Chinese Journal of Geophysics  2010, Vol. 53 Issue (3): 700-707   PDF    
Algebraic multigrid method for 3D DC resistivity modelling
LU Jing-Jin1, WU Xiao-Ping1, Klaus Spitzer2     
1. Mengcheng National Geophysical Observatory, School of Earth and Space Science, University of Science and Technology of China, Hefei 230026, China;
2. Institute of Geophysics, TU Bergakademie Freiberg, Gustav-Zeuner-Straße12, 09599, Freiberg, Germany
Received: 2009-05-20; Revised: 2009-11-20; Accepted: 2010-12-03
Corresponding author: WuXiaoping, E-mail: wxp@ustc.edu.cn
Abstract: Multigrid method is of high numerical efficiency in solving linear equations arisen from boundary value problem of partial differential equation (PDE).The usual geometrical multigrid has some defects which restrict its application in PDE with jumping coefficient.In this paper, algebraic multigrid (AMG) method is used to solve finite difference linear equations which are derived from 3D DC resistivity modelling.We solve the secondary potential to remove the singularity of the primary potential caused by source current, resulting in an accurate 3D resistivity modelling. Two models with high conductivity contrast are used to demonstrate convergence and efficiency of the AMG method.Our results show that AMG methods are very efficient and robust in comparison with incomplete cholesky conjugate gradient (ICCG) methods. Moreover, the AMG method becomes more efficient as the number of 3D grid nodes increases..
Key words: Algebraic multigrid      3D resistivity modeling      Conductivity contrast     
DOI: 10.3969/j.issn.0001-5733.2010.03.025
直流电阻率三维正演的代数多重网格方法
鲁晶津1 , 吴小平1 , KlausSpitzer2     
1. 中国科学技术大学蒙城地球物理国家野外观测站, 地球与空间科学学院, 合肥 230026;
2. Institut für Geophysik, TU Bergakademie Freiberg, Germany
摘要: 多重网格方法在求解由偏微分方程的边值问题离散所得线性系统时,具有非常高的计算效率.但常用的几何多重网格法在处理带跃变系数的偏微分方程时存在一定缺陷,限制了其应用.本文应用代数多重网格(AMG)方法求解三维直流电阻率法正演模拟形成的有限差分线性方程组,通过求解二次场的方法消除了总场中由点电源导致的奇异性,从而获得快速、精确的三维电阻率数值模拟.对两个存在大的电性差异的模型进行了模拟计算,以验证代数多重网格法的收敛效率.计算结果表明,与不完全Cholesky共轭梯度(ICCG)方法相比,代数多重网格方法具有更高的计算效率及稳定性.而且,随着三维网格节点数的增加,代数多重网格方法计算的高效性更加明显.
关键词: 代数多重网格      三维电阻率数值模拟      电性差异     
1 Introduction

As a set of widely used near-surface investigation tools in geophysics, direct current (DC) resistivity methods are based on detections of anomalous electric field, which is caused by conductivity contrast between underground target and background rocks when a stable underground electric field is inspired by a DC source[1]. The interpretation of obtained apparent resistivity data is complicated by the existence of anomalies, which are hard to be analyzed by the analytical method. It is why numerical modeling has played an important role in geoelectrics since late1960s[1~17]. Despite the seemingly ever growing power of computers, full 3D resistivity inversions are still challenging and time-consuming. Since the inversion processes are based on forward modeling, efficiency of 3D resistivity modeling is always an interesting topic.

