OPTIMAL TIKHONOV APPROXIMATION FOR A SIDEWAYS PARABOLIC EQUATION

We consider an inverse heat conduction problem with convection term which appears in some applied subjects. This problem is ill posed in the sense that the solution (if it exists) does not depend
continuously on the data. A generalized Tikhonov regularization method for this problem is given, which realizes the best possible accuracy.


Introduction
In many industrial applications one wants to determine the temperature on the surface of a body, where the surface itself is inaccessible to measurement [2,4].In this case it is necessary to determine surface temperature from a measured temperature history at a fixed location inside the body.This problem is called an inverse heat conduction problem (IHCP).In a one-dimensional setting, assuming that the body is large, the following model problem or the standard sideways heat equation: u t = u xx , x > 0, t > 0, u(x,0) = 0, x ≥ 0, u(1,t) = g(t), t ≥ 0, u(x,t)| x→∞ bounded (1.1) has been discussed by many authors [4,6,7,10,16,18,19,20].But when a fluid is flowing through the solid, for example, a gas is travelling from the rear surface, there must be a convection term in heat conduction equation [1,17].A model problem in this case is the following sideways parabolic equation with nondivergence type in the quarter plane [13]: We want to know the solution u(x,t) for 0 ≤ x < 1.This problem is ill posed in the sense that the solution (if it exists) does not depend continuously on the data.Some regularization methods and error estimates have been given in [12,13,23]; we have even obtained some results for the following more general sideways parabolic equation: u t = a(x)u xx + b(x)u x + c(x)u, x > 0, t > 0, u(x,0) = 0, x ≥ 0, u(1,t) = g(t), t ≥ 0, (1.3) where a, b, and c are known functions which satisfy some specified conditions [9,11,14].Especially, the uniqueness of solution of problem (1.3) was explained in [14].The uniqueness of solution of problem (1.2) can also be similarly established according to [3].But all the error estimates between the regularized approximate solutions based on above methods and exact solutions are at most of optimal order.
There are some papers, for example [5,15], in which the a priori choice of the regularizing parameters is generally given for the "optimal convergence rate" based on the conditional stability.But their optimal convergence rate is only the convergence with optimal order (cf.Definition 2.1(ii)), but not optimal (cf.Definition 2.1(i)).In addition, the conditions of "conditional stability" is stronger than our priori assumption (1.6).Now our interest is to give a new regularization method for problem (1.2), which will be called generalized Tikhonov regularization method, such that the error estimate of this method possesses the best possible accuracy, that is, the error estimate of this method is optimal.It is well known that it is much more difficult to prove results about optimality instead of just order optimality [8, page 75].So far, as far as we know, the unique result about optimality of IHCP is obtained only for the standard sideways heat equation (1.1) [20,21].
As we consider the problem (1.2) in L 2 (R) or H p (R) with respect to the variable t, we extend the domain of definition of the functions u(x,•), g(•) := u(1,•), f (•) := u(0,•) and other functions appearing in the paper to the whole real t-axis by defining them to be zero for t < 0. The notation • , (•,•) denotes L 2 -norm and L 2 -inner product, respectively, is the Fourier transform of function h(t), and is the H p -norm of the function h(t).We assume that there exists a priori bound for f (t) := u(0,t): Let g(t) and g δ (t) denote the exact and measured (noisy) data at x = 1 of solution u(x,t), respectively, which satisfy For the uniqueness of solution, we require that u(x,•) be bounded [14].The solution of problem (1.2) has been given in [12,13,17,23] by or equivalently, where ) (1.12) The following representations will be useful and it is easy to see from (1.9) that u(x,ξ) = e −xθ(ξ) f (ξ). (1.14)

