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A direct solver with O（N） complexity for integral equations on one-dimensional domains

Frontiers of Mathematics in China 2012年第2期7卷 217-247页

作者： Adrianna GILLMAN Patrick M. YOUNG Per-Gunnar MARTINSSON Department of Applied Mathematics University of Colorado at Boulder BoulderCO 80309-0526 USA

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.

关键词： direct solver integral equation fast direct solver boundary value problem boundary integral equation hierarchically semi-separable matrix MSC 65R20,65F05

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An Efficient Neural-Network and Finite-Difference Hybrid Method for Elliptic Interface Problems with Applications

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Communications in Computational Physics 2023年第4期33卷 1090-1105页

作者： Wei-Fan Hu Te-Sheng Lin Yu-Hau Tseng Ming-Chih Lai Department of Mathematics National Central UniversityTaoyuan 32001Taiwan Department of Applied Mathematics National Yang Ming Chiao Tung UniversityHsinchu 30010Taiwan Department of Applied Mathematics National University of KaohsiungKaohsiung 81148Taiwan National Center for Theoretical Sciences National Taiwan UniversityTaipei 10617Taiwan

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 *** 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 ***,we aim to elevate such an extra effort to ease our implementation by machine learning *** key idea is to decompose the solution into singular and regular *** 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 *** 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 *** 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 *** an application,we solve the Stokes equations with singular forces to demonstrate the robustness of the present method.

关键词： Neural networks sharp interface method fast direct solver elliptic interface problem Stokes equations

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