Pub Date : 2026-01-13DOI: 10.1016/j.cam.2026.117366
J.~A. Ezquerro, M.~A. Hernández-Verón
By using fixed point techniques, we analyze the existence and location of solution of a nonlinear integral equation of Volterra type. These techniques are used to approximate solutions to integral equations of this type, which also allow us to approximate them with quadratic convergence. Besides, we improve this analysis by applying collocation methods based on interpolation polynomials of Lagrange that use zeros of orthogonal Chebyshev polynomials as collocation points. In addition, to solve the sensitivity and ill-conditioning problems that might arise, we apply an Ulm-type iterative method that does not use inverses in the algorithm, but only matrix products.
{"title":"A quadratic approximation for solving nonlinear integral equations of Volterra type","authors":"J.~A. Ezquerro, M.~A. Hernández-Verón","doi":"10.1016/j.cam.2026.117366","DOIUrl":"10.1016/j.cam.2026.117366","url":null,"abstract":"<div><div>By using fixed point techniques, we analyze the existence and location of solution of a nonlinear integral equation of Volterra type. These techniques are used to approximate solutions to integral equations of this type, which also allow us to approximate them with quadratic convergence. Besides, we improve this analysis by applying collocation methods based on interpolation polynomials of Lagrange that use zeros of orthogonal Chebyshev polynomials as collocation points. In addition, to solve the sensitivity and ill-conditioning problems that might arise, we apply an Ulm-type iterative method that does not use inverses in the algorithm, but only matrix products.</div></div>","PeriodicalId":50226,"journal":{"name":"Journal of Computational and Applied Mathematics","volume":"482 ","pages":"Article 117366"},"PeriodicalIF":2.6,"publicationDate":"2026-01-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145979376","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 : 2026-01-12DOI: 10.1016/j.cam.2026.117352
Shaojiu Bi , Minmin Li , Guangcheng Cai
This study proposes a nonconvex fractional-order variational model for Gaussian noise removal, and a corresponding solution method is designed based on the primal-dual algorithm. Due to the significant advantages of combining the nonconvex Lq quasi-norm and the fractional-order total variation, the proposed model can better suppress the staircase effect, thereby improving the quality of the recovered image. An adaptive regularization parameter is designed based on the Morozov discrepancy principle to ensure the denoised image remains in a particular set. The algorithm’s convergence is analyzed using various mathematical theories, such as sampling saddle point theory. Simulation and comparison experiments verify the effectiveness of the algorithm in image denoising.
{"title":"Adaptive fractional-order primal-dual image denoising algorithm based on Lq quasi-norm","authors":"Shaojiu Bi , Minmin Li , Guangcheng Cai","doi":"10.1016/j.cam.2026.117352","DOIUrl":"10.1016/j.cam.2026.117352","url":null,"abstract":"<div><div>This study proposes a nonconvex fractional-order variational model for Gaussian noise removal, and a corresponding solution method is designed based on the primal-dual algorithm. Due to the significant advantages of combining the nonconvex <em>L<sub>q</sub></em> quasi-norm and the fractional-order total variation, the proposed model can better suppress the staircase effect, thereby improving the quality of the recovered image. An adaptive regularization parameter is designed based on the Morozov discrepancy principle to ensure the denoised image remains in a particular set. The algorithm’s convergence is analyzed using various mathematical theories, such as sampling saddle point theory. Simulation and comparison experiments verify the effectiveness of the algorithm in image denoising.</div></div>","PeriodicalId":50226,"journal":{"name":"Journal of Computational and Applied Mathematics","volume":"482 ","pages":"Article 117352"},"PeriodicalIF":2.6,"publicationDate":"2026-01-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145979375","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}
In this paper, we develop efficient preconditioning techniques for distributed optimal control problems governed by partial differential equations with Caputo fractional derivative in time. By employing a discretize-then-optimize approach combining mixed all-at-once schemes of finite-difference for temporal and finite-element for spatial discretizations, we derive a large-scale and ill-conditioned Kronecker structured block two-by-two linear system with distinct pivot blocks. A block approximate factorization preconditioning method that is well-suited for approximating the Schur complement is considered by utilizing the so called matching strategy. A distinctive feature of the proposed preconditioner is its computational efficiency arising from its practical Schur complement-free implementation manner. Furthermore, the eigenvalues of the preconditioned system are demonstrated to lie within parameter-free positive real intervals, ensuring fast convergence independent of problem parameters under Krylov subspace acceleration. Motivated by the inherent block-Toeplitz structures, circulant-based inexact variants of the proposed preconditioner are explored and implemented within diagonalization strategies by fast Fourier transformation (FFT). Numerical experiments are conducted to validate the effectiveness and robustness of our proposed preconditioners compared with some optimal preconditioning strategies.
