A unified approach to identifying an unknown spacewisedependent source in a variable coefficient parabolic equationfrom final and integral overdeterminations
Alemdar Hasanov ∗, Burhan Pektas
Department of Mathematics and Computer Science, Izmir University, Üçkuyular 35350, Izmir, Turkey
a r t i c l e i n f o a b s t r a c t
Article history:Received 13 May 2013Received in revised form 10 October 2013Accepted 27 November 2013Available online 14 December 2013
An adjoint problem approach with subsequent conjugate gradient algorithm (CGA) for aclass of problems of identification of an unknown spacewise dependent source in a variablecoefficient parabolic equation ut = (k(x)ux)x + F (x)H(t), (x, t) ∈ (0, l) × (0, T ] is proposed.The cases of final time and time-average, i.e. integral type, temperature observations areconsidered. We use well-known Tikhonov regularization method and show that the adjointproblems, corresponding to inverse problems ISPF1 and ISPF2 can uniquely be derived bythe Lagrange multiplier method. This result allows us to obtain representation formula forthe unique solutions of each regularized inverse problem. Using standard Fourier analysis,we show that series solutions for the case in which the governing parabolic equation hasconstant coefficient, coincide with the Picard’s singular value decomposition. It is shownthat use of these series solutions in CGA as an initial guess substantially reduces thenumber of iterations. A comparative numerical analysis between the proposed versionof CGA and the Fourier method is performed using typical classes of sources, includingoscillating and discontinuous functions. Numerical experiments for variable coefficientparabolic equation with different smoothness properties show the effectiveness of theproposed version of CGA.
Heat source identification problems are the most commonly encountered inverse problems in heat conduction. Theseproblems have been studied over several decades due to their significance not only in a variety of scientific and engineeringapplications, but also to their significance in the theory of inverse source problems for PDEs (see [1–6,8–11,13,15,17,16,18–21,28–34,37] and references therein). The final overdetermination for one-dimensional heat equation has first been con-sidered by Tikhonov [35] in study of geophysical problems. In this work the heat equation with prescribed lateral and finaldata is studied in half-plane and the uniqueness of the bounded solution is proved. For parabolic equations in a boundeddomain, when in addition to usual initial and boundary condition, a solution is given at the final time, well-posedness ofinverse source problem has been proved by Isakov [19,20]. Certain existence, uniqueness and conditional stability questionsfor various inverse source problems have been analyzed in [3,8–11,28,31,34,38]. An adjoint problem approach for the inversesource problems related to linear and nonlinear parabolic equations has been proposed in [4–6]. This approach is then devel-oped in [10,15,17], where, in particular, Fréchet differentiability of the cost functional and Lipschitz continuity of its gradientis proved. Some uniqueness theorems for inverse spacewise dependent source problems related to nonlinear parabolic and
50 A. Hasanov, B. Pektas / Applied Numerical Mathematics 78 (2014) 49–67
hyperbolic equations are given in [33]. Numerical algorithms for determination of a spacewise dependent source of theconstant coefficient parabolic equation ut = kuxx + F (x) are proposed in [13,16,21] (see also references therein). Howeverthese and other numerical algorithms can be implemented either when the thermal conductivity k(x) > 0 is a constant orthe source term of a parabolic equation is a spacewise dependent only.
In this paper we study the inverse source problems of determining the unknown spacewise dependent source F (x) inthe parabolic problem⎧⎪⎨
⎪⎩ut = (
k(x)ux)
x + F (x)H(t), (x, t) ∈ ΩT := (0, l) × (0, T ];u(x,0) = u0(x), x ∈ (0, l);(−k(x)ux(x, t)
)x=0 = g(t), u(l, t) = 0, t ∈ (0, T ),
(1.1)
from the following types of observations. In the first inverse source problem, subsequently defined as ISPF1, the unknownspacewise dependent source F (x) needs to be identified from the measured output data
uT (x) := u(x, T ; F ), x ∈ (0, l), (1.2)
i.e. the supplementary temperature measurement uT (x) given at the final time T > 0.In the second inverse source problem, subsequently defined as ISPF2, the unknown spacewise dependent source F (x)
needs to be identified from the integral of u(x, t) over the time variable t ∈ [0, T ]:
U T (x) :=T∫
0
u(x, t)dt, x ∈ (0, l), (1.3)
i.e. from the time-average temperature observation U T (x). Note that this kind of nonlocal or integral type observations arise,in particular, in mathematical modeling of settling mechanism, where U T (x) means an average density of sludge particlesduring the time [0, T ] (see [7] and references therein).
The source term F (x)H(t) of the parabolic equation is assumed to be multiplicatives separable form, with the unknownspacewise dependent source F (x) and known time-dependent heat source H(t). Source terms of this separable form arisein various physical models. In particular, the equation ut = uxx + F (x)H(t), with H(t) = exp(κt), describes the heat processof radioisotope decay, with the decay rate κ > 0 [29]. These types of source terms appear also as a control term for theheat equation. Inverse source problems for separable form source terms parabolic and hyperbolic equations have first beenstudied in [11], then in [38,39].
We define the weak solution of the direct problem (1.1) as the function u ∈ V 1,0(ΩT ), satisfying the following integralidentity [23]:
∫ ∫ΩT
(−uvt + kux vx)dx dt =∫ ∫ΩT
F (x)H(t)v(x, t)dx dt +T∫
0
g(t)v(0, t)dt, ∀v ∈ H1,1(ΩT ), (1.4)
with v(x, T ) = 0. Here V 1,0(ΩT ) := C([0, T ]; L2[0, l]) ∩ L2((0, T ); H1(0, l)) is the Banach space of functions with the norm
A. Hasanov, B. Pektas / Applied Numerical Mathematics 78 (2014) 49–67 51
Note that the weak solution in V 1,0(ΩT ) requires minimal regularity conditions (1.5) on the input data. This is importantin inverse problems theory and applications, since the initial data u0, as well as the source functions g(t), F (x) and H(t),are of square integrable functions as a noise.
Let us define by u(x, t; F ) ∈ V 1,0(ΩT ) the (unique) solution of the direct problem (1.1), for a given source F ∈ L2[0, l].Then the input–output operators corresponding to the inverse problems ISPF1 and ISPF2 are defined as follows:
Φ F : L2[0, l] �→ L2[0, l], (Φ F )(x) := u(x, t; F )|t=T ;
Ψ F : L2[0, l] �→ L2[0, l], (Ψ F )(x) :=T∫
0
u(x, t; F )dt. (1.7)
Hence the above inverse problems can be reformulated as the following operator equations:
Φ F = uT , F ∈ L2[0, l];Ψ F = U T , F ∈ L2[0, l]. (1.8)
Evidently the operator equations (1.8) cannot be satisfied exactly, due to errors in the measured output data uT and U T .Hence one needs to define least-square solutions (or quasi-solutions) of the problems ISPF1 and ISPF2, according to [36],introducing the cost functionals:
J1(F ) = ‖Φ F − uT ‖2L2[0,l] :=
l∫0
[u(x, T ; F ) − uT (x)
]2dx, F ∈ L2[0, l];
J2(F ) = ‖Ψ F − U T ‖2L2[0,l] :=
l∫0
{ T∫0
u(x, t; F )dt − U T (x)
}2
dx, F ∈ L2[0, l]. (1.9)
Since the direct problem (1.1) is a linear one, we may assume, without loss of generality, that u0(x) = g(t) ≡ 0. Then theinput–output operators defined by (1.7) are linear compact operators acting from L2[0, l] to L2[0, l]. Then we may use mostprominent regularization method for ill-posed problems, Tikhonov regularization [36]:
where α > 0 is the parameter of regularization and J1(F ), J2(F ) are the cost functionals defined by (1.9). Applying nowthe theory developed in [12, Ch. 5] (see also [22, Theorem 2.11]), we conclude that each Tikhonov functional J1α(F ) andJ2α(F ) has a unique minimum Fα(x) ∈ L2[0, l]. This minimum is also the unique solution of each normal equation(
Φ∗Φ + α I)
Fα = Φ∗uT ,(Ψ ∗Ψ + α I
)Fα = Ψ ∗U T , (1.11)
where Φ∗ : L2[0, l] �→ L2[0, l] and Ψ ∗ : L2[0, l] �→ L2[0, l] are corresponding adjoint operators. These unique solutions can bewritten as follows [12,22]
Fα = RαuT ,
Fα = RαU T , (1.12)
where Rα := (Φ∗Φ + α I)−1Φ∗ : L2[0, l] �→ L2[0, l] for ISPF1 and Rα := (Ψ ∗Ψ + α I)−1Ψ ∗ : L2[0, l] �→ L2[0, l] for ISPF2, andthe operator Rα : L2[0, l] �→ L2[0, l], α > 0, is a regularization strategy [22].
