A comparative study of Domain Embedding Methods for regularized solutions of inverse Stefan problems

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, VOL. 40, 3579–3600 (1997)

A COMPARATIVE STUDY OF DOMAIN EMBEDDINGMETHODS FOR REGULARIZED SOLUTIONS OF

INVERSE STEFAN PROBLEMS

JUN LIU∗ AND B�EATRICE GUERRIER

FAST Laboratory, URA 871 CNRS-UPMC-UPS, Batiment 502, Campus Universitaire, 91405 Orsay C�edex, France

ABSTRACT

In this paper, various Domain Embedding Methods (DEMs) for an inverse Stefan problem are presented andcompared. These DEMs extend the moving boundary domain to a larger, but simple and �xed domain. Theoriginal unknown interface position is then replaced by a new unknown, which can be a boundary temperatureor heat ux, or an internal heat source. In this way, the non-linear identi�cation problem is transformed intoa linear one in the enlarged domain. Using di�erent physical quantities as the new unknown leads to di�erentDEMs. They are analysed from various points of view (accuracy, e�ciency, etc.) through two test problems,by a comparison with a common Front-Tracking Method (FTM). The �rst test has a smooth temperature �eldand the second one has some singularities. The advantage of the DEMs in solving the inverse problem andin computing the corresponding direct mapping is shown. In the direct problem, high-order accurate schemescould be obtained more easily with the DEMs than with the FTM. In the inverse problem, an iterativeregularization and a Tikhonov regularization have been employed. For the FTM, the iterative regularizationis not e�cient—the solution oscillates when the data are noisy. As for the Tikhonov regularization, it requestsspecial care to choose an adequate penalty term. In contrast, both the regularizations give good results withall the considered DEMs, except for the second test problem at the beginning (t=0+) when the value of theheat ux and the heat source tends to ∞. Slightly di�erent regularization e�ects have been obtained whenusing di�erent DEMs. Finally, an automatic choice of the optimal regularization parameter is also discussed,using data with di�erent noise levels. We propose the use of the curve of the residual norm against theregularization parameter. ? 1997 John Wiley & Sons, Ltd.

Int. J. Numer. Meth. Engng., 40, 3579–3600 (1997)

No. of Figures: 11. No. of Tables: 6. No. of References: 32.

KEY WORDS: domain embedding; illposed; inverse Stefan problem; phase-change interface identi�cation; regularization

1. INTRODUCTION

Solid–liquid phase-change problems arise in many industrial applications of material processing,such as casting, welding, crystal growth, etc., and have been under active study from di�erent pointsof view.1; 2 In a phase-change process, it is important to determine the velocity and the pro�leof the solid–liquid interface, since the quality of the obtained material depends on the controlof the heat transfer and of the phase-change interface motion. The phenomena occurring in the

∗ Correspondence to: J. Liu, Fluides, Automatique et Systems Thermiques, Unite Associee au CNRS No 871, Batiment502, Campus Universitaire, 91405 Orsay Cedex, France

CCC 0029–5981/97/193579–22$17.50 Received 12 October 1995? 1997 John Wiley & Sons, Ltd. Revised 9 December 1996

3580 J. LIU AND B. GUERRIER

liquid phase are very complex, in general, due to the existence of convections, chemical reactions,surface e�ects, etc. One way to avoid modelling these complex phenomena is to estimate theinterface motion from measurements taken in the solid phase only. The identi�cation of the inter-face location from temperature measurements in the solid phase is often referred to as an inverseStefan problem (identi�cation problem). This problem, particularly during welding processes, isstudied in References 3 and 4 for one-dimensional cases and in References 5 and 6 for 2-D cases.Because of the important number of unknowns to be determined, which result from the temporaldiscretization of the interface s(t) or the space and time discretization in 2-D cases, a sequentialprocedure as proposed in Reference 7 was used in most published works. Another inverse Stefanproblem studied by many authors in the literature8; 9 is a control (or design) problem that looksfor the temperature or the heat ux on some boundary of the solid phase so as to get a prescribedvelocity and=or heat ux on the interface. Both the control and the identi�cation inverse Stefanproblems are ill-posed in the Hadamard’s sense,10 i.e., the solution does not exist or is not unique,or is not stable with respect to data perturbations.In this paper, we restrict our attention to the 1-D identi�cation inverse Stefan problem and

put emphasis on a comparative numerical study of three Embedding Domain Methods (DEMs).DEMs are also called �ctitious domain methods or capacitance matrix methods11; 12 in the litera-ture, and are widely used for solving PDE numerically (often with �nite di�erence methods) indomains with a complicated geometry or in unbounded domains.13; 14 A DEM was introduced byBlum15 for the identi�cation of the free boundary of plasma in a Tokamac. For the numericalsolution of one or two-phase Stefan problems, an invariant embedding with a method of lines(in time) is proposed by Meyer.16; 17 More recently, based on controllability results, a DEM wasused to study an optimal shape design problem by Haslinger et al.18 and Tiba.19 The identi�-cation of the moving boundary in a Stefan problem using a DEM was considered by Mannikkoet al.20 A numerical study is also presented in B�enard et al.21

The DEMs presented here consist in embedding the moving boundary domain (t) into a �cti-tious domain F with a larger but simple and �xed geometry, i.e., (t)⊂F for any time t. Theheat conduction problem de�ned in the moving boundary domain (t) is extended to the �xeddomain F. The unknown moving boundary s(t) of (t) is replaced by a new unknown parameterp(t), which can be a condition to be determined at a �xed boundary or a heat source term to bedetermined in a �xed subdomain of F. Once the unknown parameter p(t) is obtained, the temper-ature �eld in the domain F is available and then the interface location s(t) can be calculated byinterpolating the temperature. In this way, the original non-linear free boundary identi�cation prob-lem becomes a linear one in the �xed domain F. These DEMs have several advantages in solvingthe inverse problem as well as in computing the corresponding direct mapping. Since the trans-formed problem becomes a linear optimal control problem, a solution to the inverse problem with-out iteration can be computed by solving a Ricatti’s equation.21 If an iterative optimization methodis used, a rapid convergence can be ensured. For the corresponding direct problem to be solvedat each iteration of the optimization, since the coe�cients of the equation are constant and thecomputation domain has a simple and �xed geometry, some high-order schemes can be more easilyrealized; if implicit schemes are used, the matrix factorization is needed at the �rst time step only.Formally, a DEM decomposes the non-linear equation �(s)= z(t) into a linear equation F(p)=

z(t) and an interpolation s= I(p). The question is whether the inverse solution I ◦F−1( z(t))obtained by such a decomposition is equal or at least close to the inverse solution �−1( z(t)) ofthe original problem, for given noisy data z(t) which are the same in the original and transformedproblems. The unknown interface position s(t) is replaced by the unknown parameter p(t), but it is,

Int. J. Numer. Meth. Engng., 40, 3579–3600 (1997) ? 1997 John Wiley & Sons, Ltd.

