determination of an optimal regularization factor in system identification with tikhonov...

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERINGInt. J. Numer. Meth. Engng 2001; 51:1211–1230 (DOI: 10.1002/nme.219)

Determination of an optimal regularization factor in systemidenti0cation with Tikhonov regularization for

linear elastic continua

Hyun Woo Park1;†, Soobong Shin2;∗;‡;§ and Hae Sung Lee1;¶

1Department of Civil Engineering; Seoul National University; Seoul; Korea2Department of Civil Engineering; Dong-A University; Busan; Korea

SUMMARY

This paper presents a geometric mean scheme (GMS) to determine an optimal regularization factor forTikhonov regularization technique in the system identi0cation problems of linear elastic continua. Thecharacteristics of non-linear inverse problems and the role of the regularization are investigated by thesingular value decomposition of a sensitivity matrix of responses. It is shown that the regularizationresults in a solution of a generalized average between the a priori estimates and the a posteriori solution.Based on this observation, the optimal regularization factor is de0ned as the geometric mean betweenthe maximum singular value and the minimum singular value of the sensitivity matrix of responses.The validity of the GMS is demonstrated through two numerical examples with measurement errorsand modelling errors. Copyright ? 2001 John Wiley & Sons, Ltd.

KEY WORDS: system identi0cation; Tikhonov regularization; optimal regularization factor; geometricmean scheme; singular value decomposition; nonlinear inverse problem

1. INTRODUCTION

System identi0cation (SI) algorithms have been widely used for the last few decades in thearea of structural engineering to identify mechanical systems [1–3] and to detect damage instructures [4–7]. However, most of the applications have been limited to discrete structuressuch as trusses or frames, and few works on continua are available in the literature [1–4].This paper presents an SI algorithm using the Tikhonov regularization technique for linearelastic continua.

It is well known that system identi0cation (SI) algorithms based on the minimization of theleast-squared error between measured and computed responses suDer from inherent instabilities

∗Correspondence to: Soobong Shin, Department of Civil Engineering, Dong-A University, 840 Hadan-dong,Saha-ku, Busan 604-714, Korea

†Graduate student‡E-mail: [email protected]§Assistant Professor¶Associate Professor

Contract=grant sponsor: Korea Research Foundation

Received 17 December 1999Copyright ? 2001 John Wiley & Sons, Ltd. Revised 28 December 2000

1212 H. W. PARK, S. SHIN AND H. S. LEE

caused by the ill-posedness of inverse problems. The instabilities are characterized by the non-uniqueness and discontinuity of solutions [1–3; 5; 8; 9]. In particular, when measured data arepolluted with noise or when a 0nite element model used for the SI does not represent actualsituations, the instabilities become very severe [5; 9; 10].

To overcome the instabilities of inverse problems, Tikhonov regularization technique, inwhich a suitable regularization function is added to an error function, has been utilized [9; 10].In a regularization technique, the most important issue is to keep consistent regularizationeDect on the parameter estimation, which is controlled by a regularization factor included inthe regularization function. Therefore, it is crucial to determine a well-balanced regularizationfactor in order to obtain a physically meaningful and numerically stable solution of an inverseproblem with the regularization technique. Extensive researches have been performed on char-acterizing ill-posedness and determining the regularization factor in linear inverse problems[10–12]. Although some intuitive schemes have been proposed [1; 2; 5; 13; 14], works onnon-linear inverse problems are limited. The sources of instabilities in SI algorithms, whichare a type of non-linear inverse problems, are investigated by examining the singular valuesdecomposed from the sensitivity matrix of responses.

This paper illustrates that the SI algorithm with the regularization results in a solution ofa generalized average between the a priori estimates and the a posteriori solution. Here,the a priori estimates represent known baseline properties of system parameters, and thea posteriori solution denotes the solution obtained by given measured data. A new idea ofthe geometric mean scheme (GMS) is proposed to select optimal regularization factors innon-linear inverse problems for linear elastic continua. In the GMS, the optimal regularizationfactor is de0ned as the geometric mean between the maximum and minimum singular valuesfor balancing the maximum and minimum eDects of the a priori estimates and the a posteriorisolution in a generalized average sense.

Two examples are presented to demonstrate the eDectiveness of the GMS. Two types oferrors associated with the identi0cation, i.e., measurement errors and modelling errors, areconsidered. Detailed discussions on the behaviours of the GMS are presented and comparedwith identi0cation results from other schemes.

2. SYSTEM IDENTIFICATION PROBLEMS IN LINEAR ELASTIC CONTINUA

Figure 1 shows a two-dimensional 0nite body, for which the geometry and the boundaryconditions of the exterior boundary are known. Prescribed traction is applied on It , and dis-placement is speci0ed on Iu. It is assumed that only small parts of a given body have diDerentmaterial properties from the original, known material properties, which will be referred to asbaseline properties hereafter. The variation in the material properties may be caused by eitheran inclusion of a foreign material or degradation of material. Damage such as a crack canalso be approximately represented by reducing the elastic material properties around damagewithout modifying the 0nite element model [4].

