Pub Date : 2026-01-06DOI: 10.1016/j.cam.2026.117349
Qinian Jin , Wei Wang , Min Zhong
We consider the augmented Lagrangian method for solving linear ill-posed inverse problems in Banach spaces under the statistical framework that multiple unbiased independent identically distributed measurement data can be acquired. We use the average of these data in the method to reconstruct a solution whose feature is captured by a convex function. After analyzing the method under a priori stopping rules, which are of limited use in practice, we propose a statistical discrepancy principle, which is purely data driven, to terminate the method. We establish the convergence in expectation under general conditions and derive the convergence rates in probability when the sought solution satisfies variational source conditions. Numerical simulations are reported to test our method.
{"title":"Augmented Lagrangian method for linear inverse problems with repeated measurement data","authors":"Qinian Jin , Wei Wang , Min Zhong","doi":"10.1016/j.cam.2026.117349","DOIUrl":"10.1016/j.cam.2026.117349","url":null,"abstract":"<div><div>We consider the augmented Lagrangian method for solving linear ill-posed inverse problems in Banach spaces under the statistical framework that multiple unbiased independent identically distributed measurement data can be acquired. We use the average of these data in the method to reconstruct a solution whose feature is captured by a convex function. After analyzing the method under <em>a priori</em> stopping rules, which are of limited use in practice, we propose a statistical discrepancy principle, which is purely data driven, to terminate the method. We establish the convergence in expectation under general conditions and derive the convergence rates in probability when the sought solution satisfies variational source conditions. Numerical simulations are reported to test our method.</div></div>","PeriodicalId":50226,"journal":{"name":"Journal of Computational and Applied Mathematics","volume":"482 ","pages":"Article 117349"},"PeriodicalIF":2.6,"publicationDate":"2026-01-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145979374","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-01-06DOI: 10.1016/j.cam.2025.117310
Yang Liu , Raúl Tempone
This study analyzes the non-asymptotic convergence behavior of the randomized quasi-Monte Carlo (RQMC) method with applications to linear elliptic partial differential equations (PDEs) with lognormal coefficients. Building upon the error analysis presented in (Owen, 2006), we derive a non-asymptotic convergence estimate depending on the specific integrands, the input dimensionality, and the RQMC quadrature size. We discuss the effects of the variance and dimensionality of the input random variable. Then, we apply the RQMC method with importance sampling (IS) to approximate deterministic, real-valued, bounded linear functionals that depend on the solution of a linear elliptic PDE with a lognormal diffusivity coefficient in bounded domains of , where the random coefficient is modeled as a stationary Gaussian random field parameterized by the trigonometric and wavelet-type basis. We propose two types of IS distributions, analyze their effects on the RQMC convergence rate, and observe the improvements.
{"title":"Error analysis of randomized quasi-Monte Carlo: Non-asymptotic error bound, importance sampling and application to linear elliptic PDEs with lognormal coefficients","authors":"Yang Liu , Raúl Tempone","doi":"10.1016/j.cam.2025.117310","DOIUrl":"10.1016/j.cam.2025.117310","url":null,"abstract":"<div><div>This study analyzes the non-asymptotic convergence behavior of the randomized quasi-Monte Carlo (RQMC) method with applications to linear elliptic partial differential equations (PDEs) with lognormal coefficients. Building upon the error analysis presented in (Owen, 2006), we derive a non-asymptotic convergence estimate depending on the specific integrands, the input dimensionality, and the RQMC quadrature size. We discuss the