An efficient approximation method for the nonhomogeneous backward heat conduction problems

Transcription

1 International Journal o Engineering and Applied Sciences IJEAS) ISSN: Volume-5 Issue-3 March 018 An eicient appromation method or the nonhomogeneous bacward heat conduction problems Jin Wen Xiuen Cheng Abstract This paper presents a new improved meshless numerical scheme to solve the nonhomogeneous bacward heat conduction problems. Fundamental solutions o heat equations and radial basis unctions o both and are employed to obtain a numerical ill-conditioned the Tihonov regularization method is adopted to solve the resulting system o linear equations. Thereore generalized cross-validation GCV) method is used to choose a regularization parameter. The accuracy and eiciency o our proposed method is illustrated by several typical 1-D and -D numerical examples. Index Terms nonhomogeneous bacward heat conduction problems Fundamental solutionsradial basis unctions. I. INTRODUCTION In many real application areas it is sometimes impossible to speciy the boundary conditions or the initial temperature while dealing with a heat conductor. For instance in practice one may have to investigate the temperature distribution and the heat lux history rom the nown data at a particular time. In other words it may be possible to speciy the temperature distribution at a particular time say t t 0 and either the 1991 Mathematics Subject Classiication. Primary 47A5 Secondary 65N80. Key words and phrases. Method o undamental solution Radial basis unction Nonhomogeneous bacward heat conduction Tihonov regularization Generalized cross-validation. Corresponding author. The wor described in this article was supported by the NNSF o China ) NSF o Gansu Province 145RJZA099) Scientiic research project o Higher School in Gansu Province 014A-01) and Innovation and Entrepreneurship project o college students in Gansu Province017-6). temperature u or the heat lux on the boundary o the domain and rom this data the question arises as to whether the temperature distribution at any earlier time t can be obtained. This is usually reerred to as the t bacward heat conduction problem BHCP) or the inal boundary value problem which is compared to initial boundary value problem. In this paper an improved appromation method or solving nonhomogeneous bacward heat conduction problem NBHCP) is investigated. The determination o the unnown Jin Wen male Associate Proessor PhD master tutor research direction: Numerical Solution o Partial Dierential Equation. Xiuen Cheng emale China Master graduate student research direction: Numerical Solution o Partial Dierential Equation. initial temperature rom observable scattered inal temperature data is required in many practical situations. However the solution process or BHCP is in nature unstable because the unnown solutions or parameters have to be determined rom indirect observable data which contain measurement error. The major diiculty in establishing any numerical algorithm or appromating the solution is due to the severe ill-posedness o the problem and the ill-conditioning o the resultant discretized matrix. Another diiculty to establish any numerical solution or the BHCP is due to the nature o its physical phenomena. Although heat conduction process is very smooth the process is irreversible. In other words the characteristic o the solution or instance the shape o the interior heat low) is not aected by the observed inal data. For the NBHCP we can reer to [1 ]. In article [] Trong et. al. Improved the quasi-boundary value method to regularize the nonlinear 1-D bacward heat problem and to obtain some error estimates and the appromation solution is calculated by the contraction principle. In [1] Dang and Nguyen investigated the inverse time problem or the nonhomogeneous heat equation and they regularized this problem by the quasi-reversibility method and then obtained error estimates on the appromate solutions. Solutions were also calculated by the contraction principle. The MFS was irst introduced by Kupradze and Alesidze [3] and it appromates the solution o the problem by linear combination o undamental solutions o the governing dierential operator. Thereore it is an inherently meshless integration ree technique or solving partial dierential equations and it has been used extensively or solving various types o partial