The 0nite element method is used to obtain the discretized stiDness equation of the body

K(x)ui=Pi (1)

where K; x; ui and Pi are the stiDness matrix, system parameters, nodal displacement vectorof the structure, and the equivalent nodal load vector of the ith load case, respectively. To

Copyright ? 2001 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 2001; 51:1211–1230

DETERMINATION OF AN OPTIMAL REGULARIZATION FACTOR 1213

Figure 1. Problem de0nition and element groups.

represent stiDness properties of the body, the given domain is divided into a 0nite number ofsubdomains as shown in Figure 1, and the Young’s moduli of the subdomains are selectedas the system parameters. The Poisson’s ratios of all the subdomains are 0xed at the baselineproperty. Each subdomain may consist of a 0nite element or a prede0ned element group,which contains several 0nite elements of the same system parameter. An adaptive parametergrouping technique [4; 6] may be used to form element groups. However, since the adaptiveparameter grouping technique is out of the scope of the current paper, it is assumed that anelement group for each subdomain is prede0ned. The number of prede0ned element groupsshould be selected so that rank-de0ciency of the sensitivity matrix does not occur.

The unknown system parameters of the 0nite body are identi0ed by minimizing a least-squared error between computed and measured displacements at some discrete observationpoints located on It as shown in Figure 1.

Minimizex

ME =12

nlc∑i=1

‖uci (x)− umi ‖2 subject to R(x)60 (2)

where uci ; umi ; R and nlc are the computed displacement vector by a 0nite element model, themeasured displacement vector at observation points for the ith load case, a constraint vector forthe system parameters, and the number of load cases, respectively, with ‖ · ‖ representing theEuclidean norm of a vector. Linear constraints are used to set physically signi0cant upper andlower bounds of the system parameters [6]. The minimization problem de0ned in Equation (2)is a constrained non-linear optimization problem because the displacement vector uci is a non-linear implicit function of the system parameters x.

The error function de0ned in Equation (2) is rewritten in a single vector form as

ME = 12‖Uc(x)−Um‖2 (3)

where Uc and Um are vectors obtained by arranging the vectors of the computed displace-ments and the measured displacements for each load case in a row. The error function isnormalized by the square of the Euclidean norm of the measured displacement vector, whilesystem parameters are normalized with respect to the corresponding baseline properties. The



normalized quantities corresponding to ME , Uc, Um and x are denoted as �E , U, PU and ^,respectively. The normalized minimization problem is written in the following form:

Minimize^

�E =12‖Uc(x)−Um‖2

‖Um‖2 =12‖U(^)− PU‖2 subject to R(^)60 (4)

The solution of the above minimization problem is obtained by solving the followingquadratic sub-problem iteratively:

MinimizeQ^

[12Q^

THk−1Q^−Q^TSTk−1U

rk−1

]subject to R(^k−1 + Q^)60 (5)

where the subscript k denotes the iteration count, and Sk−1 and Hk−1 are the sensitivitymatrix of Uk−1 and the hessian matrix of the error function, respectively. The displacementresidual Urk−1 is de0ned as Urk−1 = PU − Uk−1, and Q^ is the increment of normalized systemparameters at the current iteration step. The hessian matrix in Equation (5) is approximated bythe Gauss–Newton hessian to avoid the computational complexity of calculating the second-order sensitivities of displacements.

Hk−1 ≈STk−1Sk−1 (6)

To simplify the expressions, the subscript (k − 1) of all the variables in the incrementalformulation presented hereafter is omitted.

The linear constraints of Equation (5) on the upper and lower bounds of system parame-ters can alleviate the ill-posedness of inverse problems to some extent. However, the inher-ent instabilities of an ill-posed problem cannot be suppressed in general by imposing linearconstraints on the upper and lower bounds of system parameters, which has been reportedby several researchers [5; 15–17]. This is because the instabilities of inverse problems arisefrom the characteristics of the hessian and the errors in measurements. Therefore, the insta-bilities of the SI algorithm should be investigated before the constraints are imposed, andthus the constraints are not considered for discussions on the stability of the SI algorithmhereafter. In other words, the instabilities of the SI algorithm are presented in the originalsolution space, not in the solution space reduced by the constraints for the remaining parts ofthis paper.

The 0rst-order necessary optimality condition for Equation (5) without the constraints isgiven by the following linear equation:

STSQ^u − STUr =0 (7)

Here, Q^u denotes the solution of the unconstrained quadratic sub-problem of Equation (5).By the singular value decomposition (SVD) [18], the m× n sensitivity matrix S with m¿ncan be written as a product of an m × n matrix Z, an n × n diagonal matrix �, and thetranspose of an n× n V as expressed in Equation (8). In the de0nition, m is the total numberof measured degrees of freedom for all the applied loads and n is the number of systemparameters

S=Z�VT (8)



where

ZTZ= In

VTV=VVT = In

�=diag(!j)

(9)

in which In is the identity matrix of order n, and !j is a singular value of S which hasthe descending order of !n=!max¿ · · ·¿!2¿!1 =!min¿0. The columns of Z are referredto as the left singular vectors (LSV) while the columns of V are referred to as the rightsingular vectors (RSV). In case some of the singular values are zero, the sensitivity matrixS is rank-de0cient, and Equation (7) cannot be solved in the usual manner. To avoid therank-de0ciency of the sensitivity matrix, the number of independent measured displacementsshould be larger than the number of system parameters; i.e. m¿n. This paper assumes thatthe sensitivity matrix always possesses a suTcient rank.

