Pub Date : 2024-05-01DOI: 10.4208/eajam.2024-006.060424
Che-Chia Chang,Chen-Yang Dai,Wei-Fan Hu,Te-Sheng Lin, Ming-Chih Lai
In this paper, we present a hybrid neural-network and MAC (Marker-And-Cell) scheme for solving Stokes equations with singular forces on an embedded interface in regular domains. As known, the solution variables (the pressure and velocity) exhibit non-smooth behaviors across the interface so extra discretization efforts must be paid near the interface in order to have small order of local truncation errors in finite difference schemes. The present hybrid approach avoids such additional difficulty. It combines the expressive power of neural networks with the convergence of finite difference schemes to ease the code implementation and to achieve good accuracy at the same time. The key idea is to decompose the solution into singular and regular parts. The neural network learning machinery incorporating the given jump conditions finds the singular part solution, while the standard MAC scheme is used to obtain the regular part solution with associated boundary conditions. The two- and three-dimensional numerical results show that the present hybrid method converges with second-order accuracy for the velocity and first-order accuracy for the pressure, and it is comparable with the traditional immersed interface method in literature.
本文提出了一种混合神经网络和 MAC（标记和单元）方案，用于求解规则域中嵌入界面上具有奇异力的斯托克斯方程。众所周知，求解变量（压力和速度）在整个界面上表现出非光滑行为，因此必须在界面附近付出额外的离散化努力，以便在有限差分方案中获得小阶的局部截断误差。本混合方法避免了这种额外的困难。它将神经网络的表现力与有限差分方案的收敛性结合起来，既简化了代码执行，又达到了良好的精度。其关键思路是将解分解为奇异和规则部分。神经网络学习机制结合给定的跃迁条件找到奇异部分解，而标准 MAC 方案则用于获得带有相关边界条件的规则部分解。二维和三维数值结果表明，本混合方法的速度收敛精度为二阶，压力收敛精度为一阶，与文献中的传统沉浸界面方法不相上下。
{"title":"A Hybrid Neural-Network and MAC Scheme for Stokes Interface Problems","authors":"Che-Chia Chang,Chen-Yang Dai,Wei-Fan Hu,Te-Sheng Lin, Ming-Chih Lai","doi":"10.4208/eajam.2024-006.060424","DOIUrl":"https://doi.org/10.4208/eajam.2024-006.060424","url":null,"abstract":"In this paper, we present a hybrid neural-network and MAC (Marker-And-Cell) scheme for solving Stokes equations with singular forces on an embedded interface in regular domains. As known, the solution variables (the pressure and velocity)\u0000exhibit non-smooth behaviors across the interface so extra discretization efforts must be\u0000paid near the interface in order to have small order of local truncation errors in finite\u0000difference schemes. The present hybrid approach avoids such additional difficulty. It\u0000combines the expressive power of neural networks with the convergence of finite difference schemes to ease the code implementation and to achieve good accuracy at the same\u0000time. The key idea is to decompose the solution into singular and regular parts. The\u0000neural network learning machinery incorporating the given jump conditions finds the\u0000singular part solution, while the standard MAC scheme is used to obtain the regular part\u0000solution with associated boundary conditions. The two- and three-dimensional numerical results show that the present hybrid method converges with second-order accuracy\u0000for the velocity and first-order accuracy for the pressure, and it is comparable with the\u0000traditional immersed interface method in literature.","PeriodicalId":48932,"journal":{"name":"East Asian Journal on Applied Mathematics","volume":"36 1","pages":""},"PeriodicalIF":1.2,"publicationDate":"2024-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141256799","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 : 2024-05-01DOI: 10.4208/eajam.2023-322.250224
Zheyue Fang, Xiaoping Wang
We present a moving mesh finite element method to study the finite-time blowup solution of the Landau-Lifshitz-Gilbert (LLG) equation, considering both the heat flow of harmonic map and the full LLG equation. Our approach combines projection methods for solving the LLG equation with an iterative grid redistribution method to generate adaptive meshes. Through iterative remeshing, we successfully simulate blowup solutions with maximum gradient magnitudes up to $10^4$ and minimum mesh sizes of $10^{−5}.$ We investigate the self-similar patterns and blowup rates of these solutions, and validate our numerical findings by comparing them to established analytical results from a recent study
