Directory UMM :Data Elmu:jurnal:A:Advances In Water Resources:Vol23.Issue8.2000:

A least-squares penalty method algorithm for inverse problems of

steady-state aquifer models

Hailong Li

*

_{, Qichang Yang}

Institute of Biomathematics, Anshan Normal College, Anshan 114005, People's Republic of China

Abstract

Based on the generalized Gauss±Newton method, a new algorithm to minimize the objective function of the penalty method in (Bentley LR. Adv Wat Res 1993;14:137±48) for inverse problems of steady-state aquifer models is proposed. Through detailed analysis of the ``built-in'' but irregular weighting e€ects of the coecient matrix on the residuals on the discrete governing equations, a so-called scaling matrix is introduced to improve the great irregular weighting e€ects of these residuals adaptively in every Gauss± Newton iteration. Numerical results demonstrate that if the scaling matrix equals the identity matrix (i.e., the irregular weighting e€ects of the coecient matrix are not balanced), our algorithm does not perform well, e.g., the computation cost is higher than that of the traditional method, and what is worse is the calculations fail to converge for some initial values of the unknown parameters. This poor situation takes a favourable turn dramatically if the scaling matrix is slightly improved and a simple preconditioning technique is adopted: For naturally chosen simple diagonal forms of the scaling matrix and the preconditioner, the method performs well and gives accurate results with low computational cost just like the traditional methods, and improvements are obtained on: (1) widening the range of the initial values of the unknown parameters within which the minimizing iterations can converge, (2) re-ducing the computational cost in every Gauss±Newton iteration, (3) improving the irregular weighting e€ects of the coecient matrix of the discrete governing equations. Consequently, the example inverse problem in Bentley (loc. cit.) is solved with the same accuracy, less computational e€ort and without the regularization term containing prior information on the unknown parameters. Moreover, numerical example shows that this method can solve the inverse problem of the quasilinear Boussinesq equation almost as fast as the linear one.

In every Gauss±Newton iteration of our algorithm, one needs to solve a linear least-squares system about the corrections of both the parameters and the groundwater heads on all the discrete nodes only once. In comparison, every Gauss±Newton iteration of the traditional method has to solve the discrete governing equations as many times as one plus the number of unknown parameters or head observation wells (Yeh WW-G. Wat Resour Res 1986;22:95±108).

All these facts demonstrate the potential of the algorithm to solve inverse problems of more complicated non-linear aquifer models naturally and quickly on the basis of ®nding suitable forms of the scaling matrix and the preconditioner. Ó _{2000 Elsevier}

Science Ltd. All rights reserved.

Keywords: Penalty method; Inverse problem; Scaling matrix; Preconditioner; Aquifer models; LSQR algorithm; Parameter identi®cation; Gauss±Newton method; Finite element method

1. Introduction

The traditional least-square methods for distributed parameter identi®cation of groundwater models mini-mize a quadratic objective function with respect to the parameters subject to the discrete governing equations of an elliptic or parabolic aquifer model (see e.g., [4±

6,8,9,14±16,21]). This means that the discrete equations of the aquifer models are regarded as an exact descrip-tion of the real aquifer. But in practice there exist not only measurement errors of heads but also errors of the conceptual models, of the discretization of the models, of the determination of boundary conditions and source terms etc. Hence the concept of measurement error must be generalized such that it includes also the other errors mentioned above. Such a generalization of the measurement error concept is particularly important in least-squares and maximum likelihood methods of groundwater inverse problems. For this reason, exact enforcement of the discrete governing equations may

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*

Corresponding author. Present address: Department of Earth Sciences, The University of Hong Kong, Hong Kong SAR, People's Republic of China. Tel.: +852-2857-8278; fax: +852-2517-6912.

E-mail addresses:hlica@hkusua.hku.hk, hailong@mail.asptt.ln.cn (H. Li).

not be necessarily appropriate, and may actually de-grade the head and parameter estimation [2,15]. This is the ®rst motivation of this paper.

Bentley [2] formed a penalty method least-squares objective function by adding a general positive de®nite quadratic form of the discrete governing equation re-siduals into the traditional objective function and min-imized the objective function without any constraints using a ``damped direction-search and linesearch al-gorithm''. Then he suggested at the end of his paper on p. 147 that

Convergence, as well as selecting an optimum solu-tion technique, remain topics for future investiga-tion.

Hence, it is necessary to ®nd an e€ective algorithm to minimize the penalty method objective function. This is the second motivation of this paper.

The algorithm proposed in this paper can be brie¯y introduced as follows.

First, di€erent from [2], residuals of the discrete governing equations are scaled by multiplying them with a ``scaling matrix'' determined adaptively by the current coecient matrix of the discrete governing equations. With the scaled residuals, a similar penalty method objective function to that in [2] is formed. Detailed nu-merical analysis shows that the discrete governing equation residuals are not distributed uniformly on the discrete nodes due to the wide value range of the discrete governing equation coecients and their ``built-in'' but

Notation

A _{nncoecient matrix of the discrete}

aquifer model (see (9), (10a)±(10c)).

C _{n‡np† …n‡np† precondition}

matrix.

D_{c2 a diagonal form of}C _{de®ned by (37)}

and (38).

H the piezometric head or watertable vector (see (5)).

h…x;y† the piezometric head or watertable,m.

J _{n‡nm‡np†…n‡np†Jacobian matrix}

de®ned by (32) and (39).

Jg weighted square sum of the discrete

governing equation residuals (see (15) and (20)).

Jgmp the objective function of the penalty

method de®ned in (19).

Jm weighted square sum of the head

observation residuals (see (13) and (17)).

Jp weighted square sum of the

par-ameter prior information residuals (see (14) and (18)).

Jt traditional objective function de®ned

in (16).

n the number of the discrete nodes. nm the number of observation wells.

np the number of the unknown

par-ameters (see (4a), (4b) and (11)).

P _nnp Jacobian matrix de®ned in (39), (41) and (42).

P np-dimensional unknown parameter

(log-transimissivity) vector de®ned by (11).

Rg n-dimensional discrete governing

equation residual vector de®ned by (15).

Rgm an nm-dimensional vector composed

of the components of Rg whose

in-dices equal those of the nodes co-incident with the head observation wells.

Rgr an…nÿnm†-dimensional vector

com-posed of the rest of the components ofRg except those ofRgm.

Rm nm-dimensional head observation

residual vector de®ned by (13). T…x;y†andTi unknown transmissivity, m2_{/day (see}

(1), (3), (4a), (4b) and (11)).

U _nnscaling matrix (see (15)).

U_E a diagonal form of U de®ned by (25).

