Ill-Posedness of Backward Heat Conduction Problem 1

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1 Ill-Posedness of Backward Heat Conduction Problem 1 M.THAMBAN NAIR Department of Mathematics, IIT Madras Chennai , INDIA, mtnair@iitm.ac.in 1. Ill-Posedness of Inverse Problems Problems that appear in practical situations are usually categorized into two, namely, the direct problems and inverse problems. The inverse problems are those in which one would like to determine causes for a desired or observed effect. One of the characterizing properties of many of the inverse problems is that they are usually ill-posed, in the sense that a solution that depend continuously on the data does not exist. This is in contrast with direct problems, which are usually well-posed. The concept of well-posedness is first formulated by Hadamard in the early 190 s. More specifically, a problem is said to be well-posed if it has a solution, it has at most one solution, and the solution depends continuously on the data. A problem which is not well-posed is called an ill-posed problem. Thus, an operator equation Af = g, (1.1) where one wants to find f for a given linear operator A : X Y between normed linear spaces X and Y and for any given g Y, is ill-posed if A does not have a continuous inverse. There is a large class of operators A, namely compact operators of infinite 1 A Key-Note Address: National Seminar on Mathematical Modelling at PSGR Krishnammal College for Women, Coimbatore on August 6,

2 rank, for which the equation (1.1) is ill-posed, for a compact operator of infinite rank cannot have a continuous inverse (See Nair [], Chapter 9). This fact can be illustrated by using the singular value representation whenever X and Y are Hilbert spaces. Suppose X and Y are Hilbert spaces. If {u n : n N} and {v n : n N} are orthonormal sets in X and Y respectively, and (σ n ) is a sequence of positive real numbers such that σ n 0 as n, then it can be shown that the map A : X Y defined by Af = σ n f, u n v n, f X, is a compact operator of infinite rank. Conversely, if A : X Y is a compact operator with infinite rank, then A can be represented as Af = σ n f, u n v n, f X, (1.) where {u n : n N} and {v n : n N} are orthonormal sets in X and Y respectively, and (σ n ) is a sequence of positive real numbers, called the singular values of A, such that σ n 0 as n. The representation (1.) of a compact operator A between Hilbert spaces is called the singular representation of A (See Nair [], Chapter 13). We may observe that Au n = σ n v n, A v n = σ n u n, n N. Here A denotes the adjoint of A. Suppose g R(A). Then we see that for f X, Af = g σ n f, u n = g, v n n N. This shows that the convergence of the series g, v n σ n is necessary for the existence of a solution f. Moreover, f defined by satisfies the equation Af = g. f = g, v n u n σ n

3 3 Now, if we take g k := g + σ k v k, then it follows that satisfies Af k = g k. Note that f k := g k, v n u n σ n g g k = σ k 0, but f f k = 1 σk as k. Thus a compact operator equation (1.1) is ill-posed whenever A is of infinite rank. A prototype of a compact operator equation is an integral equation of the first kind, Ω k(x, y)f(y)dy = g(x), x Ω, where k(, ) is a non-degenerate kernel (See Nair [], Chapter 9 ). Many inverse problems in applications, such as problems in Geological prospecting, Computer tomography backward heat equation lead to an integral equation of the first kind. In this talk we shall briefly describe how ill-posedness arise in backward heat conduction problem.. Backward Heat Conduction Problem Consider the heat equation u t = u, x 0 < x < π, 0 < t < τ, (.1) with boundary condition u(0, t) = 0 = u(π, t), 0 t τ. (.) Suppose the initial condition is given by u(x, 0) = f(x), 0 x π. (.3) Here u(x, t) represents the temperature at time t at the point x of a thin metallic wire of length π.

