Research Progress

• A two-level preconditioned Helmholtz-Jacobi-Davidson method for the Maxwell eigenvalue problem (Xuejun Xu)

05 12, 2022

In this paper, based on a domain decomposition (DD) method, we shall propose an efficient two-level preconditioned Helmholtz-Jacobi-Davidson (PHJD) method for solving the algebraic eigenvalue problem resulting from the edge element approximation of the Maxwell eigenvalue problem. In order to eliminate the components in orthogonal complement space of the eigenvalue, we shall solve a parallel preconditioned system and a Helmholtz projection system together in fine space. After one coarse space correction in each iteration and minimizing the Rayleigh quotient in a small dimensional Davidson space, we finally get the error reduction of this two-level PHJD method as $\gamma=c(H)(1-C\frac{\delta^{2}}{H^{2}})$, where C is a constant independent of the mesh size h and the diameter of subdomains H, $\delta$ is the overlapping size among the subdomains, and c(H) decreasing as $H\to 0$, which means the greater the number of subdomains, the better the convergence rate. Numerical results supporting our theory shall be given.
Publication:
Mathematics of Computation 91 (2022), 623-657
Author:
Qigang Liang
School of Mathematical Science, Tongji University, Shanghai 200092, People’s Republic of China
E-mail: qigang_liang@tongji.edu.cn
Xuejun Xu
Institute of Computational Mathematics, AMSS, Chinese Academy of Sciences, Beijing 100190, People’s Republic of China，and School of Mathematical Science, Tongji University, Shanghai 200092，People’s Republic of China.
E-mail: xxj@lsec.cc.ac.cn
• Evolution of Controllers Under a Generalized Structure Encoding/Decoding Scheme with Application to Magnetic Levitation System (Wenchao Xue)

05 12, 2022

Evolutionary search has been widely implemented for the adjustment of controllers’ parameters. Nevertheless, the structure of controllers, which has a more important role in control systems, has been seldom studied. To this end, an evolutionary design method of controllers is proposed to optimize both structures and parameters simultaneously in this article. A controller is made up of a combination of some basic controller components and relevant parameters. The design of controllers can be transformed into an optimization problem involving the structure (represented by discrete vectors) and parameters (represented by real numbers). A generalized structure encoding/decoding scheme is developed. Guided by the performance indicators, intelligent algorithms for both combinatorial and numerical optimization are employed to iteratively and cooperatively evolve the controller structure and parameters, respectively. In order to effectively reduce some redundant or infeasible solutions, a set of generation rules for the controller structure are put forward, which also ensures the feasibility of the structure. Furthermore, this method is applied to a magnetic levitation ball system with nonlinear dynamics and external disturbance. Both simulation and experiment results demonstrate the effectiveness and practicability of the proposed method.
Publication:
IEEE Transactions on Industrial Electronics (Volume: 69, Issue: 9, Sept. 2022)

Author:
Bin Xin
School of Automation, Beijing Institute of Technology, Beijing, China
Beijing Advanced Innovation Center for Intelligent Robots and Systems, Beijing Institute of Technology, Beijing, China
Yipeng Wang
School of Automation, Beijing Institute of Technology, Beijing, China
Peng Cheng Laboratory, Shenzhen, China
Wenchao Xue
Key Laboratory of Systems and Control, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, China
School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing, China
E-mail: wenchaoxue@amss.ac.cn
Tao Cai
School of Automation, Beijing Institute of Technology, Beijing, China
Zhun Fan
Guangdong Provincial Key Laboratory of Digital Signal and Image Processing and the Key Laboratory of Intelligent Manufacturing Technology, Ministry of Education, Shantou University, Shantou, China
Jiaoyang Zhan
School of Automation, Beijing Institute of Technology, Beijing, China
Jie Chen
School of Automation, Beijing Institute of Technology, Beijing, China
Beijing Advanced Innovation Center for Intelligent Robots and Systems, Beijing Institute of Technology, Beijing, China
• Connectedness of Kisin varieties associated to absolutely irreducible Galois representations (Sian Nie)