dierential equations. For instance the solutions or potential problems by Mathon and Johnston [4] the exterior Dirichlet problem in acoustics by Kress and Mohsen [5] the biharmonic problems by Karageorghis and Fairweather [6]. More recently the MFS has successully been applied to appromate the solutions o nonhomogenous problems [7]. The details can be ound in [8 9]. These problems are all well-posed direct problems. In 004 Hon and Wei [10] applied irstly the MFS to solve the inverse heat conduction problem IHCP). Following their wors many researchers applied this method to solve many inverse problems: the BHCP [11]; the Cauchy problem associated with the Navier system in linear elasticity [1] and Helmholtz-type equations [13 14] etc. As the authors mentioned in [15] the meshless methods require neither domain discretization as in the inite element method FEM) and inite dierence method FDM) nor boundary discretization in the boundary element method BEM) thus they improve the computational eiciency 1

2 An eicient appromation method or the nonhomogeneous bacward heat conduction problems considerably which maes them especially attractive or problems with complicated geometry in high dimensional cases. One possible disadvantage is the act that the resulting system o linear equation is always ill-conditioned even or a well-posed problem see [9]. Thereore special regularization methods are required in order to solve this system o algebraic equations. The MFS can only solve some special inds o nonhomogeneous heat equations such as t x) e x) and one can reer to [ ] or more details. Recently some authors adopt the method o particular solution to solve various types o partial dierential equations such as Helmholtz-type operators [3] Poisson s equation [4] Laplace and biharmonic operators [5] elliptic partial dierential equations [6] nonhomogeneous Cauchy problem o heat equation [7] and NBHCP [8] etc. More recently in [9] the authors proposed a new improved meshless method or solving inverse heat source problem. They combined the MFS and RBF methods to give the numerical solution about the inverse source problem. Motivated by their idea we use the method to solve NBHCP. Because the homogenization o the nonhomogeneous heat equation in our paper is very diicult and even impossible. So our proposed method can avoid the diiculty o homogenization. The rest o this paper is organized as ollows: in Section we give the mathematical ormulation o the proposed problem. Section 3 devotes to the estence and uniqueness o the proposed problem. Section 4 is devoted to solve the problem by using the method o undamental solution-radial basis unction MFS-RBF). Tihonov regularization method is given in Section 5. Section 6 presents several numerical experiments to veriy the eiciency and accuracy o our method. And inally Section 7 concludes. II. FORMULATION OF THE PROPOSED PROBLEM Consider the ollowing nonhomogeneous bacward heat conduction problem: u u 0 L) 0 t ).1) t xx with boundary conditions: u 0 g1 0 t.) u 1 g 0 t t.3) t and inal temperature distribution: u t ) u x) 0 x L.4) and the initial value u 0) u0 x) 0 x L.5) is to be determined. In [11] or the special case when 0 Mera gave the MFS to solve the problem above. However it is also a challenge to solve this problem by only MFS. Here according to the method o [9] we combine the method o undamental solutions and radial basis unctions as an improved meshless method to solve it. Remar.1. In this paper our method can be recognized as the generalization o Mera s method. Meanwhile we can give the comparison between our method and Mera s in Example 4 in Section 6. III. EXISTENCE AND UNIQUENESS According to the deinition o well-posedness given by Hadamard[30] our problem is ill-posed since the bacward heat conduction problem is unstable. However the estence and uniqueness o NBHCP are still satisied. Here we give the results about estence and uniqueness o our problem as ollows: Theorem 3.1. The solution to the NBHCP.1)-.5) is unique. Proo. I and are two dierent solutions to the problem.1)-.5) respectively. We can give a new unction u ~ u u 1 and we can easily observe that u ~ satisies the ollowing homogeneous BHCP: u ~ u~ t t xx ) 3.1) with ~ boundary conditions: 3.) u0 0 0 t u1 0 0 t t ~ 3.3) t and inal temperature distribution: u ~ t ) 00 x 1 3.4) According to the classical homogeneous BHCP such as