Pub Date : 2025-12-25DOI: 10.1016/j.cam.2025.117284
Wanwan Zhu , Guanghua Ji
Our research focused on the adaptive weak Galerkin finite element method to solve the time-dependent Poisson-Nernst-Planck (PNP) equations. Through the utilization of the Helmholtz decomposition and elliptic reconstruction operator, a comprehensive analysis of a posteriori error estimates was conducted. Both the upper and lower bound error estimators for the electrostatic potential and ion concentrations were formulated, taking into account both spatial and temporal residuals. A time-step adaptation strategy was developed to adjust the time step, followed by the development of a temporal and spatial adaptive algorithm for solving the time-dependent PNP equations using the constructed a posteriori error estimators. The validity of our methodology was confirmed through numerical simulations.
{"title":"A posteriori error estimates of the weak Galerkin finite element method for time-dependent Poisson-Nernst-Planck equations","authors":"Wanwan Zhu , Guanghua Ji","doi":"10.1016/j.cam.2025.117284","DOIUrl":"10.1016/j.cam.2025.117284","url":null,"abstract":"<div><div>Our research focused on the adaptive weak Galerkin finite element method to solve the time-dependent Poisson-Nernst-Planck (PNP) equations. Through the utilization of the Helmholtz decomposition and elliptic reconstruction operator, a comprehensive analysis of a posteriori error estimates was conducted. Both the upper and lower bound error estimators for the electrostatic potential and ion concentrations were formulated, taking into account both spatial and temporal residuals. A time-step adaptation strategy was developed to adjust the time step, followed by the development of a temporal and spatial adaptive algorithm for solving the time-dependent PNP equations using the constructed a posteriori error estimators. The validity of our methodology was confirmed through numerical simulations.</div></div>","PeriodicalId":50226,"journal":{"name":"Journal of Computational and Applied Mathematics","volume":"482 ","pages":"Article 117284"},"PeriodicalIF":2.6,"publicationDate":"2025-12-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145886046","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}
A fuzzy set lacks to determine the hesitancy of an element in terms of belongingness to a set, the same problem arises in forecasting time series data by the fuzzy set when there is the availability of multiple fuzzification methods to fuzzify the time series data to remove hesitancy in the system. In the present study, the HFS has been applied in time series forecasting and a HFTSF method is proposed by introducing essential concepts of weighted hesitant fuzzy Cartesian product, HFR, HFLRs, HFLGs and hesitant fuzzy defuzzification method. The basic steps followed in the mechanism of the proposed method are the construction of HFS by a partition of the UOD into intervals of equal and unequal length, distribution of weights based on the length of the interval, fuzzification of the data by using triangular membership function for equal and unequal intervals, construction of HFLRs and HFLGs, relation matrix obtained by weighted hesitant fuzzy Cartesian product, computation of hesitant fuzzy row vectors by max-min composition operation and finally hesitant defuzzification of the data. The proposed method is implemented over the enrollment data of the University of Alabama and the share price of SBI at Bombay stock exchange, India. Performance test, validity test, and statistical test are also examined on the forecasted value by the proposed method and well-known existing methods to examine the superiority of the proposed method. This article presents a novel prediction model that authentically captures methodological hesitancy which were absent in prior HFS based forecasting models due to reliant on aggregation operator that transform HFS into conventional fuzzy set. By directly formulating HFLRs through a weighted Cartesian product, our framework eliminates information loss. The model delivers a triple advantage: it maintain complete information integrity, guaranteeing interpretablity through a transparent calculus, and demonstrating superior accuracy and robustness against existing benchmarks in hesitant environment.