{"title":"An Adaptive Moving Mesh Method for Simulating Finite-time Blowup Solutions of the Landau-Lifshitz-Gilbert Equation","authors":"Zheyue Fang, Xiaoping Wang","doi":"10.4208/eajam.2023-322.250224","DOIUrl":"https://doi.org/10.4208/eajam.2023-322.250224","url":null,"abstract":"We present a moving mesh finite element method to study the finite-time\u0000blowup solution of the Landau-Lifshitz-Gilbert (LLG) equation, considering both the\u0000heat flow of harmonic map and the full LLG equation. Our approach combines projection methods for solving the LLG equation with an iterative grid redistribution method\u0000to generate adaptive meshes. Through iterative remeshing, we successfully simulate\u0000blowup solutions with maximum gradient magnitudes up to $10^4$ and minimum mesh\u0000sizes of $10^{−5}.$ We investigate the self-similar patterns and blowup rates of these solutions, and validate our numerical findings by comparing them to established analytical\u0000results from a recent study","PeriodicalId":48932,"journal":{"name":"East Asian Journal on Applied Mathematics","volume":"27 1","pages":""},"PeriodicalIF":1.2,"publicationDate":"2024-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141256954","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 : 2024-05-01DOI: 10.4208/eajam.2023-325.070124
Xiang Li,Yulei Liao, Pingbing Ming
We propose a deep learning based method for simulating the large bending deformation of bilayer plates. Inspired by the greedy algorithm, we propose a pretraining method on a series of nested domains, which accelerate the convergence of training and find the absolute minimizer more effectively. The proposed method exhibits the capability to converge to an absolute minimizer, overcoming the limitation of gradient flow methods getting trapped in the local minimizer basins. We showcase better performance with fewer numbers of degrees of freedom for the relative energy errors and relative $L^2$-errors of the minimizer through numerical experiments. Furthermore, our method successfully maintains the $L^2$-norm of the isometric constraint, leading to an improvement of accuracy.
{"title":"A Pre-Training Deep Learning Method for Simulating the Large Bending Deformation of Bilayer Plates","authors":"Xiang Li,Yulei Liao, Pingbing Ming","doi":"10.4208/eajam.2023-325.070124","DOIUrl":"https://doi.org/10.4208/eajam.2023-325.070124","url":null,"abstract":"We propose a deep learning based method for simulating the large bending deformation of bilayer plates. Inspired by the greedy algorithm, we propose a pretraining method on a series of nested domains, which accelerate the convergence of\u0000training and find the absolute minimizer more effectively. The proposed method exhibits the capability to converge to an absolute minimizer, overcoming the limitation of\u0000gradient flow methods getting trapped in the local minimizer basins. We showcase better performance with fewer numbers of degrees of freedom for the relative energy errors\u0000and relative $L^2$-errors of the minimizer through numerical experiments. Furthermore,\u0000our method successfully maintains the $L^2$-norm of the isometric constraint, leading to\u0000an improvement of accuracy.","PeriodicalId":48932,"journal":{"name":"East Asian Journal on Applied Mathematics","volume":"70 1","pages":""},"PeriodicalIF":1.2,"publicationDate":"2024-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141256938","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 : 2024-04-01DOI: 10.4208/eajam.2022-318.170723
Chi Young Ahn,Seongje Chae,Sangwoo Kang,Kwang-Jae Lee,Won-Kwang Park, Seong-Ho Son
In this paper, the application of the orthogonality sampling method (OSM) to the real-world microwave imaging for identifying location of small anomalies is addressed. In order to show the feasibility and limitation of the OSM, we theoretically prove that the indicator function can be represented in terms of an infinite series of Bessel functions of integer order and the transmitting and receiving signal antenna configurations. This is based on the application of the Born approximation and the reciprocity of the incident fields. Throughout real-data experiments, it was shown that the OSM works well for identifying single anomaly under the specific location of transmitter while further improvement is needed for identification of multiple anomalies. To improve the imaging performance, we consider traditional indicator function with multiple sources and design a new indicator function with multiple sources weighted by the incident field. Theoretical results are contained to demonstrate the improvement.
