This is a repository copy of An inverse problem of finding the time-dependent thermal conductivity from boundary data.

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1 This is a reposiory copy of An inverse problem of finding he ime-dependen hermal conduciviy from boundary daa. Whie Rose Research Online URL for his paper: hp://eprins.whierose.ac.uk/57/ Version: Acceped Version Aricle: Hunul, MJ and Lesnic, D (7) An inverse problem of finding he ime-dependen hermal conduciviy from boundary daa. Inernaional Communicaions in Hea and Mass Transfer, 85. pp ISSN hps://doi.org/.6/j.icheamassransfer Elsevier Ld. This manuscrip version is made available under he CC-BY-NC-ND 4. license hp://creaivecommons.org/licenses/by-nc-nd/4./ Reuse Iems deposied in Whie Rose Research Online are proeced by copyrigh, wih all righs reserved unless indicaed oherwise. They may be downloaded and/or prined for privae sudy, or oher acs as permied by naional copyrigh laws. The publisher or oher righs holders may allow furher reproducion and re-use of he full e version. This is indicaed by he licence informaion on he Whie Rose Research Online record for he iem. Takedown If you consider conen in Whie Rose Research Online o be in breach of UK law, please noify us by ing eprins@whierose.ac.uk including he URL of he record and he reason for he wihdrawal reques. eprins@whierose.ac.uk hps://eprins.whierose.ac.uk/

2 An inverse problem of finding he ime-dependen hermal conduciviy from boundary daa M.J. Hunul, and D. Lesnic Deparmen of Applied Mahemaics, Universiy of Leeds, Leeds LS 9JT, UK Deparmen of Mahemaics, Faculy of Science, Jazan Universiy, P.O. Bo 4, Jazan 454, Saudi Arabia s: (M.J. Hunul), (D. Lesnic). Absrac We consider he inverse problem of deermining he ime-dependen hermal conduciviy and he ransien emperaure saisfying he hea equaion wih iniial daa, Dirichle boundary condiions, and he hea flu as overdeerminaion condiion. This formulaion ensures ha he inverse problem has a unique soluion. However, he problem is sill ill-posed since small errors in he inpu daa cause large errors in he oupu soluion. The finie difference mehod is employed as a direc solver for he inverse problem. The inverse problem is recas as a nonlinear leas-squares minimizaion subjec o physical posiiviy bound on he unknown hermal conduciviy. Numerically, his is effecivey solved using he lsqnonlin rouine from he MATLAB oolbo. We invesigae he accuracy and sabliy of resuls on a few es numerical eamples. Keywords: Inverse problem; hermal conduciviy; hea equaion; nonlinear opimizaion. Inroducion In inverse problems, he unknown densiies or disribued source, or he coefficiens involved in he governing parial differenial equaion or in he boundary condiions for a mahemaical model under invesigaion are sough from addiional informaion on he main dependen variable soluion of he original direc iniial boundary value problem, []. In paricular, he inverse problem of idenifying he hermal diffusiviy/ conduciviy from boundary daa (emperaure and parial hea flu) has been invesigaed widely by many researchers in he pas, see [ 3,5 8,] o menion only a few. In his paper, he novely consiss in he developmen of a convergen numerical opimizaion mehod for solving his nonlinear bu well-posed inverse coefficien problem for he hea equaion. Numerically, he implemenaion is realised using he MATLAB oolbo rouine lsqnonlin. The paper is organized as follows. In Secion, he mahemaical formulaion of he inverse problem is presened. In Secion 3, he numerical soluion of he direc problem is based on he finie difference mehod wih he Crank-Nicolson scheme. In Secion 4, he minimizaion algorihm o solve he inverse problem is presened. The numerical resuls are discussed in Secion 5. Finally, conclusions are highlighed in Secion 6.