Using the properties de0ned in Equation (9), the solution of Equation (7) is expressed as

Q^u=V diag(

1!j

)ZTUr (10)

Equation (10) is de0ned as the a posteriori solution increment because it is determined purelyby the measured displacements and the analytical model of a given structure without utilizingthe a priori information on the system. The term ZTUr in Equation (10) is often referred toas the Fourier coeTcients [11].

The displacement residual Ur cannot converge to zero for noise-polluted measurementsbecause noisy displacements usually contain incompatible components that cannot be obtainedjust by adjusting the system parameters of a mathematical model. In that case, in order tomake Q^u converge to zero, each column of Z should become orthogonal to Ur in an absolutesense through the minimization iterations. Even so, the optimization iteration may diverge ifsome of the singular values become smaller than the corresponding Fourier coeTcients duringiterations.

There are two sources of noise when applying an SI algorithm; i.e. measurement errors andmodelling errors. The former represents noise caused by sensitivity of sensors or misreading oftest equipment during actual measurements. The latter occurs due to the discrepancy betweena real structure and its mathematical model employed in the SI. For example, in case a prioriinformation is not available on internal Uaws like cracks in a structure, such Uaws cannot betaken into account in the 0nite element model used for SI. The modelling errors cannot bereduced in the minimization with a prede0ned 0nite element model. The measurement errorsare probabilistic while the modelling errors are systematic in nature.

The measured displacement can be theoretically decomposed into the noise-free displace-ment PUf and the noise vector e as follows:

PU= PUf + e (11)

The modelling errors, which lead to errors in the stiDness matrix, result in noise in the com-puted displacements, but not in measured displacements. However, it is still possible to em-ploy Equation (11) by de0ning the noise-free displacements as the best-0tting displacementswith measured ones obtainable by adjusting prede0ned system parameters in the mathematical



model. This decomposition of displacement cannot be achieved explicitly, and is purely con-ceptual.

Substitution of Equation (11) into Equation (10) leads to the following expression:

Qû=V diag(

1!j

)ZT( PUf − U) +V diag

(1!j

)ZTe=Q^fu + Qêu (12)

where Q^fu and Qêu represent the solution increments contributed by the noise-free displace-ment residual and by the noise in measurement, respectively. Unless noise in measurementdata is negligible or the noise vector is nearly orthogonal to the LSV, the solution incrementfor the noisy measurement deviates from the noise-free solution mainly due to the second termof Equation (12). In particular, the components of ZTe associated with small singular valuesamplify the deviation more severely. The solution is likely to lose physical signi0cance dueto the accumulation of solution components ampli0ed by physically meaningless noise duringoptimization iterations. A small change in noise may yield a totally diDerent solution becausesmall singular values amplify the change in measurements, which is a source of discontinuitycharacteristics in SI problems.

3. REGULARIZATION

3.1. Analysis of Tikhonov regularization technique

The concept of regularization was proposed by Tikhonov to overcome the ill-posedness ofinverse problems, and has been successfully applied to various types of inverse problems[11–14; 19]. However, not much attention has been paid to the regularization technique inthe realm of structural engineering. Recently, some regularization techniques have been testedfor system identi0cation and damage detection in structures [1–3; 5].

The regularization can be interpreted as a process of mixing the a priori estimates ofsystem parameters and the a posteriori solution. The baseline properties are selected as thea priori estimates of the system parameters in this paper. The a priori estimates are takeninto account in the problem statement of inverse problems by adding a regularization functionwith the a priori estimates of the system parameters to the error function. The regularizationfunction may be de0ned diDerently for diDerent problems [1–3; 5; 9; 10; 19]. The followingregularization function proposed by Tikhonov is employed.

MR= 12�

2‖x − x0‖2 (13)

where � and x0 denote the regularization factor and the a priori estimates of system parame-ters, respectively. By adding the regularization function normalized by the a priori estimatesto the minimization problem of Equation (4), a regularized system identi0cation problem iswritten in the following form:

Minimize^

�= 12‖U(^)− PU‖2 + 1

2�2‖^− 1‖2 subject to R(^)60 (14)

where 1 denotes a column vector which has unit values in all the components. The objectivefunction in Equation (14) is referred to as the regularized error function. The regularizationfactor determines the degree of regularization in the system identi0cation; i.e. the inUuence



of the a priori estimates on the solution of Equation (14). The quadratic sub-problem ofEquation (14) is de0ned as

MinimizeQ^

[12Q^

TSTSQ^−Q^TSTUr]+ �2

[12Q^

TQ^−Q^T(1− ^)]subject to R(^+ Q^)60 (15)

The stability of Equation (15) is investigated under the unconstrained condition to clearlypresent the eDect of the regularization. Furthermore, the regularization factor should be de-termined for the unconstrained problems so that it can overcome the original sources ofinstabilities explained in the previous section. Once the regularization factor is obtained forthe unconstrained problem, the quadratic sub-problem with the active constraints de0ned inEquation (15) can be solved.

The regularized solution of the unconstrained problem of Equation (15) is obtained by useof the SVD.

Q^Ru =V diag(1− �j) diag(

1!j

)ZTUr +V diag(�j)VT(1− ^) (16)

where �j = �2=(!2j + �

2). With some mathematical manipulation of Equation (16) by use ofthe orthogonal properties of V and Z, an intuitive expression is derived as follows:

VT^Ru =diag(�j)VT1+ diag(1− �j)VTû (17)

where

^Ru = ^+ Q^Ru ; û= ^+ Qû (18)

In Equation (17), ^Ru and û represent the regularized solution and the a posteriori solutionof the unconstrained problem at the current iteration, respectively. The expression in Equation(17) implies that the projection of the regularized solution onto V is a generalized averagebetween the projections of the a priori estimates and the a posteriori solution onto V.

