Conjugate Gradient Least-Squares Algorithm for Three-Dimensional Magnetotelluric Inversion

3차원 MT 역산에서 CG 법의 효율적 적용

  • Kim, Hee-Joon (Dept. of Environmental Exploration Eng., Pukyong National University) ;
  • Han, Nu-Ree (Dept. of Civil, Urban and Geosystem Eng., Seoul National University) ;
  • Choi, Ji-Hyang (Dept. of Civil, Urban and Geosystem Eng., Seoul National University) ;
  • Nam, Myung-Jin (Dept. of Petroleum and Geosystem Eng., The University of Texas at Austin) ;
  • Song, Yoon-Ho (Korea Institute of Geoscience & Mineral Resources) ;
  • Suh, Jung-Hee (Dept. of Civil, Urban and Geosystem Eng., Seoul National University)
  • 김희준 (부경대학교 환경탐사공학과) ;
  • 한누리 (서울대학교 지구환경시스템공학부) ;
  • 최지향 (서울대학교 지구환경시스템공학부) ;
  • 남명진 ;
  • 송윤호 (한국지질자원연구원 지하수지열연구부) ;
  • 서정희 (서울대학교 지구환경시스템공학부)
  • Published : 2007.05.31

Abstract

The conjugate gradient (CG) method is one of the most efficient algorithms for solving a linear system of equations. In addition to being used as a linear equation solver, it can be applied to a least-squares problem. When the CG method is applied to large-scale three-dimensional inversion of magnetotelluric data, two approaches have been pursued; one is the linear CG inversion in which each step of the Gauss-Newton iteration is incompletely solved using a truncated CG technique, and the other is referred to as the nonlinear CG inversion in which CG is directly applied to the minimization of objective functional for a nonlinear inverse problem. In each procedure we only need to compute the effect of the sensitivity matrix or its transpose multiplying an arbitrary vector, significantly reducing the computational requirements needed to do large-scale inversion.

CG (conjugate gradient) 법은 선형 연립방정식을 반복적으로 푸는 가장 효율적인 해법 중 하나이고, 또한 비선형 최소자승문제에도 적용할 수 있다. 자기지전류(MT) 역산 문제를 풀 때에는 최소자승문제의 목적함수 자체의 최소화에 직접 CG 법을 적용하거나, Gauss-Newton 법에 기초한 반복역산의 각 반복단계에서 모형의 변화량 계산에 CG 법을 이용할 수 있다. CG 법을 적용할 경우, 임의의 벡터에 대한 감도행렬의 영향 및 그 전치행렬의 전치행렬의 영향을 감도행렬을 직접 구하지 않고 계산할 수 있다는 장점이 있기 때문에 감도행렬의 계산 규모가 방대한 3차원 역산 문제에서 계산시간을 월등히 줄일 수 있다.

Keywords

References

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