Parameter Inversion Model for Two Dimensional Parabolic Equation Using Levenberg-Marquardt Method

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1 Iteratioal Cogress o Evirometal Modellig ad Software Brigham Youg Uiversity BYU ScholarsArchive 5th Iteratioal Cogress o Evirometal Modellig ad Software - Ottawa, Otario, Caada - July Jul st, : AM Parameter Iversio Model for Two Dimesioal Parabolic Equatio Usig Leveberg-Marquardt Method Tao Mi Zhuli Hao Miqua Feg Hasog Tag Follow this ad additioal works at: Mi, Tao; Hao, Zhuli; Feg, Miqua; ad Tag, Hasog, "Parameter Iversio Model for Two Dimesioal Parabolic Equatio Usig Leveberg-Marquardt Method" (). Iteratioal Cogress o Evirometal Modellig ad Software This Evet is brought to you for free ad ope access by the Civil ad Evirometal Egieerig at BYU ScholarsArchive. It has bee accepted for iclusio i Iteratioal Cogress o Evirometal Modellig ad Software by a authorized admiistrator of BYU ScholarsArchive. For more iformatio, please cotact scholarsarchive@byu.edu, elle_amatagelo@byu.edu.

2 Iteratioal Evirometal Modellig ad Software Society (iemss) Iteratioal Cogress o Evirometal Modellig ad Software Modellig for Eviromet s Sake, Fifth Bieial Meetig, Ottawa, Caada David A. Swaye, Wahog Yag, A. A. Voiov, A. Rizzoli, T. Filatova (Eds.) Parameter Iversio Model for Two Dimesioal Parabolic Equatio Usig Leveberg-Marquardt Method Tao Mi,, Zhuli Hao, Miqua Feg, Hasog Tag. Xi a Uiversity of Techology, Xi a 754, Chia. Dept. of Civil Eg., City College, The City Uiversity of New York,New York 3 Abstract: The problem of determiig ukow parameters i the two-dimesioal heat equatio is cosidered. A method based o the Leveberg Marquardt algorithm (LMA) is examied. The approach is successfully applied to solve the iverse problem of a two-dimesioal parabolic equatio whose coefficiet is a partitio paragraph fuctio. The umerical results demostrate the effectiveess of the proposed method. Keywords: parabolic equatio; iverse problem; Leveberg Marquardt. Itroductio Iverse parabolic problems arise from various backgrouds i egieerig. For istace, i thermal systems, thermal properties, icludig the heat covectio coefficiet (Marti et al., 998) ad temperature depedet thermal coductivity(dowdig et al., 999), are ofte ukow ad eed to be recovered. Recet applicatios also iclude the optical, wherei the iterest is to recover aomalies i huma tissues(barbour et al., 995. Natioal Research Coucil, 996). I most applicatios, oe is led to excite the system by a exteral meas ad record the associated system respose. The collected data are the used to recover the sought-after ukow. These kids of problem have bee ivestigated by may researchers. Recet results iclude a aalytical method for the solutio of the over-determied iverse heat coductio(taler, 997), applicatio of eural etworks for the recoverig of electrical coductivity profiles (Glorieux et al., 999), a spectral method for solvig the lateral heat equatios(bertsso, 999), ad a discrete diffusive model for the recovery of the absorptio coefficiet from diffused reflected light(martiz et al., 998). Additioal methods iclude oliear optimizatio usig geetic algorithms(stoffa et Correspodece: mitao@xaut.edu.c

3 al., 99), Marquardt s procedure, ad thermal wave- slice tomography. Most of the discussios i the literature are devoted to the qualitative aalysis of the equatios such as existece ad uiqueess of solutio.. The Problem I this paper, we study umerical procedures for the iverse problem of simultaeously fidig the ukow fuctios p xy, ad uxyt (,, ) to satisfy: u u u axy (, ) bxy (, ) t x y u u px, y px, y dx, yu x x y y f x, y, t t T (x, y) Ω with the iitial-boudary coditios, () u x, y, = u ( x, y ) () u x, y, t u x, y, t (x,y) (3) ad the additioal coditio, uxyt (,, ) u( xyt,, ), xy, (4) where axy (, ), (, ) bxy,, d x y, f x, yt,, ad u ( x, y, t) are kow fuctios, ad are give parameters, ad the fuctios (,, ) uxyt ad, p xy are ukow. Whe p xy, is give, there is a rather satisfactory theory for the direct problem () (3) regardig posedess ad other related properties. Whe p xy, is ukow, i order to fid a solutio as well as p xy,, oe eeds additioal coditio (4), the actual form of which depeds o the cocrete physical model.. Noliear Optimizatio ad Leveberg Marquardt Algorithm I geeral, ad u ( x, y, t) are kow i the fixed, prescribed iterior poit i whose boudary is deoted by. Assumig that T is the samplig period, they ca be measured at. Further assume that p xy, is the exact solutio of, t it( i,,, I) p xy ad