effects of the variance and dimensionality of the input random variable. Then, we apply the RQMC method with importance sampling (IS) to approximate deterministic, real-valued, bounded linear functionals that depend on the solution of a linear elliptic PDE with a lognormal diffusivity coefficient in bounded domains of <span><math><msup><mi>R</mi><mi>d</mi></msup></math></span>, where the random coefficient is modeled as a stationary Gaussian random field parameterized by the trigonometric and wavelet-type basis. We propose two types of IS distributions, analyze their effects on the RQMC convergence rate, and observe the improvements.</div></div>","PeriodicalId":50226,"journal":{"name":"Journal of Computational and Applied Mathematics","volume":"482 ","pages":"Article 117310"},"PeriodicalIF":2.6,"publicationDate":"2026-01-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145928143","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-01-06DOI: 10.1016/j.cam.2025.117304
Yochay Jerby
<div><div>We introduce a novel variational numerical method for the theoretical investigation of the real zeros of the Hardy <em>Z</em>–function <em>Z</em>(<em>t</em>), designed to circumvent certain limitations of classical approaches. In particular, the traditional Riemann–Siegel formula, despite its widespread use, poses significant analytic challenges due to its complicated, non-closed-form error terms, which hinder its utility for theoretical analysis. Our approach builds on an accelerated asymptotic formula for <em>Z</em>(<em>t</em>) established in earlier work, simplifying the structure of the error term and producing a family of tractable approximants better suited to variational arguments. We construct a variational framework that connects the zeros of the core function <span><math><mrow><msub><mi>Z</mi><mn>0</mn></msub><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mo>=</mo><mi>cos</mi><mrow><mo>(</mo><mi>θ</mi><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mo>)</mo></mrow></mrow></math></span>, whose real zeros are fully understood, to those of <em>Z</em>(<em>t</em>) in the range <span><math><mrow><mn>2</mn><mi>N</mi><mo>≤</mo><mi>t</mi><mo>≤</mo><mn>2</mn><mi>N</mi><mo>+</mo><mn>2</mn></mrow></math></span>. This is achieved via continuous paths in a newly introduced high-dimensional parameter space <span><math><msub><mi>Z</mi><mi>N</mi></msub></math></span>, whose elements are generalized sections of the form<span><span><span><math><mrow><msub><mi>Z</mi><mi>N</mi></msub><mrow><mo>(</mo><mi>t</mi><mo>;</mo><mover><mrow><mi>a</mi></mrow><mo>‾</mo></mover><mo>)</mo></mrow><mo>=</mo><mi>cos</mi><mrow><mo>(</mo><mi>θ</mi><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mo>)</mo></mrow><mo>+</mo><munderover><mo>∑</mo><mrow><mi>k</mi><mo>=</mo><mn>1</mn></mrow><mi>N</mi></munderover><mfrac><msub><mi>a</mi><mi>k</mi></msub><msqrt><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msqrt></mfrac><mi>cos</mi><mrow><mo>(</mo><mi>θ</mi><mo>(</mo><mi>t</mi><mo>)</mo><mo>−</mo><mi>log</mi><mo>(</mo><mi>k</mi><mo>+</mo><mn>1</mn><mo>)</mo><mspace></mspace><mi>t</mi><mo>)</mo></mrow><mo>,</mo></mrow></math></span></span></span>where <span><math><mrow><mover><mrow><mi>a</mi></mrow><mo>‾</mo></mover><mo>=</mo><mrow><mo>(</mo><msub><mi>a</mi><mn>1</mn></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mi>a</mi><mi>N</mi></msub><mo>)</mo></mrow><mo>∈</mo><msup><mi>R</mi><mi>N</mi></msup></mrow></math></span>. This allows for a controlled deformation from <em>Z</em><sub>0</sub>(<em>t</em>) to <em>Z</em>(<em>t</em>) while preserving real zeros, drawing an analogy with spaces of real polynomials whose zeros remain real unless a collision (i.e., a double zero) occurs. We also present a detailed numerical analysis of Newton’s method applied to locating zeros <span><math><msub><mover><mrow><mi>t</mi></mrow><mo>˜</mo></mover><mi>n</mi></msub></math></span> of <em>Z</em>(<em>t</em>), using the zeros <em>t<sub>n</sub></em> of <em>Z</em><sub>0</sub>(<em>t</em>) as initial starting points.