{"title":"Hesitant fuzzy time series forecasting: A novel approach to handle the hesitancy in the system","authors":"Kamlesh Bisht , Sanjay Kumar , Manish Pant , Seema Negi","doi":"10.1016/j.cam.2025.117322","DOIUrl":"10.1016/j.cam.2025.117322","url":null,"abstract":"<div><div>A fuzzy set lacks to determine the hesitancy of an element in terms of belongingness to a set, the same problem arises in forecasting time series data by the fuzzy set when there is the availability of multiple fuzzification methods to fuzzify the time series data to remove hesitancy in the system. In the present study, the HFS has been applied in time series forecasting and a HFTSF method is proposed by introducing essential concepts of weighted hesitant fuzzy Cartesian product, HFR, HFLRs, HFLGs and hesitant fuzzy defuzzification method. The basic steps followed in the mechanism of the proposed method are the construction of HFS by a partition of the UOD into intervals of equal and unequal length, distribution of weights based on the length of the interval, fuzzification of the data by using triangular membership function for equal and unequal intervals, construction of HFLRs and HFLGs, relation matrix obtained by weighted hesitant fuzzy Cartesian product, computation of hesitant fuzzy row vectors by max-min composition operation and finally hesitant defuzzification of the data. The proposed method is implemented over the enrollment data of the University of Alabama and the share price of SBI at Bombay stock exchange, India. Performance test, validity test, and statistical test are also examined on the forecasted value by the proposed method and well-known existing methods to examine the superiority of the proposed method. This article presents a novel prediction model that authentically captures methodological hesitancy which were absent in prior HFS based forecasting models due to reliant on aggregation operator that transform HFS into conventional fuzzy set. By directly formulating HFLRs through a weighted Cartesian product, our framework eliminates information loss. The model delivers a triple advantage: it maintain complete information integrity, guaranteeing interpretablity through a transparent calculus, and demonstrating superior accuracy and robustness against existing benchmarks in hesitant environment.</div></div>","PeriodicalId":50226,"journal":{"name":"Journal of Computational and Applied Mathematics","volume":"482 ","pages":"Article 117322"},"PeriodicalIF":2.6,"publicationDate":"2025-12-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145928203","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 : 2025-12-25DOI: 10.1016/j.cam.2025.117279
Chunmei Wang , Shangyou Zhang
This paper introduces an efficient stabilizer-free weak Galerkin (WG) finite element method for solving the three-dimensional quad-curl problem. Leveraging bubble functions as a key analytical tool, the method extends the applicability of stabilizer-free WG approaches to non-convex elements in finite element partitions-a notable advancement over existing methods, which are restricted to convex elements. The proposed method maintains a simple, symmetric, and positive definite formulation. It achieves optimal error estimates for the exact solution in a discrete norm, as well as an optimal-order L2 error estimate for k > 2 and a sub-optimal order for the lowest order case . Numerical experiments are presented to validate the method’s efficiency and accuracy.
{"title":"Stabilizer-free weak galerkin methods for quad-Curl problems on polyhedral meshes without convexity assumptions","authors":"Chunmei Wang , Shangyou Zhang","doi":"10.1016/j.cam.2025.117279","DOIUrl":"10.1016/j.cam.2025.117279","url":null,"abstract":"<div><div>This paper introduces an efficient stabilizer-free weak Galerkin (WG) finite element method for solving the three-dimensional quad-curl problem. Leveraging bubble functions as a key analytical tool, the method extends the applicability of stabilizer-free WG approaches to non-convex elements in finite element partitions-a notable advancement over existing methods, which are restricted to convex elements. The proposed method maintains a simple, symmetric, and positive definite formulation. It achieves optimal error estimates for the exact solution in a discrete norm, as well as an optimal-order <em>L</em><sup>2</sup> error estimate for <em>k</em> > 2 and a sub-optimal order for the lowest order case <span><math><mrow><mi>k</mi><mo>=</mo><mn>2</mn></mrow></math></span>. Numerical experiments are presented to validate the method’s efficiency and accuracy.</div></div>","PeriodicalId":50226,"journal":{"name":"Journal of Computational and Applied Mathematics","volume":"481 ","pages":"Article 117279"},"PeriodicalIF":2.6,"publicationDate":"2025-12-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145885874","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 : 2025-12-24DOI: 10.1016/j.cam.2025.117309
S. Cacace , R. Ferretti , G. Tatafiore
We consider a scheme of Semi-Lagrangian (SL) type for the numerical solution of Hamilton–Jacobi (HJ) equations on unstructured triangular grids. As it is well known, SL schemes are not well suited for unstructured grids, due to the cost of the point location phase; this drawback is augmented by the need for repeated minimization. In this work, we consider an existing, monotone version of the scheme, that works only on the basis of node values, and adapt the algorithm to the case of an unstructured grid, using the connectivity information. Then, applying a quadratic refinement to the numerical solution, we improve accuracy at the price of some extra computational cost. The scheme can be applied to both time-dependent and stationary HJ equations; in the latter case, we also study the construction of a fast policy iteration solver and of a parallel version. We perform a theoretical analysis of the two versions, and validate them with an extensive set of examples, both in the time-dependent and in the stationary case.