U₁ a diagonal form ofUde®ned by (24).

W block diagonal matrix Diag‰k2_gW_g; W_m;kpW_pŠ.

W_{g nnweighting matrix of the residual}

vectorRg.

W_{m nmnm weighting matrix of the}

residual vectorRm.

W_{p npnp weighting matrix of the}

parameter prior information residual vectorRp.

Yi …16i6np† log-transimissivity (YiˆlogTi).

Cd andCn the Dirichlet and Neumann

boundaries.

DP the di€erence vector between the computed and true values of the parameter vectorP.

k2_g weighting factor of the penalty term Jg(see (19) and (20)).

kp weighting factor of the regularization

termJp (see (16) and (18)).

v …n‡np†-dimensional vector de®ned by…HT_;PT_†T

irregular weighting e€ects on these residuals. This ir-regularity of the residuals can be eciently improved by using the above mentioned scaling matrix.

Second, based on the generalized Gauss±Newton method [11,12,21], every iteration minimizing the pen-alty method objective function is formulated with a linear least-squares system whose unknowns are the corrections of both the unknown parameters and the groundwater heads on all the discrete nodes. The num-ber of linear equations contained in this linear system has to be more than or at least equal to the number of unknowns in it. So the number of observation wells has to be more than or at least equal to the number of un-known parameters when there is no regularization term containing prior information on the unknown par-ameters in the objective function.

Finally, a suitable algorithm is chosen to solve this linear least-squares system which is usually ill-con-ditioned. There are several sources of this ill-conditioned state, e.g., the ill-posedness of the inverse aquifer problems themselves, the wide value range of the dis-crete governing equation coecients mainly caused by the non-uniformity of the triangulation and the wide range of the transimissivity values, the deterioration of the condition number of the least-squares systems due to large weighting factor kg of the penalty term used to

enforce the discrete governing equations, etc. Moreover, this linear least-squares system is a large sparse system incorporating the discrete governing equations. In order to overcome these diculties, an iterative least-squares QR factorization (LSQR) algorithm for sparse linear least-squares problems [17,18] is chosen to solve it. At the same time, a simple but e€ective preconditioner is implemented to improve the ill-conditioned state.

Numerical results demonstrate that if either the dis-crete governing equation residuals are not scaled or no preconditioning technique is adopted, then the penalty method algorithm performs poorly due to the ill-con-ditioned state, e.g., the computation cost is higher than that of the traditional method, and what is worse is, the calculations fail to converge for some initial values of the unknown parameters. But the situation can be im-proved signi®cantly by some naturally chosen diagonal forms of the scaling matrix and the preconditioner.

Due to the sparse iterative data structure of the LSQR algorithm, the size of the discrete governing equations of our algorithm can be as large as that of the traditional methods.

In every Gauss±Newton iteration of our algorithm, one needs to solve the linear least-squares system only once. In comparison, every Gauss±Newton iteration of the traditional methods has to solve the discrete gov-erning equations as many times as one plus the number of unknown parameters if one uses the parameter per-turbation and numerical di€erence quotient method, or as many times as one plus the number of head

obser-vation wells if one uses the state adjoint sensitivity equation method (variatonal method) [21]. The total number of iterations required to minimize the penalty method objective function is less than or equal to the number required for the two traditional methods tested. All these facts demonstrate the potential of this method to solve inverse problems of more complicated non-linear aquifer models naturally and quickly on the basis of ®nding suitable forms of the scaling matrix and the preconditioner.

The paper is organized as follows. In the next part, the penalty method objective function and its minimiz-ing algorithm based on the Gauss±Newton method are given in detail both for a linear aquifer model and a non-linear model described by the elliptic Boussinesq equa-tion. In Section 3, two numerical examples are given. One of the examples is taken from [2,6]. The other is the elliptic Boussinesq equation describing an uncon®ned aquifer [1,19]. Through these numerical examples, the e€ects and utility of both the scaling matrix and the preconditioner are illustrated. At last, the conclusions are drawn on the basis of the former sections and problems that remain to investigate are stated.

2. Algorithm

2.1. Objective function of penalty method

The inverse problems of typical groundwater ¯ow equations in con®ned and uncon®ned aquifers will be used as examples to illustrate the algorithm of the least-squares penalty method. Consider two-dimensional steady ¯ow described by

ÿr …sTrh† ÿQ‡B…hÿhr† ˆ0; …x;y† 2X …1†

in inhomogeneous, isotropic aquifers subject to bound-ary conditions

hˆhd…x;y†; …x;y† 2Cd; …2†

sToh

o_nˆq…x;y†; …x;y† 2Cn; …3†

where

X: Groundwater ¯ow region. It is a given bounded horizontal domain in…x;y†plane.

Cd and Cn: The Dirichlet and Neumann boundaries

ofX, respectively.Cd[CnˆoX:

…x;y†and r: Spatial variables and the gradient operator.

h…x;y†: Groundwater head of a con®ned aquifer or watertable of an uncon®ned aquifer. Unknown. Only ®nite measurement values at given observation wells are available.

problem. For uncon®ned aquifer sˆ …hÿhb…x;y††

=M [1,19], wherehb denotes the given elevation of

the aquifer bottom and it has the same datum plane ash, andMis a positive constant describing the av-erage thickness of the uncon®ned aquifer. In this, case (1) represents the elliptic Boussinesq equation and (1)±(3) becomes a non-linear problem.

T…x;y†: Transmissivity of the aquifer. Unknown. In this paper it is assumed thatTis piecewise constant in X, i.e., there arenp given subdomains (zones) Xi

whereTi…16i6np†are unknown positive constants. Q…x;y†: Recharge term determined by factors such as source, sink, precipitation and evapotranspira-tion etc.

B…x;y†and hr: Speci®ed leakage parameters. We

assume B60 for the sake of the well-posedness of (1)±(3).

o₌on: Normal derivative with respect toCn.

hd…x;y†: Speci®ed head or watertable onCd.

q…x;y†: Speci®ed ¯ux onCn.

