Pub Date : 2023-01-09DOI: 10.48550/arXiv.2301.03200
A. Jüngel, M. Vetter
Abstract A second-order backward differentiation formula (BDF2) finite-volume discretization for a nonlinear cross-diffusion system arising in population dynamics is studied. The numerical scheme preserves the Rao entropy structure and conserves the mass. The existence and uniqueness of discrete solutions and their large-time behavior as well as the convergence of the scheme are proved. The proofs are based on the G-stability of the BDF2 scheme, which provides an inequality for the quadratic Rao entropy and hence suitable a priori estimates. The novelty is the extension of this inequality to the system case. Some numerical experiments in one and two space dimensions underline the theoretical results.
{"title":"A Convergent Entropy-Dissipating BDF2 Finite-Volume Scheme for a Population Cross-Diffusion System","authors":"A. Jüngel, M. Vetter","doi":"10.48550/arXiv.2301.03200","DOIUrl":"https://doi.org/10.48550/arXiv.2301.03200","url":null,"abstract":"Abstract A second-order backward differentiation formula (BDF2) finite-volume discretization for a nonlinear cross-diffusion system arising in population dynamics is studied. The numerical scheme preserves the Rao entropy structure and conserves the mass. The existence and uniqueness of discrete solutions and their large-time behavior as well as the convergence of the scheme are proved. The proofs are based on the G-stability of the BDF2 scheme, which provides an inequality for the quadratic Rao entropy and hence suitable a priori estimates. The novelty is the extension of this inequality to the system case. Some numerical experiments in one and two space dimensions underline the theoretical results.","PeriodicalId":48751,"journal":{"name":"Computational Methods in Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2023-01-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47747958","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A. Aimi, C. Guardasoni, L. Ortiz-Gracia, S. Sanfelici
Abstract In this work, the Fourier-cosine series (COS) method has been combined with the Boundary Element Method (BEM) for a fast evaluation of barrier option prices. After a description of its use in the Black and Scholes (BS) model, the focus of the paper is on the application of the proposed methodology to the barrier option evaluation in the Heston model, where its contribution is fundamental to improve computational efficiency and to make BEM appealing among finance practitioners as a valid alternative to Monte Carlo (MC) or other more traditional approaches. An error analysis is provided on the number of terms used in the Fourier-cosine series expansion, where the error bound estimation is based on the characteristic function of the log-asset price process. A Matlab code implementing this technique is attached at the end of the paper.
摘要在这项工作中，傅立叶余弦级数（COS）方法与边界元方法（BEM）相结合，用于快速评估障碍期权价格。在描述了其在Black and Scholes（BS）模型中的应用后，本文的重点是将所提出的方法应用于Heston模型中的障碍选择评估，其贡献对于提高计算效率和使BEM作为蒙特卡洛（MC）或其他更传统方法的有效替代方案在金融从业者中具有吸引力是至关重要的。对傅立叶余弦级数展开中使用的项的数量进行了误差分析，其中误差界估计基于对数资产价格过程的特征函数。文末附有实现该技术的Matlab代码。
{"title":"Fast Barrier Option Pricing by the COS BEM Method in Heston Model (with Matlab Code)","authors":"A. Aimi, C. Guardasoni, L. Ortiz-Gracia, S. Sanfelici","doi":"10.1515/cmam-2022-0088","DOIUrl":"https://doi.org/10.1515/cmam-2022-0088","url":null,"abstract":"Abstract In this work, the Fourier-cosine series (COS) method has been combined with the Boundary Element Method (BEM) for a fast evaluation of barrier option prices. After a description of its use in the Black and Scholes (BS) model, the focus of the paper is on the application of the proposed methodology to the barrier option evaluation in the Heston model, where its contribution is fundamental to improve computational efficiency and to make BEM appealing among finance practitioners as a valid alternative to Monte Carlo (MC) or other more traditional approaches. An error analysis is provided on the number of terms used in the Fourier-cosine series expansion, where the error bound estimation is based on the characteristic function of the log-asset price process. A Matlab code implementing this technique is attached at the end of the paper.","PeriodicalId":48751,"journal":{"name":"Computational Methods in Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2023-01-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44504945","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract Magnetic fields generated by normal or superconducting electromagnets are used to guide and focus particle beams in storage rings, synchrotron light sources, mass spectrometers, and beamlines for radiotherapy. The accurate determination of the magnetic field by measurement is critical for the prediction of the particle beam trajectory and hence the design of the accelerator complex. In this context, state-of-the-art numerical field computation makes use of boundary-element methods (BEM) to express the magnetic field. This enables the accurate computation of higher-order partial derivatives and local expansions of magnetic potentials used in efficient numerical codes for particle tracking. In this paper, we present an approach to infer the boundary data of an indirect BEM formulation from magnetic field measurements by ensemble Kálmán filtering. In this way, measurement uncertainties can be propagated to the boundary data, magnetic field and potentials, and to the beam related quantities derived from particle tracking. We provide results obtained from real measurement data of a curved dipole magnet using a Hall probe mapper system.
