Pub Date : 2025-12-06DOI: 10.1016/j.rinam.2025.100677
Jiadong Qiu , Xiang Liu , Feng Liao
In this paper, general conservative sixth- and eighth-order compact finite difference schemes are presented to solve the N-coupled nonlinear Schrödinger-Boussinesq equations numerically. The existence of the difference solution is proved by fixed-point theorem. By utilizing the discrete energy methods, the proposed difference schemes are proved to be unconditionally convergent at the order with mesh-size and time step in the discrete -norm. By using the Yoshida’s composition method, we improve the scheme (3.1)-(3.3) with a group of given time-step increments repeatedly and then obtain a temporal fourth-order difference scheme. Numerical experiments confirm the theoretical results and verify the accuracy and efficiency of our method.
{"title":"General conservative sixth- and eighth-order compact finite difference schemes for the N-coupled Schrödinger-Boussinesq equations","authors":"Jiadong Qiu , Xiang Liu , Feng Liao","doi":"10.1016/j.rinam.2025.100677","DOIUrl":"10.1016/j.rinam.2025.100677","url":null,"abstract":"<div><div>In this paper, general conservative sixth- and eighth-order compact finite difference schemes are presented to solve the N-coupled nonlinear Schrödinger-Boussinesq equations numerically. The existence of the difference solution is proved by fixed-point theorem. By utilizing the discrete energy methods, the proposed difference schemes are proved to be unconditionally convergent at the order <span><math><mrow><mi>O</mi><mrow><mo>(</mo><msup><mrow><mi>τ</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><msup><mrow><mi>h</mi></mrow><mrow><mn>8</mn></mrow></msup><mo>)</mo></mrow></mrow></math></span> with mesh-size <span><math><mi>h</mi></math></span> and time step <span><math><mi>τ</mi></math></span> in the discrete <span><math><msub><mrow><mi>L</mi></mrow><mrow><mi>∞</mi></mrow></msub></math></span>-norm. By using the Yoshida’s composition method, we improve the scheme <span><span>(3.1)</span></span>-<span><span>(3.3)</span></span> with a group of given time-step increments repeatedly and then obtain a temporal fourth-order difference scheme. Numerical experiments confirm the theoretical results and verify the accuracy and efficiency of our method.</div></div>","PeriodicalId":36918,"journal":{"name":"Results in Applied Mathematics","volume":"29 ","pages":"Article 100677"},"PeriodicalIF":1.3,"publicationDate":"2025-12-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145693047","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-11-01DOI: 10.1016/j.rinam.2025.100675
Elisabeth Halser, Elisabeth Finhold, Neele Leithäuser, Jan Schwientek, Katrin Teichert, Karl-Heinz Küfer
Multicriteria adjustable robust optimization (MARO) problems are highly relevant for a wide variety of practical problems with a two-stage-decision, typically an initial purchase decision followed by the possibility to react during operation after uncertain parameters are revealed. However, no general approaches for the definition of efficient solutions to this problem class are available in the literature for the multicriteria case. The objective of this paper is to find a meaningful definition that in particular allows the computation of solutions. By combining well-known approaches from multicriteria optimization and robust optimization in a straightforward way, we give different definitions for efficient solutions to MARO problems and look at three computation-oriented approaches to deal with the problems. These computation-oriented approaches can also be understood as additional efficiency definitions. We assess the advantages and disadvantages of the different computation-oriented approaches and analyze their connections to our initial definitions of MARO-efficiency. We observe that an -constraint inspired first-scalarize-then-robustify approach is beneficial because it provides an efficient set that is easy to understand for decision makers and provides tight bounds on the worst-case evaluation for a particular efficient solution. In contrast, a weighted sum first-scalarize-then-robustify approach keeps the problem structure more simple but the efficient set might look ambiguous. Further, we demonstrate that a first-robustify procedure only gives bad bounds and can be too optimistic as well as too pessimistic.