The discrete governing equations of (1)±(3) are the starting point of our method. One can discretize gov-erning equations (1)±(3) using various methods such as ®nite-di€erence, ®nite-volume or ®nite-element method according to one's preference. In this paper we use linear triangle ®nite element method to discretize (1)±(3). Let

f…xi;yi†;16i6ng be the position of the nodes of the triangulation, and /i…x;y†…16i6n† be the linear basis function corresponding to the ith discrete node…xi;yi†. Let

H:ˆ …h1;. . .;hn† T

…5†

be the head or watertable vector, where hi denotes the ®nite element approximation toh…xi;yi†, superscript ``T'' indicates the transpose of a vector or matrix. Then the recharge term Q, the leakage parameters B andhr,

are constants on each triangle elemente, i.e.,

Yij_eˆYe; Qj_e ˆQe; Bj_eˆBe; hrj_eˆher: …8†

By applying the standard Galerkin ®nite element pro-cedure, the discrete governing equations of (1)±(3) can be written in matrix form as follows (see e.g., [19]):

A_{H;P†HˆF; …9†}

whereAis annn matrix with elements

aijˆsij‡bij; 16i; j6n …10a†

positive weighting constant that is chosen large enough compared with all the other diagonal entries ofA_{for the}

discrete Dirichlet boundary conditions to be enforced at least as accurately as other discrete governing equations.

eis the triangle element,T_{iis the set of all the triangle} elements in the neighborhood of the ith node

…xi;yi†…16i6n), i.e., T_{iˆ fej…xi;yi†is a vertex of eg.}

F is an n-dimensional vector determined by forcing terms such as Q,B,hr and the boundary conditions of

(1)±(3), whose components can be written as

fiˆ wdhd…xi;yi†; i2 gate gradient method or the Cholesky decomposition method is used to solve the discrete aquifer model (9), otherwise sˆ …hÿhb†=M, (9) is a non-linear problem and it is solved by iterative method.

As in [2], the observation head residual vectorRmand

parameter prior information residual vector Rp can be

de®ned by

Rm…H† ˆHsÿHm; …13†

Rp…P† ˆPÿP; …14†

whereHs andHm arenm-dimensional vectors whose

re-spective components denote the simulated and measured head values at the observation wells;nmis the number of

Rg…H;P† ˆUA_{H;P†HÿF†; …15†}

whereU_{is annngiven non-singular ``scaling'' matrix}

used to scale the elements of Rg. Later we will discuss

how to choose it. Then the traditional least-squares method for aquifer parameter identi®cation can be written as [4±6,8]

matrix determined by the measurement errors of the head, W_{p is a given npnp symmetric semi-positive}

de®nite matrix determined by the errors of the prior information onP,kp is a non-negative weighting factor

used to measure the exactness and reliability of the prior information on the unknown parameters compared to those of the information on the models and head measurements.

The least-squares penalty method can be written as

min

is the penalty term to enforce the discrete governing equations,W_{gis a symmetric positive de®nite weighting}

matrix, and kg is a weighting factor to control the

en-forcement of the discrete governing equations. In this paper we will setW_gto beI_n, i.e., thenth-order identity matrix.

The least-squares penalty method objective function in [2] is the special case of (19) when U_ˆI_n. Such a choice ofUwill cause non-uniformity of the components ofRgbecause the matrixAin (15) has irregular weighting

e€ects due to the great value range in the elements ofA_.

In order to analyse the weighting e€ects ofA_whenU

andW_gequalI_{n, we de®ne 0<r1}6r26 6rnas the eigenvalues of the matrixATA _{andVi (1}6i6n) as the corresponding unit orthogonal eigenvectors. Then

V_{:ˆ …V1;. . .;Vn† …21†}

is an orthogonal matrix. De®neRh :ˆHÿAÿ1F as the

computational residual vector between the heads of least-squares method and those of direct method. Then from (15), (20) and (21) we have

Eq. (22) implies that the weighting factor of the ith component of VT_Rh is k2gri, 16i6n. Due to the

non-uniformity of the spatial discretization and the large range of the transimissivityTvalues, the value range in the elements of A_{is great. This will result in large}

con-dition number ofATA, namely the ratiorn=r1[20]. For

instance, in the simple synthetic examples in [2], our calculations show that the largest and smallest diagonal elements ofATAare respectively 4:5105_{and 37.5 with}

the latter corresponding to the node at the ``northeast corner'' of the domain in the example (Fig. 1 in [2] or this paper). It is well known thatr1is less than or equal

to the smallest diagonal element of ATA_andrngreater

than or equal to its greatest diagonal element. Hence the ratio rn=r1P4:5105=37:5ˆ1:2104. This explains

the interesting phenomenon in the numerical examples in [2] that the node with the largest residual is always at the northeast corner of the square domain, because its weighting factor corresponds to the smallest eigenvalue of ATA_.

Hence we reach the following conclusion: only by minimizing (19) with a scaling matrix U that can ap-propriately rectify the non-uniform or irregular weighting e€ects of ATA on the components of the re-sidual vector Rg, one can get a scaled ``stable'' residual

vector Rg independent of the wide value range of the

entries of Aand make it possible to exploit the deeper potential information contained in Rg. Moreover, the

scaling matrix will improve the illcondition caused by the wide value range of the entries ofA_.

Therefore, a suitable scaling matrix U _{is necessary,}

especially for realistic problems with a wide range of entries in A_{. In this paper, three simple forms of the}

scaling matrix U _{are given. Their e€ects will be shown}

and compared in the next section. These forms are: (a) the identity matrix, i.e.,

UˆI_n; …23†

where there is no ``scaling'' or improvement as in [2]; (b) the in®nite row norm form

U_ˆU_1:ˆDiag aÿ111;. . .;aÿ1nn

; …24†

where Diag‰. . .Šindicates a diagonal matrix, andaiiis theith diagonal element ofA_;

(c) the Euclidean row norm (2-norm) form

U_ˆU_E:ˆDiag sÿ1

Although the above forms ofU _{given by (24) and (25)}

scaling the irregular weighting e€ects of A _{with the}

simplest calculation without changing the original pur-pose of the least-squares penalty method. This means that the penalty termJgis corrected adaptively at every

iteration according to the change of the entries of A

mainly caused by changes in the parameter vectorP.