{"title":"Orthogonality Sampling Method for Identifying Small Anomalies in Real-World Microwave Imaging","authors":"Chi Young Ahn,Seongje Chae,Sangwoo Kang,Kwang-Jae Lee,Won-Kwang Park, Seong-Ho Son","doi":"10.4208/eajam.2022-318.170723","DOIUrl":"https://doi.org/10.4208/eajam.2022-318.170723","url":null,"abstract":"In this paper, the application of the orthogonality sampling method (OSM)\u0000to the real-world microwave imaging for identifying location of small anomalies is addressed. In order to show the feasibility and limitation of the OSM, we theoretically\u0000prove that the indicator function can be represented in terms of an infinite series of\u0000Bessel functions of integer order and the transmitting and receiving signal antenna configurations. This is based on the application of the Born approximation and the reciprocity of the incident fields. Throughout real-data experiments, it was shown that the\u0000OSM works well for identifying single anomaly under the specific location of transmitter while further improvement is needed for identification of multiple anomalies. To\u0000improve the imaging performance, we consider traditional indicator function with multiple sources and design a new indicator function with multiple sources weighted by the\u0000incident field. Theoretical results are contained to demonstrate the improvement.","PeriodicalId":48932,"journal":{"name":"East Asian Journal on Applied Mathematics","volume":"34 1","pages":""},"PeriodicalIF":1.2,"publicationDate":"2024-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140564664","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 : 2024-04-01DOI: 10.4208/eajam.2022-310.300623
Wen-Xiu Ma
We propose a kind of reduced Ablowitz-Kaup-Newell-Segur matrix spectral problems under two local group reductions by similarity transformations. Associated integrable hierarchies of matrix mKdV type integrable models are presented, which amend the complex matrix mKdV integrable hierarchies. Zero curvature equations are key objects in generating integrable models.
{"title":"Real Reduced Matrix mKdV Integrable Hierarchies Under Two Local Group Reductions","authors":"Wen-Xiu Ma","doi":"10.4208/eajam.2022-310.300623","DOIUrl":"https://doi.org/10.4208/eajam.2022-310.300623","url":null,"abstract":"We propose a kind of reduced Ablowitz-Kaup-Newell-Segur matrix spectral\u0000problems under two local group reductions by similarity transformations. Associated integrable hierarchies of matrix mKdV type integrable models are presented, which amend\u0000the complex matrix mKdV integrable hierarchies. Zero curvature equations are key objects in generating integrable models.","PeriodicalId":48932,"journal":{"name":"East Asian Journal on Applied Mathematics","volume":"102 1","pages":""},"PeriodicalIF":1.2,"publicationDate":"2024-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140564662","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 : 2024-04-01DOI: 10.4208/eajam.2022-184.030823
Huimin Miao, Xue Luo
The multivariate feedback particle filter (FPF) is formulated from the viewpoint of splitting-up methods. The essential difference between this formulation and the formal derivation is that instead of one-time control at a discrete time instant, we consider the updating stage as a stochastic flow of particles in each time interval. This allows to easily obtain a consistent stochastic flow by comparing the Kolmogorov forward equation of particles and the updating part of the Kushner’s equation in the splitting-up method. Moreover, if an optimal stochastic flow exists, the convergence of the splitting-up method can be studied by passing to an FPF with a continuous time. To guarantee the existence of a stochastic flow, we validate the Poincaré inequality for the alternating distributions, given the time discretization and the observation path, under mild conditions on the nonlinear filtering system and the initial state. Besides, re-examining the original derivation of the FPF, we show that the optimal transport map between the prior and the posterior is an $f$-divergence invariant in the abstract Bayesian inference framework.