{"title":"Multicriteria adjustable robustness","authors":"Elisabeth Halser, Elisabeth Finhold, Neele Leithäuser, Jan Schwientek, Katrin Teichert, Karl-Heinz Küfer","doi":"10.1016/j.rinam.2025.100675","DOIUrl":"10.1016/j.rinam.2025.100675","url":null,"abstract":"<div><div>Multicriteria adjustable robust optimization (MARO) problems are highly relevant for a wide variety of practical problems with a two-stage-decision, typically an initial purchase decision followed by the possibility to react during operation after uncertain parameters are revealed. However, no general approaches for the definition of efficient solutions to this problem class are available in the literature for the multicriteria case. The objective of this paper is to find a meaningful definition that in particular allows the computation of solutions. By combining well-known approaches from multicriteria optimization and robust optimization in a straightforward way, we give different definitions for efficient solutions to MARO problems and look at three computation-oriented approaches to deal with the problems. These computation-oriented approaches can also be understood as additional efficiency definitions. We assess the advantages and disadvantages of the different computation-oriented approaches and analyze their connections to our initial definitions of MARO-efficiency. We observe that an <span><math><mi>ɛ</mi></math></span>-constraint inspired first-scalarize-then-robustify approach is beneficial because it provides an efficient set that is easy to understand for decision makers and provides tight bounds on the worst-case evaluation for a particular efficient solution. In contrast, a weighted sum first-scalarize-then-robustify approach keeps the problem structure more simple but the efficient set might look ambiguous. Further, we demonstrate that a first-robustify procedure only gives bad bounds and can be too optimistic as well as too pessimistic.</div></div>","PeriodicalId":36918,"journal":{"name":"Results in Applied Mathematics","volume":"28 ","pages":"Article 100675"},"PeriodicalIF":1.3,"publicationDate":"2025-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145614873","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-11-01DOI: 10.1016/j.rinam.2025.100667
Peipei Zhao , Pengyu Zhang
Image restoration is to estimate the clean image from the recorded image, it is a highly ill-posed inverse problem. Regularization method is an important approach for solving such problem, which can usually be achieved by minimizing a cost function consisting of a data-fidelity term and a regularization term. In this paper, we consider the additive half-quadratic (HQ) regularized method for image restoration problem, and utilize the Newton method to solve the resulting minimization problem. At each Newton iteration step, a system of linear equations with symmetric positive definite coefficient matrix arises. In order to solve the linear system efficiently, we design a parameterized approximation matrix of the Schur complement inverse matrix, and construct a block preconditioner with parameter correspondingly, according to the block triangular factorization of coefficient matrix and the form of its Schur complement, then the preconditioned conjugate gradient (PCG) method is applied to solve the linear system of equations. Spectral analyses of the preconditioned matrix are also given, numerical experimental results demonstrate the effectiveness of the proposed parameterized preconditioner for solving linear system arising from additive HQ image restoration problem.
{"title":"A parameterized Schur complement preconditioner for linear system arising from additive HQ image restoration","authors":"Peipei Zhao , Pengyu Zhang","doi":"10.1016/j.rinam.2025.100667","DOIUrl":"10.1016/j.rinam.2025.100667","url":null,"abstract":"<div><div>Image restoration is to estimate the clean image from the recorded image, it is a highly ill-posed inverse problem. Regularization method is an important approach for solving such problem, which can usually be achieved by minimizing a cost function consisting of a data-fidelity term and a regularization term. In this paper, we consider the additive half-quadratic (HQ) regularized method for image restoration problem, and utilize the Newton method to solve the resulting minimization problem. At each Newton iteration step, a system of linear equations with symmetric positive definite coefficient matrix arises. In order to solve the linear system efficiently, we design a parameterized approximation matrix of the Schur complement inverse matrix, and construct a block preconditioner with parameter correspondingly, according to the block triangular factorization of coefficient matrix and the form of its Schur complement, then the preconditioned conjugate gradient (PCG) method is applied to solve the linear system of equations. Spectral analyses of the preconditioned matrix are also given, numerical experimental results demonstrate the effectiveness of the proposed parameterized preconditioner for solving linear system arising from additive HQ image restoration problem.</div></div>","PeriodicalId":36918,"journal":{"name":"Results in Applied Mathematics","volume":"28 ","pages":"Article 100667"},"PeriodicalIF":1.3,"publicationDate":"2025-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145519626","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-11-01DOI: 10.1016/j.rinam.2025.100666
Qiang Niu , Mianmian Chen , Jinheng Wu
The Lanczos algorithm is a well-known three-term recurrence that can be used to generate an orthogonal basis for a Krylov subspace derived by a symmetric matrix. In the paper, we present a statistical interpretation of the entries of the tridiagonal matrix generated by the Lanczos process with a diagonal matrix and an initial vector . We show that the entries on the main diagonal line can be interpreted as weighted mean and the entries on the super-diagonal line can be understood as weighted sum of variance. Besides, a recurrence for producing the entries on the off-diagonal entries of the tridiagonal matrix is discovered, which leads to a new implementation of the Lanczos process. Finally, numerical examples are provided to investigate the preservation of orthogonality and efficiency in data fitting.