2.2. Algorithm

Let

vˆ …v1;. . .;vn;. . .;vn‡np† T

:ˆ …HT;PT†T: …27†

Then the objective function Jgmp de®ned by (19) is a

non-linear function of the variable vector v. The minimization must be carried out iteratively. Let iit be

the iteration index andv0_{be a given initial estimate ofv.}

At every iteration we wish to determine a correction_Dv

of the current iterative value ofv, i.e.viit_{, such that}

viit‡1_ˆviit_{‡Dv; i}

itˆ0;1;2;. . . …28†

results in a reduction of the objective functionJgmp. The

Gauss±Newton method for obtaining Dv leads to the following linear system of equations [11,12,21]

JTWJ_Dvˆ ÿJTW_R;29†

where

W_ˆDiag‰k2gWg;Wm;kpWpŠ …30†

andRis the entire residual vector composed of Rg, Rm

andRp, i.e.,

J _{is the Jacobian or sensitivity matrix whenvˆv}iit_with

elements

cause the node numbernis usually very large. Hence it will be time-consuming and require substantial com-puter-storage to solve (29) directly. Note that (29) is equivalent to the following linear least-squares system

min

square root or Cholesky decomposition of the positive de®nite matrixW_{, i.e.,}

WTW _ˆW_:34†

Instead of solving (29) directly, we consider the linear least-squares system (33) which can be solved using the iterative LSQR method [17,18] or the direct QR fac-torization method [10,12]. In our numerical tests both methods are tried. For relatively small n (e.g., 6100)

both work well. When n is larger, the direct QR fac-torization method needs more storage and its speed is comparable to the iterative method. Hence the iterative LSQR method is prefered, because it can use a sparse matrix data structure to store the non-zero elements of

J. Consequently, the node numbern can be as large as that of the traditional method.

In order to solve (33) eciently and accurately, a suitable preconditioning technique is needed. LetCbe a suitable …n‡np† …n‡np† non-singular matrix which can improve the condition number of the matrixW JC _. Then instead of solving (33) directly, we ®rst solve the following system about an …n‡np†-dimensional un-known vector n

The simplest form ofCis a diagonal matrix. We wish to improve the condition number of W JC by choosing a diagonal matrix C_:ˆD_c2 such that all the diagonal el-ements of D_c2JTWJD_c2 is equal to 1. This leads to the tests this simple procedure accelerates the convergence of the iterative LSQR algorithm signi®cantly and it also widens the range of initial values of the unknown parameter P within which our algorithm can converge (see Section 4, Table 1).

The main work in implementing the above algorithm is to calculateJ_{. By calculating directly using (4a), (4b),}

(5)±(9), (10a)±(10c), (11)±(15), (27), (31) and (32), setting the matrixU_{according to the entries in}A_{of the current}

iteration and regarding U _{as a constant matrix}

wheres0

jˆ …os=oh†…xj;yj;hj†, andNi is the serial num-ber set of all the nodes in the neighborhood of the ith node…xi;yi†. For linear aquifer modelsZ_ˆ0because of s1. M_w is an nmn matrix whose ith row is an n

-dimensional vector with itsnith component being 1 and the others 0. Here ni…16i6nm† indicates the serial number of the node located at theith head observation well.I_np is thenpth-order identity matrix.Pis annnp

matrix with elements

pijˆ X

k2N_i

o_aik

o_Yjhk

ˆ

0 if i2D ln 10P

k2N_i

hk P e2Ti\T_j

R e10

Ye

ejsr/_i r/_k dxdy else; (

…41†

where

ej:ˆoY

e

o_Yj ˆ

1 wheneXj; 0 whene6Xj;

16j6np …42†

and Xj is de®ned by (4a) and (4b). Here it is assumed that each triangle element eis contained by one of the subdomainsX1;. . .;Xnp.

Based on the above discussion, the penalty method algorithm can be described as follows:

1. Initialization. Choose a suitableP0as the initial value

ofP. The initial valueH0ofHis obtained by solving

(9) when PˆP0. Set v0ˆ …H0T;P0T† T

, J0ˆ0. Choose

two suitably small numbers dv>0 and dJ >0 as the convergence criteria of the Gauss±Newton itera-tion. Choose two suitable np-dimensional vectors

Pÿ _{and P}‡ _{as the lower and upper bounds of the}

parameter vectorP.

2. Foriitˆ0;1;2;. . .repeat steps 3±5.

3. Implement Gauss±Newton method.

(a)Calculate Jacobian matrixJ_{. CalculateJijj} vˆvi_it …16i6n‡nm‡np;16j6n‡np† using (39)±

(42), while the matrix U _{in (39) is determined}

by the current entries inA. (In numerical exam-ples three forms of U de®ned by (23)±(26) are used and their utilities are compared.)

Table 1

Comparison of the e€ects of di€erent residual-scaling matrixUand the preconditionerCon the computational costa Method Uniform initial trasmissivityT(m2_/day)

1.0 10.0 100.0 200.0

(1)UˆI_n; CˆI_n‡n

p

CPU time (s) ‡1 ‡1 134.4 177.0

GN …Lsqr† ˆSlsqr Failed Failed 9(174)3342)ˆ16360 11(151)3079)ˆ21557

(2)UˆIn; CˆDc2

CPU time (s) ‡1 71.1 51.5 55.0

GN …Lsqr† ˆSlsqr Failed 14(475)710)ˆ8659 11(374)719)ˆ6264 12(180)714)ˆ6695

(3)UˆU₁; CˆIn‡np

CPU time (s) ‡1 51.4 33.0 47.0

GN …Lsqr† ˆSlsqr Failed 14(385)609)ˆ6256 10(377)434)ˆ4034 14(365)441)ˆ5732

(4)UˆUE; CˆIn‡np

CPU time (s) ‡1 51.1 32.8 47.0

GN …Lsqr† ˆSlsqr Failed 14(415)587)ˆ6221 10(370)429)ˆ3995 14(374)438)ˆ5728

(5)U_ˆU_1; C_ˆD_c2

CPU time (s) 32.6 25.8 16.3 23.5

GN …Lsqr† ˆSlsqr 18(199)279)ˆ3974 15(205)231)ˆ3142 10(187)206)ˆ1980 14(197)207)ˆ2846

(6)UˆU_E; CˆD_c2

CPU time (s) 31.5 25.5 17.9 23.0

GN …Lsqr† ˆSlsqr 18(193)265)ˆ3840 15(202)228)ˆ3104 11(183)203)ˆ2175 14(195)203)ˆ2798

(7) B & Y:

CPU time (s) 18.6 16.7 19.5 23.0

BY 12 11 13 15

(8)C&N:CN 71 54 57 90

a

GN: total Gauss±Newton iterations.Lsqr: LSQR iterations within the Gauss±Newton iteration. It di€ers for di€erent Gauss±Newton steps, hence only its range is listed in parentheses. Every iteration of LSQR includes 3…n‡np‡nm† ‡5…n‡np†multiplications and two products between matrices and vectors de®ned byW JM nand…W JM †T

(b)CalculateDv. CalculateDvby solving the lin-ear least-squares system (35) and (36) using itera-tive LSQR algorithm [17,18].

(c) Updatev. Calculateviit‡1 _{using (28).}

4. Verify the convergence. Calculate the value of Jiit‡1:ˆJgmp using (19). If

jJiit‡1ÿJiitj<dJ or jDvj1<dv …43†

holds, stop, wherejDvj₁indicates the in®nity norm of

Dv. else set iit:ˆiit‡1.

