A WAVELET METHOD FOR SOLVING BACKWARD HEAT CONDUCTION PROBLEMS

Transcription

1 Electronic Journal of Differential Equations, Vol. 7 (7), No. 9, pp. 9. ISSN: URL: or A WAVELET METHOD FOR SOLVING BACKWARD HEAT CONDUCTION PROBLEMS CHUNYU QIU, XIAOLI FENG Abstract. In this article, we consider the backward heat conduction problem (BHCP). This classical problem is more severely ill-posed than some other problems, since the error of the data will be eponentially amplified at high frequency components. The Meyer wavelet method can eliminate the influence of the high frequency components of the noisy data. The known works on this method are limited to the a priori choice of the regularization parameter. In this paper, we consider also a posteriori choice of the regularization parameter. The Hölder type stability estimates for both a priori and a posteriori choice rules are established. Moreover several numerical eamples are also provided.. Introduction Wavelet theory has been widely developed in the late of the previous century. The multiscale analysis and wavelet decomposition are now still subjects of intensive development. At the same time, the investigation of mutual interactions between wavelet theory and ill-posed problems has never stopped. Some results have been applied to the analysis of some inverse problems. Wavelet methods have advantages for use in certain inverse problems:. They allow for the decomposition of an object into multiple resolutions (or scales). This is a particular advantage for high precision inversion of the objects;. The localization of wavelets in both time and frequency makes them quite useful for analyzing local features; 3. Wavelets have ecellent data compression capabilities for spatially variable objects, such as signals characterized by singularities or images determined primarily by a set of edges; 4. Some methods of denoising, based on thresholding of the wavelet coefficients, have been proven to be nearly optimal for a number of tasks across a wide range of function classes [3]. We emphasized that Meyer wavelets possess specially important significance for solving many ill-posed problems. Meyer wavelets have the property that their Fourier transforms have compact supports. This means that they can be used to prevent high frequency noise from destroying the solution, i.e., by epanding the data and the solution in a basis of Meyer wavelets, high-frequency components can be filtered away. Within V j, which is generated by the father wavelet of Meyer, the original ill-posed problem is well-posed, and we can find a regularization parameter Mathematics Subject Classification. 65T6, 65M3, 35R5. Key words and phrases. Backward heat equation; Ill-posed problem; regularization; Meyer wavelet; error estimate. c 7 Teas State University. Submitted March 9, 7. Published September 4, 7.

2 C. QIU, X. FENG EJDE-7/9 J depending on the noise level of the data such that the solution in V j is a good approimation of the original problem. Moreover, the combination of Meyer wavelets method with the fast Fourier transform (FFT) can build high-speed algorithm. Meyer wavelet techniques have been used by Regińska et al [8, 9], Hào et al [], Eldén [6], and Wang [4] to solve the inverse heat conduction problems (IHCP), and by Vani [3], Qiu et al [7] to solve the Cauchy problem for the Laplace equation. However, they all used the a priori wavelet method, and did not consider the a posteriori error estimate. Since the numerical results for the a posteriori method do not depend on the a priori information, the a posteriori wavelet method is more effective to solve practical problems than the a priori method. In this paper, we continue to study the a priori wavelet method, and then focus on the a posteriori wavelet method and its numerical solution. Although the application of wavelet theory in differential equation has been mostly focused on numerical computation, the connection between the wavelet theory and differential equation is also searched all the time. Shen and Strang in [] have introduced the concept of heatlets in order to solve the heat equation using wavelet epansions of the initial data. The heatlet is a fundamental solution to the heat equation, when the initial data is epanded in terms of the wavelet basis, the solution to the heat equation is then obtained from an epansion using the heatlets and the corresponding wavelet coefficients of the data. In [] the authors combined heatlets with quasi-reversibility method to regularize the backward heat equation, and obtained some theoretical error estimates. However, there are no numerical results. In the present paper, we consider the following backward heat equation in a strip domain by a Meyer wavelet method. u t u, < <, t < T, u(, T ) ϕ T (), < <, where we want to determine the temperature distribution u(, t) on the interval t [, T ) from the data ϕ T (). Backward heat conduction problem is a classical ill-posed problem [, 4], and is known as the most severely ill-posed problem, which has been studied by many authors by different methods [,, 4,,, 5]. Let ĝ(ξ) denote the Fourier transform of g() defined by ĝ(ξ) : π (.) e e iξ g()d, (.) e and f H s denote the norm on the Sobolev space H s defined by /. f H s : f(ξ) ( + ξ ) dξ) s (.3) e3 When s, H : denotes the L (R)-norm, and (, ) denotes the L (R)- inner product. It is easy to know that L (R) H s (R) for s. We assume that there eists a solution u(, t) satisfying (.) in the classical sense and u(, t) L (R) for < t < T. Using the Fourier transform technique to problem (.) with respect to variable, we can get the Fourier transform û(ξ, t) of the eact solution u(, t) of problem (.): û(ξ, t) e ξ (T t) ϕ T (ξ), (.4) e4