{"title":"Multivariate Feedback Particle Filter Rederived from the Splitting-Up Scheme","authors":"Huimin Miao, Xue Luo","doi":"10.4208/eajam.2022-184.030823","DOIUrl":"https://doi.org/10.4208/eajam.2022-184.030823","url":null,"abstract":"The multivariate feedback particle filter (FPF) is formulated from the viewpoint of splitting-up methods. The essential difference between this formulation and\u0000the formal derivation is that instead of one-time control at a discrete time instant, we\u0000consider the updating stage as a stochastic flow of particles in each time interval. This allows to easily obtain a consistent stochastic flow by comparing the Kolmogorov forward\u0000equation of particles and the updating part of the Kushner’s equation in the splitting-up\u0000method. Moreover, if an optimal stochastic flow exists, the convergence of the splitting-up method can be studied by passing to an FPF with a continuous time. To guarantee the\u0000existence of a stochastic flow, we validate the Poincaré inequality for the alternating distributions, given the time discretization and the observation path, under mild conditions\u0000on the nonlinear filtering system and the initial state. Besides, re-examining the original\u0000derivation of the FPF, we show that the optimal transport map between the prior and\u0000the posterior is an $f$-divergence invariant in the abstract Bayesian inference framework.","PeriodicalId":48932,"journal":{"name":"East Asian Journal on Applied Mathematics","volume":"44 1","pages":""},"PeriodicalIF":1.2,"publicationDate":"2024-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140564659","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 : 2024-04-01DOI: 10.4208/eajam.2023-019.100823
Wenlong Zeng, Jianzhou Liu
We introduce a new subclass of H-matrices called partitioned Dashnic-Zusmanovich type (DZT) matrices and present the corresponding scaling matrices for this kind of matrices. There are three major applications. The first application is to provide equivalent eigenvalue localization related to index partition by using the nonsingularity of the new subclass. By taking some specific partitions, we provide other forms of eigenvalue localization sets that generalize and improve some well-known eigenvalue localization sets. The second application is to obtain an upper bound on the infinite norm of the inverse of partitioned DZT matrices using scaling matrices. The third application is to give an error bound of the linear complementarity problems (LCPs) by using scaling matrices. Additionally, we give another upper bound of the infinite norm and error bound of the LCPs by a reduction method, which transforms the given partitioned DZT matrix into the corresponding DZT matrix by partition and summation. The results obtained by the reduction method are generalizations of some known conclusions.
我们引入了一种新的 H 矩阵子类，称为分区达什尼奇-祖斯玛诺维奇型（DZT）矩阵，并提出了这类矩阵的相应缩放矩阵。主要应用有三个。第一个应用是利用新子类的非奇异性，提供与索引分区相关的等效特征值定位。通过一些特定的分区，我们提供了其他形式的特征值定位集，这些特征值定位集概括并改进了一些著名的特征值定位集。第二个应用是利用缩放矩阵获得分区 DZT 矩阵逆的无穷级数上限。第三个应用是利用缩放矩阵给出线性互补问题（LCP）的误差约束。此外，我们还通过还原法给出了线性互补问题的无限规范和误差约束的另一个上限。还原法通过分治和求和将给定的分治 DZT 矩阵转化为相应的 DZT 矩阵。还原法得到的结果是对一些已知结论的概括。
{"title":"Partitioned Dashnic-Zusmanovich Type Matric with Applications","authors":"Wenlong Zeng, Jianzhou Liu","doi":"10.4208/eajam.2023-019.100823","DOIUrl":"https://doi.org/10.4208/eajam.2023-019.100823","url":null,"abstract":"We introduce a new subclass of H-matrices called partitioned Dashnic-Zusmanovich type (DZT) matrices and present the corresponding scaling matrices for this\u0000kind of matrices. There are three major applications. The first application is to provide\u0000equivalent eigenvalue localization related to index partition by using the nonsingularity of the new subclass. By taking some specific partitions, we provide other forms of\u0000eigenvalue localization sets that generalize and improve some well-known eigenvalue\u0000localization sets. The second application is to obtain an upper bound on the infinite\u0000norm of the inverse of partitioned DZT matrices using scaling matrices. The third application is to give an error bound of the linear complementarity problems (LCPs) by using\u0000scaling matrices. Additionally, we give another upper bound of the infinite norm and\u0000error bound of the LCPs by a reduction method, which transforms the given partitioned\u0000DZT matrix into the corresponding DZT matrix by partition and summation. The results\u0000obtained by the reduction method are generalizations of some known conclusions.","PeriodicalId":48932,"journal":{"name":"East Asian Journal on Applied Mathematics","volume":"2013 1","pages":""},"PeriodicalIF":1.2,"publicationDate":"2024-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140564446","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 : 2024-04-01DOI: 10.4208/eajam.2023-045.090523
Eleuterio F. Toro,Annunziato Siviglia,Alessandra Spilimbergo, Lucas O. Müller
We present a class of simple advection-pressure splitting numerical methods to solve the blood flow equations in compliant arterial vessels. The schemes are inspired by the TV flux vector splitting approach for conservative systems, proposed by Toro and Vázquez [30]. But the reformulated TV-type splitting schemes of this paper have a wider range of applicability, including systems of equations in non-conservative form. The spatial differential operator is split into advection terms, which may be in conservative form, from pressure terms in conservative or non-conservative form. Additionally, unlike the original TV scheme, the reformulated splitting of this paper fully preserves the continuity equation as part of the pressure system. This last feature is consistent with zero-dimensional models for blood flow that are based on neglecting the inertial term in the momentum equation. The schemes are also well suited for systems in which geometric and biomechanical parameters of the problem vary discontinuously. The splitting schemes of this paper are systematically assessed on a carefully designed suite of test problems and compared with several existing, mainstream methods. Overall, the proposed numerical methods perform very satisfactorily and suggest themselves as attractive computational tools for modelling the dynamics of bodily fluids under realistic conditions.
