerfeld-type outgoing radiation condition for scattering ofcylindrical waves by a one-dimensional rough surface of Dirichlet kind.This talk was prepared based on joint works with Wangtao Lu (Zhejiang University,China) and Andreas Rathesfeld (WIAS, Germany).

Phase field approach for simulating solid-state dewetting problems

蒋维（武汉大学）

Thin Solid films are usually thermodynamically unstable in the as-deposited state. Heating can lead to fragmentation (or pinch-off) of a thin film and the formation of micro-/nano- solid particles. This process is well-known as solid-state dewetting in materials science, and it is often driven by the minimization of the total interfacial energy of the system. In this talk, we propose a phase field approach for simulating solid-state dewetting and the morphological evolution of patterned islands on a substrate. The evolution is governed by the Cahn–Hilliard equation with degenerate mobilities coupled with contact line boundary conditions. The proposed approach can include the surface energy anisotropy into the models. Several important features observed in experiments can be reproduced by numerically solving the proposed models.

Stabilized integrating factor Runge-Kutta method and unconditional preservation of maximum bound principle

李精伟（北京师范大学）

Maximum bound principle (MBP) is an important property for a large class of semilinear parabolic equations, in the sense that the time-dependent solution of the equation with appropriate initial and boundary conditions and nonlinear operator preserves for all time a uniform pointwise bound in absolute value. It has been a challenging problem on how to design unconditionally MBP-preserving high-order accurate time-stepping schemes for these equations. In this paper, we combine the integrating factor Runge-Kutta (IFRK) method with the linear stabilization technique to develop a stabilized IFRK (sIFRK) method, and successfully derive sufficient conditions for the proposed method to preserve MBP unconditionally in the discrete setting. We then elaborate some sIFRK schemes with up to the third-order accuracy, which are proven to be unconditionally MBP-preserving by verifying these conditions. In addition, it is shown that many classic strong stability preserving sIFRK schemes do not satisfy these conditions except the first-order one. Extensive numerical experiments are also carried out to demonstrate the performance of the proposed method. This work is joint with Xiao Li, Lili Ju and Xinlong Feng.

Singular perturbed renormalization group method and it’s dynamic significances

黎文磊（吉林大学）

In this talk, I will present several our recent results related to the Singular perturbed renormalization group method, including a systematic formula of RG strategy up to any order, application to normal form and center manifolds theory, etc.

GCGE: A Package for Solving Large Scale Eigenvalue Problems by Parallel Block Damping Inverse Power Method

李瑜（天津财经大学）

In this paper,we introduce an eigensolver based on the parallel block damping inverse power method and the corresponding package for solving large scale eigenvalue problems. The numerical methods, some implementing techniques and the structure of the package will be introduced. Finally, some numerical experiments will also be provided to validate the efficiency of the concerned numerical method and the corresponding package. Ten thousand eigenpaires are approximated for a fourteen million order matrix. The package GCGE can be downloaded in web site:https://github.com/Matrials-Of-Numerical- Algebra/GCGE.

A regularized and unconditionally convergent Galerkin FEM for the logarithmic Schrödinger equation

王廷春（南京信息工程大学）

In this talk, we aim to derive and analyze a regularized three-level Galerkin finite element method (FEM) for solving the logarithmic Schrödinger equation (LogSE) in d dimensions (d=1,2,3). To avoid numerical blow-up and/or to suppress round-off error due to the logarithmic nonlinearity in the LogSE, a traditional way is to introduce a regularized logarithmic Schrödinger equation (RLogSE) with a small regularized parameter to approximate the LogSE with linear convergence rate . Then, under a weak assumption on the exact solution for the RLogSE, we establish the optimal L2 error estimate of the proposed numerical method without any grid-ratio restriction, while the previous works always require a certain constraint on the grid ratios or assume that the exact solution has a higher regularity, especially in the two- or three-dimensional cases. Numerical results are carried out to test our theoretical analysis and simulate the wave collision with time evolution.

On Nonconvex Regularized Models for Image and surface Restoration Problems

吴春林（南开大学）

Variational methods with regularization techniques have become an important kind of methods image restoration. The convex total variation (TV) regularization, although achieved great successes, suffers from a contrast reduction effect. Recently nonconvex regularization techniques become popular. In this talk, I will mainly present three parts. The first one is a motivation of using nonconvex regularizations and a general truncated regularization framework. The second is a lower bound theory for nonconvex regularized models, which shows the good edge recovery property. The third one is an extension of total variation for surface denoising.

