Pub Date : 2024-12-10DOI: 10.1016/j.cam.2024.116406
P. Minakowski, T. Richter
The proof of Lemma 1 contains careless errors and is based on a faulty approach. We provide a correction here.
{"title":"Erratum to “A priori and a posteriori error estimates for the Deep Ritz method applied to the Laplace and Stokes problem” [J. Comput. Appl. Math. 421 (2023) 114845 ]","authors":"P. Minakowski, T. Richter","doi":"10.1016/j.cam.2024.116406","DOIUrl":"10.1016/j.cam.2024.116406","url":null,"abstract":"<div><div>The proof of Lemma 1 contains careless errors and is based on a faulty approach. We provide a correction here.</div></div>","PeriodicalId":50226,"journal":{"name":"Journal of Computational and Applied Mathematics","volume":"460 ","pages":"Article 116406"},"PeriodicalIF":2.1,"publicationDate":"2024-12-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143098802","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 : 2024-12-10DOI: 10.1016/j.cam.2024.116427
Yuncheng Xu, Sanyang Liu, Lixia Liu, Kewei Jie
This paper is concerned with the monotone stochastic tensor complementarity problem, where the expectation of the involved stochastic tensor is a strictly positive semi-definite tensor. At first, a new class of restricted nonlinear complementarity problem (NCP) function is defined by using the special structure of strictly semi-definite tensor. Then the conditional value at risk stochastic programming (CVaR-SP) model of monotone stochastic tensor complementarity problem (STCP) is established by taking the minimum value of the stochastic residual defined by the modified restricted NCP function as objective function, the nonnegativity of the variable and the CVaR inequality representing the feasibility conditions as constraint conditions. Next, the sample average approximation problem of the CVaR-SP model is presented by using the Monte Carlo method and the smoothing method. Subsequently, the conditions for the convergence of the sample average approximation problem are analyzed. Finally, the penalized sample average approximation algorithm is used to solve the problem, the related numerical results further verify the validity of the method.
{"title":"CVaR stochastic programming model for monotone stochastic tensor complementarity problem by using its penalized sample average approximation algorithm","authors":"Yuncheng Xu, Sanyang Liu, Lixia Liu, Kewei Jie","doi":"10.1016/j.cam.2024.116427","DOIUrl":"10.1016/j.cam.2024.116427","url":null,"abstract":"<div><div>This paper is concerned with the monotone stochastic tensor complementarity problem, where the expectation of the involved stochastic tensor is a strictly positive semi-definite tensor. At first, a new class of restricted nonlinear complementarity problem (NCP) function is defined by using the special structure of strictly semi-definite tensor. Then the conditional value at risk stochastic programming (CVaR-SP) model of monotone stochastic tensor complementarity problem (STCP) is established by taking the minimum value of the stochastic residual defined by the modified restricted NCP function as objective function, the nonnegativity of the variable and the CVaR inequality representing the feasibility conditions as constraint conditions. Next, the sample average approximation problem of the CVaR-SP model is presented by using the Monte Carlo method and the smoothing method. Subsequently, the conditions for the convergence of the sample average approximation problem are analyzed. Finally, the penalized sample average approximation algorithm is used to solve the problem, the related numerical results further verify the validity of the method.</div></div>","PeriodicalId":50226,"journal":{"name":"Journal of Computational and Applied Mathematics","volume":"460 ","pages":"Article 116427"},"PeriodicalIF":2.1,"publicationDate":"2024-12-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143098735","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 : 2024-12-09DOI: 10.1016/j.cam.2024.116428
Xiao-Min Cai , Yi-Fen Ke , Lin-Jie Chen
In this paper, the singular value decomposition of reduced biquaternion matrix is studied and its expression is obtained based on the complex representation of reduced biquaternion matrix. In addition, the expression of the minimal norm least squares solution for the reduced biquaternion matrix equation is derived by the generalized inverse. The proposed algorithms are very simple and convenient, because they only involve operations on the field of complex numbers. Finally, in order to prove the authenticity of our algorithms, some numerical examples are presented including color image restoration and comparison with existing results.