{"title":"Lanczos algorithm explained in statistics","authors":"Qiang Niu , Mianmian Chen , Jinheng Wu","doi":"10.1016/j.rinam.2025.100666","DOIUrl":"10.1016/j.rinam.2025.100666","url":null,"abstract":"<div><div>The Lanczos algorithm is a well-known three-term recurrence that can be used to generate an orthogonal basis for a Krylov subspace derived by a symmetric matrix. In the paper, we present a statistical interpretation of the entries of the tridiagonal matrix generated by the Lanczos process with a diagonal matrix <span><math><mi>X</mi></math></span> and an initial vector <span><math><mi>e</mi></math></span>. We show that the entries on the main diagonal line can be interpreted as <em>weighted mean</em> and the entries on the super-diagonal line can be understood as <em>weighted sum of variance</em>. Besides, a recurrence for producing the entries on the off-diagonal entries of the tridiagonal matrix is discovered, which leads to a new implementation of the Lanczos process. Finally, numerical examples are provided to investigate the preservation of orthogonality and efficiency in data fitting.</div></div>","PeriodicalId":36918,"journal":{"name":"Results in Applied Mathematics","volume":"28 ","pages":"Article 100666"},"PeriodicalIF":1.3,"publicationDate":"2025-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145519627","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-11-01DOI: 10.1016/j.rinam.2025.100663
Zhikun Tian , Jianyun Wang , Zixin Zhong
In this paper, we investigate the two-step backward differentiation formula (BDF2) finite element method for a two-dimensional time-dependent Schrödinger equation. By applying the finite element method for space discretization and the BDF2 for time discretization, we derive a fully discrete scheme for the Schrödinger equation. The errors of the exact solution with the finite element solution are divided into temporal and spatial errors for separate analysis. We obtain the optimal error estimate in both space and time for the fully discrete scheme. Finally, a numerical experiment is performed to demonstrate the accuracy and efficiency of the proposed numerical scheme.
{"title":"Optimal error estimates of BDF2 finite element method for the two-dimensional time-dependent Schrödinger equation","authors":"Zhikun Tian , Jianyun Wang , Zixin Zhong","doi":"10.1016/j.rinam.2025.100663","DOIUrl":"10.1016/j.rinam.2025.100663","url":null,"abstract":"<div><div>In this paper, we investigate the two-step backward differentiation formula (BDF2) finite element method for a two-dimensional time-dependent Schrödinger equation. By applying the finite element method for space discretization and the BDF2 for time discretization, we derive a fully discrete scheme for the Schrödinger equation. The errors of the exact solution with the finite element solution are divided into temporal and spatial errors for separate analysis. We obtain the optimal error estimate in both space and time for the fully discrete scheme. Finally, a numerical experiment is performed to demonstrate the accuracy and efficiency of the proposed numerical scheme.</div></div>","PeriodicalId":36918,"journal":{"name":"Results in Applied Mathematics","volume":"28 ","pages":"Article 100663"},"PeriodicalIF":1.3,"publicationDate":"2025-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145466098","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-11-01DOI: 10.1016/j.rinam.2025.100655
Ikha Magdalena , Anwar Efendi Nasution
Wave resonance in coastal basins can cause significant environmental and structural damage, highlighting the need for effective mitigation strategies in coastal engineering. This study explores resonance phenomena in a rectangular semi-closed basin protected by submerged triangular breakwaters. The wave dynamics are modeled utilizing modified Linear Shallow Water Equations (LSWEs), which incorporate a breakwater-induced friction factor. Analytical and numerical approaches are employed in determining the basin’s natural period—a key parameter governing the onset of resonance. Numerical simulations, formulated utilizing the finite volume method, are conducted to identify the conditions that trigger resonance. The findings reveal that submerged triangular breakwaters substantially affect the natural period and help attenuate resonance effects, providing valuable insights for the design of resilient coastal infrastructure.