The weighting factor �j, which varies with the regularization factor � from 0 to 1, adjuststhe relative magnitude between the a posteriori solution and the a priori estimates in theregularized solution. The weighing factor approaches zero as the regularization factor becomessmaller, and one as the regularization factor becomes larger. Therefore, the solution convergesto the a priori estimates for a large regularization factor while the solution converges to thea posteriori solution for a small regularization factor. In case the regularization factor is0xed, the weighting factors become larger for smaller singular values. This fact implies thatthe stronger eDect of the a priori estimates is included in a solution component correspondingto the smaller singular value, and vice versa.

Unlike Equation (10), the orthogonality of the displacement residual Ur to each LSV isnot required for the convergence of Equation (16) because non-vanishing components in the0rst term can be cancelled out by the second term. By decomposing the a posteriori solutionincrement into the noise-free components and error components using Equation (12), thefollowing expression is obtained.

VT^Ru ={diag(�j)VT1+ diag(1− �j)VT^fu

}+ diag(1− �j) diag

(1!j

)ZTe (19)



where ^fu denotes the noise-free a posteriori solution. Since the weighting factors range from0 to 1 for all singular values, the eDect of noise on the solution can be reduced. In particular,the components of ZTe associated with small singular values, which are responsible for thediscontinuity and deviation from the noise-free solution, are mostly suppressed in the regular-ized solution by the regularization eDect. This is because the weighting factors correspondingto smaller singular values become almost one for a properly selected regularization factor.

3.2. Optimal regularization factor

Several well-de0ned methods have been proposed to determine an optimal regularization factorin linear inverse problems. The L-curve method (LCM) proposed by Hansen [10; 11] and thegeneralized cross validation (GCV) method proposed by Golub et al. [12] are well-knownschemes. Kaller and M. Bertrant utilized the GCV for medical image enhancing problems [13].While the aforementioned schemes have been proven to be eDective in linear inverse problems,no rigorous schemes for non-linear inverse analysis have been proposed yet. Regularizationfactors of non-linear inverse problems can be determined by applying the LCM and the GCVat each minimization iteration, where a linearized quadratic sub-problem is solved. Erikssonet al. reported that the LCM yields non-convergent results for a non-linear inverse problemwith an explicit non-linear function model [14], which is also observed in the current research.It has also been found through our extensive numerical experiments that the GCV yields toosmall regularization factors, and is unable to eDectively control the instabilities of the SIalgorithms for continua.

To identify geometric shapes of inclusions in 0nite bodies, another scheme, referred to as thevariable regularization factor scheme (VRFS), was proposed by Lee and coworkers [1; 2; 5].In VRFS, the regularization factor is adjusted sequentially so that the regularization functionis always smaller than the error function during the optimization process. Although the VRFSworks pretty well for shape identi0cation problems [1; 2] and for damage detection problemsof structures [5], a rigorous theoretical background of the VRFS has not been presented.

A new scheme, de0ned as a geometric mean scheme (GMS), is proposed to overcomedrawbacks of existing schemes in the determination of the regularization factor. As shownin Equation (17), the regularization eDect on each component of the solution depends on therelative magnitude of the corresponding singular value to the regularization factor. Figure 2illustrates the variation of weighting factors for the maximum and minimum singular valueswith the normalized regularization factor with respect to !max in case !min=!max =0:1. In theregularized solution of Equation (17), the maximum eDect of the a priori estimates and thea posteriori solution occurs with the smallest and largest singular values, respectively. Onthe other hand, the minimum eDect of the a priori estimates and the a posteriori solutionoccurs for the largest singular value and the smallest singular value, respectively. Based onthis observation, the optimal regularization factor is de0ned as the one that balances the samemaximum and minimum eDect of the a priori estimates and the a posteriori solution, whichcan be stated as

1− �max = �min; 1− �min = �max (20)

where �max and �min are the weighting factors corresponding to the maximum and minimumsingular values, respectively. The two equations given in Equation (20) are identical and yield



Figure 2. Schematic drawing for an optimal regularization factor in the GMS.

the geometric mean between the smallest and largest singular values for the optimal solutionof �

�opt =√!max!min (21)

The weighting factor �j with the optimal regularization factor de0ned in Equation (21)becomes 1

2 at !j = �opt. Since the weighting factors become larger than 12 for singular values

smaller than �opt, the solution components corresponding to singular values smaller than �optare dominated by the a priori estimates in the regularized solution. The regularization eDectbecomes stronger in the regularized solution as the number of singular values smaller than�opt increases.

4. EXAMPLES

The behaviour of the GMS is investigated through numerical simulation studies. Two examplesare presented with two types of errors associated with the identi0cation; i.e., the measurementand modelling errors. Noise caused by measurement error is simulated by adding random noisegenerated from a uniform probability function to displacements calculated by a 0nite elementmodel [5]. The uniform probability function is selected because it generates more widelydistributed errors than the normal distribution for a given amplitude of error. The Monte-Carlo simulation is carried out to illustrate the enhancement of continuity of the solution byregularization for both examples.