Various numerical techniques have been used in the discretization of PDE, such as integral equations[2, 4, 5, 14, 16], finite element (FE)[3, 9, 11, 13, 17~19], finite difference (FD)[6~8, 12, 15, 20], and boundary element1[10, 21~23]. This paper focuses on a proper solver for the resulted linear equation system, which determines the efficiency of whole modeling process. The coefficient matrix of the equation system is often very large, sparse, symmetric and positive definite. This linear system of equations is usually solved using an iterative method, while the direct solvers are both time-and memory-consuming. Over the past decades, pre-conditioned conjugate gradient method turns out to be a good choice as an iterative solver for 3D resistivity modeling[20, 24], especially with incomplete Cholesky conjugate gradient (ICCG) methods [8, 19, 25, 26]. However, the convergence rate and calculation time of ICCG method may be affected when the number of unknowns grows into hundreds of thousands for large problem. This drawback gives way to multigrid (MG) method, since, if well designed, its convergence rate is independent of the number of grid nodes[27~30].

According to Fourier analysis, solution error can be divided into two components: smooth error component and oscillatory error component. When basic iterative method is applied for solving the equations, the oscillatory error component can be smoothed efficiently, while the smooth error component decreases very slowly and can’t be eliminated. The main idea of MG is to approximate the smooth error component on a coarser grid, where the error will become oscillatory again. After smoothed on the coarser level, the error can be transferred back onto the original fine grid. Correct the solution with the new error, an updated solution is then obtained. Repeat the procedures until a given tolerance s reached. More details about the procedure of MG method[27~30] are presented in Session 3.

Standard geometric multigrid (GMG) is popular due to its simplicity of implementation. Usually rectangular grids are used for discretization. Coarse grids are simply the coarsened version of fine grids, while its coefficient matrices come from the re-discretization of the PDE. Interpolation operator is defined either directly or linearly. Although it’s easy to be implemented, but when applied to 3D DC resistivity modeling, its convergence rate may be affected by irregular grids and high conductivity contrasts[31] (or discontinuous coefficient). Matrix-dependent operators for MG method are proposed by Zeeuw[32] to deal with the discontinuous coefficient in 2D problems. Zhebel[33] extended the operators into 3D geoelectric problems, however, the performance s not as good as expected, since the coarse grids are still formed geometrically.

Algebraic Multigrid (AMG) is first described and investigated in [34~36]. Interest in AMG methods grows in the past decade. But application of the algorithm in geoelectric problems remains untouched. Unlike geometric multigrid, interpolation and restriction operators as well as coarsening process in AMG method are based on the coefficient matrix. This essential difference gives it a great flexibility for complex geometric and physical situations, including irregular grids and high conductivity contrasts.

The setup of linear equation system for 3D DC resistivity modeling is briefly introduced first, and then AMG method is presented. The forward calculations using AMG method are carried out for a three-layer model and a cubic model, which shows good performance in comparison with ICCG algorithm.

2 3D DC Resistivity Modelling 2.1 Boundary value problem

The basic PDE for 3D DC resistivity modelling is written as,

(1)

where σ is the 3D conductivity distribution, I is electric current, (x0, y0, z0) is the position of source point. δ is the Dirac’s Delta function. u is the electrical potential subject to the following model domain[8]:

(2)

where Γs is the air-earth interface, Γ is the external surface of model boundaries, θ is the angle between the radial vector r from the source point to the boundary and the outward normal direction n at the boundary surface.

2.2 The secondary potential approach

The total potential u is singular at the source point and can’t be represented well by discrete approximation, resulting in large error, especially in the vicinity of the source point. The secondary potential approach is considered here to remove the singularity, which is first proposed by Lowry et el[37], and improved by Zhao & Yedlin[38]. Assume the conductivity around the source electrode is σ0, the primary potential up caused by the source current in a uniform half-space of σ0 can be analytically calculated by

(3)

The secondary potential is defined as

(4)

The boundary value problem for us can be written as

(5)

where σ’=σ -σ0 is the variation in the conductivity.

2.3 Finite difference m odelling

Discretization of boundary value problem (5) on a non-equidistant grid with finite difference (FD) method results in a seven-point scheme (Fig. 1).