{"title":"Analytical and numerical investigation of wave resonance over rectangular basin with submerged triangular breakwaters","authors":"Ikha Magdalena , Anwar Efendi Nasution","doi":"10.1016/j.rinam.2025.100655","DOIUrl":"10.1016/j.rinam.2025.100655","url":null,"abstract":"<div><div>Wave resonance in coastal basins can cause significant environmental and structural damage, highlighting the need for effective mitigation strategies in coastal engineering. This study explores resonance phenomena in a rectangular semi-closed basin protected by submerged triangular breakwaters. The wave dynamics are modeled utilizing modified Linear Shallow Water Equations (LSWEs), which incorporate a breakwater-induced friction factor. Analytical and numerical approaches are employed in determining the basin’s natural period—a key parameter governing the onset of resonance. Numerical simulations, formulated utilizing the finite volume method, are conducted to identify the conditions that trigger resonance. The findings reveal that submerged triangular breakwaters substantially affect the natural period and help attenuate resonance effects, providing valuable insights for the design of resilient coastal infrastructure.</div></div>","PeriodicalId":36918,"journal":{"name":"Results in Applied Mathematics","volume":"28 ","pages":"Article 100655"},"PeriodicalIF":1.3,"publicationDate":"2025-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145417120","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-11-01DOI: 10.1016/j.rinam.2025.100672
Yanyan Song , Yanqiu Wang , Qiao Xin
In this paper, we construct two finite element discretizations on convex polygonal meshes for the Stokes equations, using the generalized barycentric coordinates (GBCs). Both constructions use a bubble-enhanced second-order Floater-Lai GBC to discretize the velocity field. The pressure field is discretized by discontinuous piecewise constants and discontinuous piecewise linears, respectively. They can be viewed as the generalization of the - and the - elements to convex polygonal meshes. We prove the inf–sup stability and the optimal convergence for both discretizations. Supporting numerical results are presented.
本文利用广义质心坐标在凸多边形网格上构造了Stokes方程的两个有限元离散化。两种结构都使用气泡增强的二阶float - lai GBC来离散速度场。压力场分别采用不连续分段常数和不连续分段线性进行离散。它们可以看作是P2-P0和Q2-P1元素对凸多边形网格的推广。证明了这两种离散化方法的稳定性和最优收敛性。给出了相应的数值结果。
{"title":"Second-order finite element discretization of Stokes equations on convex polygonal meshes","authors":"Yanyan Song , Yanqiu Wang , Qiao Xin","doi":"10.1016/j.rinam.2025.100672","DOIUrl":"10.1016/j.rinam.2025.100672","url":null,"abstract":"<div><div>In this paper, we construct two finite element discretizations on convex polygonal meshes for the Stokes equations, using the generalized barycentric coordinates (GBCs). Both constructions use a bubble-enhanced second-order Floater-Lai GBC to discretize the velocity field. The pressure field is discretized by discontinuous piecewise constants and discontinuous piecewise linears, respectively. They can be viewed as the generalization of the <span><math><msub><mrow><mi>P</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>-<span><math><msub><mrow><mi>P</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span> and the <span><math><msub><mrow><mi>Q</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>-<span><math><msub><mrow><mi>P</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> elements to convex polygonal meshes. We prove the inf–sup stability and the optimal convergence for both discretizations. Supporting numerical results are presented.</div></div>","PeriodicalId":36918,"journal":{"name":"Results in Applied Mathematics","volume":"28 ","pages":"Article 100672"},"PeriodicalIF":1.3,"publicationDate":"2025-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145614876","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-11-01DOI: 10.1016/j.rinam.2025.100676
Li Wang , Jiawei Wang , Ren-Cang Li
This paper is concerned with sparse PCA via the matrix (2,1)-norm regularization (). It can produce a row-sparse projection, a useful tool in machine learning when it comes to, for example, feature selection, that aims to choose most relevant features. Mathematically, is a non-smooth optimization problem on the Stiefel manifold. For a suitably chosen regularization parameter, the optimal projection matrix has many negligible rows. A practical NEPv approach (nonlinear eigenvalue problem with eigenvector dependency) is proposed to iteratively compute the optimal projection matrix. It is shown that the approach is globally convergent in the sense that the objective is monotonically increasing during the iterative process and any accumulation point of the iterates is a stationary point to the optimization problem. Extensive numerical experiments, with an application to feature selection, have been conducted to demonstrate the performance of the practical NEPv approach, with comparison against existing feature selection methods in terms of classification accuracy. The numerical results demonstrate that is highly effective and often produces superior classification results to existing feature selection methods that are in use today.