{"title":"Color image restoration based on reduced biquaternion matrix singular value decomposition","authors":"Xiao-Min Cai , Yi-Fen Ke , Lin-Jie Chen","doi":"10.1016/j.cam.2024.116428","DOIUrl":"10.1016/j.cam.2024.116428","url":null,"abstract":"<div><div>In this paper, the singular value decomposition of reduced biquaternion matrix is studied and its expression is obtained based on the complex representation of reduced biquaternion matrix. In addition, the expression of the minimal norm least squares solution for the reduced biquaternion matrix equation is derived by the generalized inverse. The proposed algorithms are very simple and convenient, because they only involve operations on the field of complex numbers. Finally, in order to prove the authenticity of our algorithms, some numerical examples are presented including color image restoration and comparison with existing results.</div></div>","PeriodicalId":50226,"journal":{"name":"Journal of Computational and Applied Mathematics","volume":"460 ","pages":"Article 116428"},"PeriodicalIF":2.1,"publicationDate":"2024-12-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143098737","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 : 2024-12-09DOI: 10.1016/j.cam.2024.116415
Kyler Howard , Chris Rocheleau , Trevor Overton , Joel Barraza Nava , Mason Faldet , Kristina Moen , Summer Soller , Tyler Stephens , Esther van de Lagemaat , Natalie Wijesinghe , Kaylee Wong Dolloff , Nilton Barbosa da Rosa , Jennifer L. Mueller
Inherently low spatial resolution is a well-known challenge in electrical impedance tomography image reconstruction. Various approaches such as the use of spatial priors and post-processing techniques have been proposed to improve the resolution, but in the literature, comparisons using a common dataset representative of clinical images have not been considered. Here, we consider a database of 81,710 simulated EIT datasets constructed from pulmonary CT scans of 89 infants. Four techniques for improved image resolution and several combinations thereof are proposed and compared quantitatively on 16,341 known test cases reserved from the database. The techniques include an end-to-end deep learning reconstruction approach, post-processing of real-time one-step Gauss–Newton (GN) reconstructions using machine learning, post-processing using the Schur complement method, the use of an initial guess for the one-step GN method derived from the image database, and a method that makes use of the eigenfunctions of the principal component analysis of image vectors in the database. All methods resulted in improved metrics of error measurement compared to the Newton one-step error reconstruction method used as the basis for comparison.
{"title":"A comparison of techniques to improve pulmonary EIT image resolution using a database of simulated EIT images","authors":"Kyler Howard , Chris Rocheleau , Trevor Overton , Joel Barraza Nava , Mason Faldet , Kristina Moen , Summer Soller , Tyler Stephens , Esther van de Lagemaat , Natalie Wijesinghe , Kaylee Wong Dolloff , Nilton Barbosa da Rosa , Jennifer L. Mueller","doi":"10.1016/j.cam.2024.116415","DOIUrl":"10.1016/j.cam.2024.116415","url":null,"abstract":"<div><div>Inherently low spatial resolution is a well-known challenge in electrical impedance tomography image reconstruction. Various approaches such as the use of spatial priors and post-processing techniques have been proposed to improve the resolution, but in the literature, comparisons using a common dataset representative of clinical images have not been considered. Here, we consider a database of 81,710 simulated EIT datasets constructed from pulmonary CT scans of 89 infants. Four techniques for improved image resolution and several combinations thereof are proposed and compared quantitatively on 16,341 known test cases reserved from the database. The techniques include an end-to-end deep learning reconstruction approach, post-processing of real-time one-step Gauss–Newton (GN) reconstructions using machine learning, post-processing using the Schur complement method, the use of an initial guess for the one-step GN method derived from the image database, and a method that makes use of the eigenfunctions of the principal component analysis of image vectors in the database. All methods resulted in improved metrics of error measurement compared to the Newton one-step error reconstruction method used as the basis for comparison.</div></div>","PeriodicalId":50226,"journal":{"name":"Journal of Computational and Applied Mathematics","volume":"460 ","pages":"Article 116415"},"PeriodicalIF":2.1,"publicationDate":"2024-12-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143150049","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 : 2024-12-08DOI: 10.1016/j.cam.2024.116425
Xin He , Rong Hu , Ya-Ping Fang
In this paper, we propose an inertial accelerated augmented Lagrangian algorithm with a scaling coefficient tailored for solving linearly constrained convex optimization problems. Under suitable scaling conditions, we show that the iteration sequence generated by the algorithm converges to a saddle point of the Lagrangian function. Moreover, we prove fast convergence rates of the Lagrangian residual, the objective residual and the feasibility violation. In the case where the scaling coefficient grows exponentially, we show that the algorithm can achieve linear convergence rates without requiring assumptions of strong convexity or metric subregularity. Additionally, we consider an inexact variant of the proposed algorithm, wherein we find an -stationary solution of the subproblem. Under an additional assumption regarding the error sequence and the scaling coefficient, we prove the preservation of these convergence properties for the inexact variant. Finally, we present some numerical experiments to validate the effectiveness of the proposed algorithms.