5. Implement the upper and lower bound constraints to P. LetPiit‡1_{be the last correspondingn}

pcomponents of

viit‡1_{. Judge if P}ÿ_6Piit‡1_6P‡ _{holds, where the}

in-equality refers to the corresponding components of the three vectors. If there are components of Piit‡1

which are less than their lower bounds or greater than their upper bounds, then set them to equal their lower or upper bound values, respectively. Then set the last correspondingnp components ofviit‡1 to equal those

ofPiit‡1 _{and go to step 3.}

3. Numerical examples

3.1. Example of linear aquifer model

In order to demonstrate the utility of our algorithm, the steady-state test problem used by Carrera and Neuman in [6] and Bentley in [2] will be solved. It can be described by specifying the factors in the aquifer model (1)±(3) as follows (see [2,6]):

X. A square domain of 6000 m6000 m (see Fig. 1).

Cd and hd.Cd is the southern side of the square

do-main, hd100 m.

Cnand q. The other three sides of the domain areCn.

On the western side the ¯uxqˆ0:25 m2_{/day, on the}

northern and eastern sides there is no ¯ow, i.e., qˆ0:

TransmissivityT…x;y†. The domainXis divided into npˆ9 constant-transmissivity zones of

2000 m2000 m with the corresponding transmis-sivity values Ti…16i69† ranging from 5 to 150 m2_{/day (see Fig. 1).}

s…x;y;h†,B…x;y†andhr. The aquifer is assumed to be

con®ned, i.e., s…x;y;h† 1. The stratum overlying and underlying the aquifer are supposed to be im-pervious, i.e., B…x;y† 0. HenceBhr0.

Q…x;y†. In the upper half part of the northern three zones Qˆ0:27410ÿ3 _{m/day, in the lower part of}

them Qˆ0:13710ÿ3 _{m/day, in other six zones}

Qˆ0.

Head measurements. They are available at the 18 ob-servation wells shown as points in Fig. 1.

In computation, the ¯ow region is discretized into 1212 squares, each of which is divided into two

tri-angle elements. All the head observation wells coincide with the discrete nodes. In order to obtain head mea-surements at observation wells, as in [2,6], the ``true'' head values at discrete nodes, which range from 100.0 m on the southern Direchlet boundary to 205.88 m at the north-east corner, were numerically calculated using the true transmissivity values shown in Fig. 1. Then head values at the 18 observation wells were corrupted by uncorrelated Gaussian noise of variance 1. Conse-quently the head measurement weighting matrix W_m in (17) is set to be thenmth-order identity matrix as in [2,6].

To test the utility of our algorithm, the regulation termJpcontaining prior information onPis not used in

all our calculation, i.e.,kp0. ButP is restricted from

below and above by choosing Pÿ _{with uniform}

com-ponents log 0:01ˆ ÿ2 andP‡_{with uniform components}

log 103_{ˆ3. The Gauss±Newton iteration stop criteria}

are chosen to be dJ ˆ10ÿ7anddvˆ10ÿ4.

The numerical tests are composed of two parts. The ®rst part is designed to investigate the e€ects of di€erent forms of the residual scaling matrixU_{and the}

preconditionerC _{on the calculation speed and}

numeri-cal results. For this purpose, three di€erent forms ofU, i.e., I_n, U₁ and U_E de®ned by (23)±(26), two di€erent forms ofC, i.e.,I_n‡np andD_c2 de®ned by (37) and (38), and four di€erent but uniform initial values of Ti…16i6npˆ9† as in [6], namely, 1, 10, 100 and

200 m2_{/day, are used combinatorially. The weighting}

factor kg is respectively ®xed to be 1:0 when UˆIn

considering the great equivalent weighting e€ects ofATA

and to be 10.0 for the other two forms of U_{de®ned by}

(24)±(26) because of their counteracting the weighting e€ects of ATA_{. Hence there are altogether}

423ˆ24 di€erent cases.

In Table 1 the relevant data on the computational costs corresponding to the 24 cases are summarized.

The ®rst eight rows corresponding to methods 1±4 show that if either the discrete governing equation re-siduals are not scaled (i.e.,U_ˆI_n), or no precondition technique is adopted (i.e., C_ˆI_n‡n

p), our algorithm

performs poorly, e.g., the computation cost is higher than that of the traditional method, and what is worse is the Gauss±Newton iterations fail to converge for some initial values of the unknown parameters. But the situ-ation is always improved with the gradual improvement ofU _andC_.

The successive four rows corresponding to methods 5 and 6 demonstrate the improvements due to setting

U_ˆU_1orU_{E (Eqs. (24)±(26)) and}C_ˆD_{c2 (Eqs. (37)}

and (38)). The range of initial values ofTfor which the Gauss±Newton iteration converges has increased, there are no longer ``Failed'' calculations for all the four dif-ferent initial values ofT; and the computational e€ort is also signi®cantly reduced. For example, when the initial values of Pare uniform log 100 m2_{/day, S}

lsqr, the

num-ber of LSQR iterations is reduced from 16 360 for

U_ˆI_n and C_ˆI_n‡np to 1980 for U_ˆU₁ and

C_ˆD_c2. And the CPU time is reduced proportionally. In the next two rows that begin with ``B & Y'' we show the computational e€ort of the method described by Becker and Yeh [3,21], which uses a Gauss±Newton method to minimize the objective functionJtde®ned by

(16) means of by the parameter perturbation and nu-merical di€erence quotient method. Comparison of the CPU times and the number of iterations required by the ``B & Y'' method (i.e.,BY) to the results from the im-proved scaling matrixes demonstrates that the compu-tational cost of our algorithm is reduced to a level similar to that of the traditional method. This trend shows potential of our algorithm to solve inverse problems more quickly than the traditional method on the basis of ®nding more suitable forms ofU andC.

In the last row,CN, the number of iterations required by the ``conjugate gradient algorithm coupled with Newton's method'' used by Carrera and Neuman in [6] to minimize the traditional objective functionJtde®ned

by (16), is quoted. One can see thatGN, the number of Gauss±Newton iterations required by our algorithm, is less thanCN. Unfortunately, the computational costs of our algorithm and that used in [6] in every iteration minimizing the objective functions are dicult to com-pare due to the lack of relevant data.

The ®ve Failed cases in Table 1, which mean that the iterations fail to converge, can be improved so that the iterations converge if the Marquardt's method [7,12] is incorporated in our algorithm. But it takes more itera-tions and CPU times than those of the other unfailed cases listed in Table 1 where the Marquardt's method is not used. The detail will not be stated here due to space limitation.