Preliminary
Most facts here are known [8,21,22].We consider arbitrary ill-posed inverse problems where A ∈ ᏸ(X,Y ) is a linear injective bounded operator between infinite dimensional Hilbert spaces X and Y with nonclosed range R(A) of A. We assume that y δ ∈ Y are the available noisy data with y − y δ ≤ δ.Any operator R : Y → X can be considered as a special method for solving (2.1), the approximate solution to (2.1) is then given by Ry δ .Let M ⊂ X be a bounded set.We introduce the worst-case error ∆(δ,R) for identifying x from y δ ∈ Y under the assumption x ∈ M according to This worst-case error characterizes the maximal error of the method R if the solution x of the problem (2.1) varies in the set M.
where the infimum is taken over all methods R : Y → X, (ii) order optimal on the set Now we review some optimality results if the set M has been given by or equivalently, where the operator function ϕ(A * A) is well-defined via the spectral representation is the spectral decomposition of A * A, {E λ } denotes the spectral family of operator A * A, and a is a constant such that A * A ≤ a.In the case when A : A generalized Tikhonov regularized approximation x δ α is determined by solving the minimization problem or equivalently, by solving the Euler equation ) and (2.8) are identical with classical Tikhonov regularization.
In order to obtain optimality result, we assume as in [22] that the function ϕ in (2.3) and (2.4) satisfies the following assumption.

Chu-Li Fu et al. 1225
Theorem 2.3.Let M ϕ,E be given by (2.3), let Assumption 2.2 be satisfied, and let (2.9) The general Tikhonov regularization method appears to be optimal on the set M ϕ,E given by (2.3) provided the regularization parameter α is chosen properly.For this method there holds [22, Theorem 5.1] the following.

Optimal Tikhonov approximation for the problem (1.2)
In this section, we consider the generalized Tikhonov regularization method (2.8) for problem (1.2), and based on Theorems 2.3 and 2.4 show how to choose the regularization parameter such that it is optimal.We introduce the Sobolev scale (H p ), p ∈ R + of positive real order p according to where v p is given by (1.5).For problem (1.2) we require a priori smoothness condition concerning the unknown solution u(x,t) according to Denote the best possible worst-case error as where R(x) : g δ (•) → u δ (x,•) and with g δ (t) = u δ (1,t) and g(t) = u(1,t) satisfy (1.7), and the supremum is taken over u(x,t) ∈ M p,E given by (3.1).
Firstly, we formulate the problem (1.2) of identifying u(x,t) from (unperturbed) data u(1,t) = g(t) as an operator equation It is easy to know that (3.4) is equivalent to the following operator equation [12]: where is the (unitary) Fourier operator that maps any L 2 (R) function h(t) into its Fourier transform h(ξ) given by (1.4), and we know from (1.9) that where A(x) : L 2 (R) → L 2 (R) is a linear normal operator (multiplication operator) and which is the conjugated operator of A(x), so The a priori smoothness condition (3.1) and the general source set (2.3) can be transformed into their equivalent condition in the frequency domain as follows: ) By their equivalence, it is easy to know that the representation of ϕ(λ) is given (in parameter representation) by Due to (3.8) there holds that is, Note that from (1.14), we know that the condition in (3.9), is equivalent to Comparing this with (3.10), it is easy to see that  Here and following, we always denote by λ the independent variable and r the parameter.
Due to Theorem 2.3, Propositions 3.2 and 3.3 we have proved in [12] the following optimal error bounds for problem (1.2), that is, the best possible worst-case error ω(δ,x) defined by (3.2) for identifying the solution u(x,t) of the problem (1.2) from noisy data u δ (1,t) = g δ (t) ∈ L 2 (R) under the condition (1.7) and u(x,t) ∈ M p,E given by (3.1).Theorem 3.4 [12].Let δ 2 /E 2 ≤ 1, then the following stability results hold: (i) in case p = 0 and 0 < x < 1, there holds (Hölder stability), (ii) in case p > 0 and 0 ≤ x < 1, there holds Now we consider the method of generalized Tikhonov regularization, apply it to problem (1.2), and show how to choose the regularization parameter such that it guarantees Chu-Li Fu et al. 1229 the optimal error bounds given by (3.23) and (3.24).This optimality result will be obtained by applying Theorem 2.4 to our transformed problem (3.5), it yields an optimal regularized approximation u δ α (x,ξ) in the frequency domain.Due to Parseval formula ) is optimal regularized approximations in the original domain.
The method of generalized Tikhonov regularization (2.7) applied to our problem (3.5) in the frequency domain consists in the determination of a regularized approximation u δ α (x,ξ) by solving the minimization problem where θ(ξ) is given by (1.10), and From (3.28) we conclude that the Tikhonov regularized solution u δ α (x,ξ) can be written in the form The following theorem is the main result of this paper and will answer the question how to choose the regularization parameter α = α(x,δ) in (3.29), such that the Tikhonov regularized solution u δ α (x,t) = Ᏺ −1 ( u δ α (x,ξ)) is optimal on the set M p,E given by (3.1).Theorem 3.5.Let p = 0, x > 0 or p > 0, x ≥ 0, and δ 2 /E 2 ≤ 1 hold, then the Tikhonov regularized solution u δ α (x,t) = Ᏺ −1 ( u δ α (x,ξ)) with u δ α (x,ξ) given by (3.29) is optimal on 1230 Optimal Tikhonov approximation the set M p,E provided the regularization parameter α is chosen optimally by where r 0 is the (unique) solution of the equation (3.31) In the case x = 0 and p > 0 there holds and in the case 0 < x < 1, p ≥ 0, there holds Furthermore, the optimal error estimate u δ α0 (x,t) − u(x,t) ≤ ω(δ,x) holds true where ω(δ,x) is given by (3.23) and (3.24), respectively.Proof.From Theorem 2.4 it follows that the optimal regularization parameter α is given by (2.10) with ϕ(λ) given by (3.11), which is equivalent to Its parameter representation is just (3.31), so r 0 should be the solution of (3.31).Moreover, because of the strict monotonicity of functions λ(r) and ϕ(r), r 0 should be the unique solution of (3.31).Note that from (3.11), we know that φ(r and α 0 in (3.34) can be rewritten as where r 0 is the solution of (3.31), this is just the representation formula (3.30).In addition, due to (3.31), we have Note that for r 0 → +∞ when δ → 0, there holds that is, The asymptotical representation of α 0 can be given as follows.
The proof of Theorem 3.5 is complete.

