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A variational hardcut EM algorithm for the mixtures of Gaussian processes

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Science China(Information Sciences) 2023年第3期66卷 291-292页

作者： Tao LI Jinwen MA Department of Information and Computational Sciences School of Mathematical Sciences and LMAM Peking University

Gaussian process(GP) is the dominant model in the nonparametric Bayesian community, but it is not flexible enough to model non-stationary/multi-modal data, because a conventionally trained Gaussian process is stationary. To overcome this problem, the mixture of Gaussian processes(MGP) [1] was proposed. Compared with GP,

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Nonsmooth critical point theory and applications to the spectral graph theory

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Science China Mathematics 2021年第1期64卷 1-32页

作者： Kung-Ching Chang Sihong Shao Dong Zhang Weixi Zhang LMAM and School of Mathematical Sciences Peking UniversityBeijing100871China

Existing critical point theories including metric and topological critical point theories are difficult to be applied directly to some concrete problems in particular polyhedral settings,because the notions of critical sets could be either very vague or too large.To overcome these difficulties,we develop the critical point theory for nonsmooth but Lipschitzian functions defined on convex polyhedrons.This yields natural extensions of classical results in the critical point theory,such as the Liusternik-Schnirelmann multiplicity theorem.More importantly,eigenvectors for some eigenvalue problems involving graph 1-Laplacian coincide with critical points of the corresponding functions on polytopes,which indicates that the critical point theory proposed in the present paper can be applied to study the nonlinear spectral graph theory.

关键词： critical point theory nonsmooth analysis combinatorial optimization polytope spectral graph theory

Constructions of 11/2-designs from Symplectic Geometry over Finite Fields

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Acta Mathematica Sinica,English Series 2015年第9期31卷 1367-1378页

作者： Zhao CHAI Rong Quan FENG Li Wei ZENG LMAM School of Mathematical Sciences Peking University

In this paper, we construct some 1 1/2-designs, which are also known as partial geometric designs, using totally isotropic subspaces of the symplectic space and generalized symplectic graphs. Furthermore, these 1^-des... 详细信息

关键词： 1 1/2-design totally isotropic subspace generalized symplectic graph directed strongly reg-ular graph

The Riemann-Hilbert Problem for Mixed Complex Equations of First Order with Degenerate Rank 0

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Acta Mathematicae Applicatae Sinica 2015年第1期31卷 31-42页

作者： Guo-chun WEN LMAM School of Mathematical Sciences Peking University

This article deals with the Riemann-Hilbert boundary value problem for quasilinear mixed （elliptic- hyperbolic） complex equations of first order with degenerate rank 0. Firstly, we give the representation theorem and prove the uniqueness of solutions for the boundary value problem. Afterwards, by using the method of successive iteration, the existence and estimates of solutions for the boundary value problem are verified. The above problem possesses the important applications to the Tricomi problem of mixed type equations of second order. In this article, the proof of HSlder continuity of a singular double integer is very difficult and interesting as in this Section 4 below.

关键词： Riemann-Hilbert problem quasilinear mixed complex equations degenerate rank 0 unique solv-ability of the problem HSlder continuity of singulax double integer

A Branching Random Walk Method for Many-Body Wigner Quantum Dynamics

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Numerical Mathematics(Theory,Methods and Applications) 2019年第1期12卷 21-71页

作者： Sihong Shao Yunfeng Xiong LMAM and School of Mathematical Sciences Peking UniversityBeijing 100871China

A branching random walk algorithm for many-body Wigner equations and its numerical applications for quantum dynamics in phase space are proposed and ana-lyzed in this paper.Using an auxiliary function,the truncated Wigner equation and its adjoint form are cast into integral formulations,which can be then reformulated into renewal-type equations with probabilistic interpretations.We prove that the first mo-ment of a branching random walk is the solution for the adjoint equation.With the help of the additional degree of freedom offered by the auxiliary function,we are able to produce a weighted-particle implementation of the branching random walk.In contrast to existing signed-particle implementations,this weighted-particle one shows a key ca-pacity of variance reduction by increasing the constant auxiliary function and has no time discretization errors.Several canonical numerical experiments on the 2D Gaussian barrier scattering and a 4D Helium-like system validate our theoretical findings,and demonstrate the accuracy,the efficiency,and thus the computability of the proposed weighted-particle Wigner branching random walk algorithm.

