{"title":"A Parallel Iterative Probabilistic Method for Mixed Problems of Laplace Equations with the Feynman-Kac Formula of Killed Brownian Motions","authors":"Cuiyang Ding, Changhao Yan, Xuan Zeng, W. Cai","doi":"10.1137/22m1478458","DOIUrl":"https://doi.org/10.1137/22m1478458","url":null,"abstract":"","PeriodicalId":21812,"journal":{"name":"SIAM J. Sci. Comput.","volume":"15 1","pages":"3413-"},"PeriodicalIF":0.0,"publicationDate":"2022-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87378521","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 : 2022-09-24DOI: 10.48550/arXiv.2209.11922
Jianguo Huang, L. Ju, Y. Xu
In this paper, we propose an efficient exponential integrator finite element method for solving a class of semilinear parabolic equations in rectangular domains. The proposed method first performs the spatial discretization of the model equation using the finite element approximation with continuous multilinear rectangular basis functions, and then takes the explicit exponential Runge-Kutta approach for time integration of the resulting semi-discrete system to produce fully-discrete numerical solution. Under certain regularity assumptions, error estimates measured in $H^1$-norm are successfully derived for the proposed schemes with one and two RK stages. More remarkably, the mass and coefficient matrices of the proposed method can be simultaneously diagonalized with an orthogonal matrix, which provides a fast solution process based on tensor product spectral decomposition and fast Fourier transform. Various numerical experiments in two and three dimensions are also carried out to validate the theoretical results and demonstrate the excellent performance of the proposed method.
{"title":"Efficient Exponential Integrator Finite Element Method for Semilinear Parabolic Equations","authors":"Jianguo Huang, L. Ju, Y. Xu","doi":"10.48550/arXiv.2209.11922","DOIUrl":"https://doi.org/10.48550/arXiv.2209.11922","url":null,"abstract":"In this paper, we propose an efficient exponential integrator finite element method for solving a class of semilinear parabolic equations in rectangular domains. The proposed method first performs the spatial discretization of the model equation using the finite element approximation with continuous multilinear rectangular basis functions, and then takes the explicit exponential Runge-Kutta approach for time integration of the resulting semi-discrete system to produce fully-discrete numerical solution. Under certain regularity assumptions, error estimates measured in $H^1$-norm are successfully derived for the proposed schemes with one and two RK stages. More remarkably, the mass and coefficient matrices of the proposed method can be simultaneously diagonalized with an orthogonal matrix, which provides a fast solution process based on tensor product spectral decomposition and fast Fourier transform. Various numerical experiments in two and three dimensions are also carried out to validate the theoretical results and demonstrate the excellent performance of the proposed method.","PeriodicalId":21812,"journal":{"name":"SIAM J. Sci. Comput.","volume":"104 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-09-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79239240","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}
. For dense symmetric matrices with small off-diagonal (numerical) ranks and in a 5 hierarchically semiseparable form, we give a divide-and-conquer eigendecomposition method with 6 nearly linear complexity (called SuperDC) that significantly improves an earlier basic algorithm in 7 [Vogel, Xia, et al., SIAM J. Sci. Comput., 38 (2016)]. Some stability risks in the original algorithm are 8 analyzed, including potential exponential norm growth, cancellations, loss of accuracy with clustered 9 eigenvalues or intermediate eigenvalues, etc. In the dividing stage, we give a new structured low-rank 10 updating strategy with balancing that eliminates the exponential norm growth and also minimizes 11 the ranks of low-rank updates. In the conquering stage with low-rank updated eigenvalue solution, 12 the original algorithm directly uses the standard fast multipole method (FMM) to accelerate function 13 evaluations, which has the risks of cancellation, division by zero, and slow convergence. Here, we 14 design a triangular FMM to avoid cancellation. Furthermore, when there are clustered intermediate 15 eigenvalues, we design a novel local shifting strategy to integrate FMM accelerations into the solution 16 of shifted secular equations. This helps achieve both the efficiency and the reliability. We also provide 17 a deflation strategy with a user-supplied tolerance and give a precise description of the structure of 18 the resulting eigenvector matrix. The SuperDC eigensolver has significantly improved stability while 19 keeping the nearly linear complexity for finding the entire eigenvalue decomposition. Extensive 20 numerical tests are used to show the efficiency and accuracy of SuperDC.