The Young’s modulus of each element group is taken as the system parameter. Elementgroups are prede0ned to limit discussions to the regularization technique. The convergencecriterion, ‖Q^‖=‖^‖610−3, is used to terminate optimization iterations unless otherwise stated.



Figure 3. Geometry and boundary conditions ofa square plate.

Figure 4. Observation points and element groupcon0guration of a square plate.

The baseline properties are assumed to be the Young’s modulus of steel. The initial values ofthe system parameters are taken to be the same as the baseline properties for the optimization.The following upper and lower bounds are used for each system parameter:

0:1 GPa6x06630 GPa (22)

The reduction factor of the VRFS, �=0:1, is used throughout the numerical study [1; 2; 5].The recursive quadratic programming with the active set algorithm [20] is utilized for opti-mization.

4.1. Measurement error—square plate with an inclusion

To investigate the eDects of measurement errors on the identi0cation, a simulated study iscarried out with an inclusion in a square plate under the plane stress condition. Figure 3illustrates the geometry, boundary conditions and applied traction. The shadowed region inthe 0gure denotes the inclusion. Young’s modulus of the square plate is 210 GPa, which isrepresentative of steel. Two types of inclusions—a soft inclusion of aluminium (E=70GPa)and a hard inclusion of tungsten (E=380 GPa)—are considered.

Displacements are measured at the observation points located on the outer boundary ofthe square plate. Two diDerent measurement cases are considered. The observation pointsare depicted as solid circles and open squares in Figure 4 for measurement cases I andII, respectively. It is assumed that measurements are performed independently for two loadcases, tx and ty. Both x- and y-component of displacements are measured at each observationpoint. The noise amplitudes of 5 and 1 per cent are applied for measurement cases I and II,respectively [5].

The 0nite element model employed in the parameter estimation is identical to the modelused for obtaining the measured displacement, which consists of 100 8-node quadratic elementsand 384 nodes. The prede0ned element groups are shown in Figure 4, and each element groupcontains four elements.



Figure 5. Estimated Young’s moduli by dif-ferent regularization schemes (soft inclusion—

measurement case I).

Figure 6. Estimated Young’s moduli by dif-ferent regularization schemes (hard inclusion—


4.1.1. Measurement case I. Identi0ed results for the soft inclusion by diDerent regulariza-tion techniques are compared in Figure 5. Identi0cation without regularization yields resultsthat oscillate severely. It is diTcult to determine the existence of the inclusion from theidenti0ed results without regularization because the reduction in the Young’s modulus ofelement group 17 may be caused by either an actual inclusion or the oscillating results. Whena regularization technique is employed, however, the amplitude of oscillation is reduced forthe element groups in the matrix material. From the 0gure, it is seen clearly that the GMScontrols the oscillation of the identi0ed results most eDectively among the other schemes.Although the LCM and VRFS alleviate the oscillation magnitudes to some extent, they yieldrather large oscillation magnitudes compared to the GMS. Since Young’s modulus of the softinclusion reduces prominently compared with the oscillation magnitude of the other elementgroups by the GMS, the existence of a soft inclusion is clearly assured.

Figure 6 illustrates the identi0cation results for the hard inclusion with the measurementsof case I. The results by the SI without the regularization severely oscillate as in the softinclusion. The identi0ed results by the LCM are not drawn in the 0gure because optimizationby the LCM does not converge as reported by Eriksson [14]. Both the GMS and the VRFSconverge to almost the same results for the element groups in the matrix material. However,the GMS yields higher Young’s modulus of the inclusion than the VRFS. Although Young’smodulus of the inclusion is estimated to be somewhat lower than the actual value, the iden-ti0cation results by the GMS are good enough to point out the existence of a stiD materialat element group 17.

Figure 7 shows regularization factors at each iteration step obtained by the diDerent schemesfor the hard and soft inclusions, respectively. By relating regularization factors shown inFigure 7 to the identi0ed results in Figures 5 and 6, it is easily observed that a largerregularization factor yields less oscillating results. For the hard inclusion case, the LCMyields periodically oscillating regularization factors between the two values, which causes non-convergent optimization iterations. Figure 8 shows the solutions corresponding to the lowerand upper regularization factors during oscillations by the LCM together with the convergedsolution by the GMS. In the LCM, the lower regularization factor yields more oscillatingresults with sharp resolution at the hard inclusion while the upper regularization factor yields



Figure 7. Regularization factors by diDerentregularization schemes (measurement case I).

Figure 8. Two oscillating solutions by the LCMand the solution by the GMS (hard inclusion—


Figure 9. Distribution of singular values and weighting factors by GMS at the 1st iteration(hard inclusion—measurement case I).

less oscillating results with smeared resolution at the hard inclusion. The solution by the GMSseems to be a mixture of favourable aspects of the two solutions by the LCM, i.e., a lessoscillating solution with sharper resolution at the hard inclusion.

Figure 9 shows distributions of singular values of three diDerent Hessian matrices, the errorfunction, the regularization function and the regularized error function of Equation (14) at the0rst iteration step for the hard inclusion problem. In the same 0gure, the weighting factors�j associated with the singular values are also drawn. For drawing the weighting factors, theright vertical axis is used as the reference. In the 0gure, it is observed that the lowest singularvalue of the error function is very small compared with the other singular values, whichcaused the oscillations in the identi0cation without the regularization as shown in Figure 6.The singular values of the Hessian matrix of the regularized error function are shifted bythe singular value of the regularization function. However, the regularization function doesnot aDect the distribution of the singular values of the regularized error function from thesixth singular value. Therefore, the a priori estimates have a strong inUuence on the solution



Figure 10. Solution of the unconstrained sub-problem by the noise-free measurement at the 1stiteration (hard inclusion—measurement case I).