{"title":"Inertial accelerated augmented Lagrangian algorithms with scaling coefficients to solve exactly and inexactly linearly constrained convex optimization problems","authors":"Xin He , Rong Hu , Ya-Ping Fang","doi":"10.1016/j.cam.2024.116425","DOIUrl":"10.1016/j.cam.2024.116425","url":null,"abstract":"<div><div>In this paper, we propose an inertial accelerated augmented Lagrangian algorithm with a scaling coefficient tailored for solving linearly constrained convex optimization problems. Under suitable scaling conditions, we show that the iteration sequence generated by the algorithm converges to a saddle point of the Lagrangian function. Moreover, we prove fast convergence rates of the Lagrangian residual, the objective residual and the feasibility violation. In the case where the scaling coefficient grows exponentially, we show that the algorithm can achieve linear convergence rates without requiring assumptions of strong convexity or metric subregularity. Additionally, we consider an inexact variant of the proposed algorithm, wherein we find an <span><math><mi>ϵ</mi></math></span>-stationary solution of the subproblem. Under an additional assumption regarding the error sequence and the scaling coefficient, we prove the preservation of these convergence properties for the inexact variant. Finally, we present some numerical experiments to validate the effectiveness of the proposed algorithms.</div></div>","PeriodicalId":50226,"journal":{"name":"Journal of Computational and Applied Mathematics","volume":"460 ","pages":"Article 116425"},"PeriodicalIF":2.1,"publicationDate":"2024-12-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143098731","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 : 2024-12-07DOI: 10.1016/j.cam.2024.116423
Lorella Fatone , Daniele Funaro
An algorithm for the treatment of images affected by both blurring and salt&pepper noise is proposed with a cost only proportional to the number of pixels. The methodology uses an ad-hoc discretization scheme for the Laplace operator, multiplied by a suitable nonlinear term depending on the gradient. Even if this approach resembles a diffusion type algorithm, only one step of the procedure is in general needed, leading to significant time savings. The procedure is successfully tested on some standard black&white images.
{"title":"Low-cost denoising and deblurring using a novel nonlinear diffusion technique","authors":"Lorella Fatone , Daniele Funaro","doi":"10.1016/j.cam.2024.116423","DOIUrl":"10.1016/j.cam.2024.116423","url":null,"abstract":"<div><div>An algorithm for the treatment of images affected by both blurring and salt&pepper noise is proposed with a cost only proportional to the number of pixels. The methodology uses an ad-hoc discretization scheme for the Laplace operator, multiplied by a suitable nonlinear term depending on the gradient. Even if this approach resembles a diffusion type algorithm, only one step of the procedure is in general needed, leading to significant time savings. The procedure is successfully tested on some standard black&white images.</div></div>","PeriodicalId":50226,"journal":{"name":"Journal of Computational and Applied Mathematics","volume":"461 ","pages":"Article 116423"},"PeriodicalIF":2.1,"publicationDate":"2024-12-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143096259","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 : 2024-12-07DOI: 10.1016/j.cam.2024.116419
Jing Niu , Lei Du , Tomohiro Sogabe , Shao-Liang Zhang
It is well-known that a multilinear system with a nonsingular -tensor and a positive right-hand side has a unique positive solution. Tensor splitting methods generalizing the classical iterative methods for linear systems have been proposed for finding the unique positive solution. The Alternating Anderson–Richardson (AAR) method is an effective method to accelerate the classical iterative methods. In this study, we apply the idea of AAR for finding the unique positive solution quickly. We first present a tensor Richardson method based on tensor regular splittings, then apply Anderson acceleration to the tensor Richardson method and derive a tensor Anderson–Richardson method, finally, we periodically employ the tensor Anderson–Richardson method within the tensor Richardson method and propose a tensor AAR method. Numerical experiments show that the proposed method is effective in accelerating tensor splitting methods.