In Table 2 the numerical results corresponding to the 24 cases mentioned above and to the traditional method by Becker and Yeh [3,21] are summarized. Theoretically, (i.e., if there are no round o€ errors,) di€erent forms of non-singular matrix C do not in¯uence the calculated results. The numerical results support this fact: the op-timization results neither depend on the initial values of

T, nor on the change of C up to ®ve signi®cant ®gures. Hence, only results corresponding to the three di€erent forms of U are listed in Table 2, and they are valid for all the combinations of di€erent initial values of Tand forms of C _{in Table 1 which are not Failed. The last}

column is the results calculated by using the traditional method of Becker and Yeh.

The ®rst four rows in Table 2 give the values ofJt, the

traditional objective function that appeared in (19), of Jg, the weighted square sum of the discrete governing

equation residuals de®ned by (20), of Jgmp, the penalty

method objective function de®ned in (19) and of the ratio Jg=Jgmp successively. It is well-known that greater

the weighting factors of residuals in a least-squares system, smaller these residuals will be. Hence the ratio Jg=Jgmp can be regarded as a measure of the weighting

e€ects of kgU on the discrete governing equation

resid-uals. According to this measure the weighting e€ects of kgUˆ10:0UE are the weakest, and those of

kgUˆ1:0In the strongest.

The data listed under ``Identi®ed T '' and ``The Statistic Data of logT'' demonstrate that the numerical results ofTdi€er very slightly when they are calculated by using our algorithm with the three di€erent choices of

kgU and when they are obtained by the traditional

method of Becker and Yeh. For example, r…DP†, the standard deviation of the residuals between the com-puted and true log-transmissivity, ranges only from 2:7810ÿ3 _{to 3:3310}ÿ3_{, and SSE}

DP†, the standard square error between the computed and true log-trans-missivity, from 2:8110ÿ3_{to 3:3410}ÿ3_.

One may conjecture that the discrete governing equations will be less accurately satis®ed at the nodes coincident with the head observation wells than at other nodes due to the measurement error at the head obser-vation wells. This conjecture can be seen and understood clearly if ``The statistic data of Rgm and Rgr'' are

com-pared row by row, where Rgm is an nm-dimensional

vector composed of the components ofRgwhose indices

equal those of the nodes coincident with the head ob-servation wells andRgris an…nÿnm†-dimensional vector

composed of the rest of the components of Rg except

those of Rgm. For example, SSE…Rgm†, the standard

square error ofRgm, is much greater than SSE…Rgr†for

all the three choices of kgU. The ratio SSE…Rgm†/

SSE…Rgr† ranges from 2.16 to 9.69. The other pair of statisticsjRgmj₁andjRgrj₁, the respective in®nity norms ofRgmandRgr, also show the similar phenomenon with

is to say, statistically, the residuals of the discrete gov-erning equations discretized at the head-observation-well nodes are much greater than those of the discrete governing equations at other nodes. Therefore, the above-mentioned conjecture is true in the sense of statistics.

If ``The statistic data of Rgm andRgr'' are compared

column by column, the following interesting facts can be found: among the three group values listed in the three columns, the values ofk2_gSSE…Rgm†andk2_gSSE…Rgr†in the ®rst column corresponding tokg Uˆ1:0In are always

the smallest, but the values of jRgmj₁ andjRgrj₁ in the ®rst column are always the biggest. These results imply that the components of the residual vector Rg when U_ˆI_n are irregularly distributed and they have been improved whenUis set to be U₁ orU_E.

The second part of our numerical tests are designed to investigate the e€ects of the weighting factorkgon the

numerical results. To this end the initial values ofTare ®xed to be uniform 100 m2_{/day, UˆU}

E andCˆDc2,

then our algorithm is used to minimize the penalty method objective functionJgmp corresponding to

di€er-entk2_g, which is respectively set to be 10:0; 102_{; 10}4_{; 10}6

and 108_{. The results are listed in Table 3.}

First let us look at the data listed above ``The com-putation cost'' in Table 3. If they are compared column by column, the following facts can be seen:

1. Ask2_gincreases from 10.0 to 106_{, the values ofJ} t and

Jgmp tend to the same constant 16.2 increasingly, and

the values ofJgdecreases with an approximate rate of

kÿ2_g . So the ratioJg=Jgmpdecreases also with the same

approximate ratekÿ2_g . In fact, it has been theoretically proven that Jt, Jgmp and Jg have the asymptotic

properties

lim

kg!‡1Jtˆkg!‡1lim JgmpˆC1; kg!‡1lim k 2

gJgˆC2;

where C1is the value ofJt corresponding to the

tra-ditional-method solution to problem (16),C2is a

non-negative constant that vanishes if and only if C1

vanishes [13]. The numerical results in Table 3 show these properties clearly when k2g6106 with the

con-stants C1 and C2 approximating 16.2 and 94:1,

re-spectively. Unfortunately, by k2gˆ108, they begin to Table 2

Comparison of the numerical results of our algorithm and of the traditional methoda

U I_n U₁ U_E Traditional

kg 1.0 10.0 10.0 method

Jt 16.0 14.8 14.5 16.2

Jg 0.115 0.674 0.839 0.0

Jgmp 16.1 15.5 15.3

Jg=Jgmp 0.715% 4.35% 5.47% 0.0%

True T Identi®edT (m2_/day)

T1…150† 158.71 159.10 158.51 160.66

T2…150† 137.20 134.04 133.32 136.71

T3…50† 51.608 51.667 51.652 51.679

T4…150† 123.02 120.77 120.18 123.25

T5…50† 49.756 49.973 50.016 49.758

T6…15† 15.516 15.482 15.474 15.511

T7…50† 66.537 67.600 67.651 67.256

T8…15† 15.028 14.998 14.991 15.025

T9…5† 5.0254 5.0329 5.0329 5.0310

The statistic data oflogT

E…DP† 5.8910ÿ3 _4.9010ÿ3 _4.2410ÿ3 _7.0510ÿ3

r…DP† 2.7810ÿ3 _3.2510ÿ3 _3.3310ÿ3 _2.9610ÿ3 SSE…DP† 2.8110ÿ3 _3.2710ÿ3 _3.3410ÿ3 _3.0110ÿ3