我们介绍了一种新的变分数值方法，用于理论研究Hardy Z -函数Z(t)的实零点，旨在克服经典方法的某些局限性。特别是，传统的黎曼-西格尔公式尽管被广泛使用，但由于其复杂的、非封闭形式的误差项，阻碍了它在理论分析中的应用，给分析带来了重大挑战。我们的方法建立在早期工作中建立的Z(t)的加速渐近公式的基础上，简化了误差项的结构，并产生了一组更适合变分参数的易于处理的近似值。我们构造了一个变分框架，将核心函数Z0(t)=cos（θ(t)）的零点与Z(t)在2N≤t≤2N+2范围内的零点联系起来，其中Z0(t)=cos（θ(t)）的实数零点完全已知。这是通过在新引入的高维参数空间ZN中的连续路径实现的，其元素是形式ZN（t；a形式）=cos(θ(t))+∑k=1Nakk+1cos(θ(t)−log(k+1)t)的广义截面，其中a形式=(a1，…，aN)∈RN。这允许从Z0(t)到Z(t)的可控变形，同时保留实数零，与实数多项式的空间类比，其零保持实数，除非发生碰撞（即双零）。我们还提出了一个详细的数值分析，牛顿的方法应用于定位零点t ~ n的Z(t)，使用零点tn的Z0(t)作为初始起点。我们证明，对于这些零的一个不可忽略的子集，牛顿方法可能失败或变得不稳定，而我们的变分方法在探索范围内显得足够敏感，可以从相应的tn连续跟踪t ~ n。我们进一步表明，在概念层面上，将每个tn连接到唯一的实零t ~ n的非碰撞路径的存在允许人们将黎曼假设重新表述为非线性优化问题，使人想起现代机器学习中遇到的基于梯度的方法。最后，我们解释了在Riemann-Siegel公式的经典框架内构造类似变分格式的结构障碍。
{"title":"A new variational approach for the numerical location of Hardy Z-function zeros on the real line","authors":"Yochay Jerby","doi":"10.1016/j.cam.2025.117304","DOIUrl":"10.1016/j.cam.2025.117304","url":null,"abstract":"<div><div>We introduce a novel variational numerical method for the theoretical investigation of the real zeros of the Hardy <em>Z</em>–function <em>Z</em>(<em>t</em>), designed to circumvent certain limitations of classical approaches. In particular, the traditional Riemann–Siegel formula, despite its widespread use, poses significant analytic challenges due to its complicated, non-closed-form error terms, which hinder its utility for theoretical analysis. Our approach builds on an accelerated asymptotic formula for <em>Z</em>(<em>t</em>) established in earlier work, simplifying the structure of the error term and producing a family of tractable approximants better suited to variational arguments. We construct a variational framework that connects the zeros of the core function <span><math><mrow><msub><mi>Z</mi><mn>0</mn></msub><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mo>=</mo><mi>cos</mi><mrow><mo>(</mo><mi>θ</mi><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mo>)</mo></mrow></mrow></math></span>, whose real zeros are fully understood, to those of <em>Z</em>(<em>t</em>) in the range <span><math><mrow><mn>2</mn><mi>N</mi><mo>≤</mo><mi>t</mi><mo>≤</mo><mn>2</mn><mi>N</mi><mo>+</mo><mn>2</mn></mrow></math></span>. This is achieved via continuous paths in a newly introduced high-dimensional parameter space <span><math><msub><mi>Z</mi><mi>N</mi></msub></math></span>, whose elements are generalized sections of the form<span><span><span><math><mrow><msub><mi>Z</mi><mi>N</mi></msub><mrow><mo>(</mo><mi>t</mi><mo>;</mo><mover><mrow><mi>a</mi></mrow><mo>‾</mo></mover><mo>)</mo></mrow><mo>=</mo><mi>cos</mi><mrow><mo>(</mo><mi>θ</mi><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mo>)</mo></mrow><mo>+</mo><munderover><mo>∑</mo><mrow><mi>k</mi><mo>=</mo><mn>1</mn></mrow><mi>N</mi></munderover><mfrac><msub><mi>a</mi><mi>k</mi></msub><msqrt><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msqrt></mfrac><mi>cos</mi><mrow><mo>(</mo><mi>θ</mi><mo>(</mo><mi>t</mi><mo>)</mo><mo>−</mo><mi>log</mi><mo>(</mo><mi>k</mi><mo>+</mo><mn>1</mn><mo>)</mo><mspace></mspace><mi>t</mi><mo>)</mo></mrow><mo>,</mo></mrow></math></span></span></span>where <span><math><mrow><mover><mrow><mi>a</mi></mrow><mo>‾</mo></mover><mo>=</mo><mrow><mo>(</mo><msub><mi>a</mi><mn>1</mn></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mi>a</mi><mi>N</mi></msub><mo>)</mo></mrow><mo>∈</mo><msup><mi>R</mi><mi>N</mi></msup></mrow></math></span>. This