Identifiability of Piecewise Constant Conductivity

Consider the heat conduction in a nonhomogeneous insulated rod of a unit length, with the ends kept at zero temperature at all times. Our main interest is in the identification and identifiability of the discontinuous conductivity (thermal diffusivity) coefficient a(x), 0 ≤ x ≤ 1. The identification problem consists of finding a conductivity a(x) in an admissible set K for which the temperature u(x, t) fits given observations in a prescribed sense. Under a wide range of conditions one can establish the continuity of the objective function J(a) representing the best fit to the observations. Then the existence of the best fit to data conductivity follows if the admissible set K is compact in the appropriate topology. However, such an approach usually does not guarantee the uniqueness of the found conductivity a(x). Establishing such a uniqueness is referred to as the identifiability problem. For an extensive survey of heat conduction, including inverse heat conduction problems see (Beck et al., 1985; Cannon, 1984; Ramm, 2005) From physical considerations the conductivity coefficients a(x) are assumed to be in


Introduction
Consider the heat conduction in a nonhomogeneous insulated rod of a unit length, with the ends kept at zero temperature at all times.Our main interest is in the identification and identifiability of the discontinuous conductivity (thermal diffusivity) coefficient a(x),0≤ x ≤ 1.The identification problem consists of finding a conductivity a(x) in an admissible set K for which the temperature u(x, t) fits given observations in a prescribed sense.Under a wide range of conditions one can establish the continuity of the objective function J(a) representing the best fit to the observations.Then the existence of the best fit to data conductivity follows if the admissible set K is compact in the appropriate topology.However, such an approach usually does not guarantee the uniqueness of the found conductivity a(x).Establishing such a uniqueness is referred to as the identifiability problem.For an extensive survey of heat conduction, including inverse heat conduction problems see (Beck et al., 1985;Cannon, 1984;Ramm, 2005) From physical considerations the conductivity coefficients a(x) are assumed to be in The temperature u(a)=u(x, t; a) inside the rod satisfies u t − (a(x)u x ) x = f (x, t), Q =(0, 1) × (0, T), u(0, t)=q 1 (t), u(1, t)=q 2 (t), t ∈ (0, T), u(x,0)=g(x), x ∈ (0, 1), where g ∈ H = L 2 (0, 1), q 1 , q 2 ∈ C 1 [0, ∞).Suppose that one is given an observation z(t)= u(p, t; a) of the heat conduction process (2) for t 1 < t < t 2 at some observation point 0 < p < 1.From the series solution for (2) and the uniqueness of the Dirichlet series expansion (see Section 5), one can, in principle, recover all the eigenvalues of the associated Sturm-Lioville problem.If one also knows the eigenvalues for the heat conduction process with the same coefficient a and different boundary conditions, then classical results of Gelfand and Levitan (Gelfand & Levitan, 1955) show that the conductivity a(x) can be uniquely identified from the knowledge of the two spectral sequences.Alternatively, the conductivity is identifiable if the entire spectral function is known (i.e. the eigenvalues and the values of the derivatives of the normalized eigenfunctions at x = 0).However, such results have little practical value, since the observation data z(t) always contain some noise, and therefore one cannot hope to adequately identify more than just a few first eigenvalues of the problem.A different approach is taken in (Duchateau, 1995;Kitamura & Nakagiri, 1977;Nakagiri, 1993;Orlov & Bentsman, 2000;Pierce, 1979).These works show that one can identify a constant conductivity a in (2) from the measurement z(t) taken at one point p ∈ (0, 1).T h e s ew o r k s also discuss problems more general than (2), including problems with a broad range of boundary conditions, non-zero forcing functions, as well as elliptic and hyperbolic problems.
In (Elayyan & Isakov, 1997;Kohn & Vogelius, 1985) and references therein identifiability results are obtained for elliptic and parabolic equations with discontinuous parameters in a multidimensional setting.A typical assumption there is that one knows the normal derivative of the solution at the boundary of the region for every Dirichlet boundary input.For more recent work see (Benabdallah et al., 2007;Demir & Hasanov, 2008;Isakov, 2006).
In our work we examine piecewise constant conductivities a(x), x ∈ [0, 1].Suppose that the conductivity a is known to have sufficiently separated points of discontinuity.More precisely, let a ∈ PC(σ) defined in Section 2. Let u(x, t; a) be the solution of (2).The eigenfunctions and the eigenvalues for (2) are defined from the associated Sturm-Liouville problem (5).
In our approach the identifiability is achieved in two steps: First, given finitely many equidistant observation points {p m } M−1 m=1 on interval (0, 1) (as specified in Theorem 5.5), we extract the first eigenvalue λ 1 (a) and a constant nonzero multiple of the first eigenfunction G m (a)=C(a)ψ 1 (p m ; a) from the observations z m (t; a)= u(p m , t; a).This defines the M-tuple Second, the Marching Algorithm (see Theorem 5.5) identifies the conductivity a from G(a).We start by recalling some basic properties of the eigenvalues and the eigenfunctions for (2) in Section 2. Our main identifiability result is Theorem 5.5.It is discussed in Section 5.The continuity properties of the solution map a →G (a) are established in Section 4, and the continuity of the identification map G −1 (a) is proved in Section 8. Computational algorithms for the identification of a(x) from noisy data are presented in Section 10.This exposition outlines main results obtained in (Gutman & Ha, 2007;2009).
In (Gutman & Ha, 2007) the case of distributed measurements is considered as well.