05 12, 2022

We consider the Kisin variety associated to a $n$-dimensional absolutely irreducible mod $p$ Galois representation $\bar\rho$ of a $p$-adic field $K$ and a cocharacter $\mu$. Kisin conjectured that the Kisin variety is connected in this case. We show that Kisin's conjecture holds if $K$ is totally ramfied with $n=3$ or $\mu$ is of a very particular form. As an application, we also get a connectedness result for the deformation ring associated to $\bar\rho$ of given Hodge-Tate weights. We also give counterexamples to show Kisin's conjecture does not hold in general.
Publication:
Journal für die reine und angewandte Mathematik (Crelles Journal) Volume 2022 Issue 785

Author:
Miaofen Chen
Department of Mathematics, Shanghai Key Laboratory of PMMP, East China Normal University, No. 500, Dong Chuan Road, Shanghai 200241, China
E-mail: mfchen@math.ecnu.edu.cn

Sian Nie
Academy of Mathematics and Systems Science, Chinese Academy of Sciences, No. 55, Zhongguancun East Road, Beijing 100190, China
E-mail: niesian@amss.ac.cn
• Semiclassical Limit of Gross--Pitaevskii Equation with Dirichlet Boundary Condition (Ping Zhang)

05 12, 2022

In this paper, we justify the semiclassical limit of the Gross--Pitaevskii equation with Dirichlet boundary condition on the three-dimensional upper space under the assumption that the leading-order terms to both initial amplitude and initial phase function are sufficiently small in some high enough Sobolev norms. We remark that the main difficulty of the proof lies in the fact that the boundary layer appears in the leading-order terms of the amplitude functions and the gradient of the phase functions to the WKB expansions of the solutions. In particular, we partially solved the open question proposed in [D. Chiron and F. Rousset, Comm. Math. Phys., 288 (2009), pp. 503--546; C. T. Pham, C. Nore, and M. E. Brachet, Phys. D, 210 (2005), pp. 203--226] concerning the semiclassical limit of the Gross--Pitaevskii equation with Dirichlet boundary condition.
Publication:
SIAM Journal on Mathematical Analysis Vol. 54, Iss. 1

Author:
Guilong Gui
School of Mathematics, Northwest University, Xi’an 710069, China.
E-mail: glgui@amss.ac.cn

Ping Zhang
Academy of Mathematics & Systems Science and Hua Loo-Keng Key Laboratory of Mathematics, The Chinese Academy of Sciences, Beijing 100190, China, and School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China.
E-mail: zp@amss.ac.cn
• Phase Transition Analysis for Covariance-Based Massive Random Access With Massive MIMO (Ya-Feng Liu)

05 12, 2022

This paper considers a massive random access problem in which a large number of sporadically active devices wish to communicate with a base station (BS) equipped with massive multiple-input multiple-output (MIMO) antennas. Each device is preassigned a unique signature sequence, and the BS identifies the active devices by detecting which sequences are transmitted. This device activity detection problem can be formulated as a maximum likelihood estimation (MLE) problem for which the sample covariance matrix of the received signal is a sufficient statistic. The goal of this paper is to characterize the feasible set of problem parameters under which this covariance based approach is able to successfully recover the device activities in the massive MIMO regime. Through an analysis of the asymptotic behaviors of MLE via its associated Fisher information matrix, this paper derives a necessary and sufficient condition on the Fisher information matrix to ensure a vanishing probability of detection error as the number of antennas goes to infinity, based on which a numerical phase transition analysis is obtained. This condition is also examined from a perspective of covariance matching, which relates the phase transition analysis to a recently derived scaling law. Further, we provide a characterization of the distribution of the estimation error in MLE, based on which the error probabilities in device activity detection can be accurately predicted. Finally, this paper studies a random access scheme with joint device activity and data detection and analyzes its performance in a similar way.
Publication:
IEEE Transactions on Information Theory ( Volume: 68, Issue: 3, March 2022)
Author:
Zhilin Chen
University of Toronto, Toronto, ON, Canada
University of Toronto, Toronto, ON, Canada
Ya-Feng Liu
State Key Laboratory of Scientific and Engineering Computing, Institute of Computational Mathematics and Scientific/ Engineering Computing, Academy of Mathematics and Systems Science (AMSS), Chinese Academy of Sciences (CAS), Beijing, China
E-mail: yafliu@lsec.cc.ac.cn
Wei Yu
University of Toronto, Toronto, ON, Canada
• Robust Tracking Error Feedback Control for a One-Dimensional Schrodinger Equation (Baozhu Guo)