The statistic data ofRgmand Rgr k2

gSSE…Rgm† 1.3110ÿ3 2.0110ÿ2 2.5010ÿ2 0.0

k2

gSSE…Rgr† 6.0510ÿ4 2.0810ÿ3 2.5810ÿ3 0.0

SSE…Rgm†=SSE…Rgr† 2.16 9.66 9.69

jRgmj1 0.140 0.0396 0.0373 0.0

jRgrj1 0.0458 0.0170 0.0190 0.0

jRgmj1=jRgmj1 3.06 2.33 1.96

a_{Cand the initial values ofTcan be any one of the combinations in Table 1.}

DPis the di€erence vector between the computed and true values of the parameter vectorPde®ned by (11).Rgmis annm-dimensional vector composed of the components of the discrete governing equation residual vector

Rg (see (15)) whose indices equal those of the discrete nodes coincident with the head observation wells.Rgr is an…nÿnm†-dimensional vector composed of the rest of the components ofRgexcept those ofRgm. For an arbitrarym-dimensional vector:ˆ …e1;. . .;em†

T

, which can be set to beDP

or Rgm or Rgr, respectively, the vector functionsE…†, r…†, SSE…† andj j1 are de®ned as follows: E…†:ˆ …1=m† Pm

iˆ1ei indicates the mean; r…†:ˆ …1=m†Pm

iˆ1…eiÿE…†† 2

is the standard deviation; SSE…†:ˆ …1=m†Pm

lose part of their ®delity due to the serious deterio-ration of the condition number of the weighting matrixW de®ned by (30).

2. When k2_gˆ10:0, Jg takes comparatively large value

4.06 and the ratio Jg=Jgmp reaches 36.9%, implying

that much of the head measurement noise is ``ab-sorbed'' by the discrete governing equation residuals. This leads to the large errors in the estimated values of Ti…16i69† (see the second column listed under

k2_gˆ10:0). When k2_gP102_{, accurate and stable}

esti-mates ofTi…16i69†are obtained due to the sucient enforcement of the discrete governing equations, even in the case ofk2_gˆ108_{. They also di€er very slightly}

from each other. Moreover, if one compares our re-sults ofTi…16i69†when 104_6k2

g610

8_{with the}

tra-ditional method results ofTilisted in the last column of Table 2, one can easily see that eachTi…16i69†

always has the same ®rst three signi®cant ®gures with only one exception ofT7ˆ67:5 whenk2gˆ108.

3. The facts stated in the above paragraph are also sup-ported by the data listed under ``The statistic data of logT''.

In the last two rows the ``CPU Time'' and Slsqr, the

sum of LSQR iterative numbers in all the Gauss±New-ton iterations, are listed for di€erent values ofk2_g. One may see that although they di€er from each other, the average CPU time is only 18.9 s, which is a little less than 19.5 s, the CPU time of the traditional method corresponding to the method 7 B & Y and initial T ˆ100 m2_{/day in Table 1. Particularly, whenk}2

gˆ104,

the CPU time of our algorithm is only 11.1 s, 56.9% of that of the traditional method.

In short, there is a suciently wide workable range of

kg values within which the penalty method algorithm

performs well enough to give accurate estimations of the unknown parameterTwith low computational cost just like the traditional method.

3.2. Example of non-linear aquifer model described by Boussinesq equation

In order to demonstrate the advantage of our algo-rithm that it can solve inverse problem of non-linear aquifer models naturally and quickly, an uncon®ned quasilinear aquifer model described by the Boussinesq equation [1,19] is arti®cially designed by specifying the factors in the aquifer model (1)±(3) as follows:

1. The aquifer-choice function s…x;y;h† ˆ …1=M† …hÿhb…x;y††, where hb:ˆ …ÿ1=400†…x‡y†, M :ˆ

160:0 m.

2. The leakage parameters B…x;y†:ˆ ÿ5:010ÿ7

dayÿ1_{; h}

rˆ0 m.

3. All the other factors, such asX,Cd andhd,Cn andq,

transmissivity T…x;y† determined by the average thickness of the uncon®ned aquifer, rechargeQ…x;y†

and watertable measurement locations are the same as those described in the example of Section 3.1 and Fig. 1.

As before, the ¯ow region X is discretized into 1212 squares each of which is divided into two

Table 3

Comparison of the e€ects of di€erent weighting factorkgon the numerical resultsa k2

g 10:0 10

2 ₁₀4 ₁₀6 ₁₀8

Jt 6.95 14.5 16.2 16.2 16.2

Jg 4.06 8.3810ÿ1 9.3910ÿ3 9.4110ÿ5 6.6410ÿ1

Jgmp 11.01 15.3 16.2 16.2 16.9

Jg=Jgmp 36.9% 5.47% 0.058% 0.000581% 3.94%

True T Identi®edT (m2_/day)

T1…150† 130.14 158.51 160.68 160.70 160.76

T2…150† 104.24 133.32 136.69 136.72 136.71

T3…50† 50.335 51.652 51.675 51.675 51.677

T4…150† 94.447 120.18 123.12 123.15 122.96

T5…50† 50.690 50.016 49.754 49.752 49.756

T6…15† 15.254 15.474 15.511 15.511 15.511

T7…50† 70.870 67.651 67.326 67.325 67.495

T8…15† 14.779 14.991 15.024 15.024 15.021

T9…5† 5.0383 5.0329 5.0311 5.0311 5.0311

The statistic data oflogT

E…DP† )2.8510ÿ2 4.2410ÿ3 7.0110ÿ3 7.0310ÿ3 7.0910ÿ3

r…DP† 9.4410ÿ3 _3.3310ÿ3 _2.9510ÿ3 _2.9510ÿ3 _2.9910ÿ3 SSE…DP† 1.0310ÿ2 _3.3410ÿ3 _3.0010ÿ3 _3.0010ÿ3 _3.0410ÿ3

The computation cost

CPU time (s) 14.7 17.9 11.1 25.4 25.5

GN …Lsqr† ˆSlsqr 12(138)156)

ˆ1782

11(183)203)

ˆ2175

7(175)205)

ˆ1354

14(104)325)

ˆ3089

20(71)506)

ˆ3097 a

triangle elements. In order to obtain watertable mea-surements at observation wells, the true watertable val-ues at discrete nodes were numerically calculated using the true transmissivity values shown in Fig. 1. Then the watertable values at the 18 observation wells were cor-rupted by uncorrelated Gaussian noise of variance 1.

W_m,kp,Pÿ andP‡,dJ anddv are also the same as the

preceding example. The numerically calculated true watertable in X ranges from 100.0 m on the southern side to 168.43 m at the northeast corner ofX.

In Table 4 numerical results of T corresponding to two di€erent methods are compared. In the ®rst column the true transmissivities of the nine zones are listed in parentheses. The second column is the results of the penalty method algorithm when U_ˆU_E, C_ˆD_{c2 and}

kgˆ10:0. The third column is the results of the

tra-ditional method described by Yeh and Becker in [3] and [21]. One can see that the two results are almost equal and they are very close to the true values ofT. The data of E…DP†, r…DP†and SSE…DP† show that there are no

signi®cant di€erences between the two sets of values of identi®ed T.