the reerence [31] the above problem 3.1)-3.4) has a unique solution u ~ 0. Thereore our proposed problem.1)-.5) has a unique solution. Remar 3.. The above result can be also generalized to -D NBHCP. IV. METHOD OF FUNDAMENTAL SOLUTIONS-RADIAL BASIS FUNCTIONS When 0 Eq..1) can be changed as the ollowing ut uxx 4.1) The undamental solution or 4.1) is x H 4t F e 4.) t where H is the Heaviside unction. Assuming that T t is a constant the ollowing unction F t T ) 4.3) is also a non-singular solution o 4.1) in the domain 01] [0 t ] According to the paper [3] Dou and Hon [ proposed a new method o source points has been presented or solving a bacward time-ractional diusion equation BTFDE). In this paper we also choose the source points { x t ): i 1 M} as ollows: i i { ti ): R x 1 R ti t i 1 M} 4.4) Where R and t are ixed constants. Moreover the collocation points C C1 C C3 C4 are chosen as ollows: C { x t ):0 x L0 t t q 1 n p 1 } 4.5) 1 q p q p m

3 C C C { x t ): x 00 t t j 1 } 4.6) j j j j n 3 { x t ): x L0 t t n 1 n m} 4 { xl tl ):0 xl L tl t l n m1 n m s} 4.7) 4.8) This section introduces the numerical scheme or solving the problem.1).5) using the hybrid meshless method i.e. undamental solutions and radial basis unctions method. The purpose is to decide how radial basis unctions and the undamental solutions are applied to appromate the initial temperature o the nonhomogeneous bacward heat conduction problem. The choice o the location o source points greatly aect the numerical results. So in this paper we choose the source points outside the domain. We suppose that the solution o.1).5) [ 0 L] [0 t ] can be written as ollows: M u ~ 4.9) 1 M 1 M where x x t t T) in which F is the undamental solution o the heat equation that is described in 4.) and x t ) ) is a radial unction that is deined in the Table 1 which can be ound in Section 3. o [9]. We impose the appromate solution u to satisy the given partial dierential equation with the other conditions at any point C to rewrite the coeicients in 4.9) as ollows: A b 4.10) T Where T b g g ].Also [ ] 1 M [ 1 the n m n m s) M matrix A can be subdivided into eight submatrices as ollows: A11 A1 A1 A A 4.11) A31 A3 A41 A4 Where 11 A11 al 11 0 l 1 nm 1 M; 4.1) a l 1 1 l A a 1 al ) x t ) l 1 n m M 1 M; l l 4.13) t x 1 A a a 1 l x t ) l nm1 nm n 1 M; 4.14) 1 l l l a l A a x t ) l nm1 nm n M 1 M ; 4.15) l l l a l A a x t ) l nm n 1 nm n m 1 M; 4.16) 1 l l l 3 3 a l A a 3 l x t ) l nm n 1 nm n m M 1 M ; 4.17) l l International Journal o Engineering and Applied Sciences IJEAS) ISSN: Volume-5 Issue-3 March 018 u l A a a x t ) l nm n m1 nm n m s 1 M; 4.18) 41 l l l 4 4 a l A a x t ) l nm n m1 nm n m s M 1 M ; 4.19) 4 l l l V. TIKHONOV REGULARIZATION Because the NBHCP.1) to.5) is highly ill-posed the matrix A in Eq.4.10) is still ill-conditioned. That is to say most standard numerical methods cannot obtain good accuracy in solving the matrix equation 4.10) because o the bad condition number o the matrix A. As a matter o act the condition number o matrix A increases remarably with reerence to the total points. Several regularization methods have been developed or solving these illconditioned problems [33 34]. In act the measurement data g1 g and u are usually contaminated by inherent measurement errors which leads to a noisy vector b and it can be given as b b 1rand i) 1)) here randi) is a random i i number in [0 1]. For this case we use the Tihonov regularization technique [35] to solve the matrix equation 4.10) with noisy b i.e. solving the ollowing minimization problem min{ A b } 5.1) where denotes the Euclidean norm and is called a regularization parameter. The choice o a suitable regularization parameter is crucial to the accuracy o solution and is still under intensive research [35]. For the above problem one can utilize the generalized cross-validation GCV) criterion and L-curve method to choose the regularization parameter. These two methods are both popular and successul [36]. We use generalized cross-validation GCV) criterion to choose the regularization parameter. The GCV criterion is a very popular and successul method or choosing the regularization parameter [36]. The GCV method determines the optimal regularization parameter by minimizing the ollowing GCV unction: G A b ) trace I AA )) I 5.) Where A I tr tr A A 1 I) A is a matrix which produces the regularized solution when multiplied with the right hand side b i.e. A I b. In the computation we use the Matlab code developed by Hansen [37] or solving4.10). Denote the regularized solution to 4.10) by. The appromating solution problem.1).5) is then given as u M 1 M 1 M u to 5.3) So the appromate initial temperature is obtain as ollows: 3