Fig. 1 Non-equidistant grids (a) and 7-point FD scheme (b)

Suppose there are Nx, Ny, Nz grid nodes in x, y, z direction, respectively. The grids are designed to be fine near inhomogeneities and coarse towards the boundary of the model domain seen in Fig. 1a. In Fig. 1b, (i, j, k) denotes agridnode (i=1, 2, …, Nx, j=1, 2, …, Ny, k=1, 2, …, Nz), which is only connected to six neighboring nodes. Writing the discretized scheme for each node in a matrix, we get the linear equation system

(6)

A is a large, sparse, symmetric and positive definite matrix. Every matrix row can, at most, contain seven non-zero entries. The definitions of each element a(i, j, k)A can be found in Dey & Morrison[8]. More details can be seen in references [25, 26, 38].

3 Algebraic Multigrid [39~42]

Algebraic Multigrid (AMG) has two phases: setup phase and solution phase. The setup phase chooses appropriate coarse levels and interpolation for further calculations, which are based on the algebraic information from coefficient matrices. The solution phase follows standard multigrid procedures, including coarse grid correction and smoothing. We present the basic idea of multigrid for the solution phase first, and then discuss the algebraic coarsening and interpolation for the setup phase, which is the main difference between AMG and standard geometric multigrid.

3.1 Basic idea of multigrid

Let h be the grid size on fine level and H the grid size on coarse level, we have equations on the fine level,

(7)

in which A is the coefficient matrix with aijA. u is the unknown solution. f is the right-hand term. Ω is an index set while it corresponds to a grid in geometric multigrid. Superscript h denotes finer level variables while superscript H denotes coarser level variables next. On the finest level, Ah, fh are arisen directly from finite difference or finite element calculation. With a given initial solution, solve equation (7) by a simple smoother, such as Gauss-Siedel iteration, we get a solution uoldh. After the smoothing process, the error of the solution is smoothed rather than eliminated. The residual can be written as

(8)

With IHh denoting interpolation operator, restriction operator IhH is defined as the transpose of IHh,

(9)

After defining the restriction operator, the fine level residual roldh can be restricted to the coarse level as,

(10)

Using Galerkin approximation, the coarse level operator is defined as

(11)

Taking rH as the new right-hand term, we get the residual equations on coarse level,

(12)

where e is the error correction between the true solution and numerical solution. If the number of unknowns on coarse level is small enough, equation (7) can be solved accurately to get the error correction eH by using a direct method e. g. Gauss elimination. Interpolating eH back on to the fine level, a two level correction s done by

(13)

For multigrid method, equation (12) is solved by another two level correction, in which accurate solution can be obtained on the coarser grid. Recursively, the two level correction steps are repeated until the coarsest level s reached.

3.2 Algebraic coarsening

Unlike geometric multigrid, AMG bases its interpolation and coarse grid operators on the coefficient matrices. It needs no information of the geometric grid. From this point of view, the coarse level variables should be chosen carefully, otherwise the interpolation may be ineffective.

A given matrix can be associated with a graph formed by nodes and connections between nodes. Define iΩ as a node, then node i is connected to node jΩ if aij≠0(aijA). Regarding the coarse-level nodes as a subset of fine-level ones, the set of fine-level nodes can be split into two disjoint subsets, Ωh=ChFh.Ch represents the nodes which are to be contained in the coarse level (C-nodes).Fh is the complementary set (F-nodes). Define the neighborhood of node iΩh by

(14)

Nodes i, jΩh are strongly connected to each other if they satisfy the following property,

(15)

In this case, aij is called strong-connection. θ is the threshold which can be determined by the user. We can get the strong-connection neighborhood of i as follows,

(16)

Obviously, if too much nodes are chosen to be coarse nodes, more calculations are needed to define the interpolation, and more levels are needed to reach a reasonable coarsest level. Thus the numerical complexity grows. Strong-connections are taken as guideline to control the choice of coarse nodes to get a good interpolation and a reasonable description of numerical complexity. Initially, every node in Ωh is marked with the number of its strong-connections. Node with the most strong-connections is chosen to be C-node, while its strong-connection neighbors (jNis) become the F-nodes automatically. The remaining nodes are then reevaluated with their number of strong connections to F-nodes and other undetermined nodes. Continue this process until all the nodes in Ωh are either belong to Ch or Fh. A complete standard coarsening algorithm can be found in Ref.[14]

3.3 Algebraic interpolation

Define the interpolation eh=IHheH like the usual way,

(17)

Cih denotes the interpolatory nodes for node iΩh. wijh denotes the interpolatory weight from coarse node j to fine node i.