{"title":"Sparse PCA via matrix (2,1)-norm regularization with an application to feature selection","authors":"Li Wang , Jiawei Wang , Ren-Cang Li","doi":"10.1016/j.rinam.2025.100676","DOIUrl":"10.1016/j.rinam.2025.100676","url":null,"abstract":"<div><div>This paper is concerned with sparse PCA via the matrix (2,1)-norm regularization (<span><math><msub><mrow><mo>PCA</mo></mrow><mrow><mn>2</mn><mo>,</mo><mn>1</mn></mrow></msub></math></span>). It can produce a row-sparse projection, a useful tool in machine learning when it comes to, for example, feature selection, that aims to choose most relevant features. Mathematically, <span><math><msub><mrow><mo>PCA</mo></mrow><mrow><mn>2</mn><mo>,</mo><mn>1</mn></mrow></msub></math></span> is a non-smooth optimization problem on the Stiefel manifold. For a suitably chosen regularization parameter, the optimal projection matrix has many negligible rows. A practical NEPv approach (nonlinear eigenvalue problem with eigenvector dependency) is proposed to iteratively compute the optimal projection matrix. It is shown that the approach is globally convergent in the sense that the objective is monotonically increasing during the iterative process and any accumulation point of the iterates is a stationary point to the optimization problem. Extensive numerical experiments, with an application to feature selection, have been conducted to demonstrate the performance of the practical NEPv approach, with comparison against existing feature selection methods in terms of classification accuracy. The numerical results demonstrate that <span><math><msub><mrow><mo>PCA</mo></mrow><mrow><mn>2</mn><mo>,</mo><mn>1</mn></mrow></msub></math></span> is highly effective and often produces superior classification results to existing feature selection methods that are in use today.</div></div>","PeriodicalId":36918,"journal":{"name":"Results in Applied Mathematics","volume":"28 ","pages":"Article 100676"},"PeriodicalIF":1.3,"publicationDate":"2025-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145680919","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-11-01DOI: 10.1016/j.rinam.2025.100673
Elyas Shivanian , Ahmad Jafarabadi , Mousa J. Huntul
This study focuses on retrieving a time-dependent source term in the heat equation governed by two distinct nonlocal boundary conditions. The inverse problem is structured with an interior energy over-specification constraint. The proposed computational framework combines the partition of unity approach for spatial discretization with the finite difference scheme for temporal advancement. Through energy analysis, the semi-discrete time-stepping formulation is proven to be unconditionally stable and convergent at a rate of . Despite being linear and uniquely solvable, the problem is inherently ill-posed, as slight disturbances in input data can induce significant errors in the reconstructed solution. To counteract this instability, Tikhonov regularization is implemented, yielding a stable approximation even under noisy data conditions. Moreover, a novel parameter selection strategy for the regularization is introduced, which surpasses standard methods by delivering substantially improved results. Numerical simulations corroborate the scheme’s robustness, demonstrating its accuracy with noise-free inputs and its resilience when handling contaminated measurements.