求解非线性特征值问题的扩展子空间算法

谢和虎（中国科学院数学与系统科学研究院）

本报告将介绍我们最近关于求解非线性特征值问题的扩展子空间算法，该方法将定义在最细网格上的非线性特征值问题转化成在最细网格上的线性方程的求解和在一个低维空间上的非线性特征值问题的求解。由于避免了在高维空间上直接求解非线性特征值问题，扩展子空间算法可以提高求解非线性特征值问题的效率。更进一步，当非线性项是多项式的时候，结合我们最近提出的张量组装技术可以将最终计算量降低到与非线性强度几乎无关的程度，达到渐近绝对最优。本报告将介绍算法的主要思想、算法、以及在Bose-Einstein凝聚基态计算等问题中的应用。

Bose-Einstein凝聚基态解的高效有限元方法及可计算误差估计

谢满庭（天津大学）

本报告将主要介绍求解Bose-Einstein凝聚（BEC）基态解的高效有限元方法，包括基于多水平校正方法的多重网格法和自适应方法。还将介绍BEC基态解的可计算误差估计，并基于此方法得到基态能量的上下界。

Regularized Finite Difference Methods for the Logarithmic Klein-Gordon Equation

闫静叶（国防科技大学）

In order to deal with the origin singularity, we employ regularizing nonlinearity

of the logarithmic Klein-Gordon equation (LogKGE) with a regularization parameter 0 <ε≪1 and an energy regularization for the LogKGE. Four finite difference methods are applied to the regularized logarithmic Klein-Gordon equation (RLogKGE) and the energy regularized logarithmic Klein-Gordon equation (ERLogKGE). It is proven that these methods have the second order of accuracy both in space and time. Numerical experiments show that the solutions of the regularized equations converge to the solution of the initial equation as O (ε). Numerical results are reported to confirm our error estimates of the finite difference methods for the RLogKGE and ERLogKGE. Finally our results suggest that energy regularization performs better than regularizing the logarithmic nonlinearity in the LogKGE directly.

Normalized Local Minimax Methods for Finding Multiple Unstable Solutions of Semilinear Elliptic PDEs

易雯帆（湖南大学）

Inspired by the classical Goldstein line search rule and (strong)Wolfe-Powell line search rule in the optimization theory in R^m, which is aimed to guarantee the global convergence of some descent algorithms, we consider normalized Goldstein search and (strong)Wolfe-Powell rules combined with the local minimax method to be suitable for finding multiple unstable solutions of semilinear elliptic PDEs both in numerical implementation and theoretical analysis. The approach can prevent the step-size from being too small automatically and then ensure that the iterations make reasonable progress by taking full advantages of two inequalities. The feasibility of the NLMMs are verified strictly. Further, the global convergence of the NLMMs are proven rigorously. Finally, it is implemented to solve several typical semilinear elliptic boundary value problems on square or dumbbell domains for multiple solutions and the numerical results indicate that this approach performs well.

An efficient numerical method to compute ground states of a rotating Bose-Einstein condensate with impurities

袁永军（湖南师范大学）

In this talk, an efficient and accurate numerical method is developed to compute ground states of a novel rotating BEC-impurity system. The key idea is twofold. First, since the impurities are localized in a small box potential, two grids are designed to compute the ground states of the model problem, i.e., a general grid for a suitable large domain to compute the wave function of BEC and an imbedded grid for a small domain determined by the size of a box potential to calculate the wave functions of impurities. Then, a preconditioned conjugate gradient (PCG) method is applied to solve the ground states of the rotating BEC-impurity system. Extensive numerical results of ground states for the BEC-impurity system are reported to show the efficiency of our method and demonstrate interesting physical phenomena.

Fast algorithm for convolution-type potential evaluation indifferent dimensions

张勇（天津大学）

Convolution-type potential are common and important in many science andengineeringfields. Efficient and accurate evaluation of such nonlocal potentials are essential in practical simulations.In this talk, I will focus on those arising from quantum physics/chemistry and lightning-shield protection,including Coulomb, dipolar and Yukawa potential that are generated by isotropic and anisotropic smoothand fast-decaying density, as well as convolutions defined on a one-dimensional adaptive finite difference grid. The convolution kernel is usually singular or discontinuous at the origin and/or at the far field,and density might be anisotropic, which together present great challenges for numerics in both accuracyand efficiency. The state-of-art fast algorithms include Wavelet based Method(WavM), kernel truncation method(KTM), NonUniform-FFT based method(NUFFT) and Gaussian-Sum based method(GSM).Gaussian-sum/exponential-sum approximation and kernel truncation technique, combined with finiteFourier series and Taylor expansion, finally lead to a O(N log N) algorithm achieving spectral accuracy.For the one-dimensional convolutions, we shall introduce the tree and sum-of-exponential based fastalgorithm.

Numerical integrators for disordered nonlinear Schrödinger equation

赵晓飞（武汉大学）

In this talk, we will present the numerical methods for integrating a cubic nonlinear Schrödinger equation with a spatial random potential. The model is known as the continuous disordered NLS. The presence of the random potential induces roughness to the equation and to the solution, which causes convergence order reduction for classical numerical methods. We shall introduce a low-regularity integrator, where we show how to integrate the potential term and the nonlinearity by losing two spatial derivatives. Numerical results will be presented to show the accuracy of LRI compared with classical methods under random/rough potentials from applications.

时间并行算法

周涛（中国科学院数学与系统科学研究院）

本次报告将简要介绍PDE求解中时间并行算法的基本思想以及相关的数值分析理论。