Schedule of Workshop on High-Order Numerical Methods for Complex Systems in Higher Dimensions

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      May 8-9, 2015 at School of Mathematical Science, Xiamen University


    This workshop is jointly organized by Xiamen University and Peking University, and aims at bringing together researchers who are interested in solving complex systems in higher dimensions to discuss the challenges in modeling and computations of such problems.  


VenueRoom 105, Laboratory Building, Haiyun Campus


May 8






LSEC, Chinese Academy of Sciences

Tutorial on sparse spectral methods and related approximation results


 Tea break



Xiangtan University

Numerical methods for ordered patterns



May 8






Xiangtan University

Pseudo-spectral method for solving the Modified diffusion equation of the wormlike-chain statistics on spherical surface


Tea break



Tsinghua University

Defects of Liquid Crystals



May 9




Chair 许传矩



Peking University

Tensor model for bent-core molecules based on molecular theory


 Tea break


Demonstration of SG++ (by 程青, Xiamen Universityand discussion









Title, speaker and abstract

	1. A tutorial on sparse spectral methods and related approximation results
	于海军  LSEC, Chinese Academy of Science

Abstract: Sparse grid (or hyperbolic cross approximation) is one of the important tools to handle high dimensional problems.  It was first introduced by Russian mathematician Smolyak in 1960's for numerical integration problems.  Later in 1990's, serveral  ermany mathematicians extended this method to solve high-dimensional PDEs. Now, sparse grid method has been used in various scientific/engineering applications involving high dimensionality.  In this tutorial, I will cover the basic idea, efficient implementation, and applications of sparse grids, with special emphasis on spectral sparse grid methods.


	2.  Numerical methods for ordered patterns
	蒋凯 Xiangtan University

Abstract: This talk is divided into two parts. The first part is for discovering complex periodic structures in polymer systems. We will talk about the self-consistent-field theory (SCFT) for polymeric systems and our developed numerical methods. The SCFT is a class of nonlinear integro-differential equations. In particular, SCFT has multi-solutions and multi-parameters. In consideration of the features of SCFT, our proposed numerical methods include the strategies to choose initial values, iscrete scheme, and nonlinear iteration methods. At the same time, the computational box is adjusted adaptively during minimizing the SCFT energy functional. The applications for several copolymer systems, including AB diblock copolymers, ABC  linear/star triblock copolymers are provided in this talk. 
   The second part is about the quasicrystals. We will talk about our recently developed Projection Methods for quasicrystals. We have applied our methods to Laudan-type models with single order parameter, and two order parameters. These results  have been demonstrated the validity of the projection method.   
   In this talk, we will emphasize the features of our problems, the difficulties that we encounter at present, and our attempts. 


3.Pseudo-spectral method for solving the Modified diffusion equation of the wormlike-chain statistics on spherical surface

梁琴   Xiangtan University

Abstract: In this paper, we introduce a notion of ear decomposition of 3-regular polyhedral links based on the ear decomposition of the 3-regular polyhedral graphs. As a result, we obtain an upper bound for the braid index of 3-regular polyhedral links. Our results may be used to characterize and analyze the structure and complexity of protein polyhedra and entanglement in biopolymers.

	4.  Defects of Liquid Crystals

    胡煜成  Tsinghua University

Abstract: Defects in liquid crystals are of great practical importance and theoretical interest. Despite tremendous efforts, predicting the location and transition of defects under various topological constraint and external field remains to be a challenge. In this talk I will discuss the origin of defects in the Landau-de Gennes tensor model and its connection with vector models. Then I will report some of our numerical results on the defect patterns of liquid crystals under different topological constraints and different boundary conditions. Finally I will summarize our understandings of defect structure gained from these numerical results. 

5. Tensor model for bent-core molecules based on molecular theory
徐劼  Peking University

Abstract: We describe the modeling of bent-core molecules, which is an extension of a systematic modeling of rod-like molecules. The startpoint is the second virial expansion. Using the hard-core interaction in the kernel function, we obtain the molecular model. The tensor model is derived by expanding the kernel function both spatially and orientationally. Our analysis shows that the form of tensor model is determined by the molecular symmetry. The coefficients can be expressed as functions of molecular parameters, which also affect the truncation. The entropy term is handled by an extension of the Bingham closure, which is also important in dynamic model. 

   Some theoretical results are shown in the homogeneous case, which give rigorous results about the phase behavior. A phase diagram of nematic phases about the molecular parameters is presented, in which all the phases observed experimentally are included. By choosing a set of phenomenological coefficients, we obtain some novel structures. 

    We derive the interaction of a bent-core molecule and the flow field. Together with the free energy, a molecular dynamic model is obtained. Then we use the closure approximation to express high order tensors, obtaining a tensor model from the molecular model. The energy dissipation of the tensor model is consistent with the molecular model. A few flow-induced structures are obtained in the homogeneous case