4 An eicient appromation method or the nonhomogeneous bacward heat conduction problems u M M 0 x) 0) 0) 1 1 M 5.4) Table 1. Some well-nown radial basis unctions. VI. NUMERICAL VERIFICATION In this section we test six examples to demonstrate the easibility o our approach. The computations are perormed using MATLAB 6.5 and we tae L 1 and t 0. 5 or 1 without special speciication. For the RBF we choose Gaussian RBF in the ollowing examples. In order to chec the accuracy o numerical computations we compute the root mean square error RMSE) and the relative root mean squares error RRMSE) o u ) deined by 0 x N 1 t RMSE u0 ) u0 ) u0 )) N RRMSE t i1 6.1) Nt u0 ) u0 )) i1 u0) 6.) Nt u0 )) i1 where { x i} is a set o discrete points in the interval [0 1]. Example 1: Tae the exact solution to problem.1).5) as u cosx exp 6.3) u 0) cosx 6.4) And cosx exp 6.5) Here we choose t In Table we show the t RRMSEu0) with various distance parameters R and when we ix T We can observe that the numerical results are stable with respect to parameters R and t within a wide range. Thus or our proposed method the accuracy o appromate solution is relatively independent o the parameters R and t. Fig. 1 shows a comparison between the exact solution o u ) and the appromate 0 x solutions u0 x) o u 0 x ) R 5 t 0.5 T 10 C 0.1 n m s 30 M 100 and various levels o noise added into the data using MFS-RBF. It is observed that as the noise level increases the appromated unction has an acceptable accuracy. Fig. illustrates the relationship between RMSEu0) RRMSEu0)) and the parameter T and it can be seen that we can obtain better accuracy when T 10. Fig. 1 displays the comparison between exact and appromate solutions when we choose R 5 t 0.5 T 10 C 0.1 n m s 30 M 100 or Example 1. It can be observed rom this igure that our method wors very well even or or Table. The values o RRMSE u 0 ) or various values o R and t in Example 1 when 0.01 C 0.1 T. t R R=0 R=1 R= R=3 R=4 R= Figure 1. The relationship between exact and appromate solutions where R 5 t 0.5 T 10 C 0.1 n m s 30 M 100 or Example 1. CPUtime=.0184s. 4

5 International Journal o Engineering and Applied Sciences IJEAS) ISSN: Volume-5 Issue-3 March 018 Figure. The relationship between RRMSEu0) and T with R= 5 δt = 0.1 C = 0.1 n = m = s = 30 and M = 60 or Example 1. CPUtime=.57363s. Example : Consider the exact solution as ollows: 3 u x t 6.6) 3 u 0) x 6.7) and 4t 6x. 6.8) In this example we ix t 0. 5 and our numerical results are displayed in Fig. 3 when we choose the same parameters as Fig. 3 except or R. It is clear that the results about initial values are stable. The relationships between the accuracy and the parameter R is illustrated in Fig. 4. We can see rom this igure that the accuracy o numerical solutions does not change too much as the parameter R increases up to 30. Thus it is concluded that in order to obtain higher numerical accuracy we can choose parameter R rom this igure. Fig. 5 illustrates the relationships between the accuracy and the parameter M. It is observed that the accuracy o numerical results is relatively stable when the parameter M becomes larger and larger. Figure 3. The relationship between exact and appromate solutions When R t 0.5 T 10 C 0.1 n m s 30 M 100 or Example. CPUtime= s. Figure 4. The relationship between RRMSEu0) and R with t 0.1 T 5 C 0.1 n m s 30 M 30and 0.01 or Example. CPUtime= s. Figure 5. The relationship between RRMSEu0) and M with R t 0.5 T 10 C 5 n m s 30 and 0.01 or Example. CPUtime=4.5114s. Example 3: We give a nonsmooth example as ollows: 3 3 u x 0.15 t 6.9) u 0) x ) and 3t 3t 60 x ) 60.5 x 1. In this example we choose t 1 and we can see that the heat source is more complicated than the previous ones since it is nonsmooth at the point x As shown in Fig. 6 our method solves the model problem is quite satisactorily even when the noise added to the data is up to 0.1. From this igure we can see again that our method wors well in computing the initial temperature. 5