3 EJDE-7/9 BACKWARD HEAT CONDUCTION PROBLEMS 3 or equivalently, u(, t) π e iξ e ξ (T t) ϕ T (ξ)dξ. (.5) e5 For any ill-posed problem some a priori assumptions on the eact solution are needed, otherwise the convergence of the regularization approimate solution will not be obtained or the convergence rate can be arbitrarily slow [7]. When we consider problem (.) in L (R) for the variable, we assume there eists an a priori bound for ϕ () : u(, ): From the formula (.5) and the Parseval formula, we know ϕ u(, ) E. (.6) e7 ϕ e ξ ϕ T (ξ) dξ. (.7) e8 For a concrete ill-posed problem, not all regularization methods are effective. For eample, for the severely ill-posed problems with growth rate of magnitude factor reaching or eceeding O(e γξ ), γ >, ξ, the Mollification method with Gauss kernel suggested by Murio [5] cannot deal with them. Problem (.) considered in the present paper just is the case as the magnitude factor e ξ (T t). Therefore, the Mollification method is not effective both in theory and numerical computation. In addition, the convergence rate and numerical results are also not completely the same for different regularization methods. For eample, for the Modified method suggested by [6], the theoretical convergence rate for problem (.) is only logarithm type not Hölder type. So construction of specific regularization methods for different ill-posed problems is significant. The main goal of this paper is to provide a Meyer wavelet method for solving the backward heat equation (.). Our method of proving stability estimates is constructive: We construct a stable solution to the problem for both a priori and a posteriori choice rules. The outline of this paper is as follows. In Section, a brief survey on some fundamental properties of Meyer wavelet is presented. On this basis, we give the Meyer wavelet regularization method to solve the problem (.) for both a priori and a posteriori parameter choice rules, and obtain the error estimates of the a priori and a posteriori situations respectively. In Section 3, four numerical eamples are provided, and the comparison of numerical effects for a posteriori wavelet method with other methods for Eamples 3. and 3. are also taken into account.. Wavelet regularization and error estimates Let ϕ(), ψ() be Meyer scaling and wavelet functions respectively. Then from [3] we know supp ϕ [ 4 3 π, 4 3 π], supp ψ [ 8 3 π, 3 π] [ 3 π, 8 3 π], and ψ jk () j ψ( j k), j, k Z constitute an orthonormal basis of L (R) and supp ψ jk (ξ) [ 8 3 πj, 3 πj ] [ 3 πj, 8 3 πj ], k Z. (.) e