A COMPARATIVE STUDY OF DOMAIN EMBEDDING METHODS 3581

in general, di�cult to get an admissible function space for p(t) equivalent to the given admissiblefunction space for s(t). If one chooses for example L2(0; tf) as the function space for p(t) andC1(0; tf) as the function space for s(t), the original problem is not equivalent to the transformedproblem. Under the condition that there exists an exact and unique solution for both the originaland transformed problems, as it is the case for a well-posed problem, the solution obtained by usingthe DEM is equal to the solution of the original problem. For the ill-posed identi�cation problem,based on some controllability results, it is assured that approximate solutions can be obtainedby the DEM. However, these approximate solutions can be very di�erent. So, only regularizedsolutions, when the illposedness has been removed, could be compared. Our concern is whetherall the various DEMs give a good performance with a simple and suitable regularization technique.For this purpose, regularization e�ects obtained when using various DEMs are analysed in detailfrom the numerical point of view.Two regularization methods are studied—an explicit Tikhonov regularization and an implicit

iterative regularization using a conjugate gradient method. For the Tikhonov regularization, wecompare the di�erent regularization e�ects obtained for the di�erent DEMs with a similar penaltyterm on the unknown parameter p(t). For the iterative regularization, the di�erent regularizatione�ects obtained with a �xed number of iterations are compared for di�erent DEMs.The paper is organized as follows. In Section 2 the 1-D identi�cation inverse Stefan problem is

presented. In Section 3 the classical front-tracking method for the 1-D moving boundary problemis brie y recalled. In Section 4 we describe the three DEMs used to solve the inverse Stefanproblem. In Section 5 we specify two test problems in order to compare the various methods.Section 6 deals with some numerical aspects in the computation of the various direct mappingsintroduced by the DEMs. Finally, regularized solutions of the inverse problem obtained by usingthe various methods are analysed in Section 7.

2. PROBLEM DEFINITION

In a 1-D phase change problem, let us consider the moving boundary domain (t) correspondingto the solid phase:

(t)= {x|0¡x¡s(t)} (1)

where s(t) is the solid=liquid interface position.For simplicity, the variables used here are assumed to be dimensionless. The heat transfer in

the solid region (t) is governed by the following heat conduction equation:

@T@t=

@2T@x2

in (t)× ]0; tf[ (2)

with the boundary conditions:

T (0; t)=f(t) in ]0; tf[ (3)

T (s(t); t)=Tf =0 in ]0; tf[ (4)

and the initial condition:

T (x; 0)=T 0(x) in (0) (5)

Once s(t) is given, one can solve equations (2)–(5) and then determine the heat ux z(t)=Tx(x; t)at x=0, which will be used as the observation later. This is referred to as the direct problem.

? 1997 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng., 40, 3579–3600 (1997)


Since the observation obtained from a real experiment is always noisy, no particular assump-tion is made on the regularity of z(t), and the observation space Z is chosen as the usual spaceZ = L2(0; tf) in this paper. Our inverse Stefan problem (moving boundary identi�cation problem)consists in estimating the interface position s(t) from a noisy observation z(t)∈Z which is mea-sured on the back face ‘x=0’.We study a front-tracking method and various domain embedding methods to solve the iden-

ti�cation problem in this paper. For these methods, the identi�cation problem is rewritten as anoptimization problem based on a �t-to-data criterion. The main di�erence between these methodsis the di�erent associated direct mappings.

3. FRONT-TRACKING METHOD

Consider the direct mapping

� : s(·)∈ S 7→ z(·)=Tx(0; ·)∈Z (6)

that is determined by equations (2)–(5). In order to compute the mapping �(s), a classical co-ordinate transform method1 was used. The moving domain (t) is transformed into the �xeddomain ]0; 1[ by using the Landau’s coordinate transformation x=ys(t), where x and y are theold and the new space variables, respectively. Applying such a transformation to equation (2)leads to the following equation:

@T@t=

1s(t)2

@2T@y2

+ ys′(t)s(t)

@T@y

in ]0; 1[×]0; tf[ (7)

with the boundary conditions

T (0; t)=f(t); T (1; t)= 0 in ]0; tf[ (8)

and the initial condition

T (y; 0)=T 0(y) in ]0; 1[ (9)

The observation (heat ux at x=0) is then calculated by

z(t)=Ty(0; t)s(t)

(10)

The unknown parameter space S should be chosen appropriately such that the mapping � iswell de�ned. We give here a su�cient condition: If the initial condition T 0(x) and the boundarycondition f(t) satisfy the regularity and compatibility conditions: T 0(x)∈H 1(0; 1), f(t)∈H 1(0; tf)and f(0)=T 0(0), and if S =C1(0; 1), then the observation z(t)=Tx(0; t) belongs to Z = L2(0; tf).The computation of the mapping � can be performed by solving the above equations (7)–(10)

on the �xed domain ]0; 1[. The coe�cients in equation (7) depend on the interface position s(t)which is unknown and varies with time t. This shows the non-linearity of the mapping � explicitly.The identi�cation of the interface position s(t) is solved directly by minimizing the objective

functional given by

J (s(·))=∫ tf

0(�(s)(t)− z(t))2 dt (11)

in the space S.



For simplicity, the above front-tracking method is called method � later on.Because of the non-linearity of �, the objective function J (s) of equation (11) could be di�cult

to be minimized e�ciently. This di�culty will be con�rmed in our numerical experiments and isour primary motivation for developing the DEMs. The DEMs described in the following sectionallow us to replace the non-linear mapping � by a linear mapping.