6 An eicient appromation method or the nonhomogeneous bacward heat conduction problems = n = m = s = 60 and M = 30. CPUtime= and Figure 6. The exact and appromate solutions where R = δt =0.5C = 0.5 n =m = s = 30 and M = 150 or Example3.CPUtime= s. Example 4:Compared with Mera s method) The exact solution is as ollows [11]: u sinx exp 6.1) and the initial value is u 0) sin x) 6.13) Here we choose t 0. 5 which is the same as [11]. Fig. 7 shows the exact and appromated solutions by MFS-RBF and MFS respectively when we choose R 0 t 0.5 T C 0.5 n m s 60 M 30.It can be observed that the accuracy o our proposed method is better than the Mera s MFS method while the CPUtime is near to each other. In the ollowing example we introduce a more complicated case to veriy our proposed method. Example 5: 0 x 0.5. u 0) x 0.5 x ) In this example we choose t 1. We solve this direct problem by Cran-Nicolson scheme. The numerical results or u and u x) are shown in Fig.8. Fig. 9 displays the comparison between the exact solution and the numerical results. From this igure we can see that the numerical appromation is not as good as in the previous examples but it is in reasonable agreement with 6.14). In order to show that our method is also used to solve -D NBHCP we give the ollowing example: Example 6: u y exp sinx cos y). 6.15) Figs. 10a) and b) display the exact solution and appromate solution when It can be observed that the accuracy o the proposed method is stable when we adopt the noisy data to solve this problem. The dierence between exact and appromate solutions is illustrated in Fig. 10c) when we added the noise level We can see rom this igure that our method is also useul or solving -D NBHCP. Figure 7. The exact and appromated solutions by MFS-RBF and MFS respectively or Example 4 when R = 0 t = 0.5 C = 0.5 T Figure 8. The unction u and u x) obtained rom the direct problem or Example

7 International Journal o Engineering and Applied Sciences IJEAS) ISSN: Volume-5 Issue-3 March 018 where δ = 0.01 ; c) The dierence between exact and appromate solutions where R = T = C = 0. and t = 0.5 or Example 6 when δ = Figure 9. The exact and appromate solutions where R = 3 t = 1.4 n = m = s = 33 C = 0.5 T = 3 and M = 0 or Example 5. c) Figure 10. a) Exact solution o u y 0); b) The appromate solution o u y 0) with R = T = C = 0. and t = 0.5 a) b) VII. CONCLUSION This paper is devoted to a new approach based on the undamental solutions o the heat equation and the radial basis unctions with the Tihonov regularization method to solve the nonhomogeneous bacward heat conduction problem. The numerical examples or various initial values are investigated. The numerical results show that our proposed method is reasonable easible and stable even or the -D case. For our wor in the uture we will use this powerul method to solve more complicated problems such as inverse boundary problem inverse source problem and so on. REFERENCES [1] D. T. Dang H. T. Nguyen Regularization and error estimates or nonhomogeneous bacward heat problems Electron. J. Dierential Equations 006) No [] D. D. Trong P. H. Quan T. V. Khanh N. H. Tuan A nonlinear case o the 1-D bacward heat problem: regularization and error estimate Z. Anal. Anwend. 6 ) 007) [3] V. D. Kupradze M. A. Alesidze The method o unctional equations or the appromate solution o certain boundary-value problems ˇ. Vyˇisl. Mat. i Mat. Fiz ) [4] R. Mathon R. L. Johnston The appromate solution o elliptic boundary-value problems by undamental solutions SIAM J. Numer. Anal. 14 4) 1977) [5] R. Kress A. Mohsen On the simulation source technique or exterior problems in acoustics Math. Methods Appl. Sci. 8 4) 1986) [6] A. Karageorghis G. Fairweather The method o undamental solutions or the numerical solution o the biharmonic equation J. Comput. Phys. 69 ) 1987) [7] C. S. Chen The method o undamental solutions or nonlinear thermal explosions Commun. Numer. Methods Eng ) [8] G. Fairweather A. Karageorghis The method o undamental solutions or elliptic boundary value problems Adv. Comput. Math. 9 1-) 1998) 69 5 numerical treatment o boundary integral equations. [9] M. A. Golberg C. S. Chen The method o undamental solutions or potential Helmholtz and diusion problems in: Boundary integral methods: numerical and mathematical aspectsvol. 