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Mathematics, Applied
Shan-Mou Cao, Zeng-Qi Wang
Summary: The preconditioned modified Hermitian/skew-Hermitian splitting (PMHSS) iteration method and the corresponding preconditioning technique are effective for solving optimal control problems governed by Poisson's equation. Theoretical results show that the PMHSS iteration method is convergent, as the spectral radius of the iterative matrix is less than root 2/2. The proposed preconditioner is more effective as it avoids inner iterations when solving saddle point systems and remains convergent and independent of parameters and mesh size.
NUMERICAL ALGORITHMS
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Mathematics, Applied
Ivo Dravins, Maya Neytcheva
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Mathematics, Applied
Stefan Guttel, John W. Pearson
Summary: The method proposed in this study utilizes a spectral-in-time representation of the residual along with a Newton-Krylov method to achieve fast and accurate solutions for nonlinear problems across various mesh sizes and problem parameters. The outer Newton and inner Krylov iterations required for achieving the attainable accuracy of spatial discretization remain stable with respect to the number of collocation points in time, and do not significantly change with variations in other problem parameters.
IMA JOURNAL OF NUMERICAL ANALYSIS
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Computer Science, Artificial Intelligence
Liang Bai, Jiye Liang, Yunxiao Zhao
Summary: As a leading graph clustering technique, spectral clustering is widely used for capturing complex clusters in data. However, it is difficult to obtain prior information in unsupervised scenes to guide the clustering process. To solve this problem, we propose a self-constrained spectral clustering algorithm that extends the objective function by adding pairwise and label self-constrained terms. We provide theoretical analysis, an optimization model, and an iterative method to simultaneously learn the clustering results and constraints. The proposed algorithm can discover high-quality cluster structures without prior information, as demonstrated by extensive experiments on benchmark datasets.
IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE
(2023)
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Mathematics, Applied
Thirupathi Gudi, Ramesh Ch. Sau
Summary: In this paper, a finite element analysis is presented for a Dirichlet boundary control problem governed by the Stokes equation. The control is considered in a convex closed subset of the energy space H-1(Omega). The authors introduce the Stokes problem with outflow condition and control on the Dirichlet boundary to overcome the limited regularity of previous control formulations. The theoretical results are validated by numerical tests.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2023)
Article
Mathematics, Applied
V. B. Kiran Kumar, A. Lexy, M. N. N. Namboodiri, A. Noufal
Summary: In this article, an upper bound for the generalized condition number of the covariance matrix is derived, and the concept of preconditioners is extended to singular matrices. Existing preconditioners for non-singular matrices are compared with generalized preconditioners, with numerical experiments showing that the latter exhibit better performance. Additionally, the generating function of the covariance matrix is calculated, and spectral properties are derived using linear algebra techniques.
APPLIED MATHEMATICS AND COMPUTATION
(2023)
Article
Mathematics, Applied
Davod Khojasteh Salkuyeh, Hamid Mirchi
Summary: We propose a new free-parameter double splitting iteration method for solving linear equations resulting from the finite element discretization of a PDE-constrained optimization problem. We prove that the eigenvalues of the iteration matrix lie in the interval (0, 21], indicating unconditional convergence of the method. Numerical comparison against two existing methods is provided to demonstrate the effectiveness of our approach.
APPLIED MATHEMATICS LETTERS
(2023)
Article
Mathematics, Applied
Zhong-Zhi Bai, Kang-Ya Lu
Summary: This study proposes a method for optimal control problems constrained with certain time and space-fractional diffusive equations, achieving specially structured linear systems with positive definiteness. Both theoretical analysis and numerical experiments show that incorporating rotated block-diagonal preconditioners with preconditioned Krylov subspace iteration methods can exhibit optimal convergence properties.
APPLIED NUMERICAL MATHEMATICS
(2021)
Article
Mathematics, Applied
S. C. Buranay, O. C. Iyikal
Summary: This paper proposes a recursive approach to construct incomplete block-matrix factorization of M-matrices using a two-step iterative method to approximate the inverse of diagonal pivoting block matrices. The numerical results justify the effectiveness of this method in providing robust preconditioners.
MATHEMATICAL METHODS IN THE APPLIED SCIENCES
(2021)
Article
Mathematics, Applied
Andrew T. Barker, Andrei Draganescu
Summary: The study introduces an algebraic multigrid (AMG) based preconditioner for the reduced Hessian of a linear-quadratic optimization problem constrained by an elliptic partial differential equation. The construction of the preconditioner relies on a standard AMG infrastructure built for solving the forward elliptic equation, allowing for broad applicability to problems with unstructured grids, complex geometry, and varying coefficients. The method is implemented using the Hypre package and numerical examples are provided.
NUMERICAL LINEAR ALGEBRA WITH APPLICATIONS
(2021)
Article
Computer Science, Interdisciplinary Applications
Liang Fang, Stefan Vandewalle, Johan Meyers
Summary: This study proposes a new parallel-in-time multiple shooting algorithm for solving large scale optimal control problems governed by parabolic PDEs. The algorithm is validated and analyzed using different test cases, showing significant speed-ups and better performance for complex flow fields.
JOURNAL OF COMPUTATIONAL PHYSICS
(2022)
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Environmental Sciences
Junhui Qian, Ziyu Liu, Yuanyuan Lu, Le Zheng, Ailing Zhang, Fengxia Han
Summary: This paper proposes a new algorithmic framework for achieving radar-communication spectral coexistence on moving platforms and simultaneously optimizing communication rate and radar signal performance.