4. DOMAIN EMBEDDING METHODS

The DEMs presented here consist in extending the moving boundary domain (t), considered fora �nite time horizon [0; tf], to a larger but �xed domain F = ]0; �[ with �¿sm= sup06t6tf s(t),and replacing s(t)∈ S by a new identi�cation parameter p(t)∈P. The direct non-linear mapping� is changed into a linear mapping F :

p(t)∈P 7→ z(t)=Tx(0; t)∈Z = L2(0; tf) (12)

For the mapping F to be well de�ned, some regularity and compatibility on the initial and boundaryconditions should be satis�ed, and the parameter space P should be chosen appropriately. Forexample, if T 0(x)∈H 1(0; 1), f(t)∈H 1(0; tf), f(0)=T 0(0), and P= L2(0; tf), then the observationz(t)=Tx(0; t) is well in Z = L2(0; tf).The identi�cation problem becomes the minimization of the following new objective functional:

J (p)=∫ tf

0(F(p)(t)− z(t))2 dt (13)

Once the new unknown parameter p(t)∈P is found, the temperature distribution in F can bedetermined and the interface position s(t) is estimated by looking for the isotherm T = Tf =0.Various linear mappings F can be obtained by choosing di�erent identi�cation parameters p(t).

In this paper, three DEMs are studied. For the three linear mappings de�ned hereafter, the boundarycondition at x=0 is given by equation (3), while the other boundary condition (4) to be identi�edis modi�ed. The initial condition is extended to

T (x; 0)=T 0(x) in F (14)

4.1. Using a �ctitious temperature at x= � as p(t)

We extend the heat conduction equation (2) to

@T@t=

@2T@x2

in F× ]0; tf[ (15)

A Dirichlet boundary condition is imposed at x= � as follows:

T (�; t)=p(t) in ]0; tf[ (16)

Thus we get the linear mapping F1:

F1 : p(t)∈P 7→ z(t)=Tx(0; t)∈Z (17)

For convenience, this DEM is called method F1 later. In the new identi�cation problem, thetemperature T (�; t)=p(t) is the unknown instead of the interface s(t).



4.2. Using a �ctitious ux at x= � as p(t)

Similarly, we can extend the heat conduction equation (2) to F (Eq. (15)) and impose aNeumann boundary condition at x= �:

Tx(�; t)=p(t) in ]0; tf[ (18)

Now the heat ux Tx(x; t)=p(t) at x= � is the unknown parameter and we have the followinglinear mapping:

F2 : p(t)∈P 7→ z(t)=Tx(0; t)∈Z (19)

In the next sections, the above method is called method F2.

4.3. Using a �ctitious source term in the extended region [sm; �] as p(t)

We introduce a new parameter p(x; t) with

supp(p)⊂ [sm; �]× [0; tf] (20)

and extend equation (2) to

@T@t=

@2T@x2

+ p(x; t) in F× ]0; tf[ (21)

with the boundary condition at x= �

T (�; t)= 0 in ]0; tf[ (22)

The heat source p(x; t) in the region [sm; �] is now the new unknown parameter and we have thefollowing linear mapping:

F3 : p(x; t)∈P 7→ z(t)=Tx(0; t)∈Z (23)

It is obvious that the inverse solution of the above mapping is not unique, since the parameter pis a function of x and t, while the observation z is only a function of t. To reduce the variablesof the function p(x; t), we suppose that p does not vary with x. The above method is referred toas method F3.

4.4. Approximate controllability

We wonder whether the transformed problems are equivalent or at least close to the originalproblem. To answer this question, there holds the following approximate controllability result forthe three DEMs described above.

Theorem 4.1. For any given �¿0 and z ∈L2(0; tf); there exists a parameter p∈P= L2(0; tf)such that J (p)¡� where the objective functional J (·) is de�ned as (13); i.e., the image set ofmappings F1; F2 and F3 are dense in Z = L2(0; tf).

This result has been demonstrated in Reference 20 for mapping F3. With the same techniqueas in Reference 20 we give here the proof for mapping F1 by using the Mizohata’s uniquenesstheorem.22



For simplicity, we assume that f(t)= 0 and T 0(x)= 0, which can be obtained by a simpletranslation transform on the temperature T (x; t). Let us suppose that the image set F1(P) is notdense in Z . Then, there exists a non-zero function (t)∈Z such that∫ tf

0 (t)F1(p)(t) dt=

∫ tf

0 (t)Tx(0; t) dt=0; ∀p(t)∈P

For the state equation (15) and its adjoint equation

@�@t+

@2�@x2

= 0 in F× ]0; tf[ (24)

we have

0 =∫ tf

0

∫ �

0

[T(@�@t+

@2�@x2

)+ �

(@T@t

− @2T@x2

)]dx dt

=∫ �

0[�(x; tf)T (x; tf)− �(x; 0)T (x; 0)] dx +

∫ tf

0[T (x; t)�x(x; t)− �(x; t)Tx(x; t)]x= �

x= 0 dx (25)

If we introduce the conditions

�(0; t)= (t); �(�; t)= 0; �(x; tf)= 0

for the adjoint equation, we obtain∫ tf

0p(t)�x(�; t) dt=0; ∀p(t)∈P

Thus, �x(�; t)= 0. Noticing that �(�; t)= 0, it follows from the uniqueness theorem that

�(x; t)= 0 in F× ]0; tf[which contradicts the assumption of non-zero . The proof is completed for mapping F1 and issimilar for mapping F2.As shown by this theorem, one can obtain at least an approximate solution in L2(0; tf) of the

original problem by using the DEMs, even if the exact solution does not belong to the spaceL2(0; tf) (as it is the case for test B).Let us underline that, from the above proof, it can be seen that the approximate controllability

still holds when the parameter space P is a dense subspace of L2(0; tf). This means that one canchoose a priori the regularity of the approximate solution.Now, estimating the front position s(t) consists in solving the transformed problem �rst (i.e.

inverting the linear mappings F1, F2 or F3) and computing a posteriori the isotherm T = Tf ofthe temperature �eld in the �ctitious domain F.Let us also remark the di�erence between the original non-linear problem and the transformed

linear problem. The above theorem assures us of the existence of an approximate solution p(t)to the transformed linear problem, but the original non-linear problem can have no approximatesolution s(t) when the isotherm T = Tf does not exist in the domain F.