{"title":"Numerical Hopf–Lax formulae for Hamilton–Jacobi equations on unstructured geometries","authors":"S. Cacace , R. Ferretti , G. Tatafiore","doi":"10.1016/j.cam.2025.117309","DOIUrl":"10.1016/j.cam.2025.117309","url":null,"abstract":"<div><div>We consider a scheme of Semi-Lagrangian (SL) type for the numerical solution of Hamilton–Jacobi (HJ) equations on unstructured triangular grids. As it is well known, SL schemes are not well suited for unstructured grids, due to the cost of the point location phase; this drawback is augmented by the need for repeated minimization. In this work, we consider an existing, monotone version of the scheme, that works only on the basis of node values, and adapt the algorithm to the case of an unstructured grid, using the connectivity information. Then, applying a quadratic refinement to the numerical solution, we improve accuracy at the price of some extra computational cost. The scheme can be applied to both time-dependent and stationary HJ equations; in the latter case, we also study the construction of a fast policy iteration solver and of a parallel version. We perform a theoretical analysis of the two versions, and validate them with an extensive set of examples, both in the time-dependent and in the stationary case.</div></div>","PeriodicalId":50226,"journal":{"name":"Journal of Computational and Applied Mathematics","volume":"481 ","pages":"Article 117309"},"PeriodicalIF":2.6,"publicationDate":"2025-12-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145886357","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 investigate a steady system used for modelling propagation of reaction fronts in viscous fluids subject to a generalized Arrhenius law. The model consists of the incompressible Navier-Stokes equations for the fluid flow and a set of advection-diffusion equations for the temperature and degree of conversion. The resulting system is strongly coupled and presents many additional nonlinearities as the physical parameters such as the viscosity, diffusion and source terms are assumed to depend on the temperature and/or degree of conversion. Using a fixed-point method we prove the existence and uniqueness of the weak solution for the considered problem. To solve the associated fixed-point problem we consider an iterative scheme and its convergence is also studied in the present study. Here, the proposed scheme uncouples the computation of velocity, temperature and degree of conversion using the fixed-point iteration and we theoretically establish its convergence towards the unique solution of the considered model. Numerical results obtained for two test examples are presented to verify the theoretical analysis and to assess the performance of the proposed algorithm. The obtained computational results for both examples support the theoretical expectations for a good numerical convergence with the developed estimates.
{"title":"Well -posedness of coupled Navier-Stokes and advection-diffusion equations for propagation of reaction fronts in viscous fluids","authors":"Mofdi El-Amrani , Anouar Obbadi , Mohammed Seaid , Driss Yakoubi","doi":"10.1016/j.cam.2025.117277","DOIUrl":"10.1016/j.cam.2025.117277","url":null,"abstract":"<div><div>In this paper, we investigate a steady system used for modelling propagation of reaction fronts in viscous fluids subject to a generalized Arrhenius law. The model consists of the incompressible Navier-Stokes equations for the fluid flow and a set of advection-diffusion equations for the temperature and degree of conversion. The resulting system is strongly coupled and presents many additional nonlinearities as the physical parameters such as the viscosity, diffusion and source terms are assumed to depend on the temperature and/or degree of conversion. Using a fixed-point method we prove the existence and uniqueness of the weak solution for the considered problem. To solve the associated fixed-point problem we consider an iterative scheme and its convergence is also studied in the present study. Here, the proposed scheme uncouples the computation of velocity, temperature and degree of conversion using the fixed-point iteration and we theoretically establish its convergence towards the unique solution of the considered model. Numerical results obtained for two test examples are presented to verify the theoretical analysis and to assess the performance of the proposed algorithm. The obtained computational results for both examples support the theoretical expectations for a good numerical convergence with the developed estimates.</div></div>","PeriodicalId":50226,"journal":{"name":"Journal of Computational and Applied Mathematics","volume":"481 ","pages":"Article 117277"},"PeriodicalIF":2.6,"publicationDate":"2025-12-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145885862","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 : 2025-12-24DOI: 10.1016/j.cam.2025.117295
Hasanen A. Hammad , Manal Elzain Mohamed Abdalla
The existence of solutions for non-autonomous integrodifferential evolution equations with nonlocal conditions is investigated in this article. Initially, existence results for mild solutions of the proposed equation are established through the leveraging of the theory of resolvent operators, fixed point theorems, and an estimation technique grounded in the measure of noncompactness. Finally, the applicability of the findings is illustrated by means of an example concerning a class of non-autonomous nonlocal partial integrodifferential equations.