4 4 The direct problem related to the heat equation is to determine the temperature u(x, t) at time t (0, τ] from the knowledge of the initial temperature f(x) := u(x, 0), whereas, the inverse problem is to determine u(x, t) for t [0, τ) from the knowledge of the final temperature u(x, τ). by It can be shown, by using the method of separation of variables, that u(, ) defined u(x, t) := satisfies the equations (.1) (.3) if Note that a n = π π 0 a n e nt sin(nx) f(x) = u(x, 0) := f(y) sin(ny)dy, n N. a n sin(nx), the Fourier series expansion of f. Now, using the inner product ϕ, ψ = π 0 ϕ(x)ψ(x)dx, it follows that the functions u n defined by u n (x) := sin(nx), n N, satisfy π { 1, n = m, u n, u m = 0, n m, i.e., {u n : n N} is an orthonormal set with respect to the inner product,. Thus, we see that a n = f, u π n so that f(x) = f, u n u n (x), u(x, t) := e nt f, u n u n (x). For each t [0, τ], let us denote u(x, t) by ϕ t (x). Then we have ϕ t = e nt f, u n u n. Note that Thus, ϕ τ = e nτ f, u n u n = ϕ t, u n = e nt f, u n n N. e n (τ t) e nt f, u n u n = e n (τ t) ϕ t, u n u n.

5 For t [0, τ], let A t be defined by A t ϕ := e n (τ t) ϕ, u n u n. Since σ n := e n (τ t) 0 as n, we see that A t : L [0, π] L [0, π] is a compact operator with singular values σ n, n N. Thus, the problem of finding ϕ t := u(, t) from the knowledge ϕ τ := u(, τ) is same as solving the compact operator operator equation A t ϕ = ϕ τ. We may also observe that the above equation is an integral equation of the first kind. Hence, this backward heat conduction problem is ill-posed Optimal Estimate In order to obtain stable approximate solutions, one has to regularize the problem. There are many regularization procedure is available in the literature. For a comprehensive treatment of regularization methods for ill-posed inverse problems one refer the book [1] by Engl, Hanke and Nuebauer. After having considered a regularization method, the next step would be to know if such method give an error estimate of optimal order. Due to considerations explained in [1], a regularization method which gives an approximate solution f for the solution of Af = g corresponding to approximate data g with g g δ is said to be of optimal order with respect to a source set M X, if ˆf f c ω(m, δ), where ω(m, δ) := sup{ f : Af δ}. It is desirable that ω(m, δ) 0 as δ 0. But, it can be shown that, if A is not bounded below and if take M to be the closed unit ball of radius ρ, then ω(m, δ) ρ. So, in order to have the desirable property ω(m, δ) 0 as δ 0, it is necessary to choose the source set appropriately. For the back ward heat conduction problem one such source set is M ρ := {u(, t) : u(, 0) ρ}.

6 6 We shall obtain a realistic estimate for ω(m ρ, δ). Recall that Thus ϕ t (x) = u(x, t) := a n e nt sin(nx). ϕ t = u(, t) = π ϕ τ = u(, τ) = π a n e nt, a n e nτ, ϕ 0 = u(, 0) = π a n. Then, by Hölder s inequality with p = 1/(1 t/τ), we have ϕ t = π a n e n t = π a n /p a n /q e n t ( π ) 1/p ( ) 1/q a n a n = ϕ 0 (1 t/τ) ϕ τ t/τ. Hence, Thus, ϕ 0 ρ, ϕ τ δ = ϕ t ρ 1 t/τ δ t/τ. ω(m, δ) ρ 1 t/τ δ t/τ. Now the next question is whether there exists a reqularization method that yield the above order. In fact there are many regularization methods leading to the above estimate. One such method is the Tikhonov regularization (cf. [1]). In this, one finds f α that minimizes the function f A t ϕ g + α f, equivalently one solves the well-posed equation (A t A t + αi) f α = A t g.

7 7 This method yield the estimate (cf. Nair [3]) ˆf f α c ρ 1 t/τ δ t/τ, provided α is chosen either a priorily by α := c 0 δ (1 t/τ) or a posteriorily according to the Morozov s discrepancy principle, A t ϕ g = c δ. References [1] H.W. Engl, M. Hanke and A. Nuebauer, Regularization of Inverse Problems, Kluwer, Dordrect, [] M.T. Nair, Functional Analysis: A First Course, Prentice-Hall of India, New Delhi, 00. [3] On Morozov s method for Tikhonov regularization as an optimal order yielding algorithm, Zeitschrift für Analysis und ihre Anwendungen, 18 No.1 (1999)