5. TEST PROBLEMS

In order to give a comparative study of the various DEMs presented previously, we choose thefollowing two test problems, both of which have an analytical expression for the exact temperaturedistribution. The �nite time horizon [0; tf =1] will be used for our numerical experiments.



5.1. Test A

In the �rst problem, the solid–liquid interface position is given by

s∗(t)= 1− t=2 (26)

(where the superscript ∗ denotes the exact parameter to be identi�ed), and the temperature �eldin the domain (t)= ]0; s∗(t)[ is expressed as

T (x; t)= exp(t=4− (1− x)=2)− 1 (27)

The boundary condition at x=0 is

T (0; t)= exp(t=4− 1=2)− 1 (28)

and the exact observation is

z(t)=Tx(0; t)= exp(t=4− 1=2)=2 (29)

It is obvious that sm=1, so that we can choose �=1 for methods F1 and F2. The exactunknown parameters are given by

p∗(t)= exp(t=4)− 1 (30)

and

p∗(t)= exp(t=4)=2 (31)

for methods F1 and F2, respectively.A convenient value �¿1 is chosen for method F3, for which no explicit analytical expression

such as (30) is available for the exact parameter p∗(t) to be identi�ed.In this test problem, the temperature �eld in the moving domain (t) can be extended smoothly

to the �ctitious domain F (even to the whole space domain ]−∞;+∞[). The parameter spacesS and P (for mapping � and F; respectively) can be chosen as S =C1(0; tf) and P= L2(0; tf).Indeed, it can be easily checked that the conditions s∗(t)∈ S =C1(0; tf) and p∗(t)∈P= L2(0; tf)hold for the three DEMs. Therefore, it is a relative simple example, for which accurate and e�cientsolutions of both direct and inverse problems can be obtained, as shown afterwards.

5.2. Test B

In the second test problem, we consider the so-called Neumann’s analytical solution, whichcorresponds to the fusion of the semi-in�nite domain ]−∞; 1], with a discontinuity at x=1 andt=0 (T (1; 0−)¡0 and T (1; 0+)¿0).The evolution of the solid=liquid interface position is given by

s∗(t)= 1− 2a√t (32)

and the temperature �eld in (t) has the form of

T (x; t)= erfc((1− x)2√t

)/erfc(a)− 1 (33)

where a is a constant.



The boundary condition at x=0 can be expressed as

T (0; t)= �erfc(12√t

)− 1 (34)

and the exact observation is given by

z(t)=Tx(0; t)=�√�texp

(− 14 t

)(35)

with

�=1=erfc(a) (36)

For methods F1 and F2, the moving domain (t) is extended to F = ]0; 1[, and the exact tem-perature and heat ux at x=1 are, respectively, given by

p∗(t)=T (1; t)= � − 1 (37)

and

p∗(t)=Tx(1; t)=�√�t

(38)

For method F3, a �¿1 is used; there is no analytical expression for p∗(t) as in test A.We would like to underline that the conditions s(t)∈C1(0; tf) for method � and p∗(t)∈L2(0; tf)

for F2 and F3 do not hold here in contrast with test A. Indeed, s∗′(t) and p∗(t) tends to ∞

at t=0. In spite of this, �(s∗), F2(p∗) and F3(p∗) are in Z = L2(0; 1) and the approximatecontrollability result given in Section 4.4 ensures the existence of an approximate solution inL2(0; tf). Nevertheless, this allows us to compare the accuracy of the various methods in a muchmore di�cult situation than in test A.

6. NUMERICAL SOLUTIONS FOR THE DIRECT PROBLEMS

As mentioned hereinbefore, the considered inverse problem is solved by minimizing an objectivefunctional. At each iteration of the minimization procedure, the solution of the associated directproblem (or the calculation of the direct mapping) is required. Using the DEMs, the non-lineardirect mapping � is replaced by the linear mapping F . Before studying solutions of the inverseproblem, we �rst compare the linear mappings F1, F2 and F3 with the original direct mapping �from the point of view of their numerical convergence. More precisely, let F denote one of thedirect mappings �, F1, F2 and F3, and Fh the numerical solution of F with the discretizationparameter h, we will study the order of the approximation ‖Fh(q)−F(q)‖L2 as a function of h,where q= s or p. The discussion uses explicit schemes for simplicity. We recall some simple andusual discretization schemes of the direct mappings and then make a comparative study.

6.1. Discretization of the direct mappings and objective functionals

6.1.1. Mapping �. Using a forward-di�erence for the time derivative and a central-di�erencefor the space derivative (FTCS), the discretization of equation (7) can be expressed as

Tn+1i − Tn

i =�t

h2sn2(Tn

i+1 − 2Tni + Tn

i−1) + yis2n+1 − s2n4hs2n

(Tni+1 − Tn

i−1) (39)



where h is the mesh size, �t is the time step, yi=(i − 1)h, tn=(n − 1)�t, Tni = T (yi; tn) and

sn= s(tn). The boundary and initial conditions are discretized by taking their values at the dis-cretization points. The truncation error for this scheme is O(�t + h2).

6.1.2. Mappings F . The FTCS method is also used to discretize the heat equation (15) formappings F1 and F2, and the heat equation (21) for mapping F3. As well known, the trunca-tion error is in general O(�t + h2), and becomes O(h4) when �t=h2 = 1

6 . The initial conditions,Dirichlet boundary conditions and source terms are discretized by taking their values at the dis-cretization points. For the Neumann boundary condition at x=1 (when F2 is used), the formulaecorresponding to second, third and fourth order of accuracy are compared.

6.1.3. Objective functional. The observation Tx(0; t) is discretized by using various order formu-lae, similarly to the discretization of the Neumann boundary condition, and the objective functionalsare approximated by

J h(p)=nt∑2(Tx(0; t)|n − zn)2�t (40)

for the DEMs and by

J h(s)=nt∑2

(Ty(0; t)|n

sn− zn

)2�t (41)

for the FTM.