{"title":"Applying fixed point approaches for solving non-autonomous integrodifferential evolution equations under mild conditions","authors":"Hasanen A. Hammad , Manal Elzain Mohamed Abdalla","doi":"10.1016/j.cam.2025.117295","DOIUrl":"10.1016/j.cam.2025.117295","url":null,"abstract":"<div><div>The existence of solutions for non-autonomous integrodifferential evolution equations with nonlocal conditions is investigated in this article. Initially, existence results for mild solutions of the proposed equation are established through the leveraging of the theory of resolvent operators, fixed point theorems, and an estimation technique grounded in the measure of noncompactness. Finally, the applicability of the findings is illustrated by means of an example concerning a class of non-autonomous nonlocal partial integrodifferential equations.</div></div>","PeriodicalId":50226,"journal":{"name":"Journal of Computational and Applied Mathematics","volume":"481 ","pages":"Article 117295"},"PeriodicalIF":2.6,"publicationDate":"2025-12-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145885865","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 : 2025-12-24DOI: 10.1016/j.cam.2025.117263
Haneche Nabil , Hamaizia Tayeb
Fractional-order dynamic systems provide more realistic results than their ordinary counterparts due to the memory effect. This paper constructs a new hyperchaotic map using the Caputo fractional operator. A discrete dynamic system is required to describe more complex dynamical behaviors, such as chaos and hyperchaos, in the model. The exhibition of bifurcations by the fractional-order map has been investigated. Multiple positive Lyapunov exponents indicate that complex hyperchaotic dynamics have been exhibited when the fractional order or control parameters are varied. Also, the spectral entropy method (SE) is employed to measure accurately the fractional-order map’s level of complexity. It is shown that the fractional-order map has a high level of complexity compared to other discrete maps. Based on the chaotic sequences that were generated by this fractional-order map, a secure color image encryption algorithm is proposed. The proposed algorithm has superior encryption performance and high security. Experimental results and performance analysis show that this algorithm is accurate and secure for encrypting images, as it can stand up to different brute-force attacks.
{"title":"Dynamical analysis of a novel fractional-order hyperchaotic map and its application for fast color image encryption","authors":"Haneche Nabil , Hamaizia Tayeb","doi":"10.1016/j.cam.2025.117263","DOIUrl":"10.1016/j.cam.2025.117263","url":null,"abstract":"<div><div>Fractional-order dynamic systems provide more realistic results than their ordinary counterparts due to the memory effect. This paper constructs a new hyperchaotic map using the Caputo fractional operator. A discrete dynamic system is required to describe more complex dynamical behaviors, such as chaos and hyperchaos, in the model. The exhibition of bifurcations by the fractional-order map has been investigated. Multiple positive Lyapunov exponents indicate that complex hyperchaotic dynamics have been exhibited when the fractional order or control parameters are varied. Also, the spectral entropy method (SE) is employed to measure accurately the fractional-order map’s level of complexity. It is shown that the fractional-order map has a high level of complexity compared to other discrete maps. Based on the chaotic sequences that were generated by this fractional-order map, a secure color image encryption algorithm is proposed. The proposed algorithm has superior encryption performance and high security. Experimental results and performance analysis show that this algorithm is accurate and secure for encrypting images, as it can stand up to different brute-force attacks.</div></div>","PeriodicalId":50226,"journal":{"name":"Journal of Computational and Applied Mathematics","volume":"481 ","pages":"Article 117263"},"PeriodicalIF":2.6,"publicationDate":"2025-12-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145885873","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 : 2025-12-24DOI: 10.1016/j.cam.2025.117283
Liang Chen, Qiuqi Li, Hongyu Yang
This paper presents a non-intrusive model order reduction method based on nonlinear optimization for steady parameterized Stokes problems. To achieve this, we employ a weighted loss function to balance the velocity and pressure outputs to obtain a non-intrusive, data-driven algorithm utilizing only output samples. Moreover, we derive the gradients of the objective function with respect to the reduced-order matrices by resorting to the parameter-separable forms of reduced-model quantities. To enhance computational efficiency, our framework employs a two-stage offline-online decomposition. In the offline stage, we leverage gradient information to develop an optimization algorithm that computes optimal approximations for reduced-order matrices. In the online stage, the outputs can be quickly estimated for new parameter values using the reduced-order model obtained from the offline phase. Finally, we present numerical experiments to validate the effectiveness of this method, especially to demonstrate its capability to produce highly accurate approximation results.