我们提出了一类简单的平流-压力分裂数值方法来求解顺应性动脉血管中的血流方程。这些方案受到 Toro 和 Vázquez [30] 提出的保守系统 TV 通量矢量分裂方法的启发。但本文重新表述的 TV 型拆分方案具有更广泛的适用性，包括非保守形式的方程系统。空间微分算子被拆分为保守形式的平流项和保守或非保守形式的压力项。此外，与最初的 TV 方案不同，本文重新制定的拆分方案完全保留了作为压力系统一部分的连续性方程。最后一个特点与基于忽略动量方程中惯性项的零维血流模型是一致的。这些方案也非常适合问题的几何和生物力学参数变化不连续的系统。本文的拆分方案在一套精心设计的测试问题上进行了系统评估，并与几种现有的主流方法进行了比较。总体而言，所提出的数值方法性能非常令人满意，表明它们是在现实条件下模拟体液动力学的有吸引力的计算工具。
{"title":"Advection-Pressure Splitting Schemes for the Equations of Blood Flow Conservative and Non-Conservative Forms","authors":"Eleuterio F. Toro,Annunziato Siviglia,Alessandra Spilimbergo, Lucas O. Müller","doi":"10.4208/eajam.2023-045.090523","DOIUrl":"https://doi.org/10.4208/eajam.2023-045.090523","url":null,"abstract":"We present a class of simple advection-pressure splitting numerical methods\u0000to solve the blood flow equations in compliant arterial vessels. The schemes are inspired\u0000by the TV flux vector splitting approach for conservative systems, proposed by Toro and\u0000Vázquez [30]. But the reformulated TV-type splitting schemes of this paper have a wider\u0000range of applicability, including systems of equations in non-conservative form. The\u0000spatial differential operator is split into advection terms, which may be in conservative form, from pressure terms in conservative or non-conservative form. Additionally,\u0000unlike the original TV scheme, the reformulated splitting of this paper fully preserves\u0000the continuity equation as part of the pressure system. This last feature is consistent\u0000with zero-dimensional models for blood flow that are based on neglecting the inertial\u0000term in the momentum equation. The schemes are also well suited for systems in which\u0000geometric and biomechanical parameters of the problem vary discontinuously. The splitting schemes of this paper are systematically assessed on a carefully designed suite of\u0000test problems and compared with several existing, mainstream methods. Overall, the\u0000proposed numerical methods perform very satisfactorily and suggest themselves as attractive computational tools for modelling the dynamics of bodily fluids under realistic\u0000conditions.","PeriodicalId":48932,"journal":{"name":"East Asian Journal on Applied Mathematics","volume":"27 1","pages":""},"PeriodicalIF":1.2,"publicationDate":"2024-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140564663","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 : 2024-04-01DOI: 10.4208/eajam.2022-356.180923
Wenjia Xie, Zhongyi Huang
From the Black-Scholes option pricing model, this work evaluates the evolution of the mathematical modelling into the double stochastic volatility model that studies the optimization performance in partial differential equation (PDE) methods. This paper focuses on the calibration and numerical methodology processes to derive the comparison of the Heston and the double Heston models to design a more efficient numerical iterative splitting method. Through Li and Huang’s iterative splitting method, the numerical results conclude that the mixed method reduces the overall computational cost and improves the convergence of the iterative process while maintaining the simplicity, flexibility and interpretability of PDE methods.