allows for a controlled deformation from <em>Z</em><sub>0</sub>(<em>t</em>) to <em>Z</em>(<em>t</em>) while preserving real zeros, drawing an analogy with spaces of real polynomials whose zeros remain real unless a collision (i.e., a double zero) occurs. We also present a detailed numerical analysis of Newton’s method applied to locating zeros <span><math><msub><mover><mrow><mi>t</mi></mrow><mo>˜</mo></mover><mi>n</mi></msub></math></span> of <em>Z</em>(<em>t</em>), using the zeros <em>t<sub>n</sub></em> of <em>Z</em><sub>0</sub>(<em>t</em>) as initial starting points.","PeriodicalId":50226,"journal":{"name":"Journal of Computational and Applied Mathematics","volume":"482 ","pages":"Article 117304"},"PeriodicalIF":2.6,"publicationDate":"2026-01-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145979452","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-01-03DOI: 10.1016/j.cam.2026.117340
Fuad Kittaneh , Arnab Patra , Jyoti Rani
This study utilizes Orlicz functions to provide refined lower and upper bounds on the q-numerical radius of an operator acting on a Hilbert space. Additionally, the concept of q-sectorial matrices is introduced and further bounds for the q-numerical radius are established. Our results unify several existing bounds for the q-numerical radius. Suitable examples are provided to supplement the estimations.
{"title":"On the estimation of the q-numerical radius via Orlicz functions","authors":"Fuad Kittaneh , Arnab Patra , Jyoti Rani","doi":"10.1016/j.cam.2026.117340","DOIUrl":"10.1016/j.cam.2026.117340","url":null,"abstract":"<div><div>This study utilizes Orlicz functions to provide refined lower and upper bounds on the <em>q</em>-numerical radius of an operator acting on a Hilbert space. Additionally, the concept of <em>q</em>-sectorial matrices is introduced and further bounds for the <em>q</em>-numerical radius are established. Our results unify several existing bounds for the <em>q</em>-numerical radius. Suitable examples are provided to supplement the estimations.</div></div>","PeriodicalId":50226,"journal":{"name":"Journal of Computational and Applied Mathematics","volume":"482 ","pages":"Article 117340"},"PeriodicalIF":2.6,"publicationDate":"2026-01-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145928325","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-01-03DOI: 10.1016/j.cam.2026.117336
Dijana Mosić , Bibekananda Sitha
The fact that weak MPCEP and *CEPMP inverses of square matrices are generalizations of several classes of generalized inverses, inspired us to extend these concepts for rectangular matrices. Precisely, solvability of novel systems of equations is verified based on minimal rank W-weighted (right) weak Drazin inverse. As solutions of new extended systems, definitions of weighted weak MPCEP and *CEPMP inverses are presented, and some known weighted generalized inverses are unified. Characterizations and expressions for weighted weak MPCEP and *CEPMP inverses are given. Dual types of weighted weak MPCEP and *CEPMP inverses are studied too. As consequences, we get definitions and properties of dual weak MPCEP and *CEPMP inverses. Certain systems of linear equations are solved by applying weighted systems of linear equations.