3 Mahemaical formulaion In he domain Q T = {(,) < < T, < < L}, we consider he inverse problem given by he parabolic hea equaion u (,) = u a() (,)+f(,), (,) Q T, () wih known hea source f(, ), unknown emperaure u(, ) and unknown hermal conduciviy a() >, (,T], subjec o he iniial condiion u(,) = φ(), [,L], () he Dirichle emperaure boundary condiions u(,) = µ (), u(l,) = µ (), [,T], (3) and he Neumann hea flu overdeerminaion condiion a()u (,) = µ 3 (), [,T]. (4) The uniqueness of soluion of he inverse problem () (4) has been esablished in [6] and reads as follows. Theorem. (Uniqueness). If < µ 3 C[,T], hen a soluion (a(),u(,)) H +α/ [,T] H +α,+α/ (Q T ), for some α (,),a() > for [,T], o he problem () (4) is unique. In his heorem, H +α/ [,T] denoes he space of Hölder coninuously differeniable funcions on [,T] wih eponen α/. Also, H +α,+α/ (Q T ) denoes he space of coninuous funcions u along wih heir parial derivaives u, u, u in Q T, wih u being Hölder coninuous wih eponen α in [,L] uniformly wih respec o [,T], and wih u being Hölder coninuous wih eponen α/ in [,T] uniformly wih respec o [,L]. Lower-order erms b(,) u (,)+c(,)u(,), wih b and c known funcions, can also be added o he righ-hand side of equaion (), wih no qualiaive change in boh analyical and numerical analyses, [6]. 3 Numerical soluion of direc problem In his secion, we consider he direc iniial boundary value problem given by equaions () (3). We use he finie-difference mehod (FDM) wih a Crank-Nicholson scheme, [], which is uncondiionally sable and second-order accurae in space and ime. The discree form of he direc problem is as follows. We denoe u( i, j ) = u i,j,a( j ) = a j, and f( i, j ) = f i,j, where i = i, j = j for i =,M,j =,N, and = L M, = T N. Then he problem () (3) can be discreised as A j+ u i,j+ +(+B j+ )u i,j+ A j+ u i+,j+ = A j u i,j +( B j )u i,j +A j u i+,j + (f i,j +f i,j+ ), i =,(M ), j =,N, (5) u i, = φ( i ), i =,M, (6)

4 where u,j = µ ( j ), u M,j = µ ( j ), j =,N, (7) A j = ( )a j ( ), B j = ( )a j ( ). A each ime sep j+, for j =,(N ), using he Dirichle boundary condiions (7), he above difference equaion can be reformulaed as a (M ) (M ) sysem of linear equaions of he form, Du j+ = Eu j +b j, (8) where u j+ = (u,j+,u,j+,...,u M,j+,u M,j+ ) T, +B j+ A j+... A j+ +B j+ A j+... D = ,... A j+ +B j+ A j+... A j+ +B j+ B j A j... A j B j A j... E = ,... A j B j A j... A j B j and (f,j +f,j+ )+A j µ ( j )+A j+ µ ( j+ ) b j (f,j +f,j+ ) =.. (f M,j +f M,j+ ) (f M,j +f M,j+ )+A j µ ( j )+A j+ µ ( j+ ) by The numerical soluion for hea flu in equaion (4) on he inerval [,T] is given µ 3 ( j ) = a( j )u (, j ) = (4u,j u,j 3µ ( j ))a j, j =,N. (9) 4 Numerical approach o solve he inverse problem The nonlinear inverse problem () (4) can be formulaed as a nonlinear minimizaion of he leas-squares objecive funcion F(a) := a()u (,) µ 3 (), () 3

5 he discreizaions of which is F(a) = N j= [ a j u (, j ) µ 3 ( j )], () where a = (a j ) j=,n R N +. The minimizaion of () is performed using he MATLAB oolbo rouine lsqnonlin, which does no require supplying by he user he gradien of he objecive funcion, [9]. This rouine aemps o find he minimum of a sum of squares by saring from he arbirary iniial guesses a () for a. We have compiled his rouine wih he following specificaions : Algorihm is he Trus Region Reflecive (TRR) minimizaion, see [4]. Number of variables M = N. Maimum number of ieraions, (MaIer)= 4. Maimumnumberofobjecivefuncionevaluaions,(MaFunEvals)= (number of variables.) Terminaion olerance on he funcion value, (TolFun) =. Tolerance, (Tol)=. 5 Numerical resuls and discussion In his secion, we presen a few es eamples in order o es he accuracy and sabiliy of he numerical mehod inroduced in Secion 4. The roo mean square error (rmse) is used o evaluae he accuracy of he numerical resuls as follows: rmse(a()) = N N j= ( a numerical ( j ) a eac ( j )). () The inverse problem () (4) is solved subjec o boh eac and noisy hea flu measuremens (4). The noisy daa are numerically simulaed as µ ǫ 3( j ) = µ 3 ( j )+ǫ j, j =,N, (3) where ǫ j are random variables generaed from a Gaussian normal disribuion wih mean zero and sandard deviaion σ given by σ = p ma [,T] µ 3(), (4) where p represens he percenage of noise. We use he MATLAB funcion normrnd o generae he random variables ǫ = (ǫ j ) j=,n as follows: ǫ = normrnd(,σ,n). (5) 4