In this paper we propose a uniform approach to mathematical and numerical aspects of the inverse source problemsISPF1 and ISPF2 for a variable coefficient parabolic equation, from measured final data and time-average temperature obser-vations. We use well-known Tikhonov regularization method and show that the adjoint problems, corresponding to inverseproblems ISPF1 and ISPF2 can uniquely be derived by the Lagrange multiplier method. This result allows us to obtain repre-sentation formulas for unique solutions of each regularized inverse problem. Fourier method applied to the inverse problemsfor the constant coefficient parabolic equation (i.e. heat equation) shows that the proposed the representation formula is anintegral analogue of well-known Picard’s Singular Value Decomposition for compact operators.
The paper is organized as follows. In Section 2 well-known Tikhonov regularization is used to show that the adjointproblems, corresponding to inverse problems ISPF1 and ISPF2 can uniquely be derived by the Lagrange multiplier method.This allows us to derive representation formulas for each inverse problems. In Section 3 the case of constant heat conductioncoefficient is studied. Fourier method is used to show that in this case the representation formulas are equivalent to the
52 A. Hasanov, B. Pektas / Applied Numerical Mathematics 78 (2014) 49–67
Singular Value Decomposition (SVD) for the regularized solutions of the problems ISPF1 and ISPF2. Conjugate GradientAlgorithm (CGA), with the initial iteration defined by the representation formulas, applied to considered inverse problems,is given in Section 4. An estimation of the computational noise level εu > 0 corresponding to noise free and random noisyoutput data in typical classes of source terms is also discussed in this section. Comparative analysis of CGA and CollocationMethod is presented in Section 5. Some conclusions and recommendations for use of the algorithm applied to ISPF1 andISPF2, are presented in the Section 6.
2. Representation formulas for unique regularized solutions of the inverse problems ISPF1 and ISPF2
We use an idea of Lagrange multiplier in the theory of optimal control, given in [24,25], to establish a relationshipbetween the solution Fα of the normal equations (1.11) and saddle point of a regularized Lagrangian. For this aim we firstformally define the functional
L1α(F ;λ) = J1α(F ) + ⟨λ, ut − (
k(x)ux)
x − F H⟩L2(ΩT )
,
where the functional J1α(F ) is defined by (1.9) and (1.10). We assume that u0 ∈ H1[0, l] and g(t) ∈ H1[0, T ]. Then the weaksolution of (1.1) belongs to H1,1(ΩT ). Transforming the above right hand side integral we have:
L1α(F ;λ) = J1α(F ) +∫ ∫ΩT
[λut + k(x)λxux − λF (x)H(t)
]dx dt −
T∫0
g(t)λ(0, t)dt. (2.1)
The functional L1α(F ;λ), defined by (2.1), is called the regularized Lagrangian corresponding to ISPF1. Consequently, thefunction λ(x, t) ∈ V 1,0(ΩT ) is the Lagrange function (or multiplier).
Consider the dual (optimization) problem:
L1α(Fα;ψ) = supλ∈V 1,0(ΩT )
infF∈L2[0,l]
L1α(F ;λ). (2.2)
Theorem 2.1. Let, in addition to conditions (1.5), the conditions u0 ∈ H1[0, l] and g(t) ∈ H1[0, T ] hold. Then the pair 〈Fα;ψ〉 ∈L2[0, l] × V 1,0(ΩT ) is the saddle point of the Lagrangian (2.1) if and only if the pair 〈u,ψ〉 ∈ H1,1(ΩT ) × V 1,0(ΩT ) is the solution ofthe coupled system of parabolic problems⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩
ut = (k(x)ux
)x −
( T∫0
ψ(x, t)H(t)dt
)H(t)/(2α), α > 0, (x, t) ∈ ΩT ;
u(x,0) = u0(x), x ∈ (0, l);−k(0)ux(0, t) = g(t), u(l, t) = 0, t ∈ (0, T ];⎧⎪⎨
⎪⎩ψt = −(
k(x)ψx)
x, (x, t) ∈ ΩT ;ψ(x, T ) = p(x), x ∈ (0, l);ψx(0, t) = 0, ψ(l, t) = 0, t ∈ [0, T ),
(2.3)
with the input
p(x) = −2[u(x, T ) − uT (x)
]. (2.4)
Proof. Assuming that F , F + δF ∈ L2[0, l] are given arbitrary sources, we may calculate the increment δL1α(δF ;λ) :=L1α(F + δF ;λ) −L1α(F ;λ) of the Lagrangian (2.1):
δL1α(δF ;λ) = δ J1α(F ) +T∫
0
l∫0
[λδut + k(x)δuxλx − λδF (x)H(t)
]dx dt, (2.5)
where δu(x, t; δF ) := u(x, t; F + δF ) − u(x, t; F ) is the solution of the parabolic problem⎧⎪⎨⎪⎩
δut = (k(x)δux
)x + δF (x)H(t), (x, t) ∈ ΩT ;
δu(x,0) = 0, x ∈ (0, l);δux(0, t) = 0, δu(l, t) = 0, t ∈ (0, T ].
(2.6)
By (1.9) and (1.10) we conclude:
A. Hasanov, B. Pektas / Applied Numerical Mathematics 78 (2014) 49–67 53
δ J1α(F ) =l∫
0
{2[u(x, T ; F ) − uT (x)
]}δu(x, T ; δF )dx + 2α
l∫0
F (x)δF (x)dx
+ ∥∥δu(·, T ; δF )∥∥2
L2[0,l] + ‖δF‖2L2[0,l]. (2.7)
With (2.5) this yields:
δL1α(δFα;λ) =T∫
0
l∫0
[λδut + k(x)δuxλx
]dx dt +
l∫0
{2αF (x) −
T∫0
λ(x, t)H(t)dt
}δF (x)dx
+l∫
0
{2[u(x, T ; F ) − uT (x)
]}δu(x, T ; δF )dx + ∥∥δu(·, T ; δF )
∥∥2L2[0,l] + ‖δF‖2
L2[0,l].
Applying the integration by parts formula to the first right-hand side integral we obtain:
δL1α(δFα;λ) = −T∫
0
l∫0
[λt + (
k(x)λx)
x
]δu dx dt
+l∫
0
{2[u(x, T ; F ) − uT (x)
] + (λ(x, t)
)t=T
}δu(x, T ; δF )dx −
l∫0
(λδu)t=0 dx +T∫
0
(k(x)λxδu
)x=lx=0 dt
+l∫
0
{2αF (x) −
T∫0
λ(x, t)H(t)dt
}δF (x)dx + ∥∥δu(·, T ; δF )
∥∥2L2[0,l] + ‖δF‖2
L2[0,l]. (2.8)
Let now the pair 〈Fα;ψ〉 ∈ L2[0, l]× V 1,0(ΩT ) be the saddle point of the Lagrangian (2.1). Then, δL1α(δFα;ψ) = 0. By thearbitrarity of δu ∈ H1,1(ΩT ) and the homogeneous initial and boundary conditions in (2.6) we conclude that the functionψ(x, t) = λ(x, t) is the weak solution of the backward parabolic problem (2.3) and u ∈ H1,1(ΩT ) is the weak solution of theparabolic equation in (2.3) with the spacewise source term
Fα(x) = 1
2α
T∫0
ψ(x, t; Fα)H(t)dt, a.e. x ∈ (0, l). (2.9)
Conversely, if the pair 〈u,ψ〉 ∈ H1,1(ΩT ) × V 1,0(ΩT ) is the solution of the coupled system of parabolic problems (2.3),then all integrals on the right-hand side of (2.8) are zero. Hence the pair 〈Fα;ψ〉 ∈ L2[0, l] × V 1,0(ΩT ), with Fα(x) definedby (2.9), is the solution of the dual problem (2.2). �Corollary 2.1. Let 〈u,ψ〉 ∈ H1,1(ΩT ) × V 1,0(ΩT ) be the unique solution of the coupled system of parabolic problems (2.3), with theinput data p ∈ L2[0, l] given by (2.4). Then for the unique regularized solution Fα ∈ L2[0, l] of ISPF1 the representation formula (2.9)holds.
