In this paper, an one-dimensional heat conductivity equation is considered. Such an equation describes, for example, a wall which exhibits temperature changes across the thickness, whereas the temperature remains constant along the in-plane directions. The problem of recovering the unknown temperature at the left end point of the domain is studied. It is assumed that the temperature and the heat ux are measured at the right end point of the domain. Using the Laplace transform, the problem is reduced to an integral equation de ning the unknown temperature at the left end point as function of time. An approximation of the integral equation yields a linear system de ning the values of the unknown function. Additionally, the graph of the unknown function is considered as a sequence of segments or overlapping quadratic or cubic parabolas, and the condition of common tangents at common points of neighboring parabolas is imposed. The resulting overdetermined system is solved using the least square method whose tting function consists of two parts: a residual responsible for satisfying the integral equation and a term responsible for the matching of the segments/parabolas. The last term is multiplied by a regularization parameter that de nes the sti ness of the solution graph. Appropriate values of the regularization parameter are being chosen as local minimizers of a discrepancy. Numerical experiments show that one of such values provides the best choice. Numerical simulations exhibit a very exact reconstruction of solutions even in the case of large measurement errors (up to 10%).
? The paper was carried out in the framework of the grant N0. 4917/B/T02/2010/39 funded by the Ministry of Highschool Education (Poland). ?? Copyright c 2020 for this paper by its author. Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
Inverse problems, in contrast to stationary and non-stationary direct boundary value problems, are characterized by unknown boundary conditions on unreachable parts of boundaries. Such a situation is typical when studying the heat transfer in engineering objects that have complicated geometries with holes. The problem of cooling of a gas turbine casing can be mentioned as an example in this connection (see Fig. 1). Measurements give the temperature and the heat ux density on the outer boundary of a casing, whereas the temperature on channel walls of the casing should be recovered, see e.g. [1, 5]. Thus, the missing information about heat conditions in the unreachable part of the casing is compensated by a redundant condition, e.g. accounting for the heat ux on the outer casing boundary. Such problems named after Cauchy have been rst considered by Hadamard who has observed that their solutions do not depend continuously on the given boundary data. Therefore, inverse problems belong to ill-conditioned ones in the Hadamard sense, [2], which means that small disturbances of boundary data cause large errors and oscillations in solutions. Such instabilities can destroy numerical procedures since the measured values of the temperature and the heat ux are always a ected by errors. For example, the presence of temperature sensors can essentially disturb the measurement of the heat ux. To suppress instabilities typical for inverse problems, Tikhonov's regularization techniques based on the minimization of tting functionals are used. The idea of Tikhonov's method consists in including a regularization term into the tting functional. This provides the uniqueness and physically stipulated regularity of solutions.
Cauchy problems are being intensively studied because of their practical signi cance, see works [3, 4, 6, 8{11] for an exemplarily overview of stable approximation methods for solving ill-posed inverse problems. In paper [3], the problem is reduced to a linear second-kind integral Volterra equation which admits a unique solution. The method of fundamental solutions is used in paper [9] for solving a steady-state Cauchy problem. In papers [4] and [6], a nite di erence method supplemented by Fourier transform techniques is applied. Legendre polynomials are used in paper [11] for the solution of an one-dimensional Cauchy problem. Wavelet-Galerkin method supplemented by the Fourier transform is utilized in paper [10]. The questions of uniqueness of solutions of Cauchy problems are considered in paper [8].
