北京雁栖湖应用数学研究院

很多实际的应用问题都可以被归为计算数学中求解具有多个变量的非线性能量函数的极小值问题。这类多解问题通常具有多个极小，那么如何寻找全局极小和如何找到不同极小之间的关系一直是计算数学领域的两个关键问题。在报告中，我们提出了一个新的解景观 (solution landscape) 概念。解景观描述了不同的极小被相应的一阶鞍点连接，低阶鞍点被相应的高阶鞍点连接，最终连接到一个最高阶鞍点的层次结构图。我们根据解景观的特征，利用发展的鞍点动力学，结合向下搜索和向上搜索方法，可以高效地构建出完整的解景观。我们以复杂流体中的液晶和准晶为例，系统地构建了向列相液晶的缺陷景观以及发现了从晶体到准晶的形核过程。

Products of random matrices arise in many contemporary applications in the mathematics of data science. For instance, they describe the evolution of stochastic linear dynamical systems, which include popular stochastic algorithms for optimization such as Oja’s algorithm for streaming principal component analysis and the randomized Kaczmarz method for solving linear systems. In this talk, we introduce some new tools for studying random matrix products. Our argument interprets a product of independent random matrices as a martingale sequence. We then use the uniform smoothness properties of the Schatten trace classes to prove polynomial moment bounds for the random product. In particular, we develop non-asymptotic growth and concentration bounds for a product of independent random matrices. Moreover, our method can be extended to random products of contractions and random products of adaptive sequences.

Laguerre polynomials are orthogonal on the half line with respect to an exponentially decay weight.It is expected to be the natural and an efficient numerical method for partial differential equations defined on half unbounded domains. However, the applications of Laguerre polynomials in the literature are limited to small systems (mostly less than 400 bases in 1-d problems) due to the stability issue caused by the unbalanced scalings in the amplitudes of Laguerre polynomials. We present in this talk some simply but effective techniques to improve the stability of Laguerre methods,which allow us to use as many as more-than-one-thousand Laguerre bases to stably achieve close-to-machine-accuracy results for a benchmark problem. We will also present some results of using Laguerre spectral methods in computing transition pathes.

自1969年由M.R. Hestenes和M.J.D. Powell提出以来，增广拉格朗日法以其深刻的优化理论以及求解优化问题时优异的数值效果，受到数学优化、机器学习等不同领域学者的广泛关注，并已被用于许多著名优化求解器以提高求解许多大规模约束优化问题的数值效果。在本讲座中，我们将结合矩阵优化最新理论结果，介绍增广拉格朗日法在求解非线性半正定优化、黎曼流形上的非光滑优化以及随机规划等问题的研究进展。最后，我们将简要介绍增广拉格朗日法在实际问题中的若干应用，特别是我们在诸如视频防抖处理、大规模集成电路设计等问题中的尝试以及面临的挑战。

Many problems arising from communication system design can be formulated as optimization problems. In practice, one is often interested in not only the numerical solution to the problems but also the special structure of their optimal solution. In this talk, we shall use some examples from wireless communications and information theory to show that exploring the Lagrangian dual of these (convex) problems often reveal the structure of their optimal solution and the structure of the optimal solution will further lead to better algorithms for solving the corresponding problems.

本报告主要介绍超图计算理论及方法。超图是一种广义的图结构，因其具有较强的数据样本间非线性高阶关联的刻画和挖掘能力而被广泛应用于数据分类、检索等任务中。报告中将首先介绍超图的基本概念和特性。接下来围绕超图计算，介绍超图结构建模、超图结构演化及超图神经网络模型。在超图结构建模中，阐述了多类型超图的构建方式，实现面向复杂高阶关联的结构建模。在超图结构演化中，针对数据及标签数据欠定问题，介绍超图结构优化方法，实现面向复杂应用场景的超图结构更新。在超图神经网络中，介绍超图神经网络的基础模型。最后介绍超图计算在计算机视觉及数据挖掘等领域的应用。

In this talk, we develop third-order conservative sign-preserving and steady-state preserving time integrations and seek their applications in multispecies and multireaction chemical reactive flows. In this problem, the density and pressure are nonnegative, and the mass fraction should be between 0 and 1. There are four main difficulties in constructing high-order bound-preserving techniques for multispecies and multireaction detonations. First of all, most of the bound-preserving techniques available are based on Euler forward time integration. Therefore, for problems with stiff source, the time step will be significantly limited. Secondly, the mass fraction does not satisfy a maximum principle and hence it is not easy to preserve the upper bound 1. Thirdly, in most of the previous works for gaseous denotation, the algorithm relies on second-order Strang splitting methods where the flux and stiff source terms can be solved separately, and the extension to high-order time discretization seems to be complicated. Finally, most of the previous ODE solvers for stiff problems cannot preserve the total mass and the positivity of the numerical approximations at the same time. In this work, we will construct third order conservative sign-preserving Rugne-Kutta and multistep methods to overcome all these difficulties. The time integrations do not depend on the Strang splitting, i.e. we do not split the flux and the stiff source terms. Moreover, the time discretization can handle the stiff source with large time step and preserves the steady-state. Numerical experiments will be given to demonstrate the good performance of the bound-preserving technique and the stability of the scheme for problems with stiff source terms.

We study stochastic alignment of the Justh-Krishnaprasad(J-K) model under some types of noise. We present sufficient conditions leading to the nematic alignment of velocities in terms of the system parameters and initial data. For the general many-body system with a corresponding condition, we showed the accumulation of heading angles modulo and the stochastic stability of nematic alignment under the assumption of the constant communication weight, which suggests a strong evidence for the nematic alignment. We present several numerical simulations and compare them with analytical results.

Perfect imaging is one of the ultimate goals for humankind in perceiving the world, yet it is fundamentally limited by the optical aberrations resulted from the imperfect imaging systems or dynamic imaging environment. To address this long-standing problem, we develop a new framework of digital adaptive optics for universal incoherent imaging applications based on scanning light-field imaging systems. With digital measurement and synthesis of the incoherent light field with unprecedented precision, we have demonstrated a series of killer applications which are hard for traditional methods, including long-term high-speed intravital 3D imaging in mammals, gigapixel imaging with a single lens, high-speed multi-site aberration corrections for ground-based telescopes against turbulence, and real-time megapixel depth sensing. We anticipate that digital adaptive optics may facilitate broad applications in almost all fields, including industrial inspection, mobile devices, autonomous driving, surveillance, medical diagnosis, biology, and astronomy.

We consider the reconstruction of point sources in a two layered medium from the multi-frequency sparse far field patterns taken on the upper half sphere. Such a model is motivated by many applications where the measurements are only available at finitely many sensors. The point sources are located in both the upper half space and the lower half space, and consequently bring difficulty for the inverse problem because of the combination of two different types of far field patterns. After establishing the uniqueness of the point sources by the multi-frequency far field patterns at properly chosen sparse observation directions, we introduce a multi-step numerical scheme for identifying all the points sources. Numerical examples show that the proposed sampling methods work very well for locating the positions and the formulas for determining the corresponding scattering strengths are valid and stable with respect to the noises. This is a joint work with Shi Qingxiang.