The assertion of Theorem 2.1, with the input
p(x) = −2
[ T∫0
u(x, t)dt − U T (x)
], x ∈ (0, l), (2.10)
in (2.3), and the same formula (2.9) for the unique regularized solution, remain true for ISPF2.
Corollary 2.2. Let 〈u,ψ〉 ∈ H1,1(ΩT ) × V 1,0(ΩT ) be the unique solution of the coupled system of parabolic problems (2.3), with theinput data p ∈ L2[0, l] given by (2.10). Then for the unique regularized solution Fα ∈ L2[0, l] of ISPF2 the representation formula (2.9)holds.
The above results show that both regularized inverse source problems ISPF1 and ISPF2 can be interpreted as the coupledsystem of parabolic problems (2.3). Each solution 〈u,ψ〉 ∈ H1,1(ΩT ) × V 1,0(ΩT ) of this system, corresponding to the inputdata p(x), given by (2.4) or (2.10), uniquely determines the solution of the normal equation (1.11), respectively. Moreover, for
54 A. Hasanov, B. Pektas / Applied Numerical Mathematics 78 (2014) 49–67
both inverse problems the same representation formula (2.9), but with different input data p(x) in the adjoint problem (2.3),holds.
Note that the similar coupled system of two parabolic (direct and adjoint) problems has also been used in [26] to obtainan explicit formula for the quasi-solution of the Cauchy problem for Laplace’s equation.
Lemma 2.1. Let conditions of Theorem 2.1 hold. Then the Tikhonov functionals (1.10) are Fréchet differentiable, and
J ′kα(F )(x) = 2αF (x) −
T∫0
ψ(x, t; F )H(t)dt, x ∈ [0, l], k = 1,2, (2.11)
where the cases k = 1 and k = 2 correspond to the inputs (2.4) and (2.10), respectively.
Proof. Assuming that ψ ∈ V 1,0(ΩT ) is the solution of the adjoint problem (2.3), with the input (2.4), we transform the firstintegral on the right-hand side of (2.7) as follows:
2
l∫0
[u(x, T ; F ) − uT (x)
]δu(x, T ; δF )dx = −
l∫0
ψ(x, T ; F )δu(x, T ; δF )dx = −l∫
0
[ T∫0
(ψδu)t dt
]dx
=∫ ∫ΩT
[(k(x)ψx
)xδu − ψ
(k(x)δux
)x
]dx dt −
∫ ∫ΩT
ψ(x, t; F )δF (x)H(t)dx dt.
Applying now the integration by parts formula we get
2
l∫0
[u(x, T ; F ) − uT (x)
]δu(x, T ; δF )dx
=T∫
0
[k(x)ψx(x, t; F )δu(x, t; δF ) − ψ(x, t; F )k(x)δux(x, t; δF )
]x=lx=0 dt −
∫ ∫ΩT
ψ(x, t; F )δF (x)H(t)dx dt.
Taking into account the boundary conditions in (2.6) and in the adjoint problem (2.3) we obtain the following integralidentity:
2
l∫0
[u(x, T ; F ) − uT (x)
]δu(x, T ; δF )dx = −
∫ ∫ΩT
ψ(x, t; F )δF (x)H(t)dx dt, ∀F ∈ L2[0, l].
Substituting this in (2.7) and taking into account that the term ‖δu(·, T ; δF )‖L2[0,l] is of order ‖δF‖L2[0,l] we obtain formula(2.11) for the Fréchet differential of the cost functional J1α(F ). �
Assuming now that ψ ∈ V 1,0(ΩT ) is the solution of the adjoint problem (2.3), with the second input (2.10), in exactlythe similar way we can prove that the same formula (2.11) holds for the Fréchet differential of the cost functional J2α(F ).
3. The relationship between the representation formula and singular value decomposition
Consider now the constant coefficient case k(x) ≡ k = const > 0, and we assume, without loss of generality, that theinitial data and the left flux are zero: u0(x) = g(t) = 0. We will show that in this case the representation formula (2.9) isequivalent to the singular value decomposition of the regularized input–output operators corresponding to ISPF1 and ISPF2.
Theorem 3.1. Let k(x) = k = const > 0, H ∈ L2[0, T ] and u0(x) = g(t) = 0. Assume that the pair 〈u,ψ〉 ∈ V 1,0(ΩT ) × V 1,0(ΩT )
is the solution of the coupled problem (2.3) for a given α > 0 and p(x), given by (2.4). Then the solution Fα ∈ L2(0, l) of the normalequation (1.11) for ISPF1 is defined by formula (2.9) if and only if
Fα(x) =∞∑
n=0
q(α;σn)
σnuT ,nϕn(x), α � 0, (3.1)
where
q(α;σ) = σ 2
2(3.2)
σ + α
A. Hasanov, B. Pektas / Applied Numerical Mathematics 78 (2014) 49–67 55
is the filter function, and
σn =T∫
0
exp(−kλ2
n(T − t))
H(t)dt, n = 0,1,2, . . . , (3.3)
are the singular values of the input–output operator Φ , uT ,n := (uT ,ϕn)L2[0,l] is the nth Fourier coefficient of the measured output
data uT (x) and ϕn(x) := √2/l cos(λnx) are eigenfunctions corresponding to the eigenvalues λ2
n = [(n + 1/2)π/l]2 , n = 0,1,2, . . . .
Proof. Let us apply Fourier method to each parabolic problem in the coupled system (2.3), with p(x) given by (2.4). Applyingthis method to the direct problem (1.1), with the source term Fα ∈ L2[0, l] given by (2.9), we have:
u(x, t; Fα) =∞∑
n=0
un(t; Fα)ϕn(x), un(t; Fα) = Fα,n
t∫0
exp(−kλ2
n(t − τ ))
H(τ )dτ . (3.4)
Substituting in t = T integral (3.4) we find Fourier coefficients of the output data u(x, t; Fα) at t = T :
un(T ; Fα) = σn Fα,n, n = 0,1,2, . . . , (3.5)
where σn is defined by (3.3).Applying now Fourier method to the adjoint problem in (2.3), we have:
ψ(x, t; Fα) =∞∑
n=0
ψn(t; Fα)ϕn(x), (3.6)
where the function ψn(t; Fα) is the solution of the (backward) Cauchy problem{ψ ′
n(t) = kλ2nψn(t), t ∈ [0, T ),
ψn(T ) = −2[un(T ; Fα) − uT ,n
].
The solution of this problem is
ψn(t; Fα) = −2[un(T ; Fα) − uT ,n
]exp
(−kλ2n(T − t)
). (3.7)
To prove the first part of the theorem, assume that the solution Fα ∈ L2(0, l) of the normal equation (1.11) for ISPF1 isdefined by formula (2.9), via the unique solution 〈u,ψ〉 ∈ V 1,0(ΩT ) × V 1,0(ΩT ) of the coupled problem (2.3). Then for thenth Fourier coefficient of the function Fα(x) this formula implies:
Fα,n = 1
2α
T∫0
ψn(t; Fα)H(t)dt, n = 0,1,2, . . . . (3.8)
Multiplying both sides of (3.7) by H(t)/(2α) �= 0, integrating on [0, T ] and then using the definition (3.3) of the singularvalue σn , we obtain:
1
2α
T∫0
ψn(t; Fα)H(t)dt = −σn
α
[un(T ; Fα) − uT ,n
].
This, with (3.8), implies:
Fα,n = −σn
α
[un(T ; Fα) − uT ,n
].