{"title":"Weighted weak MPCEP and *CEPMP inverses","authors":"Dijana Mosić , Bibekananda Sitha","doi":"10.1016/j.cam.2026.117336","DOIUrl":"10.1016/j.cam.2026.117336","url":null,"abstract":"<div><div>The fact that weak MPCEP and *CEPMP inverses of square matrices are generalizations of several classes of generalized inverses, inspired us to extend these concepts for rectangular matrices. Precisely, solvability of novel systems of equations is verified based on minimal rank <em>W</em>-weighted (right) weak Drazin inverse. As solutions of new extended systems, definitions of weighted weak MPCEP and *CEPMP inverses are presented, and some known weighted generalized inverses are unified. Characterizations and expressions for weighted weak MPCEP and *CEPMP inverses are given. Dual types of weighted weak MPCEP and *CEPMP inverses are studied too. As consequences, we get definitions and properties of dual weak MPCEP and *CEPMP inverses. Certain systems of linear equations are solved by applying weighted systems of linear equations.</div></div>","PeriodicalId":50226,"journal":{"name":"Journal of Computational and Applied Mathematics","volume":"482 ","pages":"Article 117336"},"PeriodicalIF":2.6,"publicationDate":"2026-01-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145979370","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-01-03DOI: 10.1016/j.cam.2026.117347
Jiakuang He , Dongqing Wu
Complex conjugate matrix equations (CCME) are important in computation and antilinear systems. Existing research mainly focuses on the time-invariant version, while studies on the time-variant version and its solution using artificial neural networks are still lacking. This paper introduces zeroing neural dynamics (ZND) to solve the earliest time-variant CCME. Firstly, the vectorization and Kronecker product in the complex field are defined uniformly. Secondly, Con-CZND1 and Con-CZND2 models are proposed, and their convergence and effectiveness are theoretically proved. Thirdly, numerical experiments confirm their effectiveness and highlight their differences. The results show the advantages of ZND in the complex field compared with that in the real field, and further refine the related theory.
{"title":"Zeroing neural dynamics solving time-variant complex conjugate matrix equation X(τ)F(τ)−A(τ)X‾(τ)=C(τ)","authors":"Jiakuang He , Dongqing Wu","doi":"10.1016/j.cam.2026.117347","DOIUrl":"10.1016/j.cam.2026.117347","url":null,"abstract":"<div><div>Complex conjugate matrix equations (CCME) are important in computation and antilinear systems. Existing research mainly focuses on the time-invariant version, while studies on the time-variant version and its solution using artificial neural networks are still lacking. This paper introduces zeroing neural dynamics (ZND) to solve the earliest time-variant CCME. Firstly, the vectorization and Kronecker product in the complex field are defined uniformly. Secondly, Con-CZND1 and Con-CZND2 models are proposed, and their convergence and effectiveness are theoretically proved. Thirdly, numerical experiments confirm their effectiveness and highlight their differences. The results show the advantages of ZND in the complex field compared with that in the real field, and further refine the related theory.</div></div>","PeriodicalId":50226,"journal":{"name":"Journal of Computational and Applied Mathematics","volume":"482 ","pages":"Article 117347"},"PeriodicalIF":2.6,"publicationDate":"2026-01-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145979377","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-01-03DOI: 10.1016/j.cam.2025.117333
Ioannis Dassios , Eoin Syron
In this paper, we develop a mathematical dynamic model for a coupled hydrogen-gas-electricity network, focusing on the dynamic interactions between these systems. The model incorporates hydrogen injection into the gas network and its impact on gas flow and pressure. Additionally, gas-fired power plants consume gas to generate electricity, linking the gas and electricity networks. The model includes partial differential equations (PDEs) for the gas network, differential-algebraic equations (DAEs) for the electricity network, and numerical techniques for handling slow-fast dynamics. A mathematical Theorem is introduced to describe the coupling between hydrogen injection, gas flow, and electricity generation, demonstrating how these systems interact in a stable and efficient manner. A numerical example illustrates the theory. The proposed model and Theorem provide a robust framework for understanding and optimizing the operation of coupled energy systems.
