22 research outputs found
An improved shifted Laplace preconditioner for solving the Helmholtz equation
ํ์๋
ผ๋ฌธ (๋ฐ์ฌ) -- ์์ธ๋ํ๊ต ๋ํ์ : ์์ฐ๊ณผํ๋ํ ํ๋๊ณผ์ ๊ณ์ฐ๊ณผํ์ ๊ณต, 2021. 2. ์ ์ฐฝ์.ํฌ๋ฆํ์ธ ๋ฐฉ์ ์์ ์ฃผํ์ ์ฑ๋ถ์ ํ๋์ฅ์ ์ง๋ฐฐํ๋ ํธ๋ฏธ๋ถ ๋ฐฉ์ ์(PDE)์ด๋ค. ์ด๋ ํ์ํ PDE์ด๊ธฐ ๋๋ฌธ์, ์์ค ์ ๋ณด๋ฅผ ํฌํจํ ์ฃผ์ด์ง ์ฐ๋ณ์ ๋ํด ์ ํ ์์คํ
, ์ฆ ์ํผ๋์ค ๋งคํธ๋ฆญ์ค๋ฅผ ๊ณ์ฐํจ์ผ๋ก์จ ํด์ ์ป์ ์ ์๋ค. ํฌ๋ฆํ์ธ ๋ฐฉ์ ์์
์ํธ์๊ด์ฑ(reciorocity) ๋๋ฌธ์, ์ด์ฐํ๋ ์ํผ๋์ค ํ๋ ฌ์ ๋ณต์ ๋์นญ ํ๋ ฌ๋ก ์์ฐ์ค๋ฝ๊ฒ ํํ๋ ์ ์์ผ๋ฉฐ, ์ด๋ฅผ ํตํด Cholesky ๋ถํด๋ฅผ ์ ์ฉํ์ฌ ๋ฉ๋ชจ๋ฆฌ ์๊ตฌ๋๊ณผ ์ฐ์ ์ฐ์ฐ์ ์๋ฅผ ์ ๋ฐ์ผ๋ก ์ค์ผ ์ ์๋ค. ํ์ง๋ง PML ๊ฒฝ๊ณ์กฐ๊ฑด์ ์ฌ์ฉํ๋ฉด์
ํ๋ ฌ์ ๋์นญ ํน์ฑ์ ๋ณด์กดํ๋ ๋ฐฉ๋ฒ์ ๋๋ฆฌ ์ฌ์ฉ๋์ง ์์๋ค. ๋ฐ๋ผ์ ๋ณธ ์ฐ๊ตฌ์์๋ PML ๊ฒฝ๊ณ์กฐ๊ฑด ์ธต์ ํฌํจํ ์ ์ฒด ๊ณต๊ฐ ๋
ธ๋์ ๋ํ ๋์นญ ํ๋ ฌ๋ก์ ํฌ๋ฆํ์ธ ๋ฐฉ์ ์์ ์ด์ฐํํ๋ ๋ฐฉ๋ฒ์ ์ ์ํ๋ค. ๋ํ ํ์ฅ๋น ๊ฒฉ์ ์๊ฐ 4๋ณด๋ค ํฐ ๋ฒ์ ๋ด
์์ 0.5% ๋ฏธ๋ง์ ๋ถ์ฐ ์ค์ฐจ๋ฅผ ๋ง์กฑํ๋ ์ต์ ํ๋ ์ด์ฐํ ๊ณ์๋ฅผ ์ ์ํ๋ค. ์ ํ ์์คํ
์ ์ง์ ๋ฒ์ ์ฌ์ฉํ์ฌ 2์ฐจ์ ํน์ ์์ 3์ฐจ์ ๋ฌธ์ ๋ฅผ ํ ์ ์์ง๋ง, ๋ณต์ก๋๋ฅผ ๊ณ ๋ คํ๋ฉด ์ผ๋ฐ์ ์ธ 3์ฐจ์ ๋ฌธ์ ๋ฅผ ํ๊ธฐ์๋ ์๊ฐ์ด ์์๋, ํ๋ ฌ์ ๋ถํด์
์ ์ฅ๊ณต๊ฐ ์๊ตฌ๋ ๊ทธ๋ฆฌ๊ณ ์ํผ๋์ค ํ๋ ฌ์ ๊ณ์ฐํ๋๋ฐ ๊ทนํ ๋์ ๋น์ฉ์ด ํ์ํ๊ฒ ๋๋ค. ์ด ๊ฒฝ์ฐ, ๋ฐ๋ณต๋ฒ, ํนํ, conjugate gradient(CG), GMRES ๋ฑ๊ณผ ๊ฐ์ ํฌ๋ฆด๋ก๋ธ ๋ถ๋ถ๊ณต๊ฐ๋ฒ์ด ํจ๊ณผ์ ์ผ๋ก ์ฌ์ฉ๋ ์ ์๋ค. ๋ณธ ์ฐ๊ตฌ์์๋, ์ฌ๋ฌ ํฌ๋ฆด๋ก
๋ธ ๋ถ๋ถ๊ณต๊ฐ๋ฒ๋ค์ ๋์นญ ํ๋ ฌ์ ์ ์ฉํด ์คํ์ ์ผ๋ก ์ด๋ค ๋ฐฉ๋ฒ์ด ์ํผ๋์ค ํ๋ ฌ์ ๊ณ์ฐํ๋๋ฐ ์ ํฉํ์ง ํ์ธํ์๋ค. ์ด๋ฅผ ํตํด, ๋ค์ํ kh ์กฐ๊ฑด์ ๋ํด conjugate residual(CR)์ด ๊ฐ์ฅ ๋น ๋ฅด๊ฒ ํด์ ์ ๊ณตํ๋ ๊ฒ์ ๊ด์ฐฐํ์๋ค. ์๋ ด ์๋์๋ฅผ ๊ฐ์
ํ๊ธฐ ์ํด, preconditioner ํ๋ ฌ์ ๋ณต์ ์ํํธ(ฯต)๋ฅผ ์กฐ์ ํ์ฌ ์ต์ ์ ๊ณ ์ ๊ฐ ๊ตฐ์งํ๋ฅผ ๋ณด์ฌ์ฃผ๋ Shifted Laplace preconditioner๊ฐ ์ฌ์ฉ๋์๋ค. ์ผ๋ฐ์ ์ผ๋ก ์ด๋ฌํ preconditioner๋ฅผ ์ ์ฉํ๊ธฐ ์ํด ๋ถ์์ ๋ถํด๊ฐ ์ฃผ๋ก ์ฌ์ฉ๋๋ค. ๋ณธ ์ฐ๊ตฌ์์๋, ๋ถํด๋ ํ๋ ฌ์ p๊ฐ์ ๊ฐ์ฅ ํฐ ๊ฐ๋ง ๋จ๊ธฐ๋ ICT(p), ๋ถ์์ Cholesky ๋ถํด๋ฅผ ์ฌ์ฉํ๋ ๊ฒ์ ์ ์ํ๋ค. ์ฃผ์ด์ง kh ์กฐ๊ฑด์ ๋ํด ์ต์ ์ ฯต๊ณผ p๋ฅผ ์ ์ํ๊ณ , ์ด๋ฅผ ์ธ๊ณตํฉ์ฑ ์๋๋ชจ๋ธ์ ์ฌ์ฉํ ์์น ์คํ์ ํตํด ๊ฒ์ฆํ๋ค. ์ฌ๊ธฐ์ ์ง์ ๋ฒ๊ณผ ๋น๊ตํ์ฌ ํ๋์ฅ์ ํด๋ฅผ ๊ตฌํ๋๋ฐ ์ ์ ์๊ฐ๊ณผ ๋ฌด์ํ ์ ์๋ ์ค์ฐจ๋ฅผ ๋ณด์ฌ์ฃผ์๋ค. ํนํ, 169X169X50 ํฌ๊ธฐ์ 3์ฐจ ๋ฌธ์ ์ ๊ฒฝ์ฐ, ๊ณ ๋๋ก ์ต์ ํ๋ ํ๋ก ํ ์๋ฒ์ธ UMFPACK์ ์ฐ์ฐ ์ ๊ฐ์ ์ฝ 5%์ธ ๊ฒ์ผ๋ก ๋ํ๋ฌ๋ค.The Helmholtz equation is a partial differential equation (PDE) that governs the wavefield of a frequency component. Since it is an elliptic type PDE, it is expressed in the form that the solution can be obtained by solving a