4 4 C. QIU, X. FENG EJDE-7/9 The multiresolution analysis (MRA) {V j } j Z of Meyer wavelet is generated by V j {ϕ jk : k Z}, ϕ jk : j ϕ( j k), j, k Z, supp ϕ jk (ξ) [ 4 3 πj, 4 3 πj ], k Z. (.) e The orthogonal projection of a function g L (R) on space V J is given by P J g : k Z (g, ϕ Jk)ϕ Jk, while Q J g : k Z (g, ψ Jk)ψ Jk denotes the orthogonal projection on wavelet space W J with V J+ V J WJ. It is easy to see from (.) and (.) that P J g(ξ), for ξ 4 3 πj, (.3) e3 lem. Q j g(ξ), for j > J and ξ < 4 3 πj. (.4) e4 Since (I P J )g j J Q jg and from (.4), we know ((I P J )g) b (ξ) Q J g(ξ), for ξ < 4 3 πj. (.5) e5 Lemma. ([]). Let {V j } j Z be Meyer s MRA and suppose J N, r R. Then for all g V J, it holds the estimate D k g H r C (J )k g H r, k N, (.6) e6 where C is a positive constant and D k dk d k. or Define an operator A t : ϕ T () u(, t) by (.4), i.e., Then we have the following lemma. A t ϕ T u(, t), t < T, Â t ϕ T (ξ) e ξ (T t) ϕ T (ξ), t < T. (.7) e7 lem. Lemma.. Let {V j } j Z be Meyer s MRA and suppose J N, r R, t < T. Then for all g V J we have A t g H r C ep { (J ) (T t) } g H r, (.8) e8 where constant C is the same as in (.6). Proof. From (.6) we know that A t g H r Âtg(ξ) ( + ξ ) r dξ ) / ) e ξ (T t)ĝ(ξ) / ( + ξ ) r dξ ) / cosh(ξ (T t))ĝ(ξ) ( + ξ ) r dξ k k (T t) k ) / ξ 4k ĝ(ξ) ( + ξ ) r dξ (k)! (T t) k ) / (iξ) 4k ĝ(ξ) ( + ξ ) r dξ (k)!

5 EJDE-7/9 BACKWARD HEAT CONDUCTION PROBLEMS 5 C k k (T t) k D 4k g H r (k)! (T t) k (J )4k g H r (k)! C cosh ( (J ) (T t) ) g H r C ep { (J ) (T t) } g H r. Let ϕ T (), ϕ T () be eact and measured data, respectively, which satisfy ϕ T ϕ T H r <, for some r. (.9) e9 Since ϕ T () belongs, in general, to L (R) H r (R) for r, r should not be positive. In general, L a priori bound for eact solution as (.6) can only lead to a Hölder stability estimate for the regularization solution, but this a priori assumption can not ensure the convergence of the regularization solution at t for problem (.). To obtain a more sharp convergence for the regularization solution, here we assume, for some s r, there eists an a priori bound: ϕ H s E. (.) e Denote operator A t,j : A t P J, we can show it approimates A t in a stable way for an appropriate choice of J N depending on and E. In fact, we have A t ϕ T A t,j ϕ T H r A t ϕ T A t,j ϕ T H r + A t,j ϕ T A t,j ϕ T H r. (.) e From Lemma. and condition (.9), we can see that the second term of the righthand side of (.) satisfies A t,j ϕ T A t,j ϕ T H r A t P J (ϕ T ϕ T ) H r C ep { (J ) (T t) } P J (ϕ T ϕ T ) H r C ep { (J ) (T t) }. (.) e For the first term of the right-hand side of (.), from (.3) we have A t ϕ T A t,j ϕ T H r A t (I P J )ϕ T H r e ξ (T t) ( ) b(ξ) (I P J )ϕ T ( + ξ ) r dξ ) / / e ξ (T t) ϕ T (ξ) ( + ξ ) dξ) r ξ 4 3 πj + e ξ (T t) ( ) ) b(ξ) / (I P J )ϕ T ( + ξ ) r dξ : I + I. ξ < 4 3 πj (.3) e3

6 6 C. QIU, X. FENG EJDE-7/9 Noting (.4) and (.), we have / I e ξ (T t) ϕ T (ξ) ( + ξ ) dξ) r ξ 4 3 πj ) / ϕ (ξ) ( + ξ ) r dξ ξ 4 3 πj e tξ sup e tξ ξ 4 ( + ξ ) s r 3 πj sup e tξ ϕ ξ 4 ( + ξ ) s r H s 3 πj e t( 4 3 πj ) E ( 4 e t 3 πj ) s r (J+)(s r) ep{ t (J+) }E. ξ 4 3 πj ϕ (ξ) ( + ξ ) s dξ (J+) E (J+)(s r) ) / (.4) e4 From (.5), Lemma., and noting that Q J ϕ T W J V J+, it is easy to see that I satisfies: / I e ξ (T t) ((I P J )ϕ T ) b (ξ) ( + ξ ) dξ) r ξ < 4 3 πj / e ξ (T t) Q J ϕ T (ξ) ( + ξ ) dξ) r ξ < 4 3 πj A t Q J ϕ T H r C ep{ J (T t)} Q J ϕ T H r. Denote χ J as the characteristic function of the interval [ 3 πj, 3 πj ]. We introduce an operator M J defined by M J g ( χ J )ĝ, g L (R). Noting that ϕ T L (R), so from Parseval formula and (.) we have Q J ϕ T k Z (ϕ T, ψ Jk )ψ Jk k Z( ϕ T, ψ Jk )ψ Jk k Z(( χ J ) ϕ T, ψ Jk )ψ Jk k Z(M J ϕ T, ψ Jk )ψ Jk Q J M J ϕ T.