． 对于具有小非对角线(数值)秩和5层次半可分形式的密集对称矩阵，我们给出了具有6近线性复杂度的分治特征分解方法(称为SuperDC)，该方法显着改进了先前的基本算法[Vogel, Xia, et al.， SIAM J. Sci]。第一版。， 38(2016)]。分析了原算法存在的稳定性风险，包括指数范数增长、消去、聚类特征值或中间特征值导致的精度损失等。在划分阶段，我们给出了一种新的具有平衡的结构化低秩更新策略，该策略消除了指数范数增长，并且最小化了低秩更新的秩。在低秩更新特征值解的征服阶段，原算法直接使用标准快速多极法(FMM)加速函数13的求值，存在消去、除零、收敛慢的风险。在这里，我们设计了一个三角形FMM来避免抵消。此外，当存在聚类中间特征值时，我们设计了一种新颖的局部移位策略，将FMM加速度集成到移位长期方程的解中。这有助于实现效率和可靠性。我们还提供了一种用户提供公差的压缩策略，并给出了结果特征向量矩阵结构的精确描述。SuperDC特征求解器在保持整个特征值分解的近似线性复杂度的同时，显著提高了稳定性。通过大量的20个数值试验，验证了SuperDC的效率和准确性。
{"title":"SuperDC: Superfast Divide-And-Conquer Eigenvalue Decomposition With Improved Stability for Rank-Structured Matrices","authors":"Xiaofeng Ou, J. Xia","doi":"10.1137/21m1438633","DOIUrl":"https://doi.org/10.1137/21m1438633","url":null,"abstract":". For dense symmetric matrices with small off-diagonal (numerical) ranks and in a 5 hierarchically semiseparable form, we give a divide-and-conquer eigendecomposition method with 6 nearly linear complexity (called SuperDC) that significantly improves an earlier basic algorithm in 7 [Vogel, Xia, et al., SIAM J. Sci. Comput., 38 (2016)]. Some stability risks in the original algorithm are 8 analyzed, including potential exponential norm growth, cancellations, loss of accuracy with clustered 9 eigenvalues or intermediate eigenvalues, etc. In the dividing stage, we give a new structured low-rank 10 updating strategy with balancing that eliminates the exponential norm growth and also minimizes 11 the ranks of low-rank updates. In the conquering stage with low-rank updated eigenvalue solution, 12 the original algorithm directly uses the standard fast multipole method (FMM) to accelerate function 13 evaluations, which has the risks of cancellation, division by zero, and slow convergence. Here, we 14 design a triangular FMM to avoid cancellation. Furthermore, when there are clustered intermediate 15 eigenvalues, we design a novel local shifting strategy to integrate FMM accelerations into the solution 16 of shifted secular equations. This helps achieve both the efficiency and the reliability. We also provide 17 a deflation strategy with a user-supplied tolerance and give a precise description of the structure of 18 the resulting eigenvector matrix. The SuperDC eigensolver has significantly improved stability while 19 keeping the nearly linear complexity for finding the entire eigenvalue decomposition. Extensive 20 numerical tests are used to show the efficiency and accuracy of SuperDC.","PeriodicalId":21812,"journal":{"name":"SIAM J. Sci. Comput.","volume":"63 1","pages":"3041-"},"PeriodicalIF":0.0,"publicationDate":"2022-09-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90059527","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 : 2022-09-19DOI: 10.48550/arXiv.2209.09038
Zhaoyang Wang, P. Lin, Lei Zhang
This paper is concerned with a blood flow problem coupled with a slow plaque growth at the artery wall. In the model, the micro (fast) system is the Navier-Stokes equation with a periodically applied force and the macro (slow) system is a fractional reaction equation, which is used to describe the plaque growth with memory effect. We construct an auxiliary temporal periodic problem and an effective time-average equation to approximate the original problem and analyze the approximation error of the corresponding linearized PDE (Stokes) system, where the simple front-tracking technique is used to update the slow moving boundary. An effective multiscale method is then designed based on the approximate problem and the front tracking framework. We also present a temporal finite difference scheme with a spatial continuous finite element method and analyze its temporal discrete error. Furthermore, a fast iterative procedure is designed to find the initial value of the temporal periodic problem and its convergence is analyzed as well. Our designed front-tracking framework and the iterative procedure for solving the temporal periodic problem make it easy to implement the multiscale method on existing PDE solving software. The numerical method is implemented by a combination of the finite element platform COMSOL Multiphysics and the mainstream software MATLAB, which significantly reduce the programming effort and easily handle the fluid-structure interaction, especially moving boundaries with more complex geometries. We present some numerical examples of ODEs and 2-D Navier-Stokes system to demonstrate the effectiveness of the multiscale method. Finally, we have a numerical experiment on the plaque growth problem and discuss the physical implication of the fractional order parameter.