6 The oal amoun of noise ǫ is given by ǫ = ǫ = N (µ ǫ 3( j ) µ 3 ( j )). (6) j= In all he numerical resuls presened below we ake L = T =. We also ake he iniial guess for he unknown hermal diffusiviy a() equal o he consan a(), which from he compaibiliy of he condiions () and (4) a = is known and given by a() = µ 3 ()/φ (). 5. Eample In his eample, we consider he inverse problem given by () (4) and he inpu daa φ() = u(,) = ( + ), (7) µ () = u(,) = 4, µ () = u(,) = (+), (8) ( f(,) = + ) (+) (+) ( ( + ) 8 ( + ) )(+) (+), (9) µ 3 () = a()u (,) = (+) ( ). () I can be easily checked by direc subsiuion ha he analyical soluion of he inverse problem () (4) wih he inpu daa (7) () is given by and u(,) = + ( + ) (+), () a() = +. () Firs, we assess he convergence and accuracy of he FDM solver of Secion 3 for solving he direc problem () (3). Figure shows he numerical hea flu in equaion (4) in comparison wih he eac soluion () obained by solving he direc problem () (3) wih he inpu daa (7) (9) and () using he FDM, described in Secion 3, wih M = N {,,4}. Form his figure i can be seen ha he good agreemen beween he eac soluion () and he numerical one is obained and is order is O(( ) ), as he mesh size decreases. WenowfiM = N = 4andryorecoverheunknownhermalconduciviya()and he emperaure u(,) for eac inpu daa, i.e. p =, as well as for p {5%,%} noisy daa. The objecive funcion () is ploed, as a funcion of he number of ieraions, in Figure. From his figure, i can be seen ha a very fas convergence is achieved in abou 8 o ieraions o reach o a very low value of O( ). The relaed numerical resuls for a() and u(,) are presened in Figures 3 and 4, respecively. From hese figures i can be seen clearly ha here is good agreemen beween he numerical resuls and he analyical soluions for eac daa, i.e. p =, and is proporionae wih he errors in he inpu daa for p >. The numerical soluions for a() and u(,) converge o heir corresponding eac soluions in equaions () and (), as he percenage of noise p 5

7 decreases from % o 5% and hen o zero. For compleeness, oher he deails abou number of ieraions, he number of funcion evaluaions, he value of he objecive funcion () a final ieraion, he rmse in () and he compuaional ime are given in Table. From his able i can be seen ha accurae and sable numerical resuls are rapidly achieved by he ieraive MATLAB oolbo rouine lsqnonlin..5 µ 3 ().5 Eac Soluion M=N= M=N= M=N= Figure : The eac (equaion ()) and numerical soluions for he hea flu (4), for Eample wih M = N {,,4}, for he direc problem. Objecive funcion p= p=5% p=% Number of Ieraions Figure : Objecive funcion (), for Eample wih p {,5%,%} noise. 3.5 eac soluion p= p=5% p=% a() Figure 3: The eac (equaion ()) and numerical soluions for he hermal conduciviy a(), for Eample wih p {,5%,%} noise. 6

8 (a) Eac soluion u(,) Absolue error (b).4 Eac soluion u(,) Absolue error.3.. (c) Eac soluion u(,) Absolue error Figure 4: The eac (equaion ()) and numerical soluions for he emperaure u(, ), for Eample, wih (a) no noise, (b) p = 5% noise, and (c) p = % noise. The absolue error beween hem is also included. 7