7 EJDE-7/9 BACKWARD HEAT CONDUCTION PROBLEMS 7 So, from (.4), Q J ϕ T H r Q J M J ϕ T H r M J ϕ T H r ) / ϕ T (ξ) ( + ξ ) r dξ ξ 3 πj / ϕ (ξ) ( + ξ ) dξ) r Therefore, ξ 3 πj e ξ e ξ T ξ 3 πj sup ξ 3 πj e ξ T ξ s r ϕ H s ( + ξ ) s r (J+)(s r) ep { T (J+)} E. ) / ϕ (ξ) ( + ξ ) s dξ I C ep { J (T t) } (J+)(s r) ep { T (J+)} E C (J+)(s r) ep { (J+) (T t T (J+)} E C (J+)(s r) ep { t (J+)} E. From (.4), (.6) and (.3), we obtain A t ϕ T A t,j ϕ T H r (J+)(s r) ep{ t (J+) }E + C (J+)(s r) ep { t (J+)} E ( + C(J+)(s r) ep { t (J+)} E. Combining (.7), (.) with (.), we have (.5) e* (.6) e5 (.7) e6 A t ϕ T A t,j ϕ T H r C ep { (J ) (T t) } + ( + C(J+)(s r) ep { t (J+)} E C ep { J (T t) } + ( + C(J+)(s r) ep { t (J+)} E. Based on the above results, we will give the estimates for the a priori and a posteriori parameter choice rules, respectively. (.8) e7 thm... A-priori parameter choice. Now we can firstly give an error estimate between the wavelet regularization solution A t,j ϕ T and the eact solution u(, t) A t ϕ T in L (R). Theorem.3. Suppose that ϕ L (R) and (.9), (.) hold for r s. The problem of calculating A t,j ϕ T is stable. Furthermore, taking [ ( J : log ln ( E ) /T ) ], (.9) e8 where [a] denotes the largest integer less than or equal to a R, then the following stability estimate holds: where C is the same as in (.6). A t ϕ T A t,j ϕ T (4C + )E t T t/t, (.) e9

8 8 C. QIU, X. FENG EJDE-7/9 Proof. Note that ep { J (T t) } ep { (T t) ln ( E ) /T } (E ) T t T E t T t/t, ep { t (J +) } E ep { t ln ( E ) /T } E E t T t/t. and from estimate (.8) we have A t ϕ T A t,j ϕ T H A t ϕ T A t,j ϕ T The proof is complete. CE t T t/t + ( + C)E t T t/t (4C + )E t T t/t. rmk. Remark.4. From the result of reference [] we know estimate (.) is an order optimal Hölder stability estimate in L (R). This suggests that wavelet method must be useful for solving the considered ill-posed problem. However, from (.) we know when t +, the accuracy of the regularized solution becomes progressively lower. At t, it merely implies that the error is bounded by 4C +, i.e., the convergence of the regularized solution at t is not proved theoretically. This defect is remedied by the following result. thm. Theorem.5. Suppose that ϕ H s (R) for some s R and (.9) holds for r min{, s}. Take [ ( ( ( J : log E ) /T ( E ) (s r) ))] ln ln, (.) e where the bracket [a] is the same as in (.9). Then A t ϕ T A t,j ϕ T H r ( ln C + ( + C)( E ) s r ) T ln E + ln ( ) ln E (s r)/( ) E t T t/t ( ln E ) (T t)(s r). (.) e Proof. Note that ep { J (T t) } ep ( E { ( ( E (T t) ln ) T t ( T ln E E t T t/t ( ln E ep { t (J +) } { ( ( E E ep t ln ( E t ( T ln E E t T t/t ( ln E ) /T ( E ln (T t)(s r) (T t)(s r), ) /T ( E ln ) t(s r) E ) t(s r), ) (s r) )} ) (s r) )} E