05 12, 2022

In this article, we consider robust output tracking for a Schr？dinger equation with external disturbances in all possible channels. The challenge of the problem comes from the fact that the observation operator is unbounded and the regulated output and the control are noncollocated. An observer-based approach is adopted in investigation. We first select specially some coefficients of the disturbances to obtain a nominal system, which is a coupled PDE+ODE system. For this nominal system, we design a feedforward control by solving related regulator equation. An observer is then designed for the nominal system in terms of the tracking error only. As a result, an error feedback control is, thus, designed by replacing the state and disturbances in the feedforward control with their estimates obtained from the observer. We show that this observer based error feedback control is robust to disturbances in all possible channels and system uncertainty. The stability of the closed loop and convergence are established by the Riesz basis approach. Some numerical simulations are presented to validate the results.
Publication:
IEEE Transactions on Automatic Control (Volume: 67, Issue: 3, March 2022)
Author:
Junjun Liu
College of Mathematics, Taiyuan University of Technology, Taiyuan, China
E-mail: liujunjun@tyut.edu.cn
Baozhu Guo
School of Mathematics and Physics, North China Electric Power University, Beijing, China
Key Laboratory of System and Control, Academy of Mathematics and Systems Science, Academia Sinica, Beijing, China
E-mail: bzguo@iss.ac.cn
• Admissible ways of merging p-values under arbitrary dependence (Bin Wang)

05 12, 2022

Methods of merging several p-values into a single p-value are important in their own right and widely used in multiple hypothesis testing. This paper is the first to systematically study the admissibility (in Wald’s sense) of p-merging functions and their domination structure, without any information on the dependence structure of the input p-values. As a technical tool, we use the notion of e-values, which are alternatives to p-values recently promoted by several authors. We obtain several results on the representation of admissible p-merging functions via e-values and on (in)admissibility of existing p-merging functions. By introducing new admissible p-merging functions, we show that some classic merging methods can be strictly improved to enhance power without compromising validity under arbitrary dependence.
Publication:
Annals of Statistics 50(1): 351-375 (February 2022)
Author:
Department of Computer Science, Royal Holloway, University of London, Egham, Surrey, UK.
E-mail: v.vovk@rhul.ac.uk

Bin Wang
RCSDS, NCMIS, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, China.
E-mail: wangbin@amss.ac.cn

Ruodu Wang
Department of Statistics and Actuarial Science, University of Waterloo, Waterloo, Ontario, Canada.
E-mail: wang@uwaterloo.ca
• Convergence analysis of inexact two-grid methods: A theoretical framework (Chensong Zhang)