6.2. Numerical results

Numerical experiments are made to compare the convergence rates of the discrete mappingsdescribed above.

6.2.1. Test problem A. As discussed in Section 5, this test problem has a smooth tempera-ture �eld in the whole �ctitious domain F, and the regularity conditions s∗(t)∈C1(0; tf) andp∗(t)∈L2(0; tf) hold. The convergence rate of the considered schemes have been analysed asa function of h, �t=h2, and the discretizations of the observation Tx(0; t) and of the Neumannboundary condition.For the mapping �, a time step �t= h2=8 has been used to ensure the stability condition. The

results in Table I show that the scheme is of second order of accuracy (‖�h(s∗)−�(s∗)‖=‖�(s∗)‖∼Ch2) and that using the fourth-order discretization for the observation Tx(0; t) gives a better accu-racy than using the second-order discretization, as soon as h is small.The results for mapping F1 given in Table II show the in uence of the various schemes on

the computed value of the direct mapping. If fourth-order discretizations are used for both thestate equation and the observation, a fourth-order accuracy is obtained in the computation ofthe direct mapping. If the order of discretization for the state equation is di�erent from thatfor the observation, the accuracy obtained in the computation of the direct mapping correspondsto the lower order.Keeping the fourth-order discretization for the state equation (i.e. �t=h2 = 1=6), we compute

mapping F2, with several discretization orders for the Neumann boundary condition and for theobservation. Table III shows that a fourth-order accuracy is obtained in the computation of mapping



Table I. Relative errors ‖�h(s∗)−�(s∗)‖L2 =‖�(s∗)‖L2 computedwith the parameter s∗(t) de�ned by (26) and (32), as a functionof the mesh size h, when using the scheme (39) and di�erentdiscretizations for the observation Tx(0; t) (second order: O2,fourth order: O4)

Test A Test B1h O2 O4 O2 O4

4 3·172E−03 1·210E−04 8·012E−02 0·2288 7·688E−04 3·331E−05 1·666E−02 4·649E−0316 1·888E−04 8·521E−06 3·782E−03 9·352E−0432 4·675E−05 2·142E−06 8·939E−04 2·178E−04

Table II. Relative errors ‖F1h(p∗) − F1(p∗)‖L2 =‖F1(p∗)‖L2 computed withthe parameter p∗(t) de�ned by (30), as a function of the mesh size h,when using di�erent values of �t

h2 (�th2 =

13 :E2,

�th2 =

16 :E4) and di�erent dis-

cretizations for the observation Tx(0; t) (second order: O2, third order: O3,fourth order: O4)

1h E4, O2 E2, O2 E4, O3 E4, O4 E2, O4

4 5·703E−03 6·037E−03 5·659E−04 6·017E−05 4·174E−048 1·363E−03 1·450E−03 6·574E−05 3·393E−06 9·271E−0516 3·331E−04 3·551E−04 7·919E−06 2·013E−07 2·253E−0532 8·233E−05 8·784E−05 9·716E−07 1·226E−08 5·596E−06

Table III. Relative errors ‖F2h(p∗) − F2(p∗)‖L2 =‖F2(p∗)‖L2 computed with the parameter p∗(t) de�nedby (31), as a function of the mesh size h, when us-ing di�erent orders of discretizations for the observationz(t)= Tx(0; t) (second order: O2, third order: O3, fourthorder: O4) and for the boundary condition Tx(1; t)= 0

(second order: B2, fourth order: B4)

1h B2, O2 B2, O3 B4, O4

4 2·523E−03 5·286E−03 2·994E−058 5·810E−04 1·290E−03 1·486E−0616 1·421E−04 3·210E−04 8·559E−0832 3·538E−05 8·043E−05 5·215E−09

F2, when fourth-order discretizations are used for the state equation, the boundary condition andthe observation.In this test problem, the accuracy orders obtained in the numerical computation are in agreement

with the orders of truncation error.



The above discussion shows that the transformed linear mappings F could be computed moreaccurately than the original non-linear mapping �.

6.2.2. Test problem B. In the test problem B, the boundary condition at x=1 is not consistentwith the initial condition and the temperature �eld has a discontinuity at t=0, x=1. In this case,the order of accuracy of the numerical solution is no longer consistent with the order of truncationerror.Using fourth-order discretizations for the heat conduction equation, the Neumann boundary con-

dition and the observation, we obtain a second-order accuracy in the computation of mappingF1 and a �rst-order accuracy for mapping F2. This poor accuracy in the computation of the di-rect mappings will in uence signi�cantly the solution of the inverse problem, particularly in thenoiseless case, as shown in Section 7.The derivative of the exact parameter s∗(t) in the FTM is singular (∞) at t=0. In spite of

this singularity, the computation of � given in Table I exhibits a second-order accuracy. So, forsingular cases, the advantage of the transformed linear mappings F over the original non-linearmapping �, regarding their computation accuracy, is no longer clear.

7. REGULARISED SOLUTIONS OF THE INVERSE PROBLEM

In the previous section, a comparison has been made for numerical solutions of the direct problems.We now turn to the inverse problems, i.e., the estimation of s(t) or p(t) from z(t). The purposeof this section is to compare the methods from the viewpoint of their e�ciency and accuracyin solving the original inverse problem. The original inverse problem (identi�cation of s(t)) iswell-known to be ill-posed, as well as the three transformed inverse problems (identi�cation ofp(t)). Hence some regularizations should be applied to all these inverse problems. However, thebehaviour of the same regularization technique for the di�erent problems may not be the same,and we make a comparison between the various methods.The implicit iterative regularization and the explicit Tikhonov regularization are studied. In the

implicit iterative regularization, the objective functional (J (s) or J (p)) is not modi�ed and thesolution is stabilized by limiting the number of iterations during the minimization procedure. Theexplicit Tikhonov regularization modi�es the objective functional by adding a penalty term R. If apenalty term such as R(p)= ‖p(t)‖2L2 is used, the regularization e�ect di�ers from one method toanother. Indeed, the physical meaning of the unknown p(t) changes with the method (temperature,heat ux, heat source or interface). Let us notice that, using the state regularization conceptproposed by Chavent and Kunisch,23 the penalty term R can be chosen as a functional of thestate variable T instead of the unknown p(t) or s(t), and then the explicit regularization can beperformed independently of the method. However, this approach is not used in this study.The numerical minimization of the objective function J (s) or J (p) is performed with the cal-

culation of the gradient ∇J (s) or ∇J (p). Since the number of variables to be minimized is verylarge, the common method that approximates the gradient by �nite di�erences is expensive and anadjoint method has been used in order to calculate the gradient e�ciently and accurately. In gen-eral, the adjoint equation can be derived in three di�erent ways corresponding to direct problems(state equation and objective function) of di�erent levels:

