应6163银河线路检测中心张达治老师和刘文杰老师的邀请，受国际合作处资助，新加坡南洋理工大学WangLi-Lian副教授将于近日来访公司并做报告，欢迎感兴趣的师生参加。
报告1
时间：2019年12月15日13：30-15：00
地点：诚意楼307
Title:Jacobi Spectral Methods and Their Applications I：Jacobi Polynomials and Jacobi Approximations
Abstract：Jacobi approximations in non-uniformly Jacobi-weighted Sobolev spaces are investigated. Some results on orthogonal projections and interpolations are established. Explicit expressions describingthe dependence of approximation results on the parameters of Jacobi polynomials are given. These results serve as an important tool in the analysis of numerous quadratures and numerical methods for differential and integral equations.
报告2
时间：2019年12月15日15：10-17：20
地点：诚意楼307
Title: Jacobi Spectral Methods and Their Applications II：Generalized Jacobi polynomials and Optimal Spectral Algorithms
Abstract：We introduce a family of generalized Jacobi polynomials/functions with indexes alpha,betain R which are mutually orthogonal with respect to the corresponding Jacobi weights and which inherit selected important properties of the classical Jacobi polynomials. We establish their basic approximation properties in suitably weighted Sobolev spaces. As an example of their applications, we show that the generalized Jacobi polynomials/functions, with indexes corresponding to the number of homogeneous boundary conditions in a given partial differential equation, are the natural basis functions for the spectral approximation of this partial differential equation. Moreover, the use of generalized Jacobi polynomials/functions leads to much simplified analysis, more precise error estimates and well conditioned algorithms.
报告3
时间：2019年12月16日8：30-10：00
地点：诚意楼307
Title: Jacobi Spectral Methods and Their Applications III：Optimal Jacobi Spectral Approximations in fractional Sobolev spaces
Abstract：We define the generalised Jacobi functions of fractional degree (GJF-Fs) by allowing the (integer) degree of the Jacobi polynomials (JPs) defined by the hypergeometric functions to be real. The significance of this family of GJF-Fs resides in that they enjoy some fractional integral/derivative formulas, leading to analytical tools for both algorithm development and analysis of fractional PDEs. Indeed, the Jacobi polyfractonomials and generalised Jacobi functions/polynomials are special cases of GJF-Fs. In this talk, we highlight their impactful role in spectral approximation theory. In a nutshell, using the fractional integration by parts, the Jacobi polynomial expansion coefficients for functions with fractional Sobolev regularity can be naturally, precisely expressed in terms of GJF-Fs. We are then able to optimally estimate the decay rate of the expansion coefficients, so do the errors of the orthogonal projections, polynomial interpolation and quadratures. In particular, we can significantly improve the L infinity estimate of the Chebyshev expansion for functions with limited regularity. For example, we can recover the best possible order by polynomial polynomial approximation of functions with singularity.
报告4
时间：2019年12月16日10：10-11：40
地点：诚意楼307
Title: Towards Effective Spectral and hpMethods for PDEs with Integral Fractional Laplacian in Multiple Dimensions
Abstract：PDE with integral fractional Laplacian is a powerful tool in modelling anomalous diffusion and nonlocal interactions, but its numerical solution can be very difficult especially in multiple dimensions. In fact, many of such nonlocal models are more physically motivated and naturally set in unbounded domains. In this talk, we shall present a superfast spectral-Galerkin method with two critical components (i) based on the Dunford-Taylor formulation of fractional Laplacian operator, and (ii) using Fourier-like mapped Chebyshev functions as basis. We shall also report some of our recent attempts for integral fractional Laplacian in bounded domains, which are deemed even more notoriously difficult in effective numerical discretisations. Along this line, we work with the formulation associated with the Fouririer transformations, and derive a number of useful analytical formulas which are essential for the algorithm development.
报告5
时间：2019年12月19日8：30-10：00
地点：诚意楼307
Title: Well-conditioned Spectral Collocation Methods
Abstract：A well-conditioned collocation method is constructed for solving general pth order linear differential equations with various types of boundary conditions. Based on a suitable Birkhoff interpolation, we obtain a new set of polynomial basis functions that results in a collocation scheme with two important features: the condition number of the linear system is independent of the number of collocation points, and the underlying boundary conditions are imposed exactly. Moreover, the new basis leads to an exact inverse of the pseudospectral differentiation matrix of the highest derivative (at interior collocation points), which is therefore called the pseudospectral integration matrix (PSIM). We show that PSIM produces the optimal integration preconditioner and stable collocation solutions with even thousands of points.
报告6
时间：2019年12月19日10：10-11：40
地点：诚意楼307
Title: A Genuinely Exact and Optimal Perfect Absorbing Layer with Star-shaped Truncation for Wave Scattering Problems
Abstract：We design a truly exact and optimal perfect absorbing layer (PAL) for domain truncation of the two-dimensional Helmholtz equation in an unbounded domain with bounded scatterers. This technique is based on a complex compression coordinate transformation in polar coordinates, and a judicious substitution of the unknown field in the artificial layer. Compared with the widely-used perfectly matched layer (PML) methods, the distinctive features of PAL lie in that (i) it is truly exact in the sense that the PAL-solution is identical to the original solution in the bounded domain reduced by the truncation layer; (ii) with the substitution, the PAL-equation is free of singular coefficients and the substituted unknown field is essentially non-oscillatory in the layer; and (iii) the construction is valid for general star-shaped domain truncation. By formulating the variational formulation in Cartesian coordinates, the implementation of this technique using standard spectral-element or finite-element methods can be made easy as a usual coding practice. We provide ample numerical examples to demonstrate that this method is highly accurate, parameter-free and robust for very high wave-number and thin layer. It outperforms the classical PML and the recently advocated PML using unbounded absorbing functions. Moreover, it can fix some flaws of the PML approach.
报告人简介：
Wang Li-Lian，新加坡南洋理工大学副教授，博士生导师。主要研究领域为数值PDE的谱方法与谱元方法、电磁学中的高性能计算方法等。在Mathematics of Computation、SIAM Journal on Applied Mathematics、 SIAM Journal on Imaging Sciences 、SIAM Journal on Scientific Computings、SIAM Journal on Numerical Analysis、IEEE Transactions on Image Processing、Applied and Computational Harmonic Analysis和Automatica 等国际著名学术刊物上发表论文80余篇，由Springer出版合著书籍《Spectral Methods: Algorithms,Analysis and Applications》。