9 EJDE-7/9 BACKWARD HEAT CONDUCTION PROBLEMS 9 (J +)(s r) (J +)( s r ) and from (.8) we know A t ϕ T A t,j ϕ T H r CE t T t/t ( ln E ( ( ( E ln ( ) /T ( E ln T ln E + ln ( ln E ) (s r) ) s r (s r) ( ln E T ln E + ln ( ) ln E (s r ) s r ) s r ( ln E (T t)(s r) + ( + C)( ln E T ln E + ln ( ln E (s r), (s r) ) s r ( ln E ) T (s r E t T t/t ( ln E ) t(s r) ( ln C + ( + C)( E ) s r ) T ln E + ln ( ) ln E s r E t T t/t ( ln E ) (T t)(s r), where the factor ln E T ln E + ln ( ln E s r is bounded as +. So, the proof of estimate (.) is complete. rmk. Remark.6. When s r, estimate (.) becomes estimate (.9) and the convergence speed given by (.) is faster than the one given by (.) for s > r. Especially, when t, estimate (.) becomes A ϕ T A,J ϕ T H r ( ln C + ( + C)( E T ln E + ln ( ) ln E s r ) s r ) ( ln E (s r) E, when + and s > r. This is a logarithmical stability estimate and an important improvement to estimate (.). rmk.3 Remark.7. Noting that lim ln E T ln E + ln ( ln E s r T, estimate (.) also can be rewritten in the asymptotic form as. A t ϕ T A t,j ϕ T H r ) (C + ( + C)T s r + o() E t T t/t ( ln E (T t)(s r),

10 C. QIU, X. FENG EJDE-7/9 lem.3 thm.3.. A-posteriori parameter choice. In this subsection, we consider the a posteriori regularization parameter choice in the Morozov s discrepancy principle. This principle has been used by Feng et al [8] to solve the numerical analytic continuation, however the backward heat conduction problem is more severely ill-posed than the numerical analytic continuation. Lemma.8. Assume conditions (.9) for r and (.) for s hold. If J is chosen as the solution of the inequalities then it holds P J ϕ T ϕ T τ P J ϕ T ϕ T, τ >, (.3) e7 ep( J T ) Proof. From Equations (.3) and (.5), we know P J ϕ T ϕ T ((I P J )ϕ T ) b (ξ) dξ ) / E. (.4) e8 (τ ) ) / ϕˆ T (ξ) dξ + (Q J ϕ T ) b (ξ) dξ ξ 4 3 πj ξ < 4 3 πj : I 3 + I 4. From (.4), From (.5), ) / (.5) e9 I 3 ep( ( 4 3 πj ) T )E ep( J T )E. (.6) e3 I 4 (Q J ϕ T ) b (ξ) Js ep( T J )E ep( J T )E, s (.7) e3 Combining (.5) with (.6) and (.7), we obtain P J ϕ T ϕ T ep( J T )E. (.8) e3 On the other hand, by (.9) for r, (.3), and the triangle inequality give P J ϕ T ϕ T (I P J )ϕ T (I P J )(ϕ T ϕ T ) (τ ). (.9) From (.8) and (.9), estimate (.4) is proved. Theorem.9. Assume that conditions (.9) for r and (.) for s hold. If the regularization parameter J is chosen as the solution of inequalities (.3), then A t ϕ T A t,j ϕ T CE t T t/t, (.3) e34 where C (C( τ ) t T + (τ + ) t/t ), and the constant C is the same as in (.6). Proof. By the Parseval formula and the triangle inequality, A t ϕ T A t,j ϕ T A t,j ϕ T A t,j ϕ T + A t,j ϕ T Â t ϕ T : I 5 + I 6. (.3) e35 From Equation (.) and Lemma.8, I 5 C ep( (J ) (T t)) C(ep( (J ) T )) T t T ( C E (τ ) ) T t T. e33 (.3) e36