{"title":"On Pricing Options Under Two Stochastic Volatility Processes","authors":"Wenjia Xie, Zhongyi Huang","doi":"10.4208/eajam.2022-356.180923","DOIUrl":"https://doi.org/10.4208/eajam.2022-356.180923","url":null,"abstract":"From the Black-Scholes option pricing model, this work evaluates the evolution of the mathematical modelling into the double stochastic volatility model that\u0000studies the optimization performance in partial differential equation (PDE) methods.\u0000This paper focuses on the calibration and numerical methodology processes to derive\u0000the comparison of the Heston and the double Heston models to design a more efficient\u0000numerical iterative splitting method. Through Li and Huang’s iterative splitting method,\u0000the numerical results conclude that the mixed method reduces the overall computational\u0000cost and improves the convergence of the iterative process while maintaining the simplicity, flexibility and interpretability of PDE methods.","PeriodicalId":48932,"journal":{"name":"East Asian Journal on Applied Mathematics","volume":"33 1","pages":""},"PeriodicalIF":1.2,"publicationDate":"2024-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140603242","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 : 2024-01-01DOI: 10.4208/eajam.2022-304.070323
Fu Li, Yingxiang Xu
In this paper, we extend a diagonalization-based parallel-in-time (PinT) algorithm to the viscoelastic equation. The central difference method is used for spatial discretization, while for temporal discretization, we use the Crank-Nicolson scheme. Then an all-at-once system collecting all the solutions at each time level is formed and solved using a fixed point iteration preconditioned by an $α$-circulant matrix in parallel. Via a rigorous analysis, we find that the spectral radius of the iteration matrix is uniformly bounded by $α/(1 − α),$ independent of the model parameters (the damping coefficient $varepsilon$ and the wave velocity $sqrt{gamma}$) and the discretization parameters (the time step $tau$ and the spatial mesh size $h$). Unlike the classical wave equation with Dirichlet boundary condition where the upper bound $α/(1 − α)$ is very sharp, we find that the occurrence of the damping term $−varepsilon∆y_t,$ as well as the large final time $T,$ leads to even faster convergence of the algorithm, especially when $α$ is not very small. We illustrate our theoretical findings with several numerical examples.
{"title":"A Diagonalization-Based Parallel-in-Time Algorithm for Crank-Nicolson’s Discretization of the Viscoelastic Equation","authors":"Fu Li, Yingxiang Xu","doi":"10.4208/eajam.2022-304.070323","DOIUrl":"https://doi.org/10.4208/eajam.2022-304.070323","url":null,"abstract":"In this paper, we extend a diagonalization-based parallel-in-time (PinT) algorithm to the viscoelastic equation. The central difference method is used for spatial discretization, while for temporal discretization, we use the Crank-Nicolson scheme. Then an all-at-once system collecting all the solutions at each time level is formed and solved using a fixed point iteration preconditioned by an $α$-circulant matrix in parallel. Via a rigorous analysis, we find that the spectral radius of the iteration matrix is uniformly bounded by $α/(1 − α),$ independent of the model parameters (the damping coefficient $varepsilon$ and the wave velocity $sqrt{gamma}$) and the discretization parameters (the time step $tau$ and the spatial mesh size $h$). Unlike the classical wave equation with Dirichlet boundary condition where the upper bound $α/(1 − α)$ is very sharp, we find that the occurrence of the damping term $−varepsilon∆y_t,$ as well as the large final time $T,$ leads to even faster convergence of the algorithm, especially when $α$ is not very small. We illustrate our theoretical findings with several numerical examples.","PeriodicalId":48932,"journal":{"name":"East Asian Journal on Applied Mathematics","volume":"215 1","pages":""},"PeriodicalIF":1.2,"publicationDate":"2024-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139082096","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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