{"title":"A tensor Alternating Anderson–Richardson method for solving multilinear systems with M-tensors","authors":"Jing Niu , Lei Du , Tomohiro Sogabe , Shao-Liang Zhang","doi":"10.1016/j.cam.2024.116419","DOIUrl":"10.1016/j.cam.2024.116419","url":null,"abstract":"<div><div>It is well-known that a multilinear system with a nonsingular <span><math><mi>M</mi></math></span>-tensor and a positive right-hand side has a unique positive solution. Tensor splitting methods generalizing the classical iterative methods for linear systems have been proposed for finding the unique positive solution. The Alternating Anderson–Richardson (AAR) method is an effective method to accelerate the classical iterative methods. In this study, we apply the idea of AAR for finding the unique positive solution quickly. We first present a tensor Richardson method based on tensor regular splittings, then apply Anderson acceleration to the tensor Richardson method and derive a tensor Anderson–Richardson method, finally, we periodically employ the tensor Anderson–Richardson method within the tensor Richardson method and propose a tensor AAR method. Numerical experiments show that the proposed method is effective in accelerating tensor splitting methods.</div></div>","PeriodicalId":50226,"journal":{"name":"Journal of Computational and Applied Mathematics","volume":"461 ","pages":"Article 116419"},"PeriodicalIF":2.1,"publicationDate":"2024-12-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143096350","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 : 2024-12-06DOI: 10.1016/j.cam.2024.116424
Adán J. Serna-Reyes , Siegfried Macías , Armando Gallegos , Jorge E. Macías-Díaz
In this work, we extend the Rosenau–Kawahara equation (RKE) to the fractional scenario by using space-fractional operators of the Riesz kind. We prove that this system has functional quantities similar to the energy and the mass of the integer-order model, and we show that they are conserved. A discretized form of the model is proposed along with discretized functionals for the energy and the mass. We prove that these quantities are conserved through time. The solvability of the model is proved via Browder’s theorem. Moreover, we establish the properties of second-order convergence, stability and consistency. The numerical model is implemented using a fixed-point approach. Our computations demonstrate that the model conserves the energy and the mass, in agreement with our analysis. This is the first article in the literature in which a conservative scheme for a conservative fractional RKE is propose and rigorously analyzed for the conservation of mass and energy, positivity of energy, existence of solutions, consistency, stability and second-order convergence.
{"title":"A convergent two-step method to solve a fractional extension of the Rosenau–Kawahara system","authors":"Adán J. Serna-Reyes , Siegfried Macías , Armando Gallegos , Jorge E. Macías-Díaz","doi":"10.1016/j.cam.2024.116424","DOIUrl":"10.1016/j.cam.2024.116424","url":null,"abstract":"<div><div>In this work, we extend the Rosenau–Kawahara equation (RKE) to the fractional scenario by using space-fractional operators of the Riesz kind. We prove that this system has functional quantities similar to the energy and the mass of the integer-order model, and we show that they are conserved. A discretized form of the model is proposed along with discretized functionals for the energy and the mass. We prove that these quantities are conserved through time. The solvability of the model is proved via Browder’s theorem. Moreover, we establish the properties of second-order convergence, stability and consistency. The numerical model is implemented using a fixed-point approach. Our computations demonstrate that the model conserves the energy and the mass, in agreement with our analysis. This is the first article in the literature in which a conservative scheme for a conservative fractional RKE is propose and rigorously analyzed for the conservation of mass and energy, positivity of energy, existence of solutions, consistency, stability and second-order convergence.</div></div>","PeriodicalId":50226,"journal":{"name":"Journal of Computational and Applied Mathematics","volume":"460 ","pages":"Article 116424"},"PeriodicalIF":2.1,"publicationDate":"2024-12-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143098801","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 : 2024-12-06DOI: 10.1016/j.cam.2024.116421
Yu-Qi Niu, Bing Zheng
The randomized Kaczmarz algorithm is one of the most popular approaches for solving large-scale linear systems due to its simplicity and efficiency. In this paper, we introduce two classes of randomized Kaczmarz methods for solving linear matrix equations : the randomized block Kaczmarz (ME-RBK) and randomized average block Kaczmarz (ME-RABK) algorithms. The key feature of the ME-RBK algorithm is that the current iterate is projected onto the solution space of the sketched matrix equation at each iteration. In contrast, the ME-RABK method is pseudoinverse-free, enabling deployment on parallel computing units to significantly reduce computational time. We demonstrate that these two methods converge linearly in the mean square to the minimum norm solution of a given linear matrix equation. The convergence rates are influenced by the geometric properties of the data matrices and their submatrices, as well as by the block sizes. Numerical results indicate that our proposed algorithms are both efficient and effective for solving linear matrix equations, particularly excelling in image deblurring applications.