The definition of wijh is based on smooth error and connection between i and j. Recall that smooth error is characterized by small residuals. In order to keep this property on each level,we may assume the residuals to be zero. Then an approximate equation can be written as

(18)

Omitting the superscript h for smplicity, equation (18) can be rewritten as

(19)

The interpolatory nodes Ci can be chosen as Ci=NisCh. The non-interpolatory nodes are denoted as Di=Ni\Ci, which means Di includes all the nodes in Ni except the nodes in Ci. Furthermore, Di can be spilt into two parts according to the strength of connection: strong non-interpolatory nodes Di and weak non-interpolatory nodes Diw, which are defined as Dis=DiNis, Diw=Di\Dis. With those splitting, rewrite equation (19) as,

(20)

ej (jDiw) may be approximated by ei, since the connection is weak, we get

(21)

For ej (jDis), the following approximation is considered,

(22)

After a series of substitution with equation (21), the algebraic interpolation can be finally defined as

(23)

(24)

4 Numerical Examples

The 3D FD resistivity forward calculations are carried out using AMG method for a three-layer model and a cubic model. The l2 -norm of residual rh with a tolerance 10-8 is taken as the convergence criteria,

(25)

The calculation results are compared with those obtained by the ICCG method in term of convergence rate, which is defined by the needed iterations against the convergence criteria. Three 3D grids of different size of 49 × 49 × 25, 89 × 89 × 45 and 129 × 129 × 129 nodes are used to demonstrate the performance of the AMG method.

4.1 Three-layer model

Fig. 2 shows a three-layered model. The conductivity contrast between the anomalous layer and the background layers is 1 : 100. The numerical solutions by AMG are compared with the analytical solution. The agreement is very well (see Fig. 3). The relative error between analytical and numerical solutions s shown in Fig. 4.

Fig. 2 Three-layered model h, ρ denote thickness and resistivity of each layer respectively.
Fig. 3 Comparison of analytical and numerical solutions over a three-layered model
Fig. 4 Relative errors between analytical solution and numerical solutions

From Fig. 5, it can be seen that the number of iterations needed to meet the given convergence tolerance is almost constant (7 or 8) for three grids of different size. This means that the convergence rate of AMG algorithm is almost independent of grid size. But in contrast, the performance of ICCG algorithm in Fig. 6 shows its convergence rate deteriorates as grid size increases, ICCG method requires much more iterations to converge with increasing grid nodes.

Fig. 5 Convergence curve of AMG algorithm for three-layered model
Fig. 6 Convergence curve of ICCG algorithm for three-layered model

Fig. 7 shows the running time of AMG and ICCG methods, AMG method is obviously much more efficient than ICCG. The AMG method takes 317s for 3D modeling with a grid of 129 × 129 × 129, 2.6 times faster than ICCG method which needs 1151s. Moreover, the bigger the size of a grid is, more efficient the AMG method is.

Fig. 7 Running time of AMG and ICCG for layer model
4.2 Cubic model

Fig. 8 shows a cubic model, the numerical solutions by AMG and ICCG on a grid of 129 × 129 × 129 nodes are shown in Fig. 9. Since the analytical solution is unavailable, however, our ICCG solution for this model were proven to be very accurate compared to the results from finite element and integral equation[25]. Fig. 9 shows a good agreement between AMG and ICCG solutions.