{"title":"Efficient preconditioning techniques for time-fractional PDE-constrained optimization problems","authors":"Zhao-Zheng Liang , Guo-Feng Zhang , Lei Zhang , Mu-Zheng Zhu","doi":"10.1016/j.cam.2026.117348","DOIUrl":"10.1016/j.cam.2026.117348","url":null,"abstract":"<div><div>In this paper, we develop efficient preconditioning techniques for distributed optimal control problems governed by partial differential equations with Caputo fractional derivative in time. By employing a discretize-then-optimize approach combining mixed all-at-once schemes of finite-difference for temporal and finite-element for spatial discretizations, we derive a large-scale and ill-conditioned Kronecker structured block two-by-two linear system with distinct pivot blocks. A block approximate factorization preconditioning method that is well-suited for approximating the Schur complement is considered by utilizing the so called matching strategy. A distinctive feature of the proposed preconditioner is its computational efficiency arising from its practical Schur complement-free implementation manner. Furthermore, the eigenvalues of the preconditioned system are demonstrated to lie within parameter-free positive real intervals, ensuring fast convergence independent of problem parameters under Krylov subspace acceleration. Motivated by the inherent block-Toeplitz structures, circulant-based inexact variants of the proposed preconditioner are explored and implemented within diagonalization strategies by fast Fourier transformation (FFT). Numerical experiments are conducted to validate the effectiveness and robustness of our proposed preconditioners compared with some optimal preconditioning strategies.</div></div>","PeriodicalId":50226,"journal":{"name":"Journal of Computational and Applied Mathematics","volume":"483 ","pages":"Article 117348"},"PeriodicalIF":2.6,"publicationDate":"2026-01-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145982028","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 : 2026-01-11DOI: 10.1016/j.cam.2026.117342
Tanya V. Tafolla , Stéphane Gaudreault , Mayya Tokman
High order exponential integrators require computing linear combination of exponential-like φ-functions of large matrices A times a vector v. Krylov projection methods are the most general and remain an efficient choice for computing the matrix-function-vector-product evaluation when the matrix A is large and unable to be explicitly stored, or when obtaining information about the spectrum is expensive. The Krylov approximation relies on the Gram-Schmidt (GS) orthogonalization procedure to produce the orthonormal basis Vm. In parallel, GS orthogonalization requires global synchronizations for inner products and vector normalization in the orthogonalization process. Reducing the amount of global synchronizations is of paramount importance for the efficiency of a numerical algorithm in a massively parallel setting. We improve the strong scaling properties and parallel efficiency of exponential integrators by addressing the underlying bottleneck in the linear algebra using low-synchronization GS methods. The resulting orthogonalization algorithms have an accuracy comparable to modified Gram-Schmidt yet are better suited for distributed architecture, as only one global communication is required per orthogonalization-step. We present geophysics based numerical experiments and standard examples routinely used to test stiff time integrators, which validate that reducing global communication leads to better parallel scalability and reduced time-to-solution for exponential integrators.