{"title":"A fast front-tracking approach and its analysis for a temporal multiscale flow problem with a fractional-order boundary growth","authors":"Zhaoyang Wang, P. Lin, Lei Zhang","doi":"10.48550/arXiv.2209.09038","DOIUrl":"https://doi.org/10.48550/arXiv.2209.09038","url":null,"abstract":"This paper is concerned with a blood flow problem coupled with a slow plaque growth at the artery wall. In the model, the micro (fast) system is the Navier-Stokes equation with a periodically applied force and the macro (slow) system is a fractional reaction equation, which is used to describe the plaque growth with memory effect. We construct an auxiliary temporal periodic problem and an effective time-average equation to approximate the original problem and analyze the approximation error of the corresponding linearized PDE (Stokes) system, where the simple front-tracking technique is used to update the slow moving boundary. An effective multiscale method is then designed based on the approximate problem and the front tracking framework. We also present a temporal finite difference scheme with a spatial continuous finite element method and analyze its temporal discrete error. Furthermore, a fast iterative procedure is designed to find the initial value of the temporal periodic problem and its convergence is analyzed as well. Our designed front-tracking framework and the iterative procedure for solving the temporal periodic problem make it easy to implement the multiscale method on existing PDE solving software. The numerical method is implemented by a combination of the finite element platform COMSOL Multiphysics and the mainstream software MATLAB, which significantly reduce the programming effort and easily handle the fluid-structure interaction, especially moving boundaries with more complex geometries. We present some numerical examples of ODEs and 2-D Navier-Stokes system to demonstrate the effectiveness of the multiscale method. Finally, we have a numerical experiment on the plaque growth problem and discuss the physical implication of the fractional order parameter.","PeriodicalId":21812,"journal":{"name":"SIAM J. Sci. Comput.","volume":"4 1","pages":"646-"},"PeriodicalIF":0.0,"publicationDate":"2022-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84223858","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}
{"title":"Numerical Solutions of Quasilinear Parabolic Problems by a Continuous Space-Time Finite Element Scheme","authors":"I. Toulopoulos","doi":"10.1137/21m1403722","DOIUrl":"https://doi.org/10.1137/21m1403722","url":null,"abstract":"","PeriodicalId":21812,"journal":{"name":"SIAM J. Sci. Comput.","volume":"43 1","pages":"2944-"},"PeriodicalIF":0.0,"publicationDate":"2022-09-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86518622","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 : 2022-09-07DOI: 10.48550/arXiv.2209.02931
Tianhao Hu, Bangti Jin, Zhi Zhou
In this work, we develop an efficient solver based on neural networks for second-order elliptic equations with variable coefficients and singular sources. This class of problems covers general point sources, line sources and the combination of point-line sources, and has a broad range of practical applications. The proposed approach is based on decomposing the true solution into a singular part that is known analytically using the fundamental solution of the Laplace equation and a regular part that satisfies a suitable modified elliptic PDE with a smoother source, and then solving for the regular part using the deep Ritz method. A path-following strategy is suggested to select the penalty parameter for enforcing the Dirichlet boundary condition. Extensive numerical experiments in two- and multi-dimensional spaces with point sources, line sources or their combinations are presented to illustrate the efficiency of the proposed approach, and a comparative study with several existing approaches based on neural networks is also given, which shows clearly its competitiveness for the specific class of problems. In addition, we briefly discuss the error analysis of the approach.