A numerical example
It is easy to verify that the function is the exact solution of problem (1.2) with data 3 give the comparison of the approximation solution with exact solution at x = 0,0.1, and 0.9, respectively.The tests were performed in the following way: first, we add a normally distributed perturbation of variance 10 −4 to each function, giving vectors {g (m)   δ } 100 m=1 , then we use the formula to compute the Tikhonov approximation solution, the data δ is given according to (1.7).
It can be seen from these figures that the computational effect of the optimal Tikhonov regularization method is fairly satisfactory.For fixed δ and x, we take δ = 0.1 and x = 0, then the best possible worst error (3.24) becomes   In this experiment, we can conclude that ω(p) is a decreasing function in respect of p.The same result can be found when the various data δ is given.In theory, the monotony of the function ω(p,ω,x) = δ x [E(p)( The results of fixing δ = = 0.10, x = 0 are presented in Table 4.2, which shows that our theoretical optimal error estimate is decreasing with the index p.

(4. 2 )Figures 4 . 1 , 4 . 2 ,
Figures 4.1, 4.2, and 4.3 give the comparison of the approximation solution with exact solution at x = 0,0.1, and 0.9, respectively.The tests were performed in the following way: first, we add a normally distributed perturbation of variance 10 −4 to each function, giving vectors {g(m)

Remark 4 . 1 .
In our numerical experiment, if E is considered as a function of p, we can find the relation between E(p) and p in Table 4.1.From the latter, we conclude that the function f (•) belongs to H p (R), where p satisfies 0 ≤ p ≤ 2. The function E(p) is plotted as in Figure 4.4.Chu-Li Fu et al. 1233