Properties of the eigenvalues and the eigenfunctions
The admissible set A ad is too wide to obtain the desired identifiability results, so we restrict it as follows.
Definition 2.1.(i) a ∈P S N if function a is piecewise smooth, that is there exists a finite sequence of points 0 For definiteness, we assume that a and a ′ are continuous from the right, i.e. a(x)=a(x+) and a ′ (x)= a ′ (x+) for all x ∈ [0, 1).A l s ol e ta(1)=a(1−).(ii) Define PS = ∪ ∞ N=1 PS N .(iii) Define PC ⊂ PS as the class of piecewise constant conductivities, and For a ∈PS N the governing system (2) is given by x ∈ (0, 1). (4) The associated Sturm-Liouville problem for (4) is (5) For convenience we collect basic properties of the eigenvalues and the eigenfunctions of (5).Additional details can be found in (Birkhoff & Rota, 1978;Evans, 2010;Gutman & Ha, 2007).
(ii) Each eigenvalue is simple.For each eigenvalue λ k there exists a unique continuous, piecewise smooth normalized eigenfunction ψ k (x) such that ψ ′ k (0+) > 0, and the function a where V k varies over all subspaces of H 1 0 (0, 1) of finite dimension k.
(ii) On any subinterval (x i , x i+1 ) the coefficient a(x) has a bounded continuous derivative.Therefore, on any such interval the initial value problem (a(x)v ′ (x) )=B has a unique solution.Suppose that two eigenfunctions w 1 (x) and w 2 (x) correspond to the same eigenvalue λ k .Then they both satisfy the condition w 1 (0)=w 2 (0)=0.Therefore their Wronskian is equal to zero at x = 0. Consequently, the Wronskian is zero throughout the interval (x 0 , x 1 ), and the solutions are linearly dependent there.Thus The linear matching conditions imply that The uniqueness of solutions implies that w 2 (x)=Cw 1 (x) on (x 1 , x 2 ),e t c .T h u sw 2 (x)=Cw 1 (x) on (0, 1) and each eigenvalue λ k is simple.In particular λ 1 is a simple eigenvalue.The uniqueness and the matching conditions also imply that any solution of (a(x)v ′ (x)) ′ + λv = 0, v(0)=0, v ′ (0)=0m u s t be identically equal to zero on the entire interval (0, 1).Thus no eigenfunction ψ k (x) satisfies ψ ′ k (0)=0.Assuming that the eigenfunction ψ k is normalized in L 2 (0, 1) it leaves us with the choice of its sign for ψ ′ k (0).Letting ψ ′ k (0) > 0 makes the eigenfunction unique.
the eigenvalues of (7) with a(x)=1areπ 2 k 2 the required inequality follows.
(v) Recall that ψ 1 (x) is a continuous function on [0, 1].Suppose that there exists p ∈ (0, 1) T h e n w l , w r are continuous, and, moreover, w l , w r ∈ H 1 0 (0, 1).A l s o 1 0 w l (x)w r (x)dx = 0, and Suppose that w l is not an eigenfunction for we have This contradiction implies that w l (and w r ) must be an eigenfunction for λ 1 .However, w l (x)=0forp ≤ x ≤ 1, and as in (ii) it implies that In any case for such point p we have Since the derivative of ψ 1 is zero at any point of maximum, we have to conclude that such a maximum p is unique.