Figure 11. Solution of the unconstrainedsub-problem by the noise components in mea-surements at the 1st iteration (hard inclusion—


components corresponding to the smaller singular values, and the inUuence of the a prioriestimates decreases drastically for larger singular values. This phenomenon can be clearlyobserved by the distribution of the weighting factors in the same 0gure.

Figures 10 and 11 illustrate the solution of the unconstrained quadratic sub-problem in theRSV direction and in the system parameter direction at the 0rst iteration corresponding to thenoise-free and noise components in the measured displacements, respectively. The SI algo-rithms with the GMS and without the regularization yield almost identical solution incrementsfor the noise-free components, even though the GMS causes slightly smeared incrementscorresponding to lower singular values. However, for the noise components, the regulariza-tion develops surprising diDerences in the solution increments as demonstrated in Figure 11.Without regularization, the noise components of the measurements are ampli0ed by the low-est singular value. The solution increment caused by the noise components corresponding tothe lowest singular value is about 30 times larger than the maximum solution incrementscaused by the noise-free components in the RSV direction. The ampli0ed noise componentcontaminates the whole solution increments with the noise in the system parameter direction.Since the weighting factor for the lowest singular values in the GMS is almost 1 as shown inFigure 9, most of the noise components corresponding to the lowest singular value in Equation(17) are suppressed. Consequently, the solution increments caused by the noise componentsare very small in the SI with the GMS compared to those in the SI without the regularization.

Figures 12 and 13 show the solution of the unconstrained quadratic sub-problem in theRSV direction at the converged stage for the SI with the GMS and without regularization,respectively. In Figure 13, the absolute values of regularized solution increments are plotted inlogarithmic scale for the left vertical axis while the increments associated with the a posteriorisolution and a priori estimates are plotted in a linear scale for the right vertical axis. Boththe SI algorithms yield almost zero increments in the RSV direction at the converged state.However, two schemes exhibit diDerent patterns in reducing solution increments. The normof the displacement residual reduces only by 0.033 in the SI algorithm without regulariza-tion. Nevertheless, the norm of the solution increments is converged to the speci0ed criterion



Figure 12. Singular value, Fourier coeTcient and solution of the unconstrained sub-problem at theconverged iteration without regularization (hard inclusion—measurement case I).

Figure 13. Solution of the unconstrained sub-problem at then converged iteration by the GMS(hard inclusion—measurement case I).

because the Fourier coeTcients are reduced to below 10−5 order and the singular values main-tain relatively larger values than the Fourier coeTcients. On the other hand, the SI algorithmwith the GMS reduces the solution increments by balancing the increments associated withthe a priori estimates and a posteriori solution as shown in Figure 13.

To investigate continuity of solutions in various SI algorithms to measurement errors, aMonte-Carlo simulation with 30 trials at 5 per cent noise amplitude is carried out. A diDerent



Figure 14. Mean values and standard deviations ofestimated Young’s moduli by Monte-Carlo si-mulation (hard inclusion—measurement case I).

Figure 15. Estimated Young’s moduli by dif-ferent regularization schemes (soft inclusion—

measurement case II).

set of measured data is used for each trial by generating diDerent random noise from theuniform probability density function [6]. The relative magnitude of the standard deviationto the mean value of each system parameter obtained by the Monte-Carlo simulation is agood indicator of the continuity of solutions because the standard deviation represents thedegree of scatter of a statistical variable. The computed mean and standard deviation of eachsystem parameter from the Monte-Carlo simulations are compared in Figure 14 for diDerentregularization schemes. Results by the LCM are not presented since the LCM fails to convergein 15 out of 30 trials. When the regularization is not employed in the SI algorithm, largestandard deviations usually occur at the element groups whose estimated moduli are largerthan the baseline property. Meanwhile, the SI with a regularization technique yields small andconsistent standard deviations for all system parameters, which illustrates an enhancement ofthe continuity of solutions with a regularization technique. Both the VRFS and GMS yieldalmost identical results and smaller elastic modulus of the inclusion than the actual valuein an average sense. Despite the underestimation, the existence of an inclusion with a stiDermaterial at element group 17 is clearly distinguishable in general because oscillations in theother element groups are negligible.

4.1.2. Measurement case II. The inUuence of sparseness of measured data on estimated re-sults is studied in Figures 15 and 16. The sparseness of measured data is simulated by reducingthe number of observation points and by locating some of the observation points close to eachother as shown in Figure 4. Since the three observation points on each side of the squareplate are closely placed, the independence of information supplied by those observation pointsis reduced, which deteriorates the quality of information.

Figures 15 and 16 show the estimated Young’s modulus for the soft and hard inclusioncases with 1 per cent noise amplitude, respectively. Although the noise amplitude of thismeasurement case is much smaller than that of measurement case I, the solutions by the SIwithout the regularization oscillate more severely. This is because the lowest singular valueof the sensitivity matrix becomes much smaller in this measurement case than in the previousone due to the poor quality of information.



Figure 16. Estimated Young’s modulus by dif-ferent regularization schemes (hard inclusion—

measurement case II).