9 Table : Number of ieraions, number of funcion evaluaions, value of he objecive funcion () a final ieraion, rmse(a) and compuaional ime, for Eample. Numerical oupus p = p = 5% p = % E-6 8.E-6 7.8E-3 Number of ieraions Number of funcion evaluaions Value of objecive funcion () a final ieraion rmse(a) Compuaional ime mins.84 7 mins mins 5. Eample We now consider recovering a non-smooh hermal conduciviy, as given by equaion (5) below. We ake inpu daa given by (7), (8), ( f(,) = + ) (+) ( + )( ( + ) 8 ( + ) )(+) (+), (3) and ( µ 3 () = a()u (,) = )( + ). (4) Then he analyical soluion of he inverse problem () (4) wih his inpu daa is given by () for he emperaure u(,) and a() = + for he hermal conduciviy. Figure 5 shows he numerical hea flu in equaion (4) in comparison wih he eac soluion (4) obained by solving he direc problem () (3) wih he inpu daa (7), (8), () and (3) using he FDM, described in Secion 3, wih M = N {,,4}. From his figure i can be seen ha he good agreemen beween he eac soluion (4) and he numerical one is obained and is order is O(( ) ), as he mesh size decreases. We now fi M = N = 4 and ry o recover he hermal conduciviy a() and he emperaure u(,) for eac inpu daa, i.e. p =, as well as for p {%,3%} noisy daa. The objecive funcion () is ploed, as a funcion of he number of ieraions, in Figure 6. From his figure, i can be seen ha a very fas convergence is achieved in abou 7 o ieraions o reach o a very low value of O( 7 ). The relaed numerical resuls for a() and u(,) are presened in Figures 7 and 8, respecively. From hese figures i can be seen clearly ha here is good agreemen beween he numerical resuls and he analyical soluions for eac daa, i.e. p =, and is proporionae wih he errors in he inpu daa for p >. The numerical soluions for a() and u(,) converge o heir corresponding eac soluions in equaions (5) and (), as he percenage of noise p decreases from 3% o % and hen o zero. (5) 8

10 .8 µ 3 ().6.4. Eac Soluion M=N= M=N= M=N= Figure 5: The eac (equaion (4)) and numerical soluions for he hea flu (4), for Eample wih M = N {,,4}, for he direc problem. 5 5 Objecive funcion 5 p= p=% p=3% Number of Ieraions Figure 6: Objecive funcion (), for Eample wih p {,%,3%} noise..3. a()..9.8 eac soluion p= p=% p=3% Figure 7: The eac (equaion (5)) and numerical soluions for he hermal conduciviy a(), for Eample wih p {,%,3%} noise. 9

11 (a) 4 3 Eac soluion u(,) Absolue error 3 (b) Eac soluion u(,) Absolue error (c).8 Eac soluion u(,) Absolue error.6.4. Figure 8: The eac (equaion ()) and numerical soluions for he emperaure u(, ), for Eample, wih (a) no noise, (b) p = % noise, and (c) p = 3% noise. The absolue error beween hem is also included. Oher deails abou number of ieraions, he number of funcion evaluaions, he value of he objecive funcion () a final ieraion, he rmse in () and he compuaional imearegivenintable. Fromhisableicanbeseenhaaccuraeandsablenumerical resuls are rapidly achieved by he ieraive MATLAB oolbo rouine lsqnonlin.

12 Table : Number of ieraions, number of funcion evaluaions, value of he objecive funcion () a final ieraion, rmse(a) and he compuaional ime, for Eample. Numerical oupus p = p = % p = 3% E-6 4.E-6 9.4E-7 Number of ieraions Number of funcion evaluaions Value of objecive funcion () a final ieraion rmse(a) Compuaional ime.35 4 mins.3 4 mins mins 5.3 Eample 3 Consider he inverse problem () (4) wih he inpu daa φ() = u(,) = sin(π), µ () = µ () =, f(,) =, (6) ( µ 3 () = a()u (,) = π(.+sin(3π))ep [ π The eac soluion for he emperaure u(, ) is ( u(,) = sin(π)ep [ π and for he hermal conduciviy a() is.+ cos(3π) 3π.+ cos(3π) 3π )]. (7) )], (8) a() =.+sin(3π). (9) This eample was considered in [] and we generae he noisy hea flu measuremen (4) for his eample as in [] as muliplicaive (raher han addiive as in (3)), namely, µ ǫ 3( j ) = µ 3 ( j )(+pǫ j ), j =,N, (3) where p represens he percenage of noise and ǫ = (ǫ j ) j=,n, is a random real number beween [, ] generaed from uniform disribuion using MATLAB funcion rand as ǫ = rand(,n). (3) The objecive funcion (), as a funcion of he number of ieraions is shown in Figure 9 wih no noise and wih various mesh sizes. From his figure i can be seen ha very low converging values of he monoonically decreasing objecive funcion F in () are achieved. The corresponding numerical resuls for a() are compared wih he analyical soluion (9) in Figure, wih he numerical deails included in Table 3. From his figure and able i can be seen ha he numerical soluion for a() converges o eac soluion (9), as he mesh size decreases.