4 u x, y, t is the exact solutio to the problem () (3). Let K be a complete liear space of real umbers ad p xy, K, the x, y, x, yl x, y fuctios i K. The L is a group of basis, ii, (5) i p xy k xy We represet p, xy as a fiite sum of the form, ii, (6) i p xy k xy the size of is determied by approximate accuracy. Therefore solvig the iverse problems is to determie a -dimesioal real vector T K k, k, L, k R (7) The the fuctio T,,, p xy k xy K xy i i i satisfies the model () ~ (4), where x, yx, y, x, y x, y If, ;,, T L. u pxy xyt is the solutio of the iitial boudary value problem () ~ (3) correspodig to p xy,, the problem of determiig, coverted to fid the solutio of the followig miimizatio problem: mi (,, ), ;,, ( xyt,, ) t p xy ca be u x y t u p x y x y t (8) where t {( x, yt, ), t[, it]}( i,,, I) ad T is the samplig period of u ( x, y, t). From (7) we kow we ca fid a -dimesioal real vector which miimizes the fuctios. The istead of (8), we study the problem: where u ; x, y, t m I u xj yj ti u xj yj ti (9) j i mi ( (,, ) ( ;,, )) satisfies the model () ~ (4). Multiplyig both sides of (9) by.5 produces a geeral o-liear least squares problem:

5 mi f [ ( )] m( I) rl () l where f ; rl( ) u ( xj, yj, ti) u( ; xj, yj, ti),( i,,, I, j,,, m). The iverse problem is thus coverted ito a oliear optimizatio problem.let us itroduce the Leveberg Marquardt algorithm as follows: Iitialize: k=, : iitial guess vector of At iteratio k: ( k) ( k) ( k ), k,,, () ( k ) ca be solved by a set of liear equatios: where: ( A A I) A r T ( k) k k k () ri, A( ) [ r ( ),, r m ( I ) ( )],ad ri ( ). Solvig the above equatios, a ew iitial guess vector is obtaied whe we substitute its solutio ( k ) ito (). The above solutio process is repeated util the data meet the accuracy requiremets. I the above-metioed algorithms, each iteratio should solve the direct problem. The direct problem ivolved i this article is beig implemeted with a fiite elemet method. 3. Numerical Examples Lettig p xy, be the exact solutio, we compute u p x, y ; x, y, it by solvig the direct problem, which yields the additioal data recorded as u ; p xy, is obtaied by the above-metioed algorithm. We the compare p xy, with the true solutio, p xy. Example Cosider problem () ~ (4) o [,] [,] with the followig coditios: a( x, y) t y, b( x, y) tx, d( x, y) tx y

6 xy xy xy xy xy f( x, y, t) 5e ( ty)(( xyt)e e ) ( tx)(( xyt)e e ) x yt x y t x yt x yt xy xy xy xy 6( )e (3 3 5 )(( )e e ) ( ) e with the iitial-boudary coditios u x, y, = u ( x, y ), ( x, y) u u x x u u y y y y y y x e ( y t) e x e ( y t) e ( y, t ) x x x x y e ( x t) e y e ( x t) e ( x, t ) ad the additioal coditio uxyt (,, ) ( yt, ) x the time samplig is take as i, j,,, t., t.3, t.5,i.e. I 3. The determied parameter is p( xy, ) kx ky kt 3, where k ( i,,3) i is to be determied. The true solutio is p ( xy, ) 3x3y 5t, the direct problem is beig implemeted with the fiite elemet method. The iitial value is take as p ( xy, ) x y t ad p ( x, y) xy t, the results are show i Table : Table. Numerical simulatio results of parameter parameter K =[,,] K = [,,] Iteratios The true solutio k k k Error The error betwee the umerical solutio K [k,k,k 3] ad the true solutio K [k,k,k ] is 3 ER (k k ) (k k ) (k k ). 3 3 The error ER (k k ) (kk ) (k3 k 3) betwee the umerical solutio K [k,k,k 3] ad the true solutio K [k,k,k ] for iteratio L is show i Figure : 3