Figure 17. Geometry and boundary conditionsof a thick pipe.

All three regularization techniques yield very stable and accurate results for the soft in-clusion. However, the LCM fails to converge for the hard inclusion due to the oscillationsof the regularization factor as explained in measurement case I. The VRFS and GMS yieldalmost identical results, but underestimate the Young’s modulus of the hard inclusion as inmeasurement case I.

4.2. Modelling error—a thick pipe with three internal cracks

Behaviours of SI algorithms with respect to modelling errors are investigated in this example.A thick pipe with three cracks is subjected to internal pressure as shown in Figure 17.Measured displacements at equally spaced 80 observation points on the outer surface of thepipe are obtained by a 0nite element model with 6400 8-node quadratic elements and 19608nodes. Both x- and y-components of displacements are measured at each observation point.To simulate actual behaviours of structures realistically, elastic-perfect-plastic response of thepipe is considered with the von-Mises yield condition. For the SI, the pipe is discretized by480 8-node quadratic elements and 1520 nodes, and only the elastic behaviours are considered.The element groups used in this example are illustrated in Figure 18. A total of 60 elementgroups are used, and each element group contains 8 elements. The 0nite element model forthe identi0cation does not include cracks while the model used for calculating displacementscontains cracks. Therefore, this example contains modelling errors in the boundary conditionsin addition to errors in the constitutive law.

Identi0ed results are shown in Figure 19, in which arrows indicate the element groupswith a real crack. The SI algorithms without regularization and with the VRFS cannot yieldconverged solutions within 60 iterations, and thus only the solutions by the LCM and GMSare presented in the 0gure. The GMS and LCM yield converged solutions at 30 and 53iterations, respectively, which demonstrates the stability of the GMS over the LCM.



Figure 18. Element group con0gurationof a thick pipe.

Figure 19. Estimated Young’s moduli by dif-ferent regularization schemes (thick pipe with

three internal cracks).

As shown in Figure 19, both the LCM and GMS yield physically meaningful solutions inan overall sense. The Young’s moduli of the element groups with a crack exhibit signi0cantdrops from the baseline property compared with the oscillation amplitudes at the other elementgroups. However, the LCM predicts a large reduction in the Young’s modulus at elementgroup 7, which is located beside element group 6 and does not contain an actual crack.Both methods estimate a smaller Young’s modulus at element group 19 than that of elementgroup 12. From the physical point of view, this result may not represent the real situation ofdamage in the pipe properly because the length of the crack in element group 12 is longerthan that in element group 19. Despite such an inaccuracy in the assessment of actual damage,the existence of damage at three diDerent locations in the pipe can be clearly identi0ed bythe SI algorithms with the LCM and GMS.

Figure 20 shows a singular value distribution of each Hessian matrix and the distributionof weighting factors at the 0rst iteration step when the GMS is applied. By comparing withFigure 9, it is easily observed that this example is much more ill-posed than the hard in-clusion case presented in the previous example since the 22 singular values are smaller thanthe regularization factor obtained by the GMS. The solution components contributed by thea posteriori solution corresponding to the 22 singular values are mostly suppressed, and thea priori estimates are dominant in the solution. The contribution of the a priori estimatesto the solution rapidly decreases for singular values larger than the 22nd singular value, andmost parts of the regularized solution consist of the a posteriori solution. The distributionof the weighting factors represents the relative magnitude of regularization corresponding toeach singular value.

The non-convergence of the SI algorithm without regularization can be clearly explained byFigure 21, which shows the solution of the unconstrained quadratic sub-problem in the RSVdirection at the 60th iteration. The Fourier coeTcients are reduced to some extent in the 0gure.



Figure 20. Distribution of singular values and weighting factors by GMS at the 1st iteration(thick pipe with three internal cracks).

Figure 21. Singular value, Fourier coeTcient and solution of the unconstrained sub-problem at the60th iteration without regularization (thick pipe with three internal cracks).

However, since some of the singular values marked by solid circles in Figure 21 becomesmaller than the corresponding Fourier coeTcients, the solution increments are ampli0ed andnon-convergence of the optimization iterations is caused. Meanwhile, the SI with the GMSreduces the solution increments very eDectively by balancing the a posteriori solution and thea priori estimates as in the previous example.

To consider measurement error as well as the modelling error, 30 diDerent sets of randomnoise of 5 per cent magnitude are added to the measured displacements, and Monte-Carlo trialsare carried out for the 30 sets of simulated measurements. Since the convergence criterion,



Figure 22. Mean values and standard deviations of estimated Young’s moduli by Monte-Carlosimulation for noise-polluted measurements using GMS (thick pipe with three internal cracks).

10−3, is too tight for 30 trials with modelling errors as well as measurement errors, a newconvergence criterion of 10−2 is used for the Monte-Carlo simulation. The average numberof iterations for the new criterion is 10 for the GMS and 26 for the LCM, respectively, when10 Monte-Carlo trials are carried out. As the GMS and LCM yield almost identical resultsfor 10 trials, and the LCM requires much more iterations than the GMS, the Monte-Carlosimulation with 30 trials is performed only for the GMS.

The computed mean and standard deviation of the Young’s modulus of each group bythe GMS from 30 Monte-Carlo trials are drawn in Figure 22. In the Monte-Carlo trials, theGMS successfully converges 29 out of 30 trials. The mean values are almost identical tothe estimated Young’s moduli from measurement data without measurement errors. Since thestandard deviations are negligibly small, it can be concluded that the GMS is insensitive todiDerent noise components in the measurements, and enhances the continuity of solution veryeDectively.