(a) When the direct problem is governed by continuous PDEs, the adjoint method is explained inmany books of optimal control theory (see e.g. Reference 24). The derived adjoint equation



is still a continuous PDE. For the numerical computation of the gradient, the state andadjoint equations must be solved numerically, so the accuracy in the computation dependssigni�cantly on the discretization errors of the state equation, the adjoint equation and theobjective function.

(b) If the direct problem is discretized, the adjoint equation can be derived from the discreteobjective function and the discrete state equation. We have chosen this approach to computethe gradient. In fact, this is the Lagrange multiplier method. When the discrete state equa-tion can be solved accurately (under the oating point error), the gradient of the discreteobjective function can also be computed accurately.Although the adjoint method is very straightforward, implementing it by hand could be

human time consuming since the formulae to be calculated could be very complex. Thesymbolic computation packages, such as Maple, Macsyma, etc., can be helpful. Moreover,a symbolic di�erentiation Fortran code generator has been developed for certain classes ofdiscrete problems.25 Thanks to this Fortran code generator tool, the subroutines needed forthe computation of the objective function and its gradient with respect to the unknown canbe obtained directly from the discrete state equations and the discrete objective functiondescribed in Section 6.1.

(c) When the direct problem has been implemented in computer language, the automatic di�er-entiation techniques, which have been developed recently,26 can be used. With the automaticdi�erentiation, the code computing the gradient of the objective function can be obtaineddirectly from the code of the objective function, and the gradient is computed accuratelyunder the oating point error.

Once the objective function and its gradient are programmed, we can use a standard minimiza-tion package. All the numerical minimization results presented here are obtained with the IMSLsubroutine UMCGG, which implements a conjugate gradient algorithm.Both noiseless data and noisy data are used in order to test the regularization methods. The ob-

servation and the Neumann boundary condition are discretized by using the fourth-order formulae.Time step �t= h2=8 is used for test A when using the FTM; otherwise time step �t= h2=6 isused.

7.1. Approximation of the interface position

Using the DEMs presented in this paper, the interface position, corresponding to the isothermT =0, is computed by interpolating the discrete temperature �eld. Before comparing the inversesolutions, we �rst examine this interpolation error. More precisely we study its convergence ratefor di�erent space mesh sizes h. Let sh(t) denote the approximate interface position obtained by in-terpolating an exact discrete temperature distribution {T (xi; tn); i=1 ; : : : ; ns; n=1 ; : : : ; nt} wherexi=(i − 1)h and tn=(n − 1)�t, and s∗(t) denote the exact interface position. The interpolationused in this paper is a simple linear interpolation. By taking a su�ciently small �xed time step�t=1=210 and varying the space discretization size h, we compute the error ‖sh−s∗‖L2(0;1) for thetest problems A and B. The results are presented in Figure 1, from which we obtain approximately

‖sh − s∗‖L2(0;1)6Ch (42)

with =2·0 for test A and =2·14 for test B.For the two space discretization sizes used later, h=1=8 and h=1=16, this error is about 5·E−4

and 2·E−4, which is very small. A more accurate interpolation method could be used, however



Figure 1. Approximation L2-errors in the interface position obtained by linear interpolation from the discrete temperature�eld for test problems A and B, as a function of h

the results given in the next sections, regarding the comparison between the FTM and the DEMs,would not be changed signi�cantly.

7.2. Implicit iterative regularization

In this regularization technique, the objective functionals J (s) or J (p) are minimized iteratively.Because of the ill-posedness of the problem, limiting the number of iterations is necessary in orderto get a reasonable solution, i.e., regularization is achieved by choosing a suitable stop test. Animportant question is the choice of the optimal number of iterations which depends on the datanoise level. If the noise level is known, the procedure can be stopped when the objective functionalis equal to the noise level (this is the so-called discrepancy principle). Otherwise, an appropriatetolerance for the gradient or the number of iterations have to be �xed a priori, i.e., the calculationis stopped when the norm of the gradient is smaller than a given tolerance (‖∇J (s)‖26tol or‖∇J (p)‖26tol) or when the number of iterations is larger than a given number. The initial guessess(t)= 1 for the FTM and p(t)= 0 for the DEMs are chosen for the minimization procedure.

7.2.1. Noiseless case. In order to analyse the in uence of the discretization error on inverse solu-tions, we �rst consider the noiseless case: the exact analytical observation is used as measurementdata, and only the discretization error is concerned.Let us look at test A. This is a regular case and the computation of the direct mappings is very

accurate, as shown in Section 6. Using the (coarse) mesh size h= 18 , the relative discretization

errors are 3·4E−6, 1·48E−6 and 3·3E−5 for the mappings F1, F2 and �, respectively. With thesevery small discretization errors in data, the interface positions estimated by the various methods(with tol = 10−8) are mostly identical to the exact solution except at the end of the time horizon(t¿0·9), as shown in Figure 2 (left). A smaller tolerance tol = 10−10 and another initial guessp(t)= 1 have also been used for the estimation: the obtained results are not modi�ed, except atthe end of the time horizon. The error ampli�cation at the end of the time horizon is well-knownfor inverse heat conduction problems when the time interval of identi�cation is equal to the timehorizon of observation.7 We have not shortened the time interval of identi�cation a priori, butrelative interface position errors will be computed in the shortened interval ]0; 0·8[ unless otherwisestated.In contrast, for test B, because of the singularity at t=0 the direct mappings are di�cult to be

computed accurately with a coarse mesh size such as h= 18 , as shown in Section 6. The relative