7 ER ER L L ( K =[,,]) ( K =[,,]) Figure. The relatioship betwee the computatio accuracy ad the umber of iteratios. I the followig sectio, the data i Example were modified by radom perturbatios (errors) of %, 5% ad %: u = ( ) u where: = the radom variable betwee [-, ]; =.,.5, ad., respectively. For the differet iitial guess ad differet radom measuremet error, simulatio results are show i Table: Table. Noise r% 5 Iitial guess K (,,) (,,) (,,) (,,) (,,) (,,) Iteratios The resultig value ER The error betwee the umerical solutio K [k,k,k 3] ad the true solutio K [k,k,k ] iser (k k ) (k k ) (k k ) Example I order to demostrate the performace of the proposed scheme i dealig with discotiuous coefficiets i the equatios, cosider problem ()-(4) o [,] [,] ad subject to the followig coditios: a( x, y) x y, b( x, y) x y, p( x, y) x y f ( x, y, t) ye xye ( x ) ye ( x y) ye with the iitial-boudary coditios x uxy (,,) ey xt xt xt xt

8 u e x t y ad the additioal coditio uxyt (,, ) ( yt, ) x the time samplig is take as i, j,,, t., t.3, t.5,i.e. I 3. The determied parameter is kx ky x, y/ dxy (, ) k3 x, /y,where ki ( i,,3) is to be determied. The true solutio is a piecewise fuctio x4y x, y/ d (, x y), the direct problem is beig implemeted with the 4 x, /y.5 x.5 y x, y/ fiite elemet method. The iitial value is take as d(, x y).5 x, / y xy x, y/ add (, x y), the results are show i Tables 3: x, /y Table. 3 Numerical simulatio results of parameter( K =[.5,.5,.5]) parameter K =[.5,.5,.5] K =[,,] Iteratios The true solutio k k k Error The error ER (k k ) (k k ) (k k ) betwee the umerical solutio K [k,k,k 3] 3 3 ad the true solutio K [k,k,k ] for iteratio L is show i Figure : 3 ER ER L L

9 ( K =[.5,.5,.5]) ( K =[,,]) 4. Coclusio The results illustrate that the Leveberg Marquardt method is applicable ad efficiet i the case of determiatio of a ukow parameter i the two-dimesioal heat equatio. I particular, the algorithm was steady for data with a radom disturbace. The method is successfully applied the iverse problem of a two-dimesioal parabolic equatio whose coefficiet is a partitio paragraph fuctio. It turs out that the best perturbatio method is oe of the efficiet methods to solve this kid of problems. Its rate of covergece is high ad its domai of covergece is wide eough to be successfully used i practice. Note that the algorithm has a certai depedece o the iitial guess of the iverse parameter. Refereces T.J. Marti, Dulikravich, Iverse determiatio of steady heat covectio coefficiet distributio, Joural of Heat Tradfer, ,998. K.J.Dowdig, J.V. Beck ad B.F. Blackwell, Estimatig temperature-depedet thermal properties, Joural of Thermophysics ad Heat Tradfer3 (3), ,999. R. Barbour, H. Graber, J. Chag, S. Barbour, S. Koo ad R. Aroso, MRI guided optical tomography, IEEE Comp. Sci. Egg. (4), 63-77,995. Natioal Research Coucil, Mathematics ad Physics of Emergig Biomedical Imagig, Natioal Academic Press, Washigto, DC,996. J. Taler, Aalytical solutio of the overdetermied iverse heat coductio problem with a applicatio to moitorig thermal stresses, Heat ad Muss Trasfer 33, 9-8,997. C. Glorieux, J. Moulder, J. Basart ad J. Thoe, The determiatio of electrical coductivity profiles usig eural etworks iversio of multi-frequecy eddy-curret data, Joural of Physics D: Appl. Phys. 3, 6-6,999. F.Bertsso, A spectral method for solvig the sideways heat equatio, Iverse Problems 5, 89-96,999. M.F. Martiz, G.T. Herma ad C. Yee, Recovery of the absorptio coefficiet from diffused reflected light usig a discrete diffusive model, SIAM Joural o Applied Mathematics 59, 58-7,998. P.L. Stoffa ad M.K. Se, Noliear multiparameter optimizatio usig geetic algorithms: Iversio of plae-wave seismograms, Geophysics 56, 794-8,99.