5. CONCLUSIONS

A new criterion called the geometric mean scheme (GMS) is proposed to determine an op-timal regularization factor for an SI with regularization. The eDect of the regularization isinvestigated through theoretical formulations combined with singular values decomposed fromthe sensitivity matrix of responses. There are three main features in developing the proposedscheme. First, the instabilities of inverse problems caused by noise in measurements are inves-tigated rigorously using the singular values of the sensitivity matrix of responses. Secondly,it is shown that the regularization can be interpreted as the generalized average between thea priori estimates and the a posteriori solution, and relative inUuence of the regularization isadjusted by the regularization factor. Finally, the GMS is proposed for selecting the optimalregularization factor, which is de0ned as the geometric mean of the maximum and minimumsingular values of the sensitive matrix of responses.

Numerical simulation studies are performed to demonstrate the validity and eDectivenessof the GMS. The results identi0ed by the GMS are compared with those by the LCM andVRFS. An SI algorithm without regularization yields severely oscillating results, and fails



to converge in the example with modelling errors. The GMS yields the most accurate andreliable results regardless of types of errors among the three schemes. The LCM frequentlyfails to converge for the examples with measurement errors, and requires much more iterationsto converge than the GMS for the case with modelling errors. The VRFS is unable to yieldconverged results for the case with modelling errors, and leads to either more oscillating ormore smeared results than the GMS for the cases with measurement errors.

The GMS provides a rigorous method for determining optimal regularization factors fornonlinear inverse problems. As the GMS is formulated from general characteristics of a sen-sitivity matrix of response variables, the GMS can be applied to various types of inverseproblems.

ACKNOWLEDGEMENTS

The authors wish to acknowledge the 0nancial support of the Korea Research Foundation oDered in theprogram year of 1997.

REFERENCES

1. Lee HS, Kim YH, Park CJ, Park HW. A new spatial regularization scheme for the identi0cation of geometricshapes of inclusions in 0nite bodies. International Journal for Numerical Methods in Engineering 1999;46(7):973–992.

2. Lee HS, Park CJ, Park HW. Identi0cation of geometric shapes and material properties of inclusions intwo dimensional 0nite bodies by boundary parameterization. Computer Methods in Applied Mechanics andEngineering 2000; 181(1–3):1–20.

3. Schnur DS, Zabaras N. Finite element solution of two dimensional inverse elastic problems using spatialsmoothing. International Journal for Numerical Methods in Engineering 1990; 30:57–75.

4. Shin S, Koh HM. A numerical study on detecting defects in a plane stressed body by system identi0cation.Nuclear Engineering and Design 1999; 190:17–28.

5. Yeo IH, Shin S, Lee HS, Chang SP. Statistical damage assessment of framed structures from static responses.Journal of Engineering Mechanics ASCE 2000; 126(4):414–421.

6. Hjelmstad KD, Shin S. Damage detection and assessment of structures from static response. Journal ofEngineering Mechanics ASCE 1996; 123(6):568–576.

7. Sanayei M, Onipede O. Damage assessment of structures using static test data. AIAA Journal 1991; 29(7):1174–1179.

8. Hjelmstad KD. On the uniqueness of modal parameter estimation. Journal of Sound and Vibration 1996;192(2):581–598.

9. Bui HD. Inverse Problems in the Mechanics of Materials: An Introduction. CRC Press: Boca Raton, 1994.10. Hansen PC. Rank-de6cient and Discrete Ill-Posed Problems: Numerical Aspects of Linear Inversion. SIAM:

Philadelphia, 1998.11. Hansen PC. Analysis of discrete ill-posed problems by means of the L-curve. SIAM Review 1992; 34(4):

561–580.12. Golub GH, Heath M, Wahba G. Generalized cross-validation as a method for choosing a good ridge parameter.

Technometrics 1978; 21:215–223.13. Kaller F, Bertrant M. Tissue elasticity reconstruction using linear perturbation method. IEEE Transaction of

Medical Imaging 1996; 15(3):299–313.14. Eriksson J. Optimization and regularization of nonlinear least squares problems. Ph.D. Thesis, Department of

Computing Science, UmeXa University, Sweden, 1996.15. Neuman SP, Yakowitz S. A statistical approach to the inverse problem of aquifer hydrology. Water Resources

Research 1979; 15(4):845–860.16. Neuman SP. Calibration of distributed parameter groundwater Uow models viewed as a multiple-objective

decision process under uncertainty. Water Resources Research 1973; 9(4):1006–1021.17. Neuman SP. Role of subjective value judgement in parameter identi0cation. In Modeling and Simulation of

Water Resources Systems, Vansteenkiste GC (ed.). North-Holland: Amsterdam, 1975, 59–84.18. Golub GH, Van Loan CF. Matrix Computations (3rd edn). The Johns Hopkins University Press: London, 1996.19. Becks JV, Murio DA. Combined function speci0cation—regularization procedure for the solution of inverse

heat conduction problem. AIAA Journal 1984; 24:180–185.20. Luenberger DG. Linear and Nonlinear Programming (2nd edn). Addison-Wesley: Reading, 1989.


determination of an optimal regularization factor in system identification with tikhonov...

Documents