{"title":"BEM-Based Magnetic Field Reconstruction by Ensemble Kálmán Filtering","authors":"M. Liebsch, S. Russenschuck, S. Kurz","doi":"10.1515/cmam-2022-0121","DOIUrl":"https://doi.org/10.1515/cmam-2022-0121","url":null,"abstract":"Abstract Magnetic fields generated by normal or superconducting electromagnets are used to guide and focus particle beams in storage rings, synchrotron light sources, mass spectrometers, and beamlines for radiotherapy. The accurate determination of the magnetic field by measurement is critical for the prediction of the particle beam trajectory and hence the design of the accelerator complex. In this context, state-of-the-art numerical field computation makes use of boundary-element methods (BEM) to express the magnetic field. This enables the accurate computation of higher-order partial derivatives and local expansions of magnetic potentials used in efficient numerical codes for particle tracking. In this paper, we present an approach to infer the boundary data of an indirect BEM formulation from magnetic field measurements by ensemble Kálmán filtering. In this way, measurement uncertainties can be propagated to the boundary data, magnetic field and potentials, and to the beam related quantities derived from particle tracking. We provide results obtained from real measurement data of a curved dipole magnet using a Hall probe mapper system.","PeriodicalId":48751,"journal":{"name":"Computational Methods in Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2022-12-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47308358","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-12-12DOI: 10.48550/arXiv.2212.06093
Gabriel Acosta, Francisco M. Bersetche, J. Rossi
Abstract We study a natural alternating method of Schwarz type (domain decomposition) for a certain class of couplings between local and nonlocal operators. We show that our method fits into Lions’s framework and prove, as a consequence, convergence in both the continuous and the discrete settings.
{"title":"A Domain Decomposition Scheme for Couplings between Local and Nonlocal Equations","authors":"Gabriel Acosta, Francisco M. Bersetche, J. Rossi","doi":"10.48550/arXiv.2212.06093","DOIUrl":"https://doi.org/10.48550/arXiv.2212.06093","url":null,"abstract":"Abstract We study a natural alternating method of Schwarz type (domain decomposition) for a certain class of couplings between local and nonlocal operators. We show that our method fits into Lions’s framework and prove, as a consequence, convergence in both the continuous and the discrete settings.","PeriodicalId":48751,"journal":{"name":"Computational Methods in Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2022-12-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45441883","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract The stochastic multi-group susceptible–infected–recovered (SIR) epidemic model is nonlinear, and so analytical solutions are generally difficult to obtain. Hence, it is often necessary to find numerical solutions, but most existing numerical methods fail to preserve the nonnegativity or positivity of solutions. Therefore, an appropriate numerical method for studying the dynamic behavior of epidemic diseases through SIR models is urgently required. In this paper, based on the Euler–Maruyama scheme and a logarithmic transformation, we propose a novel explicit positivity-preserving numerical scheme for a stochastic multi-group SIR epidemic model whose coefficients violate the global monotonicity condition. This scheme not only results in numerical solutions that preserve the domain of the stochastic multi-group SIR epidemic model, but also achieves the “ order - 1 2 {mathrm{order}-frac{1}{2}} ” strong convergence rate. Taking a two-group SIR epidemic model as an example, some numerical simulations are performed to illustrate the performance of the proposed scheme.