{"title":"Solving Elliptic Problems with Singular Sources using Singularity Splitting Deep Ritz Method","authors":"Tianhao Hu, Bangti Jin, Zhi Zhou","doi":"10.48550/arXiv.2209.02931","DOIUrl":"https://doi.org/10.48550/arXiv.2209.02931","url":null,"abstract":"In this work, we develop an efficient solver based on neural networks for second-order elliptic equations with variable coefficients and singular sources. This class of problems covers general point sources, line sources and the combination of point-line sources, and has a broad range of practical applications. The proposed approach is based on decomposing the true solution into a singular part that is known analytically using the fundamental solution of the Laplace equation and a regular part that satisfies a suitable modified elliptic PDE with a smoother source, and then solving for the regular part using the deep Ritz method. A path-following strategy is suggested to select the penalty parameter for enforcing the Dirichlet boundary condition. Extensive numerical experiments in two- and multi-dimensional spaces with point sources, line sources or their combinations are presented to illustrate the efficiency of the proposed approach, and a comparative study with several existing approaches based on neural networks is also given, which shows clearly its competitiveness for the specific class of problems. In addition, we briefly discuss the error analysis of the approach.","PeriodicalId":21812,"journal":{"name":"SIAM J. Sci. Comput.","volume":"20 1","pages":"2043-"},"PeriodicalIF":0.0,"publicationDate":"2022-09-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89457152","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 : 2022-09-01DOI: 10.48550/arXiv.2209.00713
Longfei Gao
Acoustic and elastic wave equations are routinely used in geophysical and engineering studies to simulate the propagation of waves, with a broad range of applications, including seismology, near surface characterization, non-destructive structural evaluation, etc. Finite difference methods remain popular choices for these simulations due to their simplicity and efficiency. In particular, the family of finite difference methods based on the summation-by-parts operators and the simultaneous-approximation-terms technique have been proposed for these simulations, which offers great flexibility in addressing boundary and interface conditions. For the applications mentioned above, surface of the earth is usually associated with the free surface boundary condition. In this study, we demonstrate that the weakly imposed free surface boundary condition through the simultaneous-approximation-terms technique can have issue when the source terms, which introduces abrupt disturbances to the wave field, are placed too close to the surface. In response, we propose to build the free surface boundary condition into the summation-by-parts finite difference operators and hence strongly and automatically impose the free surface boundary condition to address this issue. The procedure is very simple for acoustic wave equation, requiring resetting a few rows and columns in the existing difference operators only. For the elastic wave equation, the procedure is more involved and requires special design of the grid layout and summation-by-parts operators that satisfy additional requirements, as revealed by the discrete energy analysis. In both cases, the energy conserving property is preserved. Numerical examples are presented to demonstrate the effectiveness of the proposed approach.
{"title":"Strongly imposing the free surface boundary condition for wave equations with finite difference operators","authors":"Longfei Gao","doi":"10.48550/arXiv.2209.00713","DOIUrl":"https://doi.org/10.48550/arXiv.2209.00713","url":null,"abstract":"Acoustic and elastic wave equations are routinely used in geophysical and engineering studies to simulate the propagation of waves, with a broad range of applications, including seismology, near surface characterization, non-destructive structural evaluation, etc. Finite difference methods remain popular choices for these simulations due to their simplicity and efficiency. In particular, the family of finite difference methods based on the summation-by-parts operators and the simultaneous-approximation-terms technique have been proposed for these simulations, which offers great flexibility in addressing boundary and interface conditions. For the applications mentioned above, surface of the earth is usually associated with the free surface boundary condition. In this study, we demonstrate that the weakly imposed free surface boundary condition through the simultaneous-approximation-terms technique can have issue when the source terms, which introduces abrupt disturbances to the wave field, are placed too close to the surface. In response, we propose to build the free surface boundary condition into the summation-by-parts finite difference operators and hence strongly and automatically impose the free surface boundary condition to address this issue. The procedure is very simple for acoustic wave equation, requiring resetting a few rows and columns in the existing difference operators only. For the elastic wave equation, the procedure is more involved and requires special design of the grid layout and summation-by-parts operators that satisfy additional requirements, as revealed by the discrete energy analysis. In both cases, the energy conserving property is preserved. Numerical examples are presented to demonstrate the effectiveness of the proposed approach.","PeriodicalId":21812,"journal":{"name":"SIAM J. Sci. Comput.","volume":"34 1","pages":"731-"},"PeriodicalIF":0.0,"publicationDate":"2022-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85003687","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}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：