05 12, 2022

Multigrid is one of the most efficient methods for solving large-scale linear systems that arise from discretized partial differential equations. As a foundation for multigrid analysis, two-grid theory plays an important role in motivating and analyzing multigrid algorithms. For symmetric positive definite problems, the convergence theory of two-grid methods with exact solution of the Galerkin coarse-grid system is mature, and the convergence factor of exact two-grid methods can be characterized by an identity. Compared with the exact case, the convergence theory of inexact two-grid methods (i.e., the coarse-grid system is solved approximately) is of more practical significance, while it is still less developed in the literature (one reason is that the error propagation matrix of inexact coarse-grid correction is not a projection). In this paper, we develop a theoretical framework for the convergence analysis of inexact two-grid methods. More specifically, we present two-sided bounds for the energy norm of the error propagation matrix of inexact two-grid methods, from which one can readily obtain the identity for exact two-grid convergence. As an application, we establish a unified convergence theory for multigrid methods, which allows the coarsest-grid system to be solved approximately.
Publication:
SIAM Journal on Numerical Analysis, Vol. 60, Iss. 1
Author:
Xuefeng Xu
Department of Mathematics, Purdue University, West Lafayette, IN 47907, USA
E-mail: xuxuefeng@lsec.cc.ac.cn , xu1412@purdue.edu

Chensong Zhang
LSEC & NCMIS, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China, and School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China
E-mail: zhangcs@lsec.cc.ac.cn
• Energy-preserving fully-discrete schemes for nonlinear stochastic wave equations with multiplicative noise (Jialin Hong)

05 12, 2022

In this paper, we focus on constructing numerical schemes preserving the averaged energy evolution law for nonlinear stochastic wave equations driven by multiplicative noise. We first apply the compact finite difference method and the interior penalty discontinuous Galerkin finite element method to discretize space variable and present two semi-discrete schemes, respectively. Then we make use of the discrete gradient method and the Padé approximation to propose efficient fully-discrete schemes. These semi-discrete and fully-discrete schemes are proved to preserve the discrete averaged energy evolution law. In particular, we also prove that the proposed fully-discrete schemes exactly inherit the energy evolution law almost surely if the considered model is driven by additive noise. Numerical experiments are given to confirm theoretical findings.
Publication:
Journal of Computational Physics, Volume 451, 15 February 2022
Author:
Jialin Hong
Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China
School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China
E-mail: hjl@lsec.cc.ac.cn

Baohui Hou
Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China
E-mail: houbaohui@lsec.cc.ac.cn
Liying Sun
Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China
E-mail: liyingsun@lsec.cc.ac.cn
• Analysis and approximations of Dirichlet boundary control of Stokes flows in the energy space (Wei Gong)

05 12, 2022

We study Dirichlet boundary control of Stokes flows in 2D polygonal domains. We consider cost functionals with two different boundary control regularization terms: the ^2(\Gamma)$-norm and an energy space seminorm. We prove well-posedness, provide first order optimality conditions, derive regularity results, and develop finite element discretizations for both problems, and we also prove finite element error estimates for the latter problem. The motivation to study the energy space problem follows from our analysis: we prove that the choice of the control space ^2(\Gamma)$ can lead to an optimal control with discontinuities at the corners, even when the domain is convex. This phenomenon is also observed in numerical experiments. This behavior does not occur in Dirichlet boundary control problems for the Poisson equation on convex polygonal domains, and it may not be desirable in real applications. For the energy space problem, we show that the solution of the control problem is more regular than the solution of the problem with the ^2(\Gamma)\$-regularization. The improved regularity enables us to prove a priori error estimates for the control in the energy norm. We present several numerical experiments for both control problems on convex and nonconvex domains.
Publication:
SIAM Journal on Numerical Analysis Vol. 60, Iss. 1

Author:
Wei Gong
The State Key Laboratory of Scientific and Engineering Computing, Institute of Computational Mathematics & National Center for Mathematics and Interdisciplinary Sciences, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, 100190 Beijing, China.
E-mail: wgong@lsec.cc.ac.cn

Mariano Mateos
Dpto. de Matemáticas. Universidad de Oviedo, Campus de Gijón, Spain.
E-mail: mmateos@uniovi.es

John R. Singler
Department of Mathematics and Statistics, Missouri University of Science and Technology, Rolla, MO.
E-mail: singlerj@mst.edu

Yangwen Zhang
Department of Mathematical Science, University of Delaware, Newark, DE.
E-mail: ywzhangf@udel.edu