Representation of solutions
First, we derive the solution of (4) with f = q 1 = q 2 = 0. Then we consider the general case.
Proof.(i) Note that the eigenvalues and the eigenfunctions satisfy and Bessel's inequality implies that the sequence of Fourier coefficients g, ψ k is bounded.Therefore, denoting by C various constants and using the fact that the function By Weierstrass M-test the series converges absolutely and uniformly on [0, 1]. (ii is analytic in the part of the complex plane {s ∈ C : Re s > t 0 }, and the result follows. Next we establish a representation formula for the solutions u(x, t; a) of ( 4) under more general conditions.Suppose that u(x, t; a) is a strong solution of (4), i.e. the equation and the initial condition in (4) are satisfied in H = L 2 (0, 1).L e t Φ(x, t; a)= q 2 (t) − q 1 (t) Accordingly, the weak solution u of ( 4) is defined by u(x, t; a)=v(x, t; a)+Φ(x, t; a) where v is the weak solution of (11).For the existence and the uniqueness of the weak solutions for such evolution equations see (Evans, 2010;Lions, 1971).

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Heat Conduction -Basic Research www.intechopen.com(ii) Let {λ k , ψ k } ∞ k=1 be the eigenvalues and the eigenfunctions of ( 5).Let where (iii) For each t > 0 and a ∈P Sthe series in ( 12) converges in X.Moreover, this convergence is uniform with respect to t in 0 < t 0 ≤ t ≤ Tanda∈PS.
Proof.Under the conditions specified in the Theorem the existence and the uniqueness of the weak solution v ∈ C([0, T]; H) ∩ L 2 ([0, T]; V) of ( 11) is established in (Evans, 2010;Lions, 1971).By the definition u = v + Φ.Thus the existence and the uniqueness of the weak solution u of ( 4) is established as well.
Let {ψ k } ∞ k=1 be the orthonormal basis of eigenfunctions in H corresponding to the conductivity a ∈P S .L e tB k (t)= v(•, t), ψ k .To simplify the notation the dependency of B k on a is suppressed.Then v = ∑ ∞ k=1 B k (t)ψ k in H for any t ≥ 0, and Therefore B k (t) has the representation stated in (13).Let 0 .F o r this purpose we establish that this series converges in X = C[0, 1] uniformly with respect to t ∈ [t 0 , T] and a ∈ A ad .Note that V is continuously embedded in X.F u r t h e r m o r e ,s i n c e0< ν ≤ a(x) ≤ µ the original norm in V is equivalent to the norm • V a defined by w 2 V a = 1 0 a|w ′ | 2 dx.Thusitisenough to prove the uniform convergence of the series for v in V a .The uniformity follows from the fact that the convergence estimates below do not depend on a particular t ∈ [t 0 , T] or a ∈ A ad .By the definition of the eigenfunctions ψ k one has aψ ′ k , ψ ′ j = λ k ψ k , ψ j for all k and j.Thus the eigenfunctions are orthogonal in V a .In fact, {ψ k / √ λ k } ∞ k=1 is an orthonormal basis in V a , see (Evans, 2010).Therefore the series This convergence follows from the fact that the function s → √ se −σs is bounded on [0, ∞) for any σ > 0, see (Gutman & Ha, 2009).

Continuity of the solution map
In this section we establish the continuous dependence of the eigenvalues λ k ,eigenfunctions ψ k and the solution u of (4) on the conductivities a ∈PS⊂A ad ,whenA ad is equipped with the L 1 (0, 1) topology.For smooth a see (Courant & Hilbert, 1989).to Theorem 2.2 the eigenfunctions form a complete orthonormal set in H.S in c e 1 0 aψ ′ j ψ ′ dx = λ j 1 0 ψ j ψdx for any ψ ∈ H 1 0 (0, 1) we have where • ∞ is the norm in L ∞ (0, 1).Estimates from Theorem 3.1 and the Cauchy-Schwarz inequality give and the desired continuity is established.
The following theorem is established in (Gutman & Ha, 2007).
Theorem 4.2.Let a ∈PS, PS ⊂ A ad be equipped with the L 1 (0, 1) topology, and {ψ k (x; a)} ∞ k=1 be the unique normalized eigenfunctions of the associated Sturm-Liouville system (5) satisfying the condition Theorem 4.3.Let a ∈PS⊂A ad equipped with the L 1 (0, 1) topology, and u(a) be the solution of the heat conduction process (4), under the conditions of Theorem 3.2.Then the mapping a → u(a) from PS into C([0, T]; X) is continuous.
Proof.According to Theorem 3.2 the solution u(x, t; a) is given by u(x, t; a)=v(x, t; a)+ Φ(x, t; a),w h e r ev(x, t; a)=∑ ∞ k=1 B k (t; a)ψ k (x) with the coefficients B k (t; a) given by ( 13).Let  By Theorems 4.1 and 4.2 the eigenvalues and the eigenfunctions are continuously dependent on the conductivity a.Therefore, according to (13), the coefficients B k (t, a) are continuous as functions of a from PS into C([0, T]; X).This implies that a → v N (a) is continuous.By Theorem 3.2 the convergence v N → v is uniform on A ad as N → ∞ and the result follows.