{"title":"On the randomized block Kaczmarz algorithms for solving matrix equation AXB=C","authors":"Yu-Qi Niu, Bing Zheng","doi":"10.1016/j.cam.2024.116421","DOIUrl":"10.1016/j.cam.2024.116421","url":null,"abstract":"<div><div>The randomized Kaczmarz algorithm is one of the most popular approaches for solving large-scale linear systems due to its simplicity and efficiency. In this paper, we introduce two classes of randomized Kaczmarz methods for solving linear matrix equations <span><math><mrow><mi>A</mi><mi>X</mi><mi>B</mi><mo>=</mo><mi>C</mi></mrow></math></span>: the randomized block Kaczmarz (ME-RBK) and randomized average block Kaczmarz (ME-RABK) algorithms. The key feature of the ME-RBK algorithm is that the current iterate is projected onto the solution space of the sketched matrix equation at each iteration. In contrast, the ME-RABK method is pseudoinverse-free, enabling deployment on parallel computing units to significantly reduce computational time. We demonstrate that these two methods converge linearly in the mean square to the minimum norm solution <span><math><mrow><msub><mrow><mi>X</mi></mrow><mrow><mo>∗</mo></mrow></msub><mo>=</mo><msup><mrow><mi>A</mi></mrow><mrow><mi>†</mi></mrow></msup><mi>C</mi><msup><mrow><mi>B</mi></mrow><mrow><mi>†</mi></mrow></msup></mrow></math></span> of a given linear matrix equation. The convergence rates are influenced by the geometric properties of the data matrices and their submatrices, as well as by the block sizes. Numerical results indicate that our proposed algorithms are both efficient and effective for solving linear matrix equations, particularly excelling in image deblurring applications.</div></div>","PeriodicalId":50226,"journal":{"name":"Journal of Computational and Applied Mathematics","volume":"460 ","pages":"Article 116421"},"PeriodicalIF":2.1,"publicationDate":"2024-12-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143098729","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 : 2024-12-05DOI: 10.1016/j.cam.2024.116417
Jens Lang , Bernhard A. Schmitt
This paper is concerned with the theory, construction and application of implicit Peer two-step methods that are super-convergent for variable stepsizes, i.e., preserve their classical order achieved for uniform stepsizes when applied to ODE constrained optimal control problems in a first-discretize-then-optimize setting. We upgrade our former implicit two-step Peer triplets constructed in Lang and Schmitt (Algorithms 2022) to get ready for dynamical systems with varying time scales without loosing efficiency. Peer triplets consist of a standard Peer method for interior time steps supplemented by matching methods for the starting and end steps. A decisive advantage of Peer methods is their absence of order reduction since they use stages of the same high stage order. The consistency analysis of variable-stepsize implicit Peer methods results in additional order conditions and severe new difficulties for uniform zero-stability, which intensifies the demands on the Peer triplet. Further, we discuss the construction of 4-stage methods with order pairs (4,3) and (3,3) for state and adjoint variables in detail and provide four Peer triplets of practical interest. We rigorously prove convergence of order for -stage Peer methods applied on grids with bounded or smoothly changing stepsize ratios. Numerical tests show the expected order of convergence for the new variable-stepsize Peer triplets.
{"title":"Variable-stepsize implicit Peer triplets in ODE constrained optimal control","authors":"Jens Lang , Bernhard A. Schmitt","doi":"10.1016/j.cam.2024.116417","DOIUrl":"10.1016/j.cam.2024.116417","url":null,"abstract":"<div><div>This paper is concerned with the theory, construction and application of implicit Peer two-step methods that are super-convergent for variable stepsizes, i.e., preserve their classical order achieved for uniform stepsizes when applied to ODE constrained optimal control problems in a first-discretize-then-optimize setting. We upgrade our former implicit two-step Peer triplets constructed in Lang and Schmitt (Algorithms 2022) to get ready for dynamical systems with varying time scales without loosing efficiency. Peer triplets consist of a standard Peer method for interior time steps supplemented by matching methods for the starting and end steps. A decisive advantage of Peer methods is their absence of order reduction since they use stages of the same high stage order. The consistency analysis of variable-stepsize implicit Peer methods results in additional order conditions and severe new difficulties for uniform zero-stability, which intensifies the demands on the Peer triplet. Further, we discuss the construction of 4-stage methods with order pairs (4,3) and (3,3) for state and adjoint variables in detail and provide four Peer triplets of practical interest. We rigorously prove convergence of order <span><math><mrow><mi>s</mi><mo>−</mo><mn>1</mn></mrow></math></span> for <span><math><mi>s</mi></math></span>-stage Peer methods applied on grids with bounded or smoothly changing stepsize ratios. Numerical tests show the expected order of convergence for the new variable-stepsize Peer triplets.</div></div>","PeriodicalId":50226,"journal":{"name":"Journal of Computational and Applied Mathematics","volume":"460 ","pages":"Article 116417"},"PeriodicalIF":2.1,"publicationDate":"2024-12-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143098732","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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