Using on the right hand side formula (3.5) for un(T ; Fα) we find the nth Fourier coefficient of the solution Fα ∈ L2(0, l), viathe nth Fourier coefficient uT ,n of the measured output data and the filter function (3.2):
Fα,n = q(α;σn)
σnuT ,n. (3.9)
This implies (3.1).To prove the second part of the theorem, assume that the solution Fα ∈ L2(0, l) of the normal equation (1.11) for ISPF1 is
defined by formula (3.1), with σn defined by (3.3). Then for the nth Fourier coefficient of this solution formula (3.9) holds.On the other hand, for the nth Fourier coefficient un(T ; Fα) of the output data u(x, T ; Fα), formula (3.5) holds, since the
56 A. Hasanov, B. Pektas / Applied Numerical Mathematics 78 (2014) 49–67
pair 〈u,ψ〉 ∈ V 1,0(ΩT ) × V 1,0(ΩT ) is the solution of the coupled problem (2.3). Eliminating Fα,n from formulas (3.5) and(3.9), we find:
Taking into account formula (3.3) for the singular values σn we obtain the formula
Fα,n = 1
2α
{−2
[un(T ; Fα) − uT ,n
] T∫0
exp(−kλ2
n(T − t))
H(t)dt
}, n = 0,1,2, . . . ,
which right hand side is the right hand side of (3.8), due to formula (3.7). This completes the proof of the theorem. �Corollary 3.1. Let conditions of Theorem 3.1 hold. Then the nth Fourier coefficients un(T ; Fα) and uT ,n of the output data u(x, T ; Fα),corresponding to ISPF1, and the measured output data uT (x), respectively, are related via the filter function q(α;σ) by formula (3.10).
In particular, it follows from formula (3.10) that, if the parameter of regularization is zero, this result implies:u(x, T ; Fα) = uT (x), for all x ∈ [0, l].
The same results remain true for the unique regularized solution of ISPF2.
Theorem 3.2. Let k(x) = k = const > 0, H ∈ L2[0, T ] and u0(x) = g(t) = 0. Assume that the pair 〈u,ψ〉 ∈ V 1,0(ΩT ) × V 1,0(ΩT )
is the solution of the coupled problem (2.3) for a given α > 0 and p(x), given by (2.10). Then the solution Fα ∈ L2(0, l) of the normalequation (1.11) for ISPF2 is defined by formula (2.9) if and only if
Fα(x) =∞∑
n=0
σn
σnSn + αU T ,nϕn(x), α � 0, (3.11)
where σn := σn(T ),
σn(t) =t∫
0
exp(−kλ2
n(t − τ ))
H(τ )dτ , Sn :=T∫
0
σn(t)dt, n = 0,1,2, . . . , (3.12)
and U T ,n := (U T ,ϕn)L2[0,l] is the nth Fourier coefficient of the measured output data U T (x).
Proof. Since the proof scheme is almost the same with the proof of Theorem 3.1, we will give only outlines. Assuming thatp(x) is given by (2.10) and integrating un(t; Fα) given by (3.4) we obtain the Fourier coefficients
Un(Fα) :=T∫
0
un(t; Fα)dt = Fα,n
T∫0
t∫0
exp(−kλ2
n(t − τ ))
H(τ )dτ dt, n = 0,1,2, . . . , (3.13)
of the output data
U (x; Fα) :=T∫
0
u(x, t; Fα)dt,
corresponding to Fα(x). Then by (3.12) we conclude:
Un(Fα) = Fα,nSn, n = 0,1,2, . . . . (3.14)
A. Hasanov, B. Pektas / Applied Numerical Mathematics 78 (2014) 49–67 57
Now, the Fourier coefficients ψn(t; Fα) of the solution (3.6) of the adjoint problem is defined as the solution of theCauchy problem⎧⎪⎪⎪⎨
⎪⎪⎪⎩ψ ′
n(t) = kλ2nψn(t), t ∈ [0, T ),
ψn(T ) = −2
[ T∫0
un(t; Fα)dt − U T ,n
].
The solution of this problem, for each n = 0,1,2, . . . , is the function
ψn(t; Fα) = −2
[ T∫0
un(t; Fα)dt − U T ,n
]exp
(−kλ2n(T − t)
).
Multiply both sides by H(t)/(2α) �= 0, integrate on [0, T ] and then use formula (3.8). Then we have:
Fα,n = − 1
α
[Un(Fα) − U T ,n
]σn(T ), n = 0,1,2, . . . . (3.15)
Eliminating Un(Fα) from formulas (3.14) and (3.15) we obtain:
Fα,n = σn(T )
σn(T )Sn + αU T ,n, n = 0,1,2, . . . . (3.16)
This implies (3.1).The second part of theorem can be proved in the same way. �Let us analyze now the relationship between the nth Fourier coefficients un(T ; Fα) and uT ,n of the output data
U (x, T ; Fα) and the measured output data U T (x) in the problem ISPF2. Formulas (3.14) and (3.16) imply:
Un(T ; Fα) = σn(T )Sn
σn(T )Sn + αU T ,n, n = 0,1,2, . . . . (3.17)
We introduce the filter function
q(α;μ) = μ2
μ2 + α, μn := √
σn(T )Sn, n = 0,1,2, . . . , (3.18)
corresponding to ISPF2.
Corollary 3.2. Let conditions of Theorem 3.2 hold. Then the nth Fourier coefficients un(T ; Fα) and uT ,n of the output data U (x, T ; Fα),corresponding to ISPF2, and the measured output data U T (x), respectively, are related via the filter function (3.18) by the formula
Un(T ; Fα) = q(α;μn)U T ,n.
For α = 0 formula (3.11) implies the classical Picard’s representation formula:
Fα(x) =∞∑
n=0
1
SnU T ,nϕn(x), α � 0.
Hence the parameters Sn , n = 0,1,2, . . . , given by (3.12), are defined to be the singular values of the input–output operator Ψ ,corresponding to ISPF2.
Remark 3.1. The procedure given in the proof of Theorem 3.1 is factually the Fourier method for the solution of the coupledproblem (2.3).
4. Estimates for singular values of the input–output operators
In this section we will estimate singular values corresponding to ISPF1 and ISPF2, not only to establish the degree ofill-posedness, but also to illustrate similarities and differences between these inverse problems. An estimation of the singularvalues σn , corresponding to ISPF1, is similar to one for backward parabolic problems which is well-known in literature (see,for example, [12]). Formula (3.3) implies:
|σn| �( T∫
0
exp(−2kλ2
n(T − t))
dt
)1/2
‖H‖L2[0,T ] = 1√2kλn
[1 − exp
(−2kλ2n T
)]1/2‖H‖L2[0,T ]. (4.1)
Since λn � n, estimate (4.1) shows that σn =O(n−1), i.e. ISPF1 is moderately (or modestly) ill-posed.
58 A. Hasanov, B. Pektas / Applied Numerical Mathematics 78 (2014) 49–67
Fig. 1. Singular values of the input–output operators corresponding to ISPF1 (left figure) and ISPF2 (right figure).
However, the case when the time dependent source term H(t) vanishes in some neighborhood of the final time T > 0needs to be considered separately. Let
H(t)
{= 0, t ∈ (T − ε, T );�= 0, t ∈ (0, T − ε), T > ε > 0.
(4.2)
Then by formula (3.3) we have:
|σn| � 1√2kλn
[exp
(−2kλ2nε
) − exp(−2kλ2
n T)]1/2‖H‖L2(0,T −ε), ε > 0, n = 0,1,2, . . . .
This shows that if H(t) = 0, ∀t ∈ (T − ε, T ), ε > 0, then σn =O(exp(−n2)) and hence ISPF1 is severely ill-posed.Let us estimate now the singular values Sn , n = 0,1,2, . . . , corresponding to ISPF2. Estimating the integrals given by
(3.12) we have:
|Sn| :=∣∣∣∣∣
T∫0
t∫0
exp(−kλ2
n(t − τ ))
H(τ )dτ dt
∣∣∣∣∣ � T√2kλn
[1 − exp
(−2kλ2n T
)]1/2‖H‖L2[0,T ]. (4.3)
With (4.1) this implies |Sn| = O(n−1), n = 0,1,2, . . . . This shows ISPF2 is also moderately ill-posed, and has the same degree ofill-posedness as ISPF1.