Identifiability of piecewise constant conductivities from finitely many observations
Series of the form ∑ ∞ k=1 C k e −λ k t are known as Dirichlet series.The following lemma shows that a Dirichlet series representation of a function is unique.Additional results on Dirichlet series can be found in Chapter 9 of (Saks & Zygmund, 1965).

Proof.
In both cases the series ∑ ∞ k=1 C k e −µ k z converges uniformly in Re z > 0r e g i o no ft h e complex plane, implying that it is an analytic function there.Thus Suppose that some coefficients C k are nonzero.Without loss of generality we can assume which is a contradiction.
Remark.According to Theorem 3.1 for each fixed p ∈ (0, 1) the solution z(t)=u(p, t; a) of ( 4) is given by a Dirichlet series.The series coefficients C k = g, v k v k (p) are square summable, therefore they form a bounded sequence.The growth condition for the eigenvalues stated in (iv) of Theorem 2.2 shows that Lemma 5.1(ii) is applicable to the solution z(t).
0, in this case the governing system (4) is x ∈ (0, 1), ( 14) Identifiability of Piecewise Constant Conductivity www.intechopen.com where g ∈ L 2 (0, 1) and i The central part of the identification method is the Marching Algorithm contained in Theorem 5.5.Recall that it uses only the M-tuple G(a), see (3).That is we need only the first eigenvalue λ 1 and a nonzero multiple of the first eigenfunction ψ 1 of ( 15) for the identification of the conductivity a(x).
Suppose that p * ∈ (x i−1 , x i ).Th e nψ 1 can be expressed on (x i−1 , x i ) as with A > 0. The range for γ in the above representation follows from the fact that ψ 1 (p * )= A cos γ > 0 by Theorem 2.2(5).
The identifiability of piecewise constant conductivities is based on the following three Lemmas, see (Gutman & Ha, 2007).
Then the system of equations w(q) > 0, v(q) > 0 (18) admits at most one solution q on [p, p + η].This unique solution q can be computed as follows: Otherwise compute q 1 and q 2 according to formulas ( 19) and ( 20) and discard the one that does not satisfy the conditions of the Lemma.
By the definition of a ∈P Cthere exist N ∈ N and a finite sequence 0 The following Theorem is our main result.
Theorem 5.5.Given σ > 0 let an integer M be such that Suppose that the initial data g(x) > 0, 0 < x < 1 and the observations z m the heat conduction process ( 14) are given.
Then the conductivity a ∈ A ad is identifiable in the class of piecewise constant functions PC(σ).
Proof.The identification proceeds in two steps.In step I the M-tuple G(a) is extracted from the observations z m (t).In step II the Marching Algorithm identifies a(x).
Step I. Data extraction.By Theorem 3.1 we get where By Theorem 2.2(5) ψ 1 (x) > 0onin t erv a l(0, 1).S i n c eg is positive on (0, 1) we conclude that g 1 ψ 1 (p m ) > 0. Since z m (t) is represented by a Dirichlet

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Identifiability of Piecewise Constant Conductivity www.intechopen.comseries, Lemma 5.1 assures that all nonzero coefficients (and the first term, in particular) are defined uniquely.
An algorithm for determining the first eigenvalue λ 1 ,a n dt h ec oef fi c ien tg 1 ψ 1 (p m ) from ( 21) is given in Section 10.Repeating this process for every m one gets the values of (iii) Use Lemma 5.2 to find A i , ω i and γ i from the system ) > 0thenletm := m + 1 and repeat this step (iv).(v) Use Lemma 5.2 to find A i+1 , ω i+1 and γ i+1 from the system (vi) Use formulas in Lemma 5.4 to find the unique discontinuity point x i ∈ [p m+2 , p m+3 ).