11 EJDE-7/9 BACKWARD HEAT CONDUCTION PROBLEMS Moreover, I 6 e ξ (T t) ((I P J )ϕ T ) b (ξ) dξ e ξ ((I P J )ϕ T ) b (ξ) T t T ((I PJ )ϕ T ) b (ξ) t T dξ e ξ ((I P J )ϕ T ) b (ξ) dξ ) T t T t/t ((I P J )ϕ T ) b (ξ) dξ). By combining (.6) with (.3) and (.33), it holds From (.3), (.3) and (.34), we complete the proof. (.33) e37 I 6 E ( t T ) ((τ + )) t T. (.34) e38 3. Numerical aspects In this section, we want to discuss some numerical aspects of the proposed method. 3.. Numerical implementation. Suppose that the vector {Φ( i )} N i represents samples from the function ϕ T (), and N is even, then we add a random normal distribution to each data and obtain the perturbation data Φ Φ + ɛ randn(size(φ)), (3.) where the function randn( ) generates arrays of random numbers whose elements are normally distributed with mean, variance σ, and standard deviation σ. The total noise can be measured in the sense of root mean square error according to ( N Φ Φ l : (Φ ( i Φ( i )) ) /. (3.) N i For a given measured function ϕ T, from Section, we have A t,j ϕ T (ξ) ( ϕ T, ϕ Jk)e ξ (T t) ϕ Jk e ξ (T t) k Z k Z( ϕ T, ϕ Jk) ϕ Jk. (3.3) e3.3 If no specific instructions are assumed, we will compute the regularization parameter J according to inequalities (.3) with τ.. By using the Discrete Meyer wavelet Transform (DMT) and the Fast Fourier Transform (FFT), we can easily compute the regularized solution according to formula (3.3). Algorithms for DMT are described in [3]. These algorithms are based on the FFT, and computing the DMT of a vector in R requires O(N log N) operations. 3.. Numerical tests. In this subsection some numerical tests are presented to demonstrate the usefulness of our method. The tests are performed using Matlab 7. and the wavelet package WaveLab 85, which is downloaded from wavelab/. Eamples 3. and 3. are from [9] and [6], respectively. In theoretical aspect, the error estimates of regularization solutions for the wavelet method (WM) and Fourier method (FM) both are sharper Hölder-logarithm type, but it is only weaker

12 C. QIU, X. FENG EJDE-7/9 eamp logarithm type for the Modified method (MM). Here some comparison of numerical result for these three methods are considered. The case with no eplicit solutions is considered in Eamples 3.3 and 3.4. In the following tests, the initial time is chosen as t, and the number of the discrete points N is 8. The selection of regularization parameters for Fourier method and Modified method are chosen according to [9, (.8) with s ] and [6, (3.4)] in Eamples 3. and 3., respectively. Let u be the eact solution and v be the approimation of some regularization method. The absolute error e a (u) is defined as ( N e a (u) : u v l u(n v(n) ) /, N Eample 3. ([9]). It is easy to verify that the function u(, t) e +4t (3.4) + 4t n is the unique solution of the problem u t u, R, t >, u(, T ) ϕ T () : e (3.5) eq: +4T, R. + 4T tab: Table. Regularization parameters for different methods with ɛ T ξ ma (FM) µ(mm) J(WM) eamp Figure illustrates the eact solution and the approimation corresponding to the three methods at different times t from T., T.4, T.9 with ɛ for Eample 3. in the interval [, ]. The regularization parameters are chosen as in Table. Here ξ ma, µ and J are the regularization parameters of Fourier method, Modified method and Wavelet method, respectively. Figure shows that the reconstruction error obtained by the different methods for different numbers of discrete points with T. and ɛ 3. It shows that the number N of discrete points (i.e., the step length) also plays the role of regularization parameter [5]. According to the general regularization theory, it should be neither too small nor too large. But usually, in some regularization methods, the influence of number N is less than the regularization parameter. Eample 3. ([6]). The function is the unique solution of the problem u t u, R, t >, u(, t) e t sin() (3.6) u(, T ) ϕ T () : e T sin(), R. (3.7) eq:

13 EJDE-7/9 BACKWARD HEAT CONDUCTION PROBLEMS 3 epsilone Eact solution and approimation at t Eact solution FM WM MM. 5 5 epsilone Eact solution and approimation at t Eact solution FM WM MM. 5 5 epsilone Eact solution and approimation at t Eact solution FM WM MM. 5 5 Figure. Eample 3.. Eact solution and approimation at t from T. (top), T.4 (middle), T.9 (bottom) with ɛ. fig: Figure 3 plots the eact solution and the approimation by the three methods at t from different times T., T.4, T.9 with ɛ for Eample