linear system, which is composed of the impedance matrix and a given right hand side including source information. Due to the reciprocity property of the Helmholtz equation, the discretized impedance matrix can be naturally expressed as a complex symmetric matrix that can reduce a memory requirement and the number of arithmetic operations by half by applying the Cholesky factorization. However, it is not widely disseminated how to reserve such symmetric property of the matrix using the PML boundary condition. Therefore, firstly, we propose a method that discretizes the Helmholtz equation as a symmetric matrix for the entire spatial nodes including the PML layer. We also suggest the optimized discrete coefficient to generate impedance matrix which satisfies the dispersion error of less than 0.5\% within the range that the number of grids per wavelength is greater than 4. The system can be solved by using the direct matrix solver for 2D or small 3D problems, however, time taken and memory requirement to factorize and solve the impedance matrix is extremely high to solve general 3D problems considering the complexities. In this case, iterative methods, in particular, the Krylov subspace methods, such as conjugate gradient method, general minimal residual method, etc., can be effectively applied. In this study, many Krylov iterative solvers that can be applied to the symmetric matrix are implemented to experimentally investigate which method is advantageous for solving the impedance matrix. As a result, it was observed that for various kh conditions, the conjugate residual method solves the matrix in the fastest time consistently. To shorten the convergence speed, we applied the shifted Laplacian preconditioner that can optimally modify the eigenvalue clustering by adjusting the complex shift(ฯต) of the preconditioner matrix. To apply this preconditioner, incomplete factorization method is generally used. In this study, we suggest incomplete Cholesky decomposition, especially ICT(p) method, that leaves only p largest values of the factorized matrix. Optimal pairs of ฯต and p for given kh condition are also suggested, which is validated by numerical examples using the synthetic velocity models. It is shown that only few minutes are taken to solve the problem to yield the wavefield that has negligibly small error compared to that obtained by the direct method, the error is negligibly small. Especially, in the case of a 3D problem with a size of 169X169X50, it is shown that it is about 5\% of the computation time of UMFPACK, a highly optimized directive multifrontal solver.Abstract i