1 o Comput. Eng. WIT Press/Comput. Mech. Publ. Boston MA 1999 pp [10] Y. C. Hon T. Wei A undamental solution method or inverse heat conduction problemeng. Anal. Boundary Elem ) [11] N. S. Mera The method o undamental solutions or the bacward heat conduction problem Inverse Probl. Sci. Eng. 13 1) 005) [1] L. Marin D. Lesnic The method o undamental solutions or the cauchy problem in twodimensional linear elasticity Int. J. Solids Struct ) [13] L. Marin D. Lesnic The method o undamental solutions or the Cauchy problem associatedwith two-dimensional Helmholtz-type equations Comput. & Structures ) 005) [14] B. Jin Y. Zheng A meshless method or some inverse problems associated with the Helmholtz equation Comput. Methods Appl. Mech. Engrg ) 006) [15] B. Jin Y. Zheng Boundary not method or some inverse problems associated with the Helmholtz equation Internat. J. Numer. Methods Engrg. 6 1) 005) [16] Y. Hu T. Wei Determination o an unnown heat source term rom boundary data CMES Comput. Model. Eng. Sci. 87 4) 01) [17] C. Shi C. Wang T. Wei Numerical reconstruction o a space-dependent heat source term in a multi-dimensional heat equation CMES Comput. Model. Eng. Sci. 86 ) 01) 71. [18] T. Wei J. C. Wang Simultaneous determination or a space-dependent source and the initial data by the MFS Eng. Anal. Bound. Elem. 36 1) 01)

8 An eicient appromation method or the nonhomogeneous bacward heat conduction problems [19] J. Wen A meshless method or reconstructing the heat source and partial initial temperature in heat conduction Inverse Probl. Sci. Eng. 19 7) 011) [0] J. Wen M. Yamamoto T. Wei Simultaneous determination o a time-dependent heat source and the initial temperature in an inverse heat conduction problem Inverse Probl. Sci. Eng.1 3) 013) [1] L. Yan C.-L. Fu F.-L. Yang A undamental solution method or inverse heat conductionproblem Eng. Anal. Boundary Elem ) [] L. Yan F.-L. Yang C.-L. Fu A meshless method or solving an inverse spacewise-dependent heat source problem J. Comput. Phys. 8 1) 009) [3] A. S. Muleshov M. A. Golberg C. S. Chen Particular solutions o Helmholtz-type operators using higher order polyharmonic splines Comput. Mech ) 1999) [4] C. S. Chen A. S. Muleshov M. A. Golberg The numerical evaluation o particular solutions or Poisson s equation revisit in: Boundary elements XXI Oxord 1999) Vol. 6 o Int. Ser. Adv. Bound. Elem. WIT Press Southampton 1999 pp [5] A. R. Lamichhane C. S. Chen The closed-orm particular solutions or Laplace and biharmonic operators using a Gaussian unction Appl. Math. Lett ) [6] T. Dangal C. S. Chen J. Lin Polynomial particular solutions or solving elliptic partial dierential equations Comput. Math. Appl. 73 1) 017) [7] M. Li C. S. Chen Y. C. Hon A meshless method or solving nonhomogeneous Cauchy problems Eng. Anal. Bound. Elem. 35 3) 011) [8] M. Li T. Jiang Y. C. Hon A meshless method based on RBFs method or nonhomogeneous bacward heat conduction problem Eng. Anal. Bound. Elem. 34 9) 010) [9] M. Amirahrian M. Arghand E. J. Kansa A new appromate method or an inverse timedependent heat source problem using undamental solutions and RBFs Eng. Anal. Bound.Elem ) [30] J. Hadamard Lectures on Cauchy s problem in linear partial dierential equations Yale University Press New Haven CT U.S.A [31] L. C. Evans Partial dierential equations Vol. 19 o Graduate Studies in MathematicsAmerican Mathematical Society Providence RI [3] F. F. Dou Y. C. Hon Numerical computation or bacward time-ractional diusion equationeng. Anal. Bound. Elem ) [33] P. C. Hansen Analysis o discrete ill-posed problems by means o the L-curve SIAM Rev. 34 4) 199) [34] P. C. Hansen Numerical tools or analysis and solution o redholm integral equations o the irst ind Inverse Problems 8 6) 199) [35] A. N. Tihonov V. Y. Arsenin Solutions o ill-posed problems V. H. Winston & Sons Washington D.C.: John Wiley & Sons New Yor 1977 translated rom the Russian Preace by translation editor Fritz John Scripta Series in Mathematics. [36] P. C. Hansen Ran-deicient and discrete ill-posed problems SIAM Monographs on Mathematical Modeling and Computation Society or Industrial and Applied Mathematics SIAM) Philadelphia PA 1998 numerical aspects o linear inversion. [37] P. C. Hansen Regularization tools: a Matlab pacage or analysis and solution o discrete ill-posed problems Numer. Algorithms 6 1-) 1994) 1 5. Jin Wen male Associate Proessor PhD master tutor research direction: Numerical Solution o Partial Dierential Equation. Xiuen Cheng emale China Master graduate student research direction: Numerical Solution o Partial Dierential Equation. 8