An algorithm for the direct inversion of the linear systems arising from NystrSm discretization of integral equations on one-dimensional domains is described. The method typically has O（N） complexity when applied to...An algorithm for the direct inversion of the linear systems arising from NystrSm discretization of integral equations on one-dimensional domains is described. The method typically has O（N） complexity when applied to boundary integral equations （BIEs） in the plane with non-oscillatory kernels such as those associated with the Laplace and Stokes＇ equations. The scaling coefficient suppressed by the ＂big-O＂ notation depends logarithraically on the requested accuracy. The method can also be applied to BIEs with oscillatory kernels such as those associated with the Helmholtz and time-harmonic Maxwell equations; it is efficient at long and intermediate wave-lengths, but will eventually become prohibitively slow as the wave-length decreases. To achieve linear complexity, rank： deficiencies in the off-diagonal blocks of the coefficient matrix are exploited. The technique is conceptually related to the H- and H2-matrix arithmetic of Hackbusch and coworkers, and is closely related to previous work on Hierarchically Semi-Separable matrices.展开更多
A new and efficient neural-network and finite-difference hybrid method is developed for solving Poisson equation in a regular domain with jump discontinuities on embedded irregular interfaces.Since the solution has lo...A new and efficient neural-network and finite-difference hybrid method is developed for solving Poisson equation in a regular domain with jump discontinuities on embedded irregular interfaces.Since the solution has low regularity across the interface,when applying finite difference discretization to this problem,an additional treatment accounting for the jump discontinuities must be employed.Here,we aim to elevate such an extra effort to ease our implementation by machine learning methodology.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 solution,while the standard five-point Laplacian discretization is used to obtain the regular solution with associated boundary conditions.Regardless of the interface geometry,these two tasks only require supervised learning for function approximation and a fast direct solver for Poisson equation,making the hybrid method easy to implement and efficient.The two-and three-dimensional numerical results show that the present hybrid method preserves second-order accuracy for the solution and its derivatives,and it is comparable with the traditional immersed interface method in the literature.As an application,we solve the Stokes equations with singular forces to demonstrate the robustness of the present method.展开更多
文摘An algorithm for the direct inversion of the linear systems arising from NystrSm discretization of integral equations on one-dimensional domains is described. The method typically has O（N） complexity when applied to boundary integral equations （BIEs） in the plane with non-oscillatory kernels such as those associated with the Laplace and Stokes＇ equations. The scaling coefficient suppressed by the ＂big-O＂ notation depends logarithraically on the requested accuracy. The method can also be applied to BIEs with oscillatory kernels such as those associated with the Helmholtz and time-harmonic Maxwell equations; it is efficient at long and intermediate wave-lengths, but will eventually become prohibitively slow as the wave-length decreases. To achieve linear complexity, rank： deficiencies in the off-diagonal blocks of the coefficient matrix are exploited. The technique is conceptually related to the H- and H2-matrix arithmetic of Hackbusch and coworkers, and is closely related to previous work on Hierarchically Semi-Separable matrices.
基金the supports by National Science and Technology Council,Taiwan,under the research grants 111-2115-M-008-009-MY3,111-2628-M-A49-008-MY4,111-2115-M-390-002,and 110-2115-M-A49-011-MY3,respectivelythe supports by National Center for Theoretical Sciences,Taiwan.
文摘A new and efficient neural-network and finite-difference hybrid method is developed for solving Poisson equation in a regular domain with jump discontinuities on embedded irregular interfaces.Since the solution has low regularity across the interface,when applying finite difference discretization to this problem,an additional treatment accounting for the jump discontinuities must be employed.Here,we aim to elevate such an extra effort to ease our implementation by machine learning methodology.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 solution,while the standard five-point Laplacian discretization is used to obtain the regular solution with associated boundary conditions.Regardless of the interface geometry,these two tasks only require supervised learning for function approximation and a fast direct solver for Poisson equation,making the hybrid method easy to implement and efficient.The two-and three-dimensional numerical results show that the present hybrid method preserves second-order accuracy for the solution and its derivatives,and it is comparable with the traditional immersed interface method in the literature.As an application,we solve the Stokes equations with singular forces to demonstrate the robustness of the present method.