The singular values, of the input–output operators (1.7), defined by (3.3) and (3.12) and corresponding to ISPF1 and ISPF2,are plotted in Fig. 1. All these curves are L-curves that are concave when plotted in log–log scale. The flat parts in figuresare due to logarithmic scaling on xy coordinates.
5. The Galerkin finite element method for numerical solution of the forward problem
In all numerical examples below the synthetic output data uT ,h and U T ,h , defined by (1.2) and (1.3), are generated fromthe numerical solution uh(x, t) of the forward problem. Hence one needs to estimate the computational noise level
εu = ∥∥u(·, T ) − uh(·, T )∥∥
L2h[0,l], (5.1)
corresponding to continuous and discontinuous thermal conductivity coefficients, as well as sources. Here ‖ · ‖L2h
is the
discrete analogue of the L2-norm ‖ · ‖L2 . For this aim, we will apply Galerkin finite element method (FEM) for discretizationthe direct problem, using the piecewise-linear polynomial basis functions
ξi(x) =⎧⎨⎩
(x − xi−1)/h, x ∈ (xi−1, xi];(xi+1 − x)/h, x ∈ (xi, xi+1]; ξ1(x) =
{(x2 − x)/h, x ∈ (x1, x2];0, x /∈ [x1, x2].
0, x /∈ [xi−1, xi+1], i = 2, Nx;
A. Hasanov, B. Pektas / Applied Numerical Mathematics 78 (2014) 49–67 59
Here ωh := {xi ∈ [0, l): xi = (i −1)h, i = 1, Nx} is the uniform space mesh, with mesh parameter h = l/Nx . Then the function
uh(x, t) =Nx∑
i=1
ci(t)ξi(x)
is the piecewise-linear approximation of u(x, t) in the finite-dimensional subspace H1h(0, l) ⊂ H1(0, l) spanned by the set of
basic functions {ξi(x)}. Here ci(t) := uh(xi, t), i = 1, Nx , are the time dependent unknown parameters. For convenience wetreat here t ∈ (0, T ] as a parameter and the function u(x, t) as a mapping u : [0, T ] �→ H1
h(0, l), defined as u(t)(x) := u(x, t),x ∈ (0, l), t ∈ (0, T ].
The Galerkin finite element problem [35] is to determine uh(x, t) ∈ H1h(0, l) such that{ 〈∂t uh, ξi〉 + a(uh, ξi) = H(t)
⟨F (x), ξi
⟩, t ∈ (0, T ];
〈u0,h, ξi〉 := 〈u0, ξi〉, t = 0, i = 1, Nx,(5.2)
where⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
〈∂t uh, ξi〉 :=xi∫
xi−1
ut(x, t)ξi(x)dx +xi+1∫xi
ut(x, t)ξi(x)dx,
a(uh, ξi) := 1
h2
xi∫xi−1
k(x)(ci(t) − ci−1(t)
)dx − 1
h2
xi+1∫xi
k(x)(ci+1(t) − ci(t)
)dx,
⟨F (x), ξi
⟩ := 1
h
xi∫xi−1
F (x)(ci(t) − ci−1(t)
)dx + 1
h
xi+1∫xi
F (x)(ci+1(t) − ci(t)
)dx, i = 1, Nx.
(5.3)
By the standard FEM procedure, we then obtain from semi-discrete Galerkin discretization (5.2) the full discretizationwhich involves further discretization in time:⎧⎪⎨
⎪⎩⟨
u j+1h − u j
h
τ, ξi
⟩+ a
((u j+1
h + u jh
)/2, ξi
) = ((H(t j+1) + H(t j)
)/2
)⟨F (x), ξi
⟩, t ∈ (0, T ];
〈u0,h, ξi〉 := 〈u0, ξi〉, t = 0, i = 1, Nx.
(5.4)
Here ωT := {t j ∈ (0, T ]: t j = jτ , j = 1, Nt} is the uniform space mesh, with mesh parameter τ = T /Nt .The FEM scheme (fem 2) is used in the numerical simulation of the forward as well as corresponding backward problems
in subsequent computational experiments.To estimate the computational noise level
εu = ∥∥u(·, T ) − uh(·, T )∥∥
L2h[0,l], (5.5)
corresponding to the continuous and discontinuous conductivity coefficient cases, we consider the following examples. Inthe computational experiments here and below it is assumed that l = 1, T = 1.
Example 5.1 (Estimation of the computational noise levels corresponding to continuous and discontinuous thermal conductivities).For the given smooth spacewise source term the function
F (x) = sin(πx)(
A(x) + π2 B(x)) + π cos(πx)C(x), x ∈ [0,1],
with A(x) = −201x4 + 242x3 − 85x2 + 24x − 4, B(x) = 10x6 − 20x5 + 12x4 − 4x3 + 2x2 and C(x) = −100x5 + 160x4 − 76x3 +24x2 − 8x, the analytical solution of the direct problem (1.1) is the function
u(x, t) = exp(−0.5t)x2(x − 1)2 sin(πx), x ∈ [0,1], t ∈ [0,1].Other input data are H(t) = 0.5 exp(−0.5t), t ∈ [0,1], k(x) = 1 + 5x2 and u0(x) = x2(x − 1)2 sin(πx), x ∈ [0,1].
Then the exact output data uT (x) := u(x,1), is the function
uT (x) := u(x,1) = exp(−0.5)x2(x − 1)2 sin(πx), x ∈ [0,1].The computational noise level εu , corresponding to the continuous thermal conductivity k(x) = 1+5x2, is given in the fourthcolumn of Table 1.
60 A. Hasanov, B. Pektas / Applied Numerical Mathematics 78 (2014) 49–67
Table 1Computational noise levels for continuous and discontinuous thermal conductivities.
Nx × Nt h τ Continuous k(x) Discontinuous k(x)
εu εu
101 × 51 0.01 0.02 1.03 × 10−5 3.16 × 10−3
151 × 51 0.0067 0.02 8.62 × 10−6 3.06 × 10−3
151 × 101 0.0067 0.01 8.54 × 10−6 3.02 × 10−3
201 × 101 0.005 0.01 4.26 × 10−6 2.56 × 10−3
Assume now that the thermal conductivity k(x) is the discontinuous (piecewise constant) function:
k(x) ={
k1, x ∈ [0, ξ),
k2, x ∈ (ξ,1],where k1 = 0.9, k2 = 1.0, ξ = 0.5. The same function u(x, t) = exp(−0.5t)x2(x − 1)2 sin(πx), x ∈ [0,1], t ∈ [0,1] above isthe exact solution of the direct problem (1.1) with appropriate discontinuous spacewise source term F (x), and other inputdata. The values of the parameter εu > 0 are of order 10−3, as the fifth column of Table 1 shows. In all cases the values ofthe parameter εu > 0 are of order 10−3, as the fifth column of Table 1 shows.
The minimal mesh size, when εu > 0 is of order 10−3 for the discontinuous thermal conductivity case, is Nx × Nt =201 × 101. For this reason, this mesh is defined as an optimal mesh, and will be used in subsequent inversion algorithms innumerical solutions of the direct and adjoint problems. �
Further computational experiments show that in the continuous as well as discontinuous thermal conductivities andspacewise source cases, the values of the parameter εu > 0, defined by (5.5), satisfies the condition: εu � Mu10−3,Mu ∈ (0,1). Since the synthetic data uT (x) := uh(x, T ), generated from the finite-element solution uh(x, t) of the directproblem (1.1), is used as a measured output data, this means that this synthetic data can be assumed to be a noise freemeasured output data.
6. The Conjugate Gradient Algorithm with optimal initial guess
In this section we will illustrate performance characteristics of the Conjugate Gradient Algorithm (CGA) applied to ISPF1and ISPF2. Note that the proof scheme of Theorem 3.1 and Theorem 3.2 shows that, in the constant coefficient case(k(x) ≡ k > 0) the Fourier method of solving the coupled problem (2.3) can also be considered as an alternative methodfor approximate solutions of the problems ISPF1 and ISPF2. In the case of variable coefficient k(x), we will use this approachfor obtaining an optimal initial iteration for CGA.