In Table 5 the computational costs of the above mentioned two methods are compared. The ®rst two rows are the results of the penalty method algorithm, the next two rows the results of the traditional method by Becker and Yeh.

From Table 5 one can see the following facts: 1. For the non-linear Boussinesq aquifer model and the

four di€erent initial uniform values of Ti…16i69†, bothSlsqr, the sum of LSQR iterative numbers in all

the Gauss±Newton iterations, and the CPU time of our algorithm do not increase signi®cantly compared with those when the aquifer model is linear, i.e., those corresponding to method 6 in Table 1. For one case both even decrease, but the CPU time of the tra-ditional method by Becker and Yeh increases by 9±12. 16 times compared with that when the aquifer model is linear, i.e., that corresponding to ``B & Y'' in Table 1. On the other hand, the comparison of the CPU times of the two methods in Table 5 shows that our algorithm runs 6.5±8.6 times as fast as the traditional method. These facts demonstrate the po-tential of our algorithm to solve the inverse problems of more complicated non-linear aquifer models natu-rally and quickly on the basis of ®nding suitable forms of U _and C_{. The main reason of these facts}

is that the computational procedure of our algorithm is the same regardless of the linearity or non-linearity of the aquifer models with respect to h.

2. Unfortunately, our algorithm Failed when the initial values of T are uniform 1 m2_{/day. How to improve}

this by looking for more suitable forms of U _andC

or other techniques remains interesting open prob-lem. However, the incorporation of the Marquardt's method [7,12] in our algorithm can improve the cal-culation so that the iteration converge. But it takes more iterations and CPU times than those of the other three unfailed cases listed in the ®rst two rows in Table 5 where the Marquardt's method is not used. The detail will not be stated here due to space limitation.

Table 4

Comparison of the numerical results of our penalty method algorithm and of the traditional method by Becker and Yeh for an uncon®ned quasilinear aquifer modela

Methods Penalty method (kgUˆ10UE; CˆDc2)

B & Y (traditional)

True T Identi®edT (m2_/day)

T1…150† 154.70 157.71

T2…150† 115.71 120.98

T3…50† 53.099 53.425

T4…150† 128.80 133.71

T5…50† 54.394 53.967

T6…15† 15.758 15.796

T7…50† 49.628 58.607

T8…15† 14.923 15.119

T9…5† 5.0132 5.0164

Statistic data oflogT

E…DP† )9.5310ÿ3 4.0710ÿ3

r…DP† 2.1110ÿ3 _2.0810ÿ3 SSE…DP† 2.2010ÿ3 _2.1010ÿ3 a_{The initial values of}

Tcan be any one of those in Table 5. The signs

E…DP†, r…DP† and SSE…DP† have the same meanings as those in Table 2.

Table 5

Comparison of the computational cost of our algorithm and of the traditional method of Becker and Yeh for an uncon®ned quasilinear aquifer modela

Method Uniform initial transmissivityT(m2_/day)

1.0 10.0 100.0 200.0

Penalty (kgUˆ10UE;CˆDc2)

CPU time (s) ‡1 22.0 31.0 35.0

GN …Lsqr† ˆSlsqr Failed 13(197)216)ˆ2657 19(170)215)ˆ3841 21(178)226)ˆ4317

B & Y (traditional)

CPU time (s) 244.0 190.0 201.0 230.0

BY 16 13 14 16

a

4. Conclusion

The least-squares penalty method objective function described in [2] is generalized by scaling the discrete governing equation residuals with a scaling matrix U

determined by the current coecient matrix of the dis-crete governing equations adaptively. The utilities of three di€erent simple forms of U are shown and compared.

The generalized Gauss±Newton method [11,12,21] is used to minimize the generalized penalty method objective function. Every Gauss±Newton iteration is formulated by a linear least-squares system whose unknowns are the corrections of both the parameters and the groundwater heads on all the discrete nodes. An iterative LSQR algorithm for sparse linear least-squares problems [17,18] is chosen to solve this least-squares system. Due to the sparse iterative data structure of the LSQR algorithm, the size of the discrete governing equations can be as large as that of the traditional method. A simple but ecient preconditioner C is proposed to improve the condition number of the least-squares system.

Numerical results show that even simple and natural diagonal forms ofU and C have signi®cant e€ects on: (1) widening the range of the initial values of the un-known parameters within which the minimizing itera-tion can converge; (2) reducing the computaitera-tional cost in every Gauss±Newton iteration; (3) improving the ir-regular distribution of the discrete governing equation residuals on the nodes.

The reducing trend of computational cost with the improvement of U _and C _{and the suciently wide}

range of the workable values of kg shown in

nu-merical tests demonstrate adequate potential of our algorithm to perform better and faster than the tra-ditional method on the basis of ®nding suitable forms of U and C. Moreover, because the computation procedure of our algorithm is the same regardless of the linearity or non-linearity of the aquifer models with respect to h, numerical example shows that it can solve the inverse problem of a quasilinear Boussinesq aquifer model almost as fast as that of linear ones. This evidence indicates some promise of our algorithm to solve inverse problems of more complicated non-linear aquifer models naturally and quickly.

All the above mentioned potential of our algorithm are ultimately dependent upon the ®ndings of more suitable forms of the scaling matrix U, of the precon-dition matrix C and of the weighting matrixW_g. How to look for them according to the respective distribu-tions of the discrete governing equation residuals and the discrete aquifer model errors on di€erent nodes, to the data structure of A_{, remains interesting open}

problems.

Acknowledgements

Part of this work was done when the author vis-ited the Institute of Mathematics, Computer Science Department of the Federal Armed Forces University in Munich and Chair of Applied Mathematics, Technical University of Munich. The author thanks Prof. Dr. Ulrich Hornung, Prof. Dr. K.-H. Ho€mann and Dr. T. Lohmann for their instruction, Dr. Ste-phan, Dr. Youcef, Miss Christa and Dr. Evelina for their enthusiastic help. The author is also grateful to the public ``Meschach Library'' which is available over Internet/AARnet via netlib, or at the anony-mous ftp site thrain.anu.edu.au in the directory pub/ meschach. This library provides us a complete set of C routines for performing calculations on matrices and vectors. This work was partly supported by the fund of ``The research of the groundwater in An-shan'' awarded by the science and technology com-mittee of Anshan City. Many thanks to Prof. M.B. Allen and other anonymous referees for their instructions.

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