13 5 Objecive funcion 5 M=N= M=N=4 M=N= Number of Ieraions Figure 9: Objecive funcion (), for Eample 3 wih no noise and wih M = N {,4,8} eac soluion M=N= M=N=4 M=N=8 a() Figure : The eac (equaion (9)) and numerical soluions for he hermal conduciviy a(), for Eample 3 wih no noise and wih various mesh size M = N {,4,8}. Table 3: Number of ieraions, number of funcion evaluaions, value of he objecive funcion () a final ieraion, rmse(a) and he compuaional ime, for Eample 3 wih various mesh size M = N {,4,8} and wih no noise. Numerical oupus M = N = M = N = 4 M = N = E-3 6.4E-9 3.4E-8 Number of ieraions Number of funcion evaluaions Value of objecive funcion () a final ieraion rmse(a) Compuaional ime mins mins.7 hours When we include various levels of noise p {,3,5}% as in (3) o he hea flu measuremen (4) we obain sable resuls for he a() as hermal conduciviy shown in Figure. Furhermore, he resuls become more accurae as he amoun of noise p decreases. Numerical resuls are also comparable in erms of sabiliy and accuracy wih hose in [] obained using a oally differen mehod.

14 .5 a().5 eac soluion p=% p=3% p=5% Figure : The eac (equaion (9)) and numerical soluions for he hermal conduciviy a(), for Eample 3 wih p {%,3%,5%} noise and no regularizaion wih M = N = 4. 6 Conclusions This paper has presened he deerminaion of he ime-dependen hermal conduciviy from hea flu measuremens in he one-dimensional parabolic hea equaion. The resuling inverse problem has been reformulaed as a nonlinear leas-squares opimizaion problem, which has been solved using he MATLAB oolbo rouine lsqnonlin. The numerical resuls are shown o be sable and accurae. The inverse problem seems sable, and hence, no regularizaion was found necessary o be employed. Acknowledgmens M.J. Hunul would like o hank Jazan Universiy in Saudi Arabia and Unied Kingdom Saudi Arabian Culural Bureau (UKSACB) in London for pursuing his PhD a he Universiy of Leeds. References [] Azari, H., Allegreo, W., Lin, Y. and Zhang, S. (4) Numerical procedures for recovering a ime-dependen coefficien in a parabolic differenial equaion, Dynamics of Coninuous, Discree and Impulsive Sysems, Series B: Applicaions & Algorihms,, [] Cannon, J.R. (963) Deerminaion of an unknown coefficien in a parabolic differenial equaion, Duke Mahemaical Journal, 3, [3] Cannon, J.R. and Rundell, W. (99) Recovering a ime-dependen coefficien in a parabolic differenial equaion, Journal of Mahemaical Analysis and Applicaions, 6, [4] Coleman, T.F. and Li, Y. (996) An inerior rus region approach for nonlinear minimizaion subjec o bounds, SIAM Journal on Opimizaion, 6,

15 [5] Hussein, M.S., Lesnic, D. and Ismailov, M.I. (6) An inverse problem of finding he ime-dependen diffusion coefficien from an inegral condiion, Mahemaical Mehods in he Applied Sciences, 39, [6] Ivanchov, M.I. (998) On deerminaion of a ime-dependen leading coefficien in a parabolic equaion, Siberian Mahemaical Journal, 39, [7] Jones, B.F., Jr. (96) The deerminaion of a coefficien in a parabolic differenial equaion, Par I. Eisence and uniqueness, Journal of Mahemaical Mechanics,, [8] Jones, B.F., Jr. (963) Various mehods for finding unknown coefficiens in parabolic differenial equaions, Communicaions on Pure and Applied Mahemaics, 6, [9] Mahwoks R Documenaion Opimizaion Toolbo-Leas Squares (Model Fiing) Algorihms, available from /brnoybu.hml. [] Smih, G.D. (985) Numerical Soluion of Parial Differenial Equaions: Finie Difference Mehods, Oford Applied Mahemaics and Compuing Science Series, Third ediion. [] Soboleva, O.V. and Briziskii, R.V.(6) Numerical sudy of he inverse problem for he diffusion-reacion equaion using opimizaion mehods, IOP Conference Series: Maerials Science and Engineering, 4, 96 (5 pp). [] Yousefi, S.A., Lesnic, D. and Barikbin, Z. () Saisfier funcion in Riz-Galerkin mehod for he idenificaion of a ime-dependen diffusiviy, Journal of Inverse and Ill-Posed Problems,,