6.1. Algorithm of CGA with optimal initial guess and stopping parameters
Here we use the version of the CGA given in [22, Ch. 2.7]. The initial iteration F (0)(x) here is determined by the Fouriermethod, assuming in (1.1) k := k(l/2).
Step 1. Set n = 0, find the initial iteration F (n)α (x) by using (3.1), assuming k := k(l/2).
Step 2. Compute the discrete analogue of the gradient J ′α(F (n)
α ). Calculate the descent direction
p(n)(x) := J ′α
(F (n)α
)(x).
Step 3. Find the descent direction parameter β(n) > 0 from
β(n) = 〈Φ F (n)α − uγ
T ,Φp(n)〉 + α〈F (n)α , p(n)〉
‖Φp(n)‖2L2(0,l) + α‖p(n)‖2
L2(0,l)
≡ 1
2
〈 J ′α(F (n)
α ), p(n)〉 + α〈F (n)α , p(n)〉
‖Φp(n)‖2L2(0,l) + α‖p(n)‖2
L2(0,l)
.
Step 4. Find F (n+1)α (x) = F (n)
α (x) − β(n) p(n)(x) and compute the convergence error
e(n;α;γ ) := ∥∥u(·, T ; F (n)
α
) − uγT
∥∥L2[0,l].
Step 5. If
e(n;α;γ ) < ε J , (6.1)
then go to Step 7; otherwise, set n := n + 1.
A. Hasanov, B. Pektas / Applied Numerical Mathematics 78 (2014) 49–67 61
Step 6. Compute the new descent direction
p(n)(x) := 1
2J ′α
(F (n)α
)(x) + ‖ J ′
α(F (n)α )‖2
L2(0,l)
‖ J ′α(F (n−1)
α )‖2L2(0,l)
p(n−1)(x)
and go to Step 3.
Step 7. Stop the process and plot approximation F (n)α (x).
The number n of iterations for which inequality (6.1) holds, will be defined below as the number of iterations nCGAcorresponding to the stopping parameter ε J > 0 of CGA.
Remark that the same algorithm of CGA, with the operator Φ replaced by Ψ , is used for ISPF2.To study the performance characteristics of CGA we need to introduce also the accuracy error E(n;α;γ ), defined as
E(n;α;γ ) := ‖F − F (n)α ‖L2[0,l] , where n is the iteration number of CGA.
In all numerical experiments below, the synthetic noise free output data uT (x) and U T (x) are generated from the nu-merical solution of the direct problem (1.1) for a given source F (x). Then the noisy output data
uγT (x) = uT (x) + γ ‖uT ‖∞ randn(x),
UγT (x) = U T (x) + γ ‖U T ‖∞ randn(x) (6.2)
are obtained for ISPF1 and ISPF2, respectively, using the MATLAB “randn” function, which generates arrays of random num-bers whose elements are normally distributed with mean 0 and standard deviation σ = 1. Here γ > 0 is the noise level.The composite trapezoidal rule is used for approximating the integral in (1.3).
The value of the stopping parameter ε J > 0 in (6.1) can be defined based on an analysis of L-curve behavior of theconvergence error e(n;α;γ ). Another approach is the Morozov’s discrepancy principle. We explain this approach shortly forISPF1. Rewriting (6.2) as uγ
T (x) = uT (x) + δuγT (x) and assuming that F (x) is the exact solution of ISPF1, i.e. Φ F = uT , we
have ∥∥Φ F − uγT
∥∥L2[0,l] = ∥∥Φ F − uT − δuγ
T
∥∥L2[0,l] = ∥∥δuγ
T
∥∥L2[0,l].
Hence the residue, when the exact right hand side of the operator equation Φ F = uT is replaced by noisy uγT one, is
‖Φ F − uγT ‖L2[0,l] = ‖δuγ
T ‖L2[0,l] . The Morozov’s discrepancy principle asserts that the parameter of regularization α > 0
needs to be determined such that for the noisy solution F γ (x) this residue will be closest to ‖δuγT ‖L2[0,l]:∥∥Φ F γ − uγ
T
∥∥L2[0,l] = θ
∥∥δuγT
∥∥L2[0,l], θ > 1.
6.2. Reconstruction of an unknown spacewise dependent source by the Fourier method: the constant coefficient case
As a first step we consider the constant coefficient case (k(x) = 1) and apply the Fourier method, using the partial sum
F Nα (x) =
N∑n=0
q(α;σn)
σnuT ,nϕn(x), α � 0, (6.3)
of the series (3.1).
Example 6.1 (Reconstruction of a smooth spacewise dependent source by Fourier method). Assuming k(x) = 1 and the spacewisesource term as Gaussian normal distribution given by
F (x) = 1
σ√
2πexp
(− (x − μ)2
2σ 2
), x ∈ [0,1], (6.4)
with the standard deviation μ = 0.5 and mean σ = 0.1, we generate the noise free synthetic output data uT (x) := uh(x, T )
from the numerical solution of the direct problem (1.1), with the following data H(t) = 5 exp(−0.5t), u0(x) = 0.The left Fig. 2 shows that in the noise free output data case, the reconstructed by (6.3) spacewise source term F N
α (x) isalmost the same with the exact solution, i.e. with the function (6.4) (solid line). Here E(N) := ‖F N − F‖L2[0,l] = 1.2 × 10−3.Here the parameter N in (6.3) is taken to be N = 20. Remark that the values N = 18–25 are optimal ones in sense ofaccuracy, and the performance of the Fourier method deteriorates with an increasing values of N .
In the second stage of this example the Fourier method is applied to the same ISPF1, by using the noisy data (6.2), withthe noise level γ = 5%. Dashed and dotted lines in right Fig. 2 correspond to the reconstructed spacewise source Fα(x)without (α = 0) and with regularization (α = 2 × 10−5), respectively. In both cases the parameter N in (6.3) is taken to beN = 15. Note that the dashed line shows the sensitivity of the Fourier method with respect to the noisy data uγ
T (x), and thedotted line shows the role of the parameter of regularization α > 0, when γ �= 0. �
62 A. Hasanov, B. Pektas / Applied Numerical Mathematics 78 (2014) 49–67
Fig. 2. The reconstructed Gaussian by the Fourier method: from noise free output data (left figure) and noisy output data with noise level γ = 5% (rightfigure).
Table 2Optimal values of the parameter of regularization α > 0 depending on noise levels.
The value α = 2 × 10−5 used in the above example is of the optimal values of the parameter of regularization in senseof accuracy and stability. Note that this value can be estimated theoretically using the fundamental results given in [12](see also [22]). Another way is the spectral cutoff approach. Specifically, the representation (6.3) of the approximate solutionF Nα (x) reveals the regularizing property of adding the term α‖F‖2 to the cost functionals. Using the property
q(α;σn) ={
0, 0 ≈ σ 2n � α,
1, σ 2n � α,
(6.5)
of the filter function (3.2) we conclude that adding the term F Nα (x) acts as a filter. As result of property (6.5) we obtain:
q(α;σn)
σnδuγ
T ,n ={
0, 0 ≈ σ 2n � α,
1σn
δuγT ,n, σ 2
n � α.(6.6)
This shows that contributions from singular values which are large relative to the parameter of regularization (σ 2n � α)
are left almost unchanged whereas contributions from small singular values (σ 2n � α) are almost eliminated. This suggests
to choose the parameter of regularization α > 0 sufficiently large to ensure that errors δuγT ,n in the output data are not
magnified by small singular values.It is well-known that there are several methods (Discrepancy Principle, Generalized Cross Validation, L-curve) for the
selection of the parameter of regularization α > 0 in Tikhonov functional. In our computational experiments we used theL-curve criterion [14] to determine a reasonable choice for the regularization parameter. For the considered inverse problemsthe optimal values of the parameter of regularization α > 0 for the typical noise levels γ > 0 are given in Table 2. Withinthese optimal values, the values N = 10–20 of the parameter N in (6.3) are used in the computational experiments.