Fig. 8 Cubic model of size 2 m × 2 m × 2 m buried at depth of 0.5 m
Fig. 9 Comparison of numerical solutions on grid of 129 × 129 × 129 by AMG and ICCG for cubic model

From the convergence curve of AMG algorithm in Fig. 10, the l2 -norm of residual for the AMG method achieves convergence tolerance after 6 to 8 iterations, which is also very stable with increasing grid size. Similarly to the three-layered model, the ICCG result needs more iterations to reach convergence tolerance shown in Fig. 11. It can be seen in Fig. 12, the running time of AMG is 297s for 3D modeling on a grid of 129 × 129 × 129, 3.0 times faster than ICCG method which takes 1194s. The AMG method shows more efficient than ICCG for 3D model.

Fig. 10 Convergence curve of AMG algorithm for cubic model
Fig. 11 Convergence curve of ICCG algorithm for cubic model
Fig. 12 Running time of AMG and ICCG for cubic mode
4.3 Storage requirements

As to the storage requirement for AMG and ICCG methods, the coefficient matrices are stored only on the finest grid for ICCG, while the coefficient matrices, as well as interpolation operators, are required to be stored on each level for AMG. However, only nonzero entries of the coefficient matrices are stored for AMG and ICCG methods, and the number of grid nodes decreases rapidly in the coarser levels for AMG, the storage requirement of AMG increases a little more (i. e., about 2 times) in comparison to ICCG.

5 Conclusions

We implement 3D DC resistivity modeling using AMG method. Our results show that AMG is an efficient solver for 3D FD resistivity forward problem even high conductivity contrast exits in the models. The convergence rate of AMG algorithm is almost independent of 3D grid size which is the most attractive character of multigrid method. In comparison with ICCG method, the computation speed of AMG is much faster than ICCG. The AMG method takes around 300s for 3D modeling with a grid of 129 × 129 × 129, about 3 times faster than ICCG method. Moreover, with increasing of the number of 3D grid nodes, AMG method shows more efficient.

Acknowledgments

This work is supported by funds from the Natural Science Foundation of China (No. 40674037, 40874034). We thank Dr. Li Yuguo at UCSD and an anonymous reviewer for their constructive comments.