Figure 2. Interface positions obtained by using the various methods for test A with h= 18 ; tol = 10

−8 and for test B withh= 1

8 ; tol = 10−6, in the noiseless case

Figure 3. Unknown parameters p(t) obtained by using F1 with two mesh sizes h= 18 and h= 1

16 , for tests A and Bin the noiseless case

discretization errors with h= 18 are 6·1E−3 and 4·6E−3 for F1 and �, respectively. With these

large discretization errors, the estimated results are sensitive to the value of the tolerance tol anda larger tolerance tol = 10−6 is chosen. The interface positions estimated by the various methodsare shown in Figure 2 (right).The improvement induced by re�ning the mesh is then analysed. The parameter p(t) obtained

by using method F1 with h= 18 or h= 1

16 is shown in Figure 3. As can be seen, a re�ned meshsomewhat improves the quality of the results at the beginning (t≈ 0), however the obtained resultbegins to oscillate when t¿0·8. It could be explained by the fact that, since the number ofunknowns to be determined is increased for the �ner mesh, the degree of ill-posedness (or ofconditioning) of the problem is also increased.The above results are con�rmed by looking at the evolution of the relative interface position

error against the iteration number when using F1 for both test A and B (Figure 4). For test A, theerror does not change much after about 100 iterations for h= 1

8 and h= 116 : the error in the data

introduced by discretization is very small and the minimization is not sensitive to it. In contrast,for test B, taking h= 1

8 , the discretization error is large and the relative interface position errorbegins to increase after about 30 iterations. Similar results are obtained for F2 and F3. Amongthe various methods, the best accuracy is obtained with F1.For method �, we changed the objective functional of equation (41), by weighting it by s(t)2,

and wrote

Jh(s)=nt∑2(Ty(0; t)|n − snz n)2�t (43)



Figure 4. Evolution of the relative interface position error as a function of the iteration index during the minimizationprocedure, for tests A and B in the noiseless case, when using method F1 with h= 1

8 and h= 116

Such a modi�cation improves convergence in our numerical experiments. In particular for test A,the number of iterations needed has been reduced by about �ve times (to about 100 iterations).For test B, as expected, the convergence is more di�cult than that for test A, because of thesingularity of the unknown function at t=0.

7.2.2. Low noise level case. In this section we use noisy data. They are obtained by addinga white Gaussian noise to the analytical observation z(t). The standard deviation of the noiseis 0·01 and thus the signal-to-noise ratio is about 35. The noise is dominant compared to thediscretization error. In such a situation, a relatively coarse mesh size (h= 1

8) can be used.First, we analyse the DEMs by using the curve of the objective functional J (pk) versus the

number of iterations k. The values of J (pk) and of relative interface position error are plottedin Figure 5 against the iteration number k, when method F2 is used. From the approximatecontrollability described in Section 4.4, we know that J (pk) can be as small as expected, whenk increases (tends to ∞). Let us examine the curve of J (pk) versus k in detail: it can be seenthat there exists an ‘elbow’ point such that J (pk) decreases rapidly down to it, and then decreasesvery slowly. The value of J (pk) at the ‘elbow’ point approximates to the noise level. Figure 5(left) presents a regular situation, where the parameter to be identi�ed p∗(t) is in L2(0; 1): the‘elbow’ point is close to the minimum point of the relative interface position error. In contrast,Figure 5 (right) presents an irregular situation, where p∗(t) is not in L2: the ‘elbow’ point is farfrom the minimum point of the relative error, and more iterations are needed to get the minimumthan for the regular situation. Methods F1 and F3 have also been analysed: the results obtainedwith F3 are similar to those with F2; for F1, since p∗(t)∈L2(0; 1) in tests A and B, the ‘elbow’point corresponds to the minimum of the interface position error for both the tests. The conclusionis that the evolution of the objective functional versus the iteration number gives some usefulinformation to choose the optimal regularization when p∗(t)∈L2.To compare the di�erent DEMs, the interfaces obtained with a �xed number of iterations are

shown in Figure 6, and the relative interface position errors (on the interval [0; 0 · 8]) correspondingto the respective optimal solutions are shown in Table IV. As can be seen, the iterative regulariza-tion method is quite e�ective with the DEMs, since the corresponding inverse problems are linear.It can be seen that F1 is the best for test B. The minimal relative errors obtained for test A havethe same order of magnitude, F2 being sightly better.The conclusions are quite di�erent for the FTM. The minimization requests often many iterations

to converge, even the procedure may not converge at all for some initial guesses. This may be



Figure 5. Evolution of the relative interface position error and of the objective functional (√

J (p)), as a function of the

iteration index during the minimization procedure, using method F2 with h= 18 , for tests A and B in the low level noisecase

Figure 6. Interfaces obtained by using the DEMs with h= 18 after 20 iterations, for test A and B in the low level noisecase

Table IV. Relative interface position errors correspondingto the optimal solution of the various methods with the

iterative regularization in the low noise level case

Method F1 F2 F3 �

Test A 4·9E−03 3·9E−03 4·5E−03 1·2E−02Test B 2·8E−03 2·1E−02 5·2E−02 3·5E−02

explained by the fact that the problem is very non-linear and the number of degrees of freedomis very large. The objective functional and the relative interface position error are plotted againstthe iteration number in Figure 7 (left) and the obtained interface position is shown in Figure 7(right), for test A. Here, the minimization works well: the objective functional decreases till thenoise level when the iteration number increases. However, whatever the number of iterations, therelative error is large, and, as a result, limiting the number of iterations does not regularize theproblem. Similar results are obtained for test B. So, the iterative regularization method is di�cultto apply directly to the original non-linear problem in both regular and singular cases.



Figure 7. Evolution of the relative interface position error and the objective function (√

J (p)) as a function of theiteration index during the minimization procedure (left), and interface obtained after 20 iterations (right), using method �

with h= 18 , for test A in the low level noise case

Figure 8. Interfaces obtained by using the DEMs with h= 18 after 20 iterations, for test A and B in the high level noisecase

7.2.3. High noise level case. A noise of standard deviation 0·04 (the signal-to-noise ratio isabout 9) is also tested. Similar analysis results are obtained. Of course, as expected, the optimalnumber of iterations is reduced as compared with the low noise level case. The interfaces obtainedwith the optimal iteration number by using the DEMs are compared with the exact interface inFigure 8. The results are still satisfactory.