{"title":"Positivity-Preserving Numerical Method for a Stochastic Multi-Group SIR Epidemic Model","authors":"Han Ma, Qimin Zhang, X. Xu","doi":"10.1515/cmam-2022-0143","DOIUrl":"https://doi.org/10.1515/cmam-2022-0143","url":null,"abstract":"Abstract The stochastic multi-group susceptible–infected–recovered (SIR) epidemic model is nonlinear, and so analytical solutions are generally difficult to obtain. Hence, it is often necessary to find numerical solutions, but most existing numerical methods fail to preserve the nonnegativity or positivity of solutions. Therefore, an appropriate numerical method for studying the dynamic behavior of epidemic diseases through SIR models is urgently required. In this paper, based on the Euler–Maruyama scheme and a logarithmic transformation, we propose a novel explicit positivity-preserving numerical scheme for a stochastic multi-group SIR epidemic model whose coefficients violate the global monotonicity condition. This scheme not only results in numerical solutions that preserve the domain of the stochastic multi-group SIR epidemic model, but also achieves the “ order - 1 2 {mathrm{order}-frac{1}{2}} ” strong convergence rate. Taking a two-group SIR epidemic model as an example, some numerical simulations are performed to illustrate the performance of the proposed scheme.","PeriodicalId":48751,"journal":{"name":"Computational Methods in Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2022-12-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42594497","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The local discontinuous Galerkin (LDG) method is studied for a third-order singularly perturbed problem of convection-diffusion type. Based on a regularity assumption for the exact solution, we prove almost O(N-(k+12)){O(N^{-(k+frac{1}{2})})} (up to a logarithmic factor) energy-norm convergence uniformly in the perturbation parameter. Here, k≥0{kgeq 0} is the maximum degree of piecewise polynomials used in discrete space, and N is the number of mesh elements. The results are valid for the three types of layer-adapted meshes: Shishkin-type, Bakhvalov–Shishkin-type, and Bakhvalov-type. Numerical experiments are conducted to test the theoretical results.
{"title":"Local Discontinuous Galerkin Method for a Third-Order Singularly Perturbed Problem of Convection-Diffusion Type","authors":"Li Yan, Zhoufeng Wang, Yao Cheng","doi":"10.1515/cmam-2022-0176","DOIUrl":"https://doi.org/10.1515/cmam-2022-0176","url":null,"abstract":"The local discontinuous Galerkin (LDG) method is studied for a third-order singularly perturbed problem of convection-diffusion type. Based on a regularity assumption for the exact solution, we prove almost <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>O</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:msup> <m:mi>N</m:mi> <m:mrow> <m:mo>-</m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mrow> <m:mi>k</m:mi> <m:mo>+</m:mo> <m:mfrac> <m:mn>1</m:mn> <m:mn>2</m:mn> </m:mfrac> </m:mrow> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> </m:msup> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_cmam-2022-0176_eq_0280.png\" /> <jats:tex-math>{O(N^{-(k+frac{1}{2})})}</jats:tex-math> </jats:alternatives> </jats:inline-formula> (up to a logarithmic factor) energy-norm convergence uniformly in the perturbation parameter. Here, <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>k</m:mi> <m:mo>≥</m:mo> <m:mn>0</m:mn> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_cmam-2022-0176_eq_0414.png\" /> <jats:tex-math>{kgeq 0}</jats:tex-math> </jats:alternatives> </jats:inline-formula> is the maximum degree of piecewise polynomials used in discrete space, and <jats:italic>N</jats:italic> is the number of mesh elements. The results are valid for the three types of layer-adapted meshes: Shishkin-type, Bakhvalov–Shishkin-type, and Bakhvalov-type. Numerical experiments are conducted to test the theoretical results.","PeriodicalId":48751,"journal":{"name":"Computational Methods in Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2022-12-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138537980","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract Based on a hierarchical basis a posteriori error estimator, an adaptive weak Galerkin finite element method (WGFEM) is proposed for the Stokes problem in two and three dimensions. In this paper, we propose two novel diagonalization techniques for velocity and pressure, respectively. Using diagonalization techniques, we need only to solve two diagonal linear algebraic systems corresponding to the degree of freedom to get the error estimator. The upper bound and lower bound of the error estimator are also shown to address the reliability of the adaptive method. Numerical simulations are provided to demonstrate the effectiveness and robustness of our algorithm.
{"title":"A Posteriori Error Estimator for Weak Galerkin Finite Element Method for Stokes Problem Using Diagonalization Techniques","authors":"Jiachuan Zhang, Ran Zhang, Jingzhi Li","doi":"10.1515/cmam-2022-0087","DOIUrl":"https://doi.org/10.1515/cmam-2022-0087","url":null,"abstract":"Abstract Based on a hierarchical basis a posteriori error estimator, an adaptive weak Galerkin finite element method (WGFEM) is proposed for the Stokes problem in two and three dimensions. In this paper, we propose two novel diagonalization techniques for velocity and pressure, respectively. Using diagonalization techniques, we need only to solve two diagonal linear algebraic systems corresponding to the degree of freedom to get the error estimator. The upper bound and lower bound of the error estimator are also shown to address the reliability of the adaptive method. Numerical simulations are provided to demonstrate the effectiveness and robustness of our algorithm.","PeriodicalId":48751,"journal":{"name":"Computational Methods in Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2022-12-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45425406","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract We present a new time-adaptive FMM for a space-time boundary element method for the heat equation. The method extends the existing parabolic FMM by adding new operations that allow for an efficient treatment of tensor product meshes which are adaptive in time. We analyze the efficiency of the new operations and the approximation quality of the related kernel expansions and present numerical experiments that reveal the benefits of the new method.