References
[1] Zhou X X, Zhong B S. Numerical Simulation Techniques in Electrical Prospecting. Chengdu: Sichuan Publishing House of Science and Technology, 1986.
[2] Dieter K, Paterson N R, Grant F S. IP and resistivity type curves for three-dimensional bodies. Geophysics, 1969, 34: 615-632. DOI:10.1190/1.1440035
[3] Coggon J H. Electromagnetic and electrical modeling by the finite element method. Geophysics, 1971, 36: 132-155. DOI:10.1190/1.1440151
[4] Hohmann G W. Three-dimensional induced polarization and electromagnetic modeling. Geophysics, 1975, 40(2): 309-324. DOI:10.1190/1.1440527
[5] Lee T. An integral equation and its solution for some two-and three-dimensional problem in resistivity and induced polarization. Geophys.J.R.Astr.Soc., 1975, 42(1): 81-95.
[6] Mufti I R. Finite-difference resistivity modeling for arbitrarily shaped two-dimensional structures. Geophysics, 1976, 41(1): 62-78. DOI:10.1190/1.1440608
[7] Mufti I R. A practical approach to finite-difference resistivity modeling. Geophysics, 1978, 43(5): 930-942. DOI:10.1190/1.1440874
[8] Dey A, Morrison H F. Resistivity modeling for arbitrarily shaped three-dimensional structures. Geophysics, 1979, 44(4): 753-780. DOI:10.1190/1.1440975
[9] Fox R C, Hohmann G W, Killpack T J, et al. Topographic effects in resistivity and induced-polarization surveys. Geophysics, 1980, 45(1): 75-93. DOI:10.1190/1.1441041
[10] Okabe M. Boundary element method for the arbitrary inhomogeneities problem in electrical prospecting. Geophys.Prospect., 1981, 29: 39-59. DOI:10.1111/gpr.1981.29.issue-1
[11] Pridmore D F, Hohmann G W, Ward S H, et al. An investigation of finite-element modelling for electrical and electromagnetical data in three dimensions. Geophysics, 1981, 46(7): 1009-1024. DOI:10.1190/1.1441239
[12] Scribe H. Computations of the electrical potential in the three-dimensional structure. Geophys.Prospect., 1981, 29: 790-802. DOI:10.1111/gpr.1981.29.issue-5
[13] Holcombe H T, Jiracek G R. Three-dimensional terrain corrections in resistivity surveys. Geophysics, 1984, 49(4): 439-452. DOI:10.1190/1.1441679
[14] Oppliger G L. Three-dimensional terrain corrections for mise-a-la-masse and magnetometric resistivity surveys. Geophysics, 1984, 49(10): 1718-1729. DOI:10.1190/1.1441579
[15] James B A. Efficient microcomputer-based finite-difference resistivity modeling via Polozhii decomposition. Geophysics, 1985, 50(3): 443-465. DOI:10.1190/1.1441923
[16] Xu S Z, Gao Z C, Zhao S K. An integral formulation for 3-D terrain modeling for resistivity surveys. Geophysics, 1988, 53(4): 546-552. DOI:10.1190/1.1442486
[17] Queralt P, Pous J, Marcuello A. 2-D resistivity modeling:An approach to arrays parallel to the strike direction. Geophysics, 1991, 56(7): 941-950. DOI:10.1190/1.1443127
[18] Zhou B, Greenhalgh S A. Finite element three-dimensional direct current resistivity modelling:accuracy and efficiency considerations. Geophys.J.Int., 2001, 145: 679-688. DOI:10.1046/j.0956-540x.2001.01412.x
[19] Li Y G, Spitzer K. Three-dimensional DC resistivity forward modeling using finite elements in comparison with finite-difference solutions. Geophys.J.Int., 2002, 151: 924-934. DOI:10.1046/j.1365-246X.2002.01819.x
[20] Spitzer K. A 3-D finite-difference algorithm for dc resistivity modeling using conjugate gradient methods. Geophys.J.Int., 1995, 123: 903-914. DOI:10.1111/gji.1995.123.issue-3
[21] Xu S Z, Zhao S K, Ni Y. A boundary element method for 2-D DC resistivity modeling with a point current source. Geophysics, 1998, 63(2): 399-404. DOI:10.1190/1.1444339
[22] Ma Q Z. The boundary element method for 3-D DC resistivity modeling in layered earth. Geophysics, 2002, 67(2): 610-617. DOI:10.1190/1.1468622
[23] Xu S Z. The boundary element method in geophysics. Tulsa: Society of exploration geophysics, 2001.
[24] Spitzer K, Wurmstich B. Speed and accuracy in 3D resistivity modeling, in Three-dimensional Electromagnetics, eds Oristaglio, M.L.& Spies, B.R., SEG Book Series. eophysical Developments.Society of Exploration Geophysicists, 1997: 161-176.