{"title":"Low-synchronization Arnoldi algorithms with application to exponential integrators","authors":"Tanya V. Tafolla , Stéphane Gaudreault , Mayya Tokman","doi":"10.1016/j.cam.2026.117342","DOIUrl":"10.1016/j.cam.2026.117342","url":null,"abstract":"<div><div>High order exponential integrators require computing linear combination of exponential-like φ-functions of large matrices <em>A</em> times a vector <em>v</em>. Krylov projection methods are the most general and remain an efficient choice for computing the matrix-function-vector-product evaluation when the matrix <em>A</em> is large and unable to be explicitly stored, or when obtaining information about the spectrum is expensive. The Krylov approximation relies on the Gram-Schmidt (GS) orthogonalization procedure to produce the orthonormal basis <em>V<sub>m</sub></em>. In parallel, GS orthogonalization requires <em>global synchronizations</em> for inner products and vector normalization in the orthogonalization process. Reducing the amount of global synchronizations is of paramount importance for the efficiency of a numerical algorithm in a massively parallel setting. We improve the strong scaling properties and parallel efficiency of exponential integrators by addressing the underlying bottleneck in the linear algebra using low-synchronization GS methods. The resulting orthogonalization algorithms have an accuracy comparable to modified Gram-Schmidt yet are better suited for distributed architecture, as only one global communication is required per orthogonalization-step. We present geophysics based numerical experiments and standard examples routinely used to test stiff time integrators, which validate that reducing global communication leads to better parallel scalability and reduced time-to-solution for exponential integrators.</div></div>","PeriodicalId":50226,"journal":{"name":"Journal of Computational and Applied Mathematics","volume":"482 ","pages":"Article 117342"},"PeriodicalIF":2.6,"publicationDate":"2026-01-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145979450","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}
Large-scale quaternion matrix equations face challenges such as high dimensionality and non-commutativity of quaternion multiplication, which often result in high computational complexity and low efficiency with conventional methods. To this end, utilizing generalized Hamilton-real (GHR) calculus, we propose a quaternion random reshuffling (QRR) algorithm for solving large-scale quaternion matrix equations. We also provide a convergence analysis for the QRR algorithm. Numerical experiments show that the QRR algorithm achieves stable convergence performance and faster convergence rates in solving large-scale generalized Sylvester quaternion matrix equations. Thus, the QRR algorithm is expected to provide an efficient and robust solution for solving large-scale quaternion matrix equations.
{"title":"A random reshuffling method for generalized Sylvester quaternion matrix equations","authors":"Qiankun Diao , Yiming Jiang , Jinlan Liu , Dongpo Xu","doi":"10.1016/j.cam.2026.117346","DOIUrl":"10.1016/j.cam.2026.117346","url":null,"abstract":"<div><div>Large-scale quaternion matrix equations face challenges such as high dimensionality and non-commutativity of quaternion multiplication, which often result in high computational complexity and low efficiency with conventional methods. To this end, utilizing generalized Hamilton-real (GHR) calculus, we propose a quaternion random reshuffling (QRR) algorithm for solving large-scale quaternion matrix equations. We also provide a convergence analysis for the QRR algorithm. Numerical experiments show that the QRR algorithm achieves stable convergence performance and faster convergence rates in solving large-scale generalized Sylvester quaternion matrix equations. Thus, the QRR algorithm is expected to provide an efficient and robust solution for solving large-scale quaternion matrix equations.</div></div>","PeriodicalId":50226,"journal":{"name":"Journal of Computational and Applied Mathematics","volume":"482 ","pages":"Article 117346"},"PeriodicalIF":2.6,"publicationDate":"2026-01-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145979372","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 : 2026-01-08DOI: 10.1016/j.cam.2026.117345
Jiamin Lu, Liwen Xu, Hao Cheng
In this paper, we investigate an inverse random source problem for the fractional pseudo-parabolic equation, where the source is driven by a fractional Brownian motion (fBm). For the direct problem, we illustrate the existence and uniqueness of the mild solution. For the inverse random source problem, the uniqueness is proved and the instability is characterized. To address this instability, we apply Tikhonov regularization, achieving stable numerical solutions and giving error estimates. Finally, numerical experiments demonstrate the effectiveness of the regularization method.
{"title":"An inverse random source problem for pseudo-parabolic equation of Caputo type with fractional-order Laplacian operator","authors":"Jiamin Lu, Liwen Xu, Hao Cheng","doi":"10.1016/j.cam.2026.117345","DOIUrl":"10.1016/j.cam.2026.117345","url":null,"abstract":"<div><div>In this paper, we investigate an inverse random source problem for the fractional pseudo-parabolic equation, where the source is driven by a fractional Brownian motion (fBm). For the direct problem, we illustrate the existence and uniqueness of the mild solution. For the inverse random source problem, the uniqueness is proved and the instability is characterized. To address this instability, we apply Tikhonov regularization, achieving stable numerical solutions and giving error estimates. Finally, numerical experiments demonstrate the effectiveness of the regularization method.</div></div>","PeriodicalId":50226,"journal":{"name":"Journal of Computational and Applied Mathematics","volume":"482 ","pages":"Article 117345"},"PeriodicalIF":2.6,"publicationDate":"2026-01-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145979454","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 : 2026-01-08DOI: 10.1016/j.cam.2026.117344
Wenqi Lu , Hongmei Lin , Heng Lian
Problems involving a low-rank tensor with a Tucker format or a tensor train (TT) format, such as the tensor decomposition or tensor optimization problems, have been frequently studied in the literature. Motivated by the success of randomized algorithms for low-rank matrix decomposition, we develop randomized algorithms for these two tensor formats and present a detailed theoretical analysis of the randomized tensor decomposition as well as on its use in the optimization and regression problem. For the latter, we focus on the nonconvex projected gradient descent algorithm previously used only on the Tucker format, which we also extend to the TT format, where one key step in the computation is performing singular value decomposition of the matricized tensor variable. We provide error bounds both in expectation and bounds with high probability.