{"title":"Dynamic modeling and analysis of coupled hydrogen-Gas-Electricity networks","authors":"Ioannis Dassios , Eoin Syron","doi":"10.1016/j.cam.2025.117333","DOIUrl":"10.1016/j.cam.2025.117333","url":null,"abstract":"<div><div>In this paper, we develop a mathematical dynamic model for a coupled hydrogen-gas-electricity network, focusing on the dynamic interactions between these systems. The model incorporates hydrogen injection into the gas network and its impact on gas flow and pressure. Additionally, gas-fired power plants consume gas to generate electricity, linking the gas and electricity networks. The model includes partial differential equations (PDEs) for the gas network, differential-algebraic equations (DAEs) for the electricity network, and numerical techniques for handling slow-fast dynamics. A mathematical Theorem is introduced to describe the coupling between hydrogen injection, gas flow, and electricity generation, demonstrating how these systems interact in a stable and efficient manner. A numerical example illustrates the theory. The proposed model and Theorem provide a robust framework for understanding and optimizing the operation of coupled energy systems.</div></div>","PeriodicalId":50226,"journal":{"name":"Journal of Computational and Applied Mathematics","volume":"482 ","pages":"Article 117333"},"PeriodicalIF":2.6,"publicationDate":"2026-01-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145979451","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-01-03DOI: 10.1016/j.cam.2026.117353
Huiting Zhang, Zhanshan Wang
In 2014, Tian [1] pointed out that: “A challenging task is to give the closed-form of the general common Hermitian solution of and that satisfy X > 0 ( ≥ 0, < 0, ≤ 0), which is equivalent to solving the Hermitian matrix inequality (HMI) .” Due to the discontinuity and nonconvexity of the rank and inertia of a matrix, the approach in [1] cannot provide the closed-form solutions to the HMI. To solve this problem, we adopt an auxiliary variable substitution method, to transform the matrix inequality problem into the task of finding a common Hermitian nonnegative definite solution (common Hermitian positive definite solution) K for a constructed matrix pair. A theorem is proposed, an algorithm is established, and a closed-form solution is obtained for the first time. Moreover, the proposed method not only overcomes the limitation in [1] where no explicit solution could be provided, but also facilitates the solution of a class of output feedback controllers design problems. Utilizing the obtained results, we discuss the applications of the kind of matrix inequality from the perspectives of applied mathematics and the control theory. Finally, two numerical examples are given to validate the correctness of the derived closed-form solutions.
{"title":"Closed-form solutions to a kind of matrix inequality and its applications in control systems","authors":"Huiting Zhang, Zhanshan Wang","doi":"10.1016/j.cam.2026.117353","DOIUrl":"10.1016/j.cam.2026.117353","url":null,"abstract":"<div><div>In 2014, Tian [1] pointed out that: “A challenging task is to give the closed-form of the general common Hermitian solution of <span><math><mrow><msub><mi>B</mi><mn>2</mn></msub><mi>X</mi><msubsup><mi>B</mi><mrow><mn>2</mn></mrow><mo>*</mo></msubsup><mo>=</mo><msub><mi>A</mi><mn>2</mn></msub></mrow></math></span> and <span><math><mrow><msub><mi>B</mi><mn>3</mn></msub><mi>X</mi><msubsup><mi>B</mi><mrow><mn>3</mn></mrow><mo>*</mo></msubsup><mo>=</mo><msub><mi>A</mi><mn>3</mn></msub></mrow></math></span> that satisfy <em>X</em> > 0 ( ≥ 0, < 0, ≤ 0), which is equivalent to solving the Hermitian matrix inequality (HMI) <span><math><mrow><mi>A</mi><mo>+</mo><mi>B</mi><mi>X</mi><mo>+</mo><msup><mrow><mo>(</mo><mi>B</mi><mi>X</mi><mo>)</mo></mrow><mo>*</mo></msup><mo>+</mo><mi>C</mi><mi>Y</mi><mi>D</mi><mo>+</mo><msup><mrow><mo>(</mo><mi>C</mi><mi>Y</mi><mi>D</mi><mo>)</mo></mrow><mo>*</mo></msup><mo>≥</mo><mn>0</mn><mspace></mspace><mrow><mo>(</mo><mo>></mo><mn>0</mn><mo>,</mo><mo><</mo><mn>0</mn><mo>,</mo><mo>≤</mo><mn>0</mn><mo>)</mo></mrow></mrow></math></span>.” Due to the discontinuity and nonconvexity of the rank and inertia of a matrix, the approach in <span><span>[1]</span></span> cannot provide the closed-form solutions to the HMI. To solve this problem, we adopt an auxiliary variable substitution method, to transform the matrix inequality problem into the task of finding a common Hermitian nonnegative definite solution (common Hermitian positive definite solution) <em>K</em> for a constructed matrix pair. A theorem is proposed, an algorithm is established, and a closed-form solution is obtained for the first time. Moreover, the proposed method not only overcomes the limitation in <span><span>[1]</span></span> where no explicit solution could be provided, but also facilitates the solution of a class of output feedback controllers design problems. Utilizing the obtained results, we discuss the applications of the kind of matrix inequality from the perspectives of applied mathematics and the control theory. Finally, two numerical examples are given to validate the correctness of the derived closed-form solutions.</div></div>","PeriodicalId":50226,"journal":{"name":"Journal of Computational and Applied Mathematics","volume":"482 ","pages":"Article 117353"},"PeriodicalIF":2.6,"publicationDate":"2026-01-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145928327","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-01-03DOI: 10.1016/j.cam.2026.117343
Yebo Xiong , Jianzhou Liu
H-matrices are widely used in fields such as control theory, engineering computation, optimization theory, probability and statistics. Many important results rely on the assumption that a given matrix is an H-matrix, making accurate H-matrix identification algorithms essential. Although various methods for identifying H-matrices have been proposed, few consider the effects of round-off errors. In this paper, we present an efficient and necessary condition for H-matrices. Under the consideration of rounding errors, we propose a chase-like iterative algorithm for H-matrix identification. Additionally, the scaling factors in the positive diagonal matrices are computed to finite decimals, enabling the algorithms to produce accurate results unaffected by round-off errors in certain cases. Finally, numerical experiments demonstrate the efficiency and accuracy of our algorithms.