To compare the Fourier method with CGA in the constant coefficient case (k(x) = 1), the above example is solved also byCGA, for noise free as well as noisy output data. In the noise free data case, the obtained reconstructions were almost thesame (left Fig. 2). The accuracy error corresponding to both algorithms was E(nCGA;α;γ ) ≈ ×10−3.
Next example shows the comparison of CGA and Fourier method, when output data contains a random noise.
Example 6.2 (Comparison of CGA and Fourier method applied to ISPF1: the discontinuous spacewise source (6.7)). We consider theheat transfer in homogeneous media, assuming that, the thermal conductivity is a constant (k(x) = 1) and the spacewisedependent source is a discontinuous function
F (x) =⎧⎨⎩
0, 0 � x < 1/3,
1, 1/3 � x < 2/3, (6.7)
0, 2/3 � x � 1.
A. Hasanov, B. Pektas / Applied Numerical Mathematics 78 (2014) 49–67 63
Fig. 3. Comparison of CGA and Fourier method applied to ISPF1 γ = 5%.
Fig. 4. The reconstructed discontinuous spacewise dependent source by the Fourier method: ISPF1 (left figure) and ISPF2 (right figure).
We generate the noise free synthetic output data uT (x) := uh(x, T ) from the numerical solution of the direct problem (1.1),assuming H(t) = 5 exp(−0.5t) and u0(x) = 0. The reconstructed by CGA and Fourier method sources from noisy outputdata (6.2) with noise level γ = 5% are plotted in Fig. 3. The results show that in the noisy output data case the obtainedreconstructions by CGA are better.
Example 6.3 (Reconstruction of a discontinuous spacewise dependent source in ISPF1 and ISPF2 by Fourier method). In this example,we use the same data from Example 6.2, to reconstruct the spacewise dependent source (6.7), in ISPF1 and ISPF2. Togenerate the noise free synthetic output data U T (x) in ISPF2 Simpson formula is used. Optimal values of α > 0 are usedfrom Table 2. In case noise free data case the accuracy error E(n;α;γ ) for both problems ISPF1 (left Fig. 4) and ISPF2 (rightFig. 4) are E(n;α;γ ) ≈ 0.9 × 10−1, when N = 20 in (6.3). In the noisy data case, this error was about 0.19–0.21, for bothISPF1 and ISPF2. �
The same reconstructions are obtained by CGA. These and other computational experiments for various classes of space-wise dependent source functions show that, in the constant coefficient case the reconstructed sources by the both Fourier methodand CGA are almost the same.
64 A. Hasanov, B. Pektas / Applied Numerical Mathematics 78 (2014) 49–67
Fig. 5. The reconstructed Gaussian by CGA in ISPF1 and ISPF2 from noise free output data (left figure) and behavior of the errors (right figure).
Fig. 6. The reconstructed Gaussian by CGA in ISPF1 from noisy data with noise levels γ = 3%; 5%; 10%.
7. Reconstruction of an unknown spacewise dependent source by CGA in a variable coefficient parabolic equation
In this section, we present some benchmark test examples to illustrate the accuracy and stability of the above describedCGA, with an initial iteration defined by the Fourier method, assuming in (1.1) k := k(l/2). In all examples the coefficient k(x) > 0is not a constant, but depends upon x ∈ (0, l) and may also be a discontinuous function.
In all examples below the noise free output data uT and U T were generated from the numerical solution of the directproblem (1.1). These data are then used in (6.2) to generate the random noisy data, by adding a quite large amount ofrandom noise in the input data, to avoid an inverse crime. The mollification of the noisy data (6.2) is performed throughthe Steklov’s averaging technique.
Example 7.1 (Reconstruction of the Gaussian (6.4) in ISPF1 and ISPF2). We use the Gaussian normal distribution given by (6.4) in(1.1), assuming μ = 0.5 and σ = 0.1, and H(t) = 5 exp(−0.5t), k(x) = 1 + 5x2 and u0(x) = 0, to generate the synthetic (noisefree) output data uT and U T . Then, assuming k := k(l/2), the initial iteration F (0)(x) for CGA is determined by the Fouriermethod, using (6.3), with N = 10, and an optimal value of the parameter of regularization α > 0 given in Table 2.
The left Fig. 5 shows that the reconstructed spacewise sources by CGA in ISPF1 and ISPF2 are almost the same, and alsoare almost identical with the exact one. Moreover, the accuracy obtained for the variable coefficient case is the same with
A. Hasanov, B. Pektas / Applied Numerical Mathematics 78 (2014) 49–67 65
Fig. 7. The reconstructed Gaussian by CGA in ISPF2 for noise levels γ = 3%; 5%; 10% (left figure) and behavior of the errors (right figure).
Table 3Performance characteristics of CGA in the reconstruction of Gaussian ISPF1/ISPF2.
the case, when k(x) = 1, given in the left Fig. 2. The behavior of the convergence error and accuracy error are plotted in theright Fig. 5.
Results of computational experiments related to the reconstruction of the Gaussian in ISPF1 and ISPF2, for the noiselevels γ = 3%, 5%, 10% are shown in Fig. 6 and in Fig. 7, accordingly. The dashed lines in Fig. 6 are the initial iterationsF (0)(x), determined by the Fourier method. These figures and the performance characteristics presented in Table 3 showthat the proposed version of CGA applied to ISPF1 and ISPF2, is highly effective in sense of accuracy, stability as well asnumber of iterations. �
Remark that in all cases the reconstructions are almost the same if the time dependent source H(t) is replaced by adiscontinuous function.
Example 7.2 (Reconstruction of the discontinuous spacewise source (6.7)). We use spacewise dependent discontinuous sourcegiven by (6.7) in the direct problem (1.1) to generate the output data. Assuming other input data same as in Example 7.1,to generate the output data. The initial iteration F (0)(x) is determined by the partial Fourier sum (6.3), assuming α =1.0 × 10−5 and N = 5. The numerical results obtained by CGA are plotted in Fig. 8. These results show that for the noiselevels γ = 3–10% the reconstructions of the discontinuous spacewise source are qualitatively acceptable. �Example 7.3 (Reconstruction of the discontinuous spacewise source (6.7) in non-homogeneous media). In the final part of numer-ical experiments we consider the heat transfer in non-homogeneous media, assuming that, in addition to the spacewisedependent discontinuous source, the thermal conductivity is also a discontinuous function:
k(x) ={
k1 x ∈ [0, ξ),
k2, x ∈ (ξ,1],where k1 = 0.9, k2 = 1.0, ξ = 0.5. We assume that the spacewise dependent discontinuous source is given by (6.7), andother input data are the same as in Example 7.2. Using the synthetic output data uT and U T in (6.2) generate the noisydata uγ
T and uγT , for ISPF1 and ISPF2. The reconstructed for the noise level γ = 5% discontinuous spacewise sources in ISPF1
and ISPF2 are plotted in the left Fig. 9. Comparison these results with the previous one (i.e. give in Fig. 8) shows that, anaccuracy and stability the proposed version of CGA with the FEM scheme (5.4) are almost the same in the case of continuousas well as discontinuous thermal conductivity k(x). The difference is only that in the case when the thermal conductivity isa discontinuous function, the “stabilization” of the convergence error begins faster as the right Fig. 9 shows. �
66 A. Hasanov, B. Pektas / Applied Numerical Mathematics 78 (2014) 49–67
Fig. 8. The reconstructed by CGA spacewise dependent discontinuous source in ISPF1 (left) and ISPF2 (right), when k(x) is a continuous function.
Fig. 9. The reconstructed by CGA spacewise dependent discontinuous source in ISPF1 and ISPF2, when k(x) is a discontinuous function (left) and behaviorof the errors (right figure): γ = 5%.