{"title":"A Time-Adaptive Space-Time FMM for the Heat Equation","authors":"R. Watschinger, G. Of","doi":"10.1515/cmam-2022-0117","DOIUrl":"https://doi.org/10.1515/cmam-2022-0117","url":null,"abstract":"Abstract We present a new time-adaptive FMM for a space-time boundary element method for the heat equation. The method extends the existing parabolic FMM by adding new operations that allow for an efficient treatment of tensor product meshes which are adaptive in time. We analyze the efficiency of the new operations and the approximation quality of the related kernel expansions and present numerical experiments that reveal the benefits of the new method.","PeriodicalId":48751,"journal":{"name":"Computational Methods in Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2022-12-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48073717","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-12-06DOI: 10.48550/arXiv.2212.02954
Marius Paul Bruchhäuser, M. Bause
Abstract In this work, a cost-efficient space-time adaptive algorithm based on the Dual Weighted Residual (DWR) method is developed and studied for a coupled model problem of flow and convection-dominated transport. Key ingredients are a multirate approach adapted to varying dynamics in time of the subproblems, weighted and non-weighted error indicators for the transport and flow problem, respectively, and the concept of space-time slabs based on tensor product spaces for the data structure. In numerical examples, the performance of the underlying algorithm is studied for benchmark problems and applications of practical interest. Moreover, the interaction of stabilization and goal-oriented adaptivity is investigated for strongly convection-dominated transport.
{"title":"A Cost-Efficient Space-Time Adaptive Algorithm for Coupled Flow and Transport","authors":"Marius Paul Bruchhäuser, M. Bause","doi":"10.48550/arXiv.2212.02954","DOIUrl":"https://doi.org/10.48550/arXiv.2212.02954","url":null,"abstract":"Abstract In this work, a cost-efficient space-time adaptive algorithm based on the Dual Weighted Residual (DWR) method is developed and studied for a coupled model problem of flow and convection-dominated transport. Key ingredients are a multirate approach adapted to varying dynamics in time of the subproblems, weighted and non-weighted error indicators for the transport and flow problem, respectively, and the concept of space-time slabs based on tensor product spaces for the data structure. In numerical examples, the performance of the underlying algorithm is studied for benchmark problems and applications of practical interest. Moreover, the interaction of stabilization and goal-oriented adaptivity is investigated for strongly convection-dominated transport.","PeriodicalId":48751,"journal":{"name":"Computational Methods in Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2022-12-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42984459","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-11-30DOI: 10.48550/arXiv.2211.17011
D. Breit, Alan Dodgson
Abstract We consider the 3D stochastic Navier–Stokes equation on the torus. Our main result concerns the temporal and spatio-temporal discretisation of a local strong pathwise solution. We prove optimal convergence rates for the energy error with respect to convergence in probability, that is convergence of order (up to) 1 in space and of order (up to) 1/2 in time. The result holds up to the possible blow-up of the (time-discrete) solution. Our approach is based on discrete stopping times for the (time-discrete) solution.
{"title":"Space-Time Approximation of Local Strong Solutions to the 3D Stochastic Navier–Stokes Equations","authors":"D. Breit, Alan Dodgson","doi":"10.48550/arXiv.2211.17011","DOIUrl":"https://doi.org/10.48550/arXiv.2211.17011","url":null,"abstract":"Abstract We consider the 3D stochastic Navier–Stokes equation on the torus. Our main result concerns the temporal and spatio-temporal discretisation of a local strong pathwise solution. We prove optimal convergence rates for the energy error with respect to convergence in probability, that is convergence of order (up to) 1 in space and of order (up to) 1/2 in time. The result holds up to the possible blow-up of the (time-discrete) solution. Our approach is based on discrete stopping times for the (time-discrete) solution.","PeriodicalId":48751,"journal":{"name":"Computational Methods in Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2022-11-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47859877","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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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