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dc.description.abstractWe review preconditioning methods for solving the Poisson equation. Many numerical methods for the Poisson equation induce an ill-conditioned system whose condition number has order $O(h^{-2})$ where $h$ is the grid size. In order to enable practical computation within reasonable time, an efficient preconditioner is necessary. MILU preconditioner has been known to be the most efficient preconditioner for Poisson equation with Dirichlet boundary condition among all ILU-type preconditioners since it only reduces the order of condition number from $O(h^{-2})$ to $O(h^{-1})$. MILU preconditioner, however, cannot be defined for Neumann boundary condition. Park et al. \cite{MILU_ILU} proposed a new efficient preconditioner by mixing MILU and ILU in appropriate proportion applied with a standard finite volume method for solving the Neumann problems. In this thesis, we review preconditioners for each Dirichlet and Neumann problems focusing on incomplete factorization preconditioners. Also, we provide numerical experiments for the comparison of preconditioners for Neumann problems. Mixed preconditioning with the suggested optimal ratio achieves the condition number of order $O(h^{-1})$ while other standard preconditioning methods are fail to reduce the order. The mixture preconditioning is expected to be widely applied to many fields such as fluid dynamics and image processing.;이 논문은 포아송 방정식을 푸는데 사용되는 선조건(Preconditioning)법을 검증하는 데에 그 목적을 둔다. 선조건법은 포아송 방정식을 수치적으로 푸는데 매우 중요한 기법이다. 포아송 방정식을 근사하는 많은 수치적 방법들이 조건수(condition number)가 O(ℎ−2)인 불안정한 선형 계를 유도한다. 여기서 ℎ 는 격자의 크기이다. ILU 선조건법 중 MILU 선조건자가 디리클레 경계조건문제에 대해서 유일하게 조건수를 O(ℎ−1)으로 줄여 가장 최적의 방법으로 알려져 있다. 하지만 노이만 경계조건이 주어진 경우에는 MILU 선조건자가 정의되지 않는다. 본 논문에서는 디리클레 경계조건에서 MILU 선조건법의 최적성에 대해 알아보고, 노이만 경계조건에 대해 최근 제시된 ILU와 MILU 선조건자를 적절한 비율로 섞은 혼합 MILU방법에 대해 검토한다. 마지막으로 몇 가지 수치적 실험을 통해 본 논문에서 논의된 선조건자들의 성능을 확인한다.-
dc.description.tableofcontents1 Introduction. 1 2 Analysis for basic preconditioners 7 2.1 Spl i t t ing pr e condi t ione r s 9 2.1.1 Jacobi preconditioner 10 2.1.2 SGS preconditioner 11 2.2 Incomplete factorization preconditioners 12 2.2.1 ILU precondi t ioner 12 2.2.2 MILU preconditioner 15 2.3 Preconditioned Conjugate Gradient Method 16 3 Preconditioners for Dirichlet Problems 19 3.1 Analyses on banded preconditioners 19 3.2 Optimality of MILU preconditioner 21 3.2.1 Gustafssons Perturbed MILU 24 3.2.2 Classical MILU 28 4 Preconditioners for Neumann Problems 31 4.1 Discretization 31 4.2 ILU Preconditioner 34 4.3 MILU Preconditioner 36 4.4 Mixed MILU-ILU Preconditioner 37 5 Numerical Experiments 40 5.1 Performance comparison on unit disk 40 5.2 Tests on various domains 42 5.3 Convergence test of PCG 45 6 Conclusion 48 Reference 50 국문초록 55-
dc.format.extent1628510 bytes-
dc.publisher이화여자대학교 대학원-
dc.titlePreconditioning methods for the Poisson equation-
dc.typeMaster's Thesis-
dc.title.translated포아송 방정식에 적용되는 선조건법-
dc.creator.othernameYesom Park-
dc.format.pageⅩⅠ, 55 p,-
dc.identifier.major대학원 수학과- 2-
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