7.3. Tikhonov regularization

In this section, we study the Tikhonov regularization. The solution is stabilized by addinga penalty term in the objective functional J (q), i.e., by replacing J (q) with J�(q)= J (q) + �R(q),where q is p or s. It is clear that the interface solution s(t), which is the parameter to be identi�edultimately, depends strongly on the choice of the regularization parameter �, of the penalty termR(q), and of the direct mapping F (or �). The main purpose here is to compare the e�ects of theforegoing methods on the regularized solution s(t).As in Section 7.2, noisy data with standard deviation 0·01 are used. The regularized objective

functional J�(p) is minimized with a stop test corresponding to a gradient tolerance tol = 10−8 forall the DEMs and with a looser tolerance tol = 10−5 for the FTM due to its very slow convergence.As in Section 7.2.2, we compare �rst the various DEMs. The regularized solutions have been

computed with R(p)= ‖p(t)‖2L2 and with various values of �. Concerning the optimal choice of �,



Figure 9. Evolution of the relative interface position error, the residual and the objective function (√

J (p)) as a function

of �, when using F2 with h= 18 , for tests A and B in the low level noise case

Figure 10. Interfaces obtained by using the various DEMs with optimal �, for tests A and B in the low level noise case

many studies have been published (e.g. see Reference 27). We propose here to use the evolutionof the residual J (p�) versus � to get an approximation of the optimal �. The evolutions of therelative error, residual J (p�) and objective functional J�(p�) are plotted against the parameter �in Figure 9 for F2. As can be seen, there exists an ‘elbow’ point for the the curve of the residualJ (p�) (not the objective functional J�(p�)) versus the (decreasing) parameter � in a log–log form.The behaviour of J (p�) versus � is similar to that of J (pk) versus k in the iterative regularization:J (p�) decreases rapidly down to the ‘elbow’ point, and then decreases very slowly. The value ofJ (p�) at the ‘elbow’ point is about the noise level. Figure 9 (left) shows that the ‘elbow’ point isclose to the minimum of the relative error when the parameter to be identi�ed p∗(t) belongs toL2(0; 1) (regular situation). Figure 9 (right) shows that the ‘elbow’ point is far from the minimumof the relative error and that a smaller � should be used to get the minimum, when p∗(t) =∈L2(0; 1)(irregular situation).In test A, the optimal value of � is approximately 10−2 for F1 and 10−3 for F2 and F3. In

test B, the values 10−2, 10−4 and 10−3 are found for F1, F2 and F3, respectively. The solutionscorresponding to optimal � are given in Figure 10. Let us underline that since the regularizedsolution s�(t) depends strongly on the value of � and since the optimal parameter � di�ers fromone method to another, we must compare the various methods using their corresponding optimal �.For the FTM, if the penalty term R(s)= ‖s(t) − s(0)‖2L2 is used (which is referred to as the

�0 method here), the minimization does not converge easily and the solution obtained with optimal� is still very oscillatory (the value of relative errors in Table V does not well illustrate this



Table V. Relative interface position errors obtained with anoptimal �, when using the various methods with the Tikhonov

regularization in the low noise level case

Method F1 F2 F3 �0 �1

Test A 5·0E−03 1·8E−03 3·3E−03 7·4E−02 2·5E−03Test B 2·8E−03 2·0E−02 4·9E−02 8·4E−02 1·5E−03

Figure 11. Evolution of the relative interface position error, the residual and the objective function (√

J (p)) as a function

of � (left) and interface obtained with the optimal regularization parameter (right), when using the �1-method with h= 18

for test B in the low level noise case

Table VI. Relative interface position errors obtained withan optimal �, when using the various methods with theTikhonov regularization in the high noise level case

Method F1 F2 F3 �1

Test A 1·1E−02 4·5E−03 8·2E−03 1·4E−02Test B 6·3E−03 6·7E−02 0·12 4·1E−03

oscillatory phenomenon, since the L2-norm is used to compute the relative errors). On the contrary,the solution obtained with the penalty term R(s)= ‖s′(t)‖2L2 (which is called the �1 method here) isvery good. The results obtained with R(s)= ‖s′(t)‖2L2 are shown in Figure 11. As the minimizationproblem is strongly non-linear, the procedure is very slow.To summarize this comparison, we give in Table V the relative interface position errors using

the various methods with optimal �. Among the DEMs, F1 gives satisfactory solutions for thetwo test problems and F2 gives a better solution for test A.A noise with standard deviation 0·04 has also been tested, similar results are obtained and shown

in Table VI.

8. CONCLUSION

Three domain embedding methods for the identi�cation of solid–liquid interface position have beenanalysed and compared with the front-tracking method. Good performances for the considered in-verse Stefan problem have been obtained by using DEMs. As the transformed problem becomes



linear, a regularized solution could be easily obtained, and an optimal regularization parametercould be automatically chosen with a reasonable computation cost. Furthermore, at each iterationof optimization, the associated direct mapping can be computed e�ciently. A satisfactory regular-ization is obtained by using the iterative method or by using the Tikhonov method with a simplepenalty term in the L2 space.The front-tracking method has also some advantages. An accurate regularized solution has been

obtained by choosing an appropriate penalty term. No interpolation is needed to obtain the in-terface solution. However, in our opinion, it exhibits more important drawbacks than advantages.The convergence could not be ensured. Moreover, the iterative regularization and the Tikhonovregularization in L2 do not work, and special care should be taken to choose an adequate penaltyterm.Among the domain embedding methods, F1 is more robust; it gives satisfactory results for

smooth as well as for singular problems. F2 is slightly more accurate for the smooth problem. Ingeneral, a good choice of methods should depend on some a priori information (regularity) onthe new unknown parameter p(t) to be used.

ACKNOWLEDGEMENT

We would like to thank the referees for their many helpful suggestions.

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A comparative study of Domain Embedding Methods for regularized solutions of inverse Stefan problems

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