{"title":"A non-intrusive model order reduction method based on nonlinear optimization for parameterized Stokes problems","authors":"Liang Chen, Qiuqi Li, Hongyu Yang","doi":"10.1016/j.cam.2025.117283","DOIUrl":"10.1016/j.cam.2025.117283","url":null,"abstract":"<div><div>This paper presents a non-intrusive model order reduction method based on nonlinear optimization for steady parameterized Stokes problems. To achieve this, we employ a weighted loss function to balance the velocity and pressure outputs to obtain a non-intrusive, data-driven algorithm utilizing only output samples. Moreover, we derive the gradients of the objective function with respect to the reduced-order matrices by resorting to the parameter-separable forms of reduced-model quantities. To enhance computational efficiency, our framework employs a two-stage offline-online decomposition. In the offline stage, we leverage gradient information to develop an optimization algorithm that computes optimal approximations for reduced-order matrices. In the online stage, the outputs can be quickly estimated for new parameter values using the reduced-order model obtained from the offline phase. Finally, we present numerical experiments to validate the effectiveness of this method, especially to demonstrate its capability to produce highly accurate approximation results.</div></div>","PeriodicalId":50226,"journal":{"name":"Journal of Computational and Applied Mathematics","volume":"482 ","pages":"Article 117283"},"PeriodicalIF":2.6,"publicationDate":"2025-12-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145928222","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 : 2025-12-24DOI: 10.1016/j.cam.2025.117269
Leonidas Gkimisis , Süleyman Yıldız , Thomas Richter , Peter Benner
In this work, we investigate the data-driven inference of a discrete-time dynamical system via a sparse Full-Order Model (sFOM). We first formulate the involved Least Squares (LS) problem and discuss the need for regularization, indicating a connection between the typically employed l2 regularization and the stability of the inferred discrete-time sFOM. We then provide theoretical insights considering the consistency and stability properties of the inferred numerical schemes that form the sFOM and exemplify them via illustrative, 1D test cases of linear diffusion and linear advection. For linear advection, we analytically derive a “sampling CFL” condition, which dictates a bound for the ratio of spatial and temporal discretization steps in the training data that ensures stability of the inferred sFOM. Finally, we investigate the sFOM inference for two nonlinear problems, namely a 2D Burgers’ test case and the incompressible flow in an oscillating lid-driven cavity, and draw connections between the theoretical findings and the properties of the inferred, nonlinear sFOMs.
Novelty statement: sparse FOM inference for dynamical systems in discrete time. Theoretical insights on the analytical solution of the sparse FOM least-squares problem. Established connection between the stability of sparse FOM and the l2 regularization of the least-squares problem.