{"title":"An accurate chase-like iterative algorithm for H-matrices","authors":"Yebo Xiong , Jianzhou Liu","doi":"10.1016/j.cam.2026.117343","DOIUrl":"10.1016/j.cam.2026.117343","url":null,"abstract":"<div><div>H-matrices are widely used in fields such as control theory, engineering computation, optimization theory, probability and statistics. Many important results rely on the assumption that a given matrix is an H-matrix, making accurate H-matrix identification algorithms essential. Although various methods for identifying H-matrices have been proposed, few consider the effects of round-off errors. In this paper, we present an efficient and necessary condition for H-matrices. Under the consideration of rounding errors, we propose a chase-like iterative algorithm for H-matrix identification. Additionally, the scaling factors in the positive diagonal matrices are computed to finite decimals, enabling the algorithms to produce accurate results unaffected by round-off errors in certain cases. Finally, numerical experiments demonstrate the efficiency and accuracy of our algorithms.</div></div>","PeriodicalId":50226,"journal":{"name":"Journal of Computational and Applied Mathematics","volume":"482 ","pages":"Article 117343"},"PeriodicalIF":2.6,"publicationDate":"2026-01-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145928326","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-01-03DOI: 10.1016/j.cam.2026.117354
Yifen Ke , Changfeng Ma , Yajun Xie
For large and sparse linear systems,it is effective to reduce the original problem into a lower dimensional linear system and solve the derived equation instead. The main contribution of this paper is that a novel adaptive parameter iteration algorithm is constructed from the perspective of numerical optimization for a class of two-by-two linear systems. The new algorithm adopts a prediction-correction two-step iteration, which uses delayed information to define the iterations. Global convergence results are established, and the algorithm enjoys at least a Q-linear convergence rate under suitable conditions. Numerical experiments demonstrate the efficiency and effectiveness of the new algorithm in applications to elliptic PDE-constrained optimization problems,complex symmetric systems, and saddle point problems, in comparison with existing similar algorithms.
{"title":"An adaptive parameter iteration algorithm for a class of large and sparse linear systems","authors":"Yifen Ke , Changfeng Ma , Yajun Xie","doi":"10.1016/j.cam.2026.117354","DOIUrl":"10.1016/j.cam.2026.117354","url":null,"abstract":"<div><div>For large and sparse linear systems,it is effective to reduce the original problem into a lower dimensional linear system and solve the derived equation instead. The main contribution of this paper is that a novel adaptive parameter iteration algorithm is constructed from the perspective of numerical optimization for a class of two-by-two linear systems. The new algorithm adopts a prediction-correction two-step iteration, which uses delayed information to define the iterations. Global convergence results are established, and the algorithm enjoys at least a <em>Q</em>-linear convergence rate under suitable conditions. Numerical experiments demonstrate the efficiency and effectiveness of the new algorithm in applications to elliptic PDE-constrained optimization problems,complex symmetric systems, and saddle point problems, in comparison with existing similar algorithms.</div></div>","PeriodicalId":50226,"journal":{"name":"Journal of Computational and Applied Mathematics","volume":"482 ","pages":"Article 117354"},"PeriodicalIF":2.6,"publicationDate":"2026-01-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145928205","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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