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Operations Research & Management Science
M. V. Dolgopolik
Summary: In this two-part study, we develop a general theory of exact augmented Lagrangians for constrained optimization problems in Hilbert spaces. These augmented Lagrangians are continuously differentiable for smooth problems and do not suffer from the Maratos effect, making them attractive for numerical optimization applications. We aim to study the theoretical properties of exact augmented Lagrangians and explore their applications in constrained variational problems, problems with PDE constraints, and optimal control problems.
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Multidisciplinary Sciences
Carlo Sinigaglia, Davide E. Quadrelli, Andrea Manzoni, Francesco Braghin
Summary: This paper demonstrates an efficient approach to achieve thermal cloaking by controlling the distribution of active heat sources. By formulating the problem as a PDE-constrained optimization, the goal is to actuate the space-time control field so that the thermal field outside the obstacle is indistinguishable from the reference field. To tackle this problem rapidly and reliably, a parametrized reduced order model is utilized, providing significant computational speedups compared to a high-fidelity, full-order model.
PROCEEDINGS OF THE ROYAL SOCIETY A-MATHEMATICAL PHYSICAL AND ENGINEERING SCIENCES
(2022)
Article
Mathematics
Yan-Ran Li, Xin-Hui Shao, Shi-Yu Li
Summary: In this paper, we proposed the three-block splitting (TBS) iterative method and proved its unconditional convergence. The corresponding TBS preconditioner was derived and the spectral properties of the preconditioned matrix were studied. Numerical examples in two-dimensions demonstrated the advantages of the TBS iterative method and TBS preconditioner with the Krylov subspace method.
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Computer Science, Software Engineering
Zhong-Zhi Bai, Michele Benzi
BIT NUMERICAL MATHEMATICS
(2017)
Article
Mathematics, Applied
Shan-Mou Cao, Wei Feng, Zeng-Qi Wang
APPLIED MATHEMATICS LETTERS
(2018)
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Mathematics, Applied
Fatemeh Panjeh Ali Beik, Michele Benzi
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Mathematics, Applied
Zeng-Qi Wang
NUMERICAL LINEAR ALGEBRA WITH APPLICATIONS
(2018)
Article
Mathematics, Applied
Fatemeh Panjeh Ali Beik, Michele Benzi
SIAM JOURNAL ON MATRIX ANALYSIS AND APPLICATIONS
(2018)
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Mathematics, Applied
Zeng-Qi Wang
NUMERICAL LINEAR ALGEBRA WITH APPLICATIONS
(2018)
Article
Mathematics, Applied
Zeng-Qi Wang, Jun-Feng Yin, Quan-Yu Dou
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2020)
Article
Multidisciplinary Sciences
Galina Muratova, Tatiana Martynova, Evgeniya Andreeva, Vadim Bavin, Zeng-Qi Wang
Article
Mathematics, Applied
Shan-Mou Cao, Zeng-Qi Wang
Summary: The preconditioned modified Hermitian/skew-Hermitian splitting (PMHSS) iteration method and the corresponding preconditioning technique are effective for solving optimal control problems governed by Poisson's equation. Theoretical results show that the PMHSS iteration method is convergent, as the spectral radius of the iterative matrix is less than root 2/2. The proposed preconditioner is more effective as it avoids inner iterations when solving saddle point systems and remains convergent and independent of parameters and mesh size.
NUMERICAL ALGORITHMS
(2021)
Editorial Material
Mathematics, Applied
Zhong-Zhi Bai, Lothar Reichel, Zeng-Qi Wang
APPLIED NUMERICAL MATHEMATICS
(2021)
Article
Mathematics, Applied
Wei Feng, Zeng-Qi Wang, Ruo-Bing Zhong, Galina Muratova
Summary: The paper discusses the preconditioning techniques for saddle point linear systems arising in convection-diffusion control problems, highlighting the practicality of the preconditioned Chebyshev iteration method in dealing with large scale linear systems.
EAST ASIAN JOURNAL ON APPLIED MATHEMATICS
(2022)
Article
Mathematics, Applied
Michele Benzi, Michele Rinelli
Summary: Spectral projectors of Hermitian matrices are crucial in various applications, including electronic structure computations. Linear scaling methods for gapped systems are based on the localization property of these special matrix functions, which implies rapid decay of entries away from the main diagonal or under more general sparsity patterns. This paper presents new decay bounds using the relation with the sign function and an integral representation, which are optimal in an asymptotic sense. It also investigates the influence of isolated extremal eigenvalues on decay properties and predicts a superexponential behavior.
LINEAR ALGEBRA AND ITS APPLICATIONS
(2022)
Article
Nanoscience & Nanotechnology
You Xiao, Shuai Wei, Jiajia Xu, Ruoyan Ma, Xiaoyu Liu, Xiaofu Zhang, Tiger H. Tao, Hao Li, Zengqi Wang, Lixing You, Zhen Wang
Summary: The spectrometer combines a superconducting single-photon detector array and 3D-printed photonic-crystal filters, achieving a system sensitivity down to -108.2 dBm and a resolution of 5 nm from 1200 to 1700 nm.
Article
Mathematics
Michele Benzi
Summary: This paper illustrates the role of the field of values of a matrix in certain problems of numerical analysis, such as the approximation of matrix functions and the convergence of preconditioned Krylov subspace methods for solving large systems of equations arising from the discretization of partial differential equations.
BOLLETTINO DELLA UNIONE MATEMATICA ITALIANA
(2021)
Article
Mathematics, Applied
Rui Li, Zeng-Qi Wang
SIAM JOURNAL ON MATRIX ANALYSIS AND APPLICATIONS
(2020)