14 4 C. QIU, X. FENG EJDE-7/ T., epsilone 3 FM WM MM Error for different numbers N Figure. Eample 3.. Reconstruction error obtained by the different methods for different numbers of discrete points with T., ɛ 3. fig: 3. in the interval [ 3π, 3π]. The regularization parameters are also chosen as in Table. Figure 4 illustrates that the reconstruction error obtained by the different methods for different noisy levels with T., N 8. It shows that, for the different methods, the error between the eact solution and the approimate solution gets smaller as the noise in the data decreases. eamp3 From the Figures and 3, we can also see that, for all methods, the smaller the T, the better the approimation. This phenomenon conforms to the theory that, the larger the T, the more ill-posed the problem. Unfortunately, for general data ϕ T, it is not easy to find an eplicit analytical solution to problem (.), so we will construct new eamples as follows: take a function ϕ () L (R) and solve the well-posed problem u t u, R, t >, u(, ) ϕ (), R, to get an approimation to ϕ T (). To avoid the inverse crime, we use the finite difference to compute this well-posed problem. Here we discretize problem (3.8) only with respect to the spatial variable and leave the time variable t continuous, and then we obtain a system of ordinary differential equations and we can solve it using an eplicit Runge-Kutta method. See the details in [6]. Then we add a random noise to ϕ T () to get the noisy data ϕ T (). At last, we use the proposed regularized technique to obtain the regularized solution at t. Eample 3.3. We choose a non-smooth function + 3, 3, ϕ () 3, < 3,, > 3. (3.8) eq:b

15 EJDE-7/9 BACKWARD HEAT CONDUCTION PROBLEMS 5 Eact solution and approimation at t Eact solution FM WM MM epsilone 5 5 Eact solution and approimation at t Eact solution FM WM MM epsilone 5 5 Eact solution and approimation at t Eact solution FM WM MM epsilone 5 5 Figure 3. Eample 3.. Eact solution and approimation at t from T. (top), T.4 (middle), T.9 (bottom) with ɛ. fig:3

16 6 C. QIU, X. FENG EJDE-7/9.4. T., N8 FM WM MM Error for different noisy level ε Figure 4. Eample 3.3. Reconstruction error obtained by the different methods for different noisy levels with T., N 8. fig:4 eamp4 Eample 3.4. We consider a discontinuous function, 3, ϕ (), < 3,, > 3. Figures 5 and 6 illustrate the eact and regularized solutions corresponding to different ɛ for Eamples 3.3 and 3.4, respectively. The results show that the approimation gets better as the noise level ɛ decreases. Although ϕ () in Eample 3.3 is not smooth and ϕ () in Eample 3.4 is even non-continuous, our method is also effective for them. In summary, from the above different kinds of numerical eamples, we can conclude that the numerical solution is stable, and the Meyer wavelet method is an applicable method. This accords with our theoretical results. Acknowledgments. The authors are very grateful to the anonymous referee for their helpful comments and suggestions and to Prof. Chu-Li Fu for his constructive suggestions. The authors also thank Dr. Yuan-Xiang Zhang and Dr. Yun-Jie Ma for helpful discussion. The work is supported by the National Natural Science Foundation of China (Nos. 4456, ), and by the Natural Science Basic Research Plan in Shaani Province of China (No. 5JQ6). References s3 [] A. Carasso; Error bounds in the final value problem for the heat equation, SIAM J. Math. Anal., 7, 976, s4 [] D. Colton; The approimation of solutions to the backwards heat equation in a nonhomogeneous medium, J. Math. Anal. Appl., 7, 979, s [3] I. Daubechies; Ten Lectures on Wavelets, SIAM, Philadelphia, PA, 99. s5 [4] L. Eldén; Regularization of the backwards solution of parabolic problems, In: Inverse and Improperly posed problems in Differential Equations (G. Anger, editor), Akademie Verlag, Berlin, 979.