关键词： Wigner equation branching random walk quantum dynamics variance reduction signed-particle Monte Carlo method adjoint equation renewal-type equations importance sam-pling resampling

A Newton multigrid method for steady-state shallow water equations with topography and dry areas

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Applied Mathematics and Mechanics(English Edition) 2016年第11期37卷 1441-1466页

作者： Kailiang WU Huazhong TANG LMAM School of Mathematical Sciences Peking University HEDPS CAPT & LMAM School of Mathematical SciencesPeking University School of Mathematics and Computational Science Xiangtan University

A Newton multigrid method is developed for one-dimensional （1D） and two- dimensional （2D） steady-state shallow water equations （SWEs） with topography and dry areas. The nonlinear system arising from the well-balanced finite volume discretization of the steady-state SWEs is solved by the Newton method as the outer iteration and a geometric multigrid method with the block symmetric Gauss-Seidel smoother as the inner iteration. The proposed Newton multigrid method makes use of the local residual to regularize the Jacobian matrix of the Newton iteration, and can handle the steady- state problem with wet/dry transition. Several numerical experiments are conducted to demonstrate the efficiency, robustness, and well-balanced property of the proposed method. The relation between the convergence behavior of the Newton multigrid method and the distribution of the eigenvalues of the iteration matrix is detailedly discussed.

关键词： Newton method multigrid block symmetric Gauss-Seidel shallow waterequation （SWE） steady-state solution

Equivalence of three kinds of well-posed-ness of discontinuous Riemann-Hilbert problem for elliptic complex equation in multiply connected domains

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Applied Mathematics(A Journal of Chinese Universities) 2014年第2期29卷 183-193页

作者： WEN Guo-chun LMAM School of Mathematical Sciences Peking University

In this article, we first introduce the general linear elliptic complex equation of first order with certain conditions, and then propose discontinuous Riemann-Hilbert problem and some kinds of modified well-posed-ness for the complex equation. Then we verify the equivalence of three kinds of well-posed-ness. The discontinuous boundary value problem possesses many applications in mechanics and physics etc.

关键词： Discontinuous Riemann-Hilbert problems linear elliptic complex equation equivalence of threekinds of well-posed-ness multiply connected domains.

A Result on the Quasi-periodic Solutions of Forced Isochronous Oscillators at Resonance

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Chinese Annals of Mathematics,Series B 2015年第4期36卷 523-542页

作者： Bin LIU Yingchao TANG LMAM School of Mathematical Sciences Peking University

In this paper, the authors are concerned with the forced isochronous oscillators with a repulsive singularity and a bounded nonlinearity x'' + V'(x) + g(x) = e(t, x, x'),where the assumptions on V, g and e are regular, described precisely in the introduction.Using a variant of Moser's twist theorem of invariant curves, the authors show the existence of quasi-periodic solutions and boundedness of all solutions. This extends the result of Liu to the case of the above system where e depends on the velocity.

关键词：拟周期解振荡器同步 Moser扭转定理共振不变曲线有界性非线性

The Discontinuous Riemann–Hilbert Problem for Elliptic Complex Equations of First Order

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Acta Mathematica Sinica,English Series 2013年第12期29卷 2233-2244页

作者： Guo Chun WEN LMAM School of Mathematical Sciences Peking University

In this article, we first transform the general uniformly elliptic systems of first order equations with certain conditions into the complex equations, and propose the discontinuous Riemann- Hilbert problem and its modified well-posedness for the complex equations. Then we give a priori estimates of solutions of the modified discontinuous Riemann-Hilbert problem for the complex equations and verify its solvability. Finally the solvability results of the original discontinuous Riemann-Hilbert boundary value problem can be derived. The discontinuous boundary value problem possesses many applications in mechanics and physics etc.

关键词： Discontinuous Riemann Hilbert problems elliptic complex equations estimates and ex-istence of solutions multiply connected domains