Chapter 1 Introduction 1
1.1 Background and Importance . . . . . . . . . . . . . . . . . . . . . 2
1.1.1 Helmholtz equation . . . . . . . . . . . . . . . . . . . . . 2
1.1.2 Krylov subspace methods . . . . . . . . . . . . . . . . . . 3
1.1.3 Preconditioners . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2 Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.3 Organization of the Thesis . . . . . . . . . . . . . . . . . . . . . . 14
Chapter 2 The Helmholtz equation 15
2.1 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.2 Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
Chapter 3 Krylov subspace method and preconditioner 30
3.1 Krylov subspace methods . . . . . . . . . . . . . . . . . . . . . . 32
3.1.1 Convergence . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.2 Preconditioner . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3.2.1 Shifted Laplace preconditioner . . . . . . . . . . . . . . . 48
3.2.2 Incomplete factorization . . . . . . . . . . . . . . . . . . . 61
3.2.3 Fill-in comparison between ICT(p) and IC(k) . . . . . . . 67
3.2.4 Spectrum of preconditioned system . . . . . . . . . . . . . 69
3.3 Comparison between IC(k) and ICT(p) with Shifted Laplace preconditioner
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
Chapter 4 Numerical Examples 83
4.1 Parameter selection of preconditioner . . . . . . . . . . . . . . . . 83
4.1.1 Numerical domain size dependeny . . . . . . . . . . . . . 83
4.1.2 kh dependency . . . . . . . . . . . . . . . . . . . . . . . . 94
4.1.3 Computational time . . . . . . . . . . . . . . . . . . . . . 98
4.2 Heterogeneous model . . . . . . . . . . . . . . . . . . . . . . . . . 106
4.2.1 2D : Marmousi . . . . . . . . . . . . . . . . . . . . . . . . 106
4.2.2 2D : Pluto . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
4.2.3 3D : SEG/EAGE 3D . . . . . . . . . . . . . . . . . . . . . 118
Chapter 5 Conclusion 124
Bibliography 127
์ด๋ก 138Docto