{"title":"A local meshless technique for recovering dual forms of time-varying sources in the nonlocal inverse heat equation","authors":"Elyas Shivanian , Ahmad Jafarabadi , Mousa J. Huntul","doi":"10.1016/j.rinam.2025.100673","DOIUrl":"10.1016/j.rinam.2025.100673","url":null,"abstract":"<div><div>This study focuses on retrieving a time-dependent source term in the heat equation governed by two distinct nonlocal boundary conditions. The inverse problem is structured with an interior energy over-specification constraint. The proposed computational framework combines the partition of unity approach for spatial discretization with the finite difference scheme for temporal advancement. Through energy analysis, the semi-discrete time-stepping formulation is proven to be unconditionally stable and convergent at a rate of <span><math><mrow><mi>O</mi><mrow><mo>(</mo><mi>δ</mi><mi>t</mi><mo>)</mo></mrow></mrow></math></span>. Despite being linear and uniquely solvable, the problem is inherently ill-posed, as slight disturbances in input data can induce significant errors in the reconstructed solution. To counteract this instability, Tikhonov regularization is implemented, yielding a stable approximation even under noisy data conditions. Moreover, a novel parameter selection strategy for the regularization is introduced, which surpasses standard methods by delivering substantially improved results. Numerical simulations corroborate the scheme’s robustness, demonstrating its accuracy with noise-free inputs and its resilience when handling contaminated measurements.</div></div>","PeriodicalId":36918,"journal":{"name":"Results in Applied Mathematics","volume":"28 ","pages":"Article 100673"},"PeriodicalIF":1.3,"publicationDate":"2025-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145614874","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-11-01DOI: 10.1016/j.rinam.2025.100665
Jingwei Chen
This paper develops an -parametrized framework for analyzing the strong convergence of the stochastic theta (ST) method for stochastic differential equations driven by time-changed Lévy noise (TCSDEwLNs). The analysis accommodates time–space-dependent coefficients satisfying local Lipschitz conditions. Properties of the inverse subordinator are investigated and explicit moment bounds for the exact solution are derived with jump rate incorporated. The analysis demonstrates that the ST method converges strongly with order of , establishing a precise relationship between numerical accuracy and the time-change mechanism. This theoretical advancement extends existing results and facilitates applications in finance, physics and biology where time-changed Lévy models are prevalent.
{"title":"α-scaled strong convergence of stochastic theta method for stochastic differential equations driven by time-changed Lévy noise beyond Lipschitz continuity","authors":"Jingwei Chen","doi":"10.1016/j.rinam.2025.100665","DOIUrl":"10.1016/j.rinam.2025.100665","url":null,"abstract":"<div><div>This paper develops an <span><math><mi>α</mi></math></span>-parametrized framework for analyzing the strong convergence of the stochastic theta (ST) method for stochastic differential equations driven by time-changed Lévy noise (TCSDEwLNs). The analysis accommodates time–space-dependent coefficients satisfying local Lipschitz conditions. Properties of the inverse subordinator <span><math><mi>E</mi></math></span> are investigated and explicit moment bounds for the exact solution are derived with jump rate incorporated. The analysis demonstrates that the ST method converges strongly with order of <span><math><mrow><mo>min</mo><mrow><mo>{</mo><msub><mrow><mi>η</mi></mrow><mrow><mi>F</mi></mrow></msub><mo>,</mo><msub><mrow><mi>η</mi></mrow><mrow><mi>G</mi></mrow></msub><mo>,</mo><msub><mrow><mi>η</mi></mrow><mrow><mi>H</mi></mrow></msub><mo>,</mo><mi>α</mi><mo>/</mo><mn>2</mn><mo>}</mo></mrow></mrow></math></span>, establishing a precise relationship between numerical accuracy and the time-change mechanism. This theoretical advancement extends existing results and facilitates applications in finance, physics and biology where time-changed Lévy models are prevalent.</div></div>","PeriodicalId":36918,"journal":{"name":"Results in Applied Mathematics","volume":"28 ","pages":"Article 100665"},"PeriodicalIF":1.3,"publicationDate":"2025-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145519625","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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