17 EJDE-7/9 BACKWARD HEAT CONDUCTION PROBLEMS 7. T The eact solution ut Eact solution and approimation at t Eact solution WM with ε WM with ε WM with ε Figure 5. Eample 3.3. Computed input data ϕ T () (top); eact and regularized solutions with different ɛ from T.9 (bottom). fig:5 Larsdifference s Engl-996 Feng+Ning+5 Fu+Xiong+Qian+7 s Marie-8 s Kolaczyk s [5] L. Eldén; Numerical solution of the sideways heat equation by difference approimation in time, Inverse Probl.,, 995, [6] L. Eldén, F. Berntsson, T. Regiǹska; Wavelet and Fourier methods for solving the sideways heat equation, SIAM J. Sci. Comput.,,, [7] H. W. Engl, M. Hanke, A. Neubauer; Regularization of inverse problems, Kluwer Academic Publishers, 996. [8] X. L. Feng, W. T. Ning; A wavelet regularization method for solving numerical analytic continuation, Int. J. Comput. Math., 9, 5, [9] C. L. Fu, X. T. Xiong, Z. Qian; Fourier regularization for a backward heat equation, J. Math. Anal. Appl., 33, 7, [] D. N. Hào, A. Schneider, H. J. Reinhardt; Regularization of a non-characteristic Cauchy problem for a parabolic equation, Inverse Probl.,, 995, [] B. M. C. Hetrick, R. Hughes, E. McNabb; Regularization of the backward heat equation via heatlets, Electron. J. Differential Equations, 8 no. 3, 8, 8. [] V. Isakov; Inverse problems for partial differential equations, Springer-Verlag, New York, 998. [3] E. D. Kolaczyk; Wavelet methods for the inversion of certain homogeneous linear operators in the presence of noisy data, Ph.D. thesis, Stanford University, Stanford, Calif., USA, 994. [4] M. M. Lavrentév, V. G. Romanov, S. P. Shishatskii; Ill-posed problems of mathematical physics and analysis, AMS, Providence, Rhode Island, 986.

18 8 C. QIU, X. FENG EJDE-7/9 T The eact solution ut Eact solution and approimation at t Eact solution WM with ε WM with ε WM with ε Figure 6. Eample 3.4: Computed input data ϕ T () (top); Eact and regularized solutions with different ɛ from T.9 (bottom). fig:6 murio-993 [5] D. A. Murio; The mollification method and the numerical solution of ill-posed problems, John Wiley and Sons, Interscience, New York, 993. Qian+Fu+Shi+7 [6] Z. Qian, C. L. Fu, R. Shi; A modified method for a backward heat conduction problem, Appl. Math. Comput., 85, 7, Qiu+8 [7] C. Y. Qiu, C. L. Fu; Wavelets and regularization of the Cauchy problem for the Laplace equation, J. Math. Anal. Appl., 338 (), 8, s9 [8] T. Regiǹska; Sideways heat equation and wavelets, J. Comput. Appl. Math., 63, 995, 9 4. Reginska+Elden-997 [9] T. Regiǹska, L. Eldén; Solving the sideways heat equation by a wavelet-galerkin method, Inverse Probl., 3, 997, s6 [] T. I. Seidman; Optimal filtering for the backward heat equation, SIAM J. Numer. Anal., 33, 996, 6 7. Shen+Strong- [] J. H. Shen, G. Strang; On wavelet fundamental solutions to the heat equation heatlets, J. Differential Equations, 6,, s7 [] U. Tautenhahn, T. Schröter; On optimal regularization methods for the backward heat equation, Z. Anal. Anew., 5, 996, vani+ [3] C. Vani, A. Avudainayagam; Regularized solution of the Cauchy problem for the Laplace equation using Meyer Wavelets, Math. Comput. Model., 36,, Wang+ [4] J. R. Wang; Uniform convergence of wavelet solution to the sideways heat equation, Acta Math. Sin. (Engl. Ser.),, 6 (),

19 EJDE-7/9 BACKWARD HEAT CONDUCTION PROBLEMS 9 s8 [5] B. Yildiz, H. Yetişkin, A. Sever; A stability estimate on the regularized solution of the backward heat equation, Appl. Math. Comp., 35, 3, Chunyu Qiu School of Mathematics and Statistics, Lanzhou University, Lanzhou 73, China address: qcy@lzu.edu.cn Xiaoli Feng School of Mathematics and Statistics, Xidian University, Xi an 77, China address: iaolifeng@idian.edu.cn