8. Conclusions
This paper presents a systematic study aimed at understanding inverse source problems of identifying an unknownspacewise dependent source in a variable coefficient parabolic equation from final and integral temperature overdetermina-tions. We propose an alternative to Conjugate Gradient Method approach which leads these inverse problems the coupledsystem of parabolic (direct and adjoint) problems. Then we show that the unique regularized solution of each inverse sourceproblem can be derived by the integral representation formula, via the solution of the coupled problem. Fourier method ap-plied to the coupled problem shows that, in the constant coefficient case, the integral representation formula implies thewell-known Picard’s representation. Numerical analysis of considered inverse problems show that, when k(x) is a constant,for the noise free output data the performance characteristics of both (CGA and Fourier) methods are almost the same. Forthe noisy output data the reconstructions obtained by both methods are also close. When k(x) is a continuous or discon-tinuous function, the Fourier method is used to obtain an initial iteration for CGA, which essentially reduces the number ofiterations of CGA. Numerical results related to reconstruction of continuous as well as discontinuous spacewise dependentsource term show that the proposed version of GCA is highly effective in sense of stability, accuracy and minimality of thenumber of iterations. Extensions of Theorem 3.1 and Theorem 3.2 to the two-dimensional case, for the parabolic equationut = (k1(x, y)ux)x + (k2(x, y)u y)y + F (x, y)H(t), when ΩT := (0, lx) × (0, l y) × (0, T ], are straightforward.
A. Hasanov, B. Pektas / Applied Numerical Mathematics 78 (2014) 49–67 67
9. Funding
This work was supported by the Scientific and Technological Research Council of Turkey, and Izmir University researchfund. The work of the first author was also partially supported by the Research Grant (No. 2989/GFZ), Ministry of Educationand Science of the Republic of Kazakhstan.
Acknowledgements
The authors would like to thank Prof. Dr. Andreas Neubauer for his valuable comments and suggestions. The authorgratefully thanks the anonymous referees and the Editor for valuable suggestions.
References
[1] O.M. Alifanov, E.A. Artyukhin, S.V. Rumyantsev, Extreme Methods for Solving Ill-Posed Problems with Application to Inverse Heat Transfer Problems,Begell House Inc., New York, 1996.
[2] A.E. Badia, T.H. Duong, A. Hamdi, Identification of a point source in a linear advection–dispersion–reaction: application to a pollution source problem,Inverse Probl. 21 (2005) 1121–1136.
[3] I. Bushuyev, Global uniqueness for inverse parabolic problems with final observation, Inverse Probl. 11 (1995) L11–L16.[4] J.R. Cannon, Determination of an unknown heat source from overspecified boundary data, SIAM J. Numer. Anal. 15 (1990) 470–484.[5] J.R. Cannon, D. Zachmann, Parameter identification in parabolic partial differential equations from overspecified boundary data, Int. J. Eng. 20 (6) (1982)
779–788.[6] J.R. Cannon, P. DuChateau, Structural identification of an unknown source term in a heat equation, Inverse Probl. 14 (1998) 535–551.[7] J.P. Chancelier, M. Cohen de Lara, F. Pacard, Existence of a solution in an age-dependent transport-diffusion PDE: A model of settler, Math. Models
Methods Appl. Sci. 5 (3) (1995) 267–278.[8] M. Choulli, M. Yamamoto, Conditional stability in determining a heat source, J. Inverse Ill-Posed Probl. 12 (2004) 233–244.[9] S. D’haeyer, B.T. Johansson, M. Slodicka, Reconstruction of a spacewise-dependent heat source in a time-dependent heat diffusion process, IMA J. Eng.
Math. 4 (2012) 1–21.[10] Dinh Nho Háo, Methods for Inverse Heat Conduction Problems, Pater Lang, Berlin, 1998.[11] H.W. Engl, O. Scherzer, M. Yamamoto, Uniqueness and stable determination of forcing terms in linear partial differential equations with overspecified
boundary data, Inverse Probl. 10 (1994) 1253–1276.[12] H.W. Engl, M. Hanke, A. Neubauer, Regularization of Inverse Problems, Kluwer Academic Publishers, Dordrecht, 1996.[13] A. Farcas, D. Lesnic, The boundary-element method for the determination of a heat source dependent on one variable, J. Eng. Math. 54 (2006) 375–388.[14] P.C. Hansen, Analysis of discrete ill-posed problems by means of the L-curve, SIAM Rev. 34 (1992) 561–580.[15] A. Hasanov, Simultaneous determination of source terms in linear parabolic problem from the final data overdetermination: weak solution approach,
J. Math. Anal. Appl. 330 (2007) 766–779.[16] A. Hasanov, Identification of spacewise and time dependent source terms in 1D heat conduction equation from temperature measurement at a final
time, Int. J. Heat Mass Transf. 55 (2012) 2069–2080.[17] A. Hasanov, M. Otelbaev, B. Akpayev, Inverse heat conduction problems with boundary and final time measured output data, Inverse Probl. Sci. Eng.
19 (7) (2011) 895–1006.[18] F. Hettlich, W.A. Rundell, Identification of a discontinuous source in the heat equation, Inverse Probl. 17 (2001) 1465–1482.[19] V. Isakov, Inverse Source Problems, American Mathematical Society, Providence, 1990.[20] V. Isakov, Inverse parabolic problems with the final overdetermination, Commun. Pure Appl. Math. 44 (2) (1991) 185–209.[21] T. Johansson, D. Lesnic, Determination of a spacewise dependent heat source, J. Comput. Appl. Math. 209 (2007) 66–80.[22] A. Kirsch, An Introduction to the Mathematical Theory of Inverse Problems, Springer-Verlag, New York, 2011.[23] O.A. Ladyzhenskaya, Boundary Value Problems in Mathematical Physics, Springer–Verlag, New York, 1985.[24] J.L. Lions, Optimal Control of Systems Governed by Partial Differential Equations, Springer-Verlag, Berlin, 1971.[25] J.L. Lions, Distributed systems with uncomplete data and Lagrange multipliers, in: R. Conti, E. De Giorgi, F. Giannessi (Eds.), Optimization and Related
Fields, in: Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1986, pp. 273–305.[26] B. Mukanova, Numerical reconstruction of unknown boundary data in the Cauchy problem for Laplace’s equation, Inverse Probl. Sci. Eng. (2012) 1–13,
http://dx.doi.org/10.1080/17415977.2012.744405.[27] V.I. Plotnikov, Uniqueness and existence theorems and a priori properties of generalized solutions, Dokl. Akad. Nauk SSSR 165 (1965) 1405–1407.[28] A.I. Prilepko, D.G. Orlovsky, I.A. Vasin, Methods for Solving Inverse Problems in Mathematical Physics, Marcel Dekker, New York, 2000.[29] A. Qian, Y. Li, Optimal error bound and generalized Tikhonov regularization for identifying an unknown source in the heat equation, J. Math. Chem. 49
(2011) 765–775.[30] D.E. Reeve, M. Spivack, Determination of a source term in the linear diffusion equation, Inverse Probl. 10 (1994) 1335–1344.[31] W. Rundell, Determination of an unknown non-homogeneous term in a linear partial differential equation from overspecified boundary data, Appl.
Anal. 10 (3) (1993) 231–242.[32] E.G. Savateev, P. Duchateau, On problems of determining the source function in a parabolic equation, J. Inverse Ill-Posed Probl. 3 (1995) 83–102.[33] M. Slodicka, Some uniqueness theorems for inverse spacewise dependent source problems in nonlinear PDEs, Inverse Probl. Sci. Eng. (2013),
http://dx.doi.org/10.1080/17415977.2013.8234183, in press.[34] V.V. Solov’ev, Solvability of the inverse problem of a source, using overdetermination on the upper case for a parabolic equation, Differ. Equ. 25 (1990)
1114–1119.[35] A.N. Tikhonov, Theoréms d’unicite pour l’équation de la chaleur, Mat. Sb. 42 (1935) 199–216.[36] A. Tikhonov, V. Arsenin, Solution of Ill-Posed Problems, Wiley, New York, 1977.[37] V. Thomée, Galerkin Finite Element Methods for Parabolic Problems, Springer-Verlag, Berlin, 2006.[38] M. Yamamoto, Conditional stability in determination of force terms of heat equations in a rectangle, Math. Comput. Model. 18 (1993) 79–88.[39] M. Yamamoto, Stability, reconstruction formula and regularization for an inverse source hyperbolic problem by a control method, Inverse Probl. 11