{"title":"Randomized tensor decomposition and optimization in the tucker and tensor train formats","authors":"Wenqi Lu , Hongmei Lin , Heng Lian","doi":"10.1016/j.cam.2026.117344","DOIUrl":"10.1016/j.cam.2026.117344","url":null,"abstract":"<div><div>Problems involving a low-rank tensor with a Tucker format or a tensor train (TT) format, such as the tensor decomposition or tensor optimization problems, have been frequently studied in the literature. Motivated by the success of randomized algorithms for low-rank matrix decomposition, we develop randomized algorithms for these two tensor formats and present a detailed theoretical analysis of the randomized tensor decomposition as well as on its use in the optimization and regression problem. For the latter, we focus on the nonconvex projected gradient descent algorithm previously used only on the Tucker format, which we also extend to the TT format, where one key step in the computation is performing singular value decomposition of the matricized tensor variable. We provide error bounds both in expectation and bounds with high probability.</div></div>","PeriodicalId":50226,"journal":{"name":"Journal of Computational and Applied Mathematics","volume":"482 ","pages":"Article 117344"},"PeriodicalIF":2.6,"publicationDate":"2026-01-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145979371","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 : 2026-01-08DOI: 10.1016/j.cam.2026.117339
Corentin Bonte, Arne Bouillon, Giovanni Samaey, Karl Meerbergen
Recently, the ParaOpt algorithm was proposed as an extension of the time-parallel Parareal method to optimal control. ParaOpt uses quasi-Newton steps that each require solving a system of matching conditions iteratively. The state-of-the-art parallel preconditioner for linear problems leads to a set of independent smaller systems that are currently hard to solve. We generalize the preconditioner to the nonlinear case and propose a new, fast inversion method for these smaller systems, avoiding disadvantages of the current options with adjusted boundary conditions in the subproblems.
{"title":"Efficient parallel inversion of ParaOpt preconditioners","authors":"Corentin Bonte, Arne Bouillon, Giovanni Samaey, Karl Meerbergen","doi":"10.1016/j.cam.2026.117339","DOIUrl":"10.1016/j.cam.2026.117339","url":null,"abstract":"<div><div>Recently, the ParaOpt algorithm was proposed as an extension of the time-parallel Parareal method to optimal control. ParaOpt uses quasi-Newton steps that each require solving a system of matching conditions iteratively. The state-of-the-art parallel preconditioner for linear problems leads to a set of independent smaller systems that are currently hard to solve. We generalize the preconditioner to the nonlinear case and propose a new, fast inversion method for these smaller systems, avoiding disadvantages of the current options with adjusted boundary conditions in the subproblems.</div></div>","PeriodicalId":50226,"journal":{"name":"Journal of Computational and Applied Mathematics","volume":"482 ","pages":"Article 117339"},"PeriodicalIF":2.6,"publicationDate":"2026-01-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145979378","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}
This research is about a nonlocal dispersal host-pathogen model with Neumann boundary conditions. The nonlocal dispersion operator lacks compactness, which generates an additional challenge in showing the existence of a global compact attractor, and then the stability of the steady states. Therefore, we study the asymptotic stability of the positive steady states, where we show that depends on R0. For R0 < 1, we prove the global asymptotic stability of the pathogen-free steady state, whereas for R0 > 1, we demonstrate the model’s uniform persistence and the existence of a positive steady state. Moreover, we establish the global behavior of the positive steady state for two cases: (i) the spatially homogeneous case, with both dispersal coefficients are not a zero, (ii) the spatially heterogeneous case, with one of the dispersion rate is a zero. Finally, the asymptotic profiles of the positive steady state as one or both diffusion coefficients tend to infinity is established. This gives the most favored sites for the host and virus particles. Our results shed light on the interplay between spatial dispersal and disease dynamics and have implications for the design of effective control strategies.