[25] Wu X P, Xiao Y F, Qi C, et al. Computations of secondary potential for 3d dc resistivity modelling using an incomplete Choleski conjugate-gradient method. Geophys.Prospect., 2003, 51: 567-577. DOI:10.1046/j.1365-2478.2003.00392.x
[26] Wu X P. A 3-D finite-element algorithm for DC resistivity modeling using the shifted incomplete Cholesky conjugate gradient method. Geophys.J.Int., 2003, 154(3): 947-956. DOI:10.1046/j.1365-246X.2003.02018.x
[27] Hackbusch W. Multi-Grid Methods and Applications. Heidelberg: Springer-Verlag Berlin, 1985.
[28] Briggs W L.A Multigrid Tutorial.SIAM, Philadelphia, 1987 Briggs W L.A Multigrid Tutorial.SIAM, Philadelphia, 1987
[29] Wesseling P.An introduction to multigrid methods.Chichester:John Wiley and Sons Ltd., 1992 Wesseling P.An introduction to multigrid methods.Chichester:John Wiley and Sons Ltd., 1992
[30] Trottenberg U, Oosterlee C W, Schuller A.Multigrid.San Diego, CA:Academic Press Inc., 2001 Trottenberg U, Oosterlee C W, Schuller A.Multigrid.San Diego, CA:Academic Press Inc., 2001
[31] Moucha R, Bailey R C. An accurate and robust multigrid algorithm for 2d forward resistivity modeling. Geophys.Prospect., 2004, 52: 197-212. DOI:10.1111/gpr.2004.52.issue-3
[32] Zeeuw P M. Matrix-dependent prolongations and restrictions in a blackbox multigrid solver. J.Comput.Appl.Math., 1990, 33: 1-27. DOI:10.1016/0377-0427(90)90252-U
[33] Zhebel E.A multigrid method with Matrix-dependent transfer operators for 3D diffusion problems with jump coefficients[Ph.D.theses].Germany:TU Bergakademie Freiberg, 2006 Zhebel E.A multigrid method with Matrix-dependent transfer operators for 3D diffusion problems with jump coefficients[Ph.D.theses].Germany:TU Bergakademie Freiberg, 2006
[34] Stueben K. Algebraic multigrid (AMG):Experiences and comparisons. Appl.Math.Comput., 1983, 13: 419-452.
[35] Ruge J W, Stueben K.Efficient solution of finite difference and finite element equations by algebraic multigrid (AMG).in:D.J.Paddon, H.Holstein Eds.Multigrid Methods for Integral and Differential Equations, The Institute of Mathematics and its Applications Conference Series, Clarendon Press, Oxford, New Series, 1985, 3:169-212 Ruge J W, Stueben K.Efficient solution of finite difference and finite element equations by algebraic multigrid (AMG).in:D.J.Paddon, H.Holstein Eds.Multigrid Methods for Integral and Differential Equations, The Institute of Mathematics and its Applications Conference Series, Clarendon Press, Oxford, New Series, 1985, 3:169-212
[36] Ruge J W, Stueben K.Algebraic multigrid (AMG).In:S.F.McCormick Ed.Multigrid Methods, Frontiers in Applied Mathematics, 5, SIAM, Philadelphia, 1986 Ruge J W, Stueben K.Algebraic multigrid (AMG).In:S.F.McCormick Ed.Multigrid Methods, Frontiers in Applied Mathematics, 5, SIAM, Philadelphia, 1986
[37] Lowry T, Allen M B, Shive P N. Singularity removal:a refinement of resistivity modeling techniques. Geophysics, 1989, 54(6): 766-774. DOI:10.1190/1.1442704
[38] Zhao S, Yedlin M J. Some refinements on the finite-difference method for 3-d dc resistivity modeling. Geophysics, 1996, 61(5): 1301-1307. DOI:10.1190/1.1444053
[39] Ruge J W, Stueben K.Algebraic multigrid.In:Stephen F.McCormick, eds., Multigrid Methods, 1987 Ruge J W, Stueben K.Algebraic multigrid.In:Stephen F.McCormick, eds., Multigrid Methods, 1987
[40] Stueben K.Algebraic multigrid (AMG):an introduction with applications.In:U.Trottenberg, C.W.Oosterlee, A.Schueller Eds.Multigrid, New York:Academic Press, 2000.Also GMD Report 53, March 1999 Stueben K.Algebraic multigrid (AMG):an introduction with applications.In:U.Trottenberg, C.W.Oosterlee, A.Schueller Eds.Multigrid, New York:Academic Press, 2000.Also GMD Report 53, March 1999
[41] Wagner C.Introduction to algebraic multigrid.Course notes of an algebraic multigrid course, Universität Heidelberg, 1999 Wagner C.Introduction to algebraic multigrid.Course notes of an algebraic multigrid course, Universität Heidelberg, 1999
[42] Stueben K. A review of algebraic multigrid. J.Comput.Appl.Math., 2001, 128: 281-309. DOI:10.1016/S0377-0427(00)00516-1