{"title":"A CFL-type condition and theoretical insights for discrete-time sparse full-order model inference","authors":"Leonidas Gkimisis , Süleyman Yıldız , Thomas Richter , Peter Benner","doi":"10.1016/j.cam.2025.117269","DOIUrl":"10.1016/j.cam.2025.117269","url":null,"abstract":"<div><div>In this work, we investigate the data-driven inference of a discrete-time dynamical system via a sparse Full-Order Model (sFOM). We first formulate the involved Least Squares (LS) problem and discuss the need for regularization, indicating a connection between the typically employed <em>l</em><sub>2</sub> regularization and the stability of the inferred discrete-time sFOM. We then provide theoretical insights considering the consistency and stability properties of the inferred numerical schemes that form the sFOM and exemplify them via illustrative, 1D test cases of linear diffusion and linear advection. For linear advection, we analytically derive a “sampling CFL” condition, which dictates a bound for the ratio of spatial and temporal discretization steps in the training data that ensures stability of the inferred sFOM. Finally, we investigate the sFOM inference for two nonlinear problems, namely a 2D Burgers’ test case and the incompressible flow in an oscillating lid-driven cavity, and draw connections between the theoretical findings and the properties of the inferred, nonlinear sFOMs.</div><div><strong>Novelty statement:</strong> sparse FOM inference for dynamical systems in discrete time. Theoretical insights on the analytical solution of the sparse FOM least-squares problem. Established connection between the stability of sparse FOM and the <em>l</em><sub>2</sub> regularization of the least-squares problem.</div></div>","PeriodicalId":50226,"journal":{"name":"Journal of Computational and Applied Mathematics","volume":"482 ","pages":"Article 117269"},"PeriodicalIF":2.6,"publicationDate":"2025-12-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145886049","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 : 2025-12-24DOI: 10.1016/j.cam.2025.117255
Longze Tan, Jingrun Chen
We study large-scale, ill-conditioned, and overdetermined least squares problems arising from the discretization of partial differential equations (PDEs), especially those induced by the random feature method (RFM). To address these challenges, our methods consist of three main components: (1) a count sketch technique is used to sketch the original matrix to a smaller matrix; (2) a QR factorization or a singular value decomposition is employed for the smaller matrix to obtain the preconditioner, which is multiplied to the original matrix from the right-hand side; (3) least squares iterative solvers are employed to solve the preconditioned least squares system. This leads to two high-precision randomized preconditioned methods, namely the CSQRP-LSQR and CSSVDP-LSQR methods, which explicitly construct the preconditioned matrix and thereby avoid the numerical instabilities associated with the implicit preconditioning used in methods such as Blendenpik and LSRN. Under mild assumptions, we show that the condition number of the preconditioned system is independent of that of the original matrix and also establish error estimates for the CSQRP-LSQR method. Extensive numerical experiments on two- and three-dimensional PDE problems demonstrate that the proposed methods consistently achieve superior stability, higher accuracy, and improved computational efficiency compared to LSRN, QR-based solvers, and state-of-the-art sparse direct solvers. In particular, the CSSVDP-LSQR method remains robust for large-scale ill-conditioned least squares problems with infinite condition numbers or rank deficiencies, significantly reducing solution errors while maintaining competitive runtime performance.
{"title":"High-precision randomized preconditioned iterative methods for the random feature method","authors":"Longze Tan, Jingrun Chen","doi":"10.1016/j.cam.2025.117255","DOIUrl":"10.1016/j.cam.2025.117255","url":null,"abstract":"<div><div>We study large-scale, ill-conditioned, and overdetermined least squares problems arising from the discretization of partial differential equations (PDEs), especially those induced by the random feature method (RFM). To address these challenges, our methods consist of three main components: (1) a count sketch technique is used to sketch the original matrix to a smaller matrix; (2) a QR factorization or a singular value decomposition is employed for the smaller matrix to obtain the preconditioner, which is multiplied to the original matrix from the right-hand side; (3) least squares iterative solvers are employed to solve the preconditioned least squares system. This leads to two high-precision randomized preconditioned methods, namely the CSQRP-LSQR and CSSVDP-LSQR methods, which explicitly construct the preconditioned matrix and thereby avoid the numerical instabilities associated with the implicit preconditioning used in methods such as Blendenpik and LSRN. Under mild assumptions, we show that the condition number of the preconditioned system is independent of that of the original matrix and also establish error estimates for the CSQRP-LSQR method. Extensive numerical experiments on two- and three-dimensional PDE problems demonstrate that the proposed methods consistently achieve superior stability, higher accuracy, and improved computational efficiency compared to LSRN, QR-based solvers, and state-of-the-art sparse direct solvers. In particular, the CSSVDP-LSQR method remains robust for large-scale ill-conditioned least squares problems with infinite condition numbers or rank deficiencies, significantly reducing solution errors while maintaining competitive runtime performance.</div></div>","PeriodicalId":50226,"journal":{"name":"Journal of Computational and Applied Mathematics","volume":"481 ","pages":"Article 117255"},"PeriodicalIF":2.6,"publicationDate":"2025-12-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145885870","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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