Identifiability of piecewise constant conductivity with one discontinuity
The Marching Algorithm of Theorem 5.5 requires measurements of the system at possibly large number of observation points.Our next Theorem shows that if a piecewise constant conductivity a is known to have just one point of discontinuity x 1 ,a n di t sv a l u e sa 1 and a 2 are known beforehand, then the discontinuity point x 1 can be determined from just one measurement of the heat conduction process.

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Identifiability of Piecewise Constant Conductivity www.intechopen.comTheorem 6.1.Let p ∈ (0, 1) be an observation point, g(x) > 0 on (0, 1), and the observation z p (t)= u(x p , t; a), t ∈ (T 1 , T 2 ) of the heat conduction process ( 14) be given.Suppose that the conductivity a ∈ A ad is piecewise constant and has only one (unknown) point of discontinuity x 1 ∈ (0, 1).G i v e n positive values a 1 = a 2 such that a(x)=a 1 for 0 ≤ x < x 1 and a(x)=a 2 for x 1 ≤ x < 1 the point of discontinuity x 1 is constructively identifiable.
Proof.Arguing as in the previous Theorem where > 0 the uniqueness of the Dirichlet series representation implies that one can uniquely determine the first eigenvalue λ 1 and the value of G p = g 1 ψ 1 (p).
Without loss of generality one can assume that a 1 > a 2 .In this case we show that the first eigenvalue λ 1 is strictly increasing as a function of the discontinuity point 1 , x b 1 ).However, this is impossible, since ψ 1,b (x) > 0o n(0, 1).Thus there exists a unique conductivity of the type sought in the Theorem for which its first eigenvalue is equal to λ 1 ,i.e.a is identifiable.Now the unique discontinuity point x 1 of a can be determined as follows.Let Then the first eigenfunction ψ 1 is given by It is shown in Theorem 8.2 that the Marching Algorithm not only provides the unique identification of the conductivity a, but that the identification is also continuous (stable).This result is based on the continuity of eigenvalues, eigenfunctions, and the solutions with respect to the L 1 (0, 1) topology in the set of admissible parameters A ad ,seeSection4.Numerical experiments show that, because of the ill-posedness of the identification of eigenvalues from a Dirichlet series representation, one can only identify G(a) with some error.Thus the Marching Algorithm would not be practically useful.In Section 10 we presented algorithms for the identification of conductivities from noise contaminated data.
Its main novel point is, in agreement with the theoretical developments, the separation of the identification process into two separate parts.In part one the first eigenvalue and a multiple of the first eigenfunction are extracted from the observations.In the second part a general minimization method is used to find a conductivity which corresponds to the recovered eigenfunction.
The first eigenvalue and the eigenfunction in part one of the algorithm are found from the Dirichlet series representation of the solution of the heat conduction process.The numerical experiments in (Gutman & Ha, 2009) confirm that even for noiseless data the second eigenvalue cannot be reliably found.These experiments showed that in our tests a simple regression type algorithm identified λ 1 better than a more complex Levenberg-Marquardt algorithm.The last part of the algorithm employs Powell's nonlinear minimization method because it does not require numerical computation of the gradient of the objective function.
The numerical experiments show that the conductivity identification was achieved with a 15-18% relative error for various noise levels in the observations.

Fig. 1 .
Fig. 1.Conductivity identification by the Marching Algorithm.The dots are a multiple of the first eigenfunction at the observation points p m .The algorithm identifies the values of the conductivity a and its discontinuity points v(x)=H i+1 (x)=B cos(Ω 2 (x − p m+3 )+Γ 2 ), |Γ 2 | < π/2.Let i := i + 1, m := m + 3.If m < l then return to step (iv).If m ≥ l then go to the next step (vii).(vii) Do steps (ii)-(vi) in the reverse direction of x,a d v a n c i n gf r o mx = 1t ox = p l+1 .Identify the values and the discontinuity points of a on [p l+1 ,1], as well as determine the corresponding functions H i (x).
stop, otherwise use Lemma 5.4 with η = 2δ, and the above parameters to find the discontinuity x j ∈ [p l−1 , p l+1 ].S t o p .
25)for some A, B > 0. The matching conditions at x 1 giveA sin ω 1 x 1 = B sin ω 2 (1 − x 1 ) a)={λ 1 (a), G 1 (a), ••• , G M−1 (a)}, where he values G m (a) are a constant nonzero multiple of the first eigenfunction ψ 1 (a).In principle, if G(a) is known, then the identification of the conductivity a can be accomplished by the Marching Algorithm.Theorem 7.1 shows under what conditions the M-tuple G(a) can be extracted from the observations z m (t),t h u s assuring the identifiability of a.