{"title":"Behavior of a nonlocal species-pathogen system with varied dispersal mechanisms in heterogeneous habitats","authors":"Boumdiene Guenad , Salih Djilali , Soufiane Bentout","doi":"10.1016/j.cam.2026.117341","DOIUrl":"10.1016/j.cam.2026.117341","url":null,"abstract":"<div><div>This research is about a nonlocal dispersal host-pathogen model with Neumann boundary conditions. The nonlocal dispersion operator lacks compactness, which generates an additional challenge in showing the existence of a global compact attractor, and then the stability of the steady states. Therefore, we study the asymptotic stability of the positive steady states, where we show that depends on <em>R</em><sub>0</sub>. For <em>R</em><sub>0</sub> < 1, we prove the global asymptotic stability of the pathogen-free steady state, whereas for <em>R</em><sub>0</sub> > 1, we demonstrate the model’s uniform persistence and the existence of a positive steady state. Moreover, we establish the global behavior of the positive steady state for two cases: (i) the spatially homogeneous case, with both dispersal coefficients are not a zero, (ii) the spatially heterogeneous case, with one of the dispersion rate is a zero. Finally, the asymptotic profiles of the positive steady state as one or both diffusion coefficients tend to infinity is established. This gives the most favored sites for the host and virus particles. Our results shed light on the interplay between spatial dispersal and disease dynamics and have implications for the design of effective control strategies.</div></div>","PeriodicalId":50226,"journal":{"name":"Journal of Computational and Applied Mathematics","volume":"482 ","pages":"Article 117341"},"PeriodicalIF":2.6,"publicationDate":"2026-01-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145979453","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 : 2026-01-07DOI: 10.1016/j.cam.2026.117351
Mohamed Kamel RIAHI
In this paper, we introduce novel fast matrix inversion algorithms that leverage triangular decomposition and a recurrent formalism, incorporating Strassen’s fast matrix multiplication algorithm. Our focus lies on triangular matrices, where we propose a unique computational approach based on combinatorial techniques for directly inverting general non-singular triangular matrices. Unlike iterative methods, our combinatorial approach enables the direct construction of the inverse through nonlinear combinations of carefully selected matrix entries, allowing full parallelization and efficient implementation on parallel architectures. Although combinatorial algorithms often suffer from exponential time complexity, limiting their practical use, our method overcomes this by deriving recurrent relations that facilitate recursive triangular splitting, striking a balance between efficiency and accuracy. We provide rigorous mathematical proofs to validate the approach and present extensive numerical experiments demonstrating its effectiveness.
Additionally, we develop several innovative numerical linear algebra algorithms that directly factorize the inverse of general matrices, with significant potential for generating preconditioners that accelerate Krylov subspace iterative solvers and improve the solution of large-scale linear systems.
Our comprehensive evaluation confirms that the proposed algorithms outperform classical approaches in terms of computational efficiency, opening up new avenues for advanced matrix inversion techniques and the development of effective preconditioning strategies.
{"title":"Combinatorial and recurrent approaches for efficient matrix inversion: Sub-cubic algorithms leveraging fast matrix products","authors":"Mohamed Kamel RIAHI","doi":"10.1016/j.cam.2026.117351","DOIUrl":"10.1016/j.cam.2026.117351","url":null,"abstract":"<div><div>In this paper, we introduce novel fast matrix inversion algorithms that leverage triangular decomposition and a recurrent formalism, incorporating Strassen’s fast matrix multiplication algorithm. Our focus lies on triangular matrices, where we propose a unique computational approach based on combinatorial techniques for directly inverting general non-singular triangular matrices. Unlike iterative methods, our combinatorial approach enables the direct construction of the inverse through nonlinear combinations of carefully selected matrix entries, allowing full parallelization and efficient implementation on parallel architectures. Although combinatorial algorithms often suffer from exponential time complexity, limiting their practical use, our method overcomes this by deriving recurrent relations that facilitate recursive triangular splitting, striking a balance between efficiency and accuracy. We provide rigorous mathematical proofs to validate the approach and present extensive numerical experiments demonstrating its effectiveness.</div><div>Additionally, we develop several innovative numerical linear algebra algorithms that directly factorize the inverse of general matrices, with significant potential for generating preconditioners that accelerate Krylov subspace iterative solvers and improve the solution of large-scale linear systems.</div><div>Our comprehensive evaluation confirms that the proposed algorithms outperform classical approaches in terms of computational efficiency, opening up new avenues for advanced matrix inversion techniques and the development of effective preconditioning strategies.</div></div>","PeriodicalId":50226,"journal":{"name":"Journal of Computational and Applied Mathematics","volume":"482 ","pages":"Article 117351"},"PeriodicalIF":2.6,"publicationDate":"2026-01-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145979373","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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