The purpose of the paper presented is to propose a stable method for solving an one-dimensional Cauchy problem related to reconstructing the temperature of one surface of a wall using measurements of the temperature and the heat ux on the other wall surface. The main features of the method consist in the reduction of the problem to an integral equation using the Laplace transform, approximation of solutions by sequences of segments or overlapping quadratic or cubic parabolas, and imposing the condition of common tangents at common points of neighboring segments/parabolas, which controls the rigidity of the graphs of approximate solutions. Numerical simulations show a very exact reconstruction of solutions even in the case of large measurement errors (up to 10%). 2
The governing equation and the initial and boundary conditions are the
following:
c
=
T (x; 0) = T0 (x) ;
T ( ; t) = H (t) ;
T (0; t) = F (t) :
(
is the heat conductivity Q
Obviously, the problem described by (
For the next considerations, it is convenient to introduce non-dimensional variables # =
T Tmax ;
x = ; = c t 2 ; where
Tmax =
T (x; t);
(
> 0; #( ; 0) = #0( ) := T0(x)=Tmax;
Tmax # (0; ) = ( ) := F (t)=Tmax; > 0: Finally, the unknown boundary temperature at = 0 reads 3 3.1
Laplace Transform
In view of linearity of equations (
L# ( ; ) = # ( ; s) :=
# ( ; ) e s d
and observe that the system of equations (
For simplicity, assume that #0 ( ) = #0 = const. Then the solution of the
direct problem given by (13), (14), (16), and (17) has the form
(s) cosh ps (
s cosh ps s q (s) 1s pssinchopshsps Setting ps = i and observing that cosh(i ) = cos( ) yield the following roots of the equation cosh(ps) = 0: sn = 2n; where n = (2n 1) 2 ; n = 1; 2; : : : : 3.2
Inverse Laplace Transform Let L 1f and Resxf denote the inverse Laplace transform of f and the residue of f at x, respectively. The following auxiliary calculations are true: L 1 cosh ps (1
s cosh ps 1 1 Res s=0
1 1 + X cosh ps (1 s cosh ps
(s lim n=1 s!sn 2 X1 cos n (1 n=1 4 X1 sin n n=1 2n
1 L 1 1 sinh ps s ps cosh ps
L 1 [s q (s)] =q0 ( ) + q0 ( ) ; L 1 [s (s)] = 0 ( ) + 0 cf. e.g. L [q0 ( )] = s q (s) q0 ; ( ) ; (22) 1 X sin (2n n=1 n sin n
e Since for
4 nX1=1 sin (22nn 11) 2 = 1 > 0, the square bracket following ( ) vanishes.
n=1 4 X1 sin (2n
2n n=1 0 and #0 = 0 will be considered. Then (25) (26) (27) for the determination of ( ). 4.1
Approximation of the Integral Equation Assume that the temperature h(t) is measured with the sampling time so that the values hk = h( k); k = k ; k = 0; 1; 2; : : :; are available. Then equation (27) assumes the form
Z k 0
Using the sampling j = cient 2 (0; 1), we have (p) k(p) dp = hk; with k(p) = (1; k; p) : (28) ( j ); j = 0; : : : ; k, and choosing a mixing coe Z k 0
(p) Xk Z
j j=1 j 1 k X [ j=1 k(p)dp = Xk Z
j
k ) rkj ] =: X j=0 j kj :
rkj =
kj =
(33)
where M de nes the time horizon. The matrix form of this system reads
[ ] f g = fhg ;
In order to validate the solution of integral equation (27) via approximation (30),
use a known particular solution of equation (
Bi # (1; ) = 0: (32) Here, Bi is the Biot number.
The solution has the form # ( ; ) =Tb
1 2Tb 2Tb 1 X wn ( ) n=1 1 X wn ( ) n=1
1 where Thus, if Bi = 0, we can set r( ) = @#(1; )=@ = 0, h( ) = #(1; ), and ( ) = Tb 1 e . Therefore, the solution given by approximation (30) can be compared with ( ). Computer experiments show that system (30) is numerically unstable (see Fig. 2). The instability occurs near to the time horizon M . Thus, the regularization of solutions of (30) is necessary. Remember that approximate solutions are searched as grid functions assuming values j at grid points j , j = 0; : : : M . The idea of the regularization is to introduce some rigidity to the graph of the approximation. This can be done by accounting for approximate rst, second, third, or fourth derivatives. Regularization with the 2nd derivative. Consider two adjacent intervals [ i 1; i] and [ i; i+1] (see Fig. 3) and denote the nite di erence approximations of the left and right rst derivatives at i by ~ and ~~, respectively. Impose the condition i = ~ ~~ = i h i 1 i+1h i = h1 ( i+1 2 i + i 1) that expresses a small jump of the approximate rst derivative at i. 0 (34) and therefore i+1 M 1 X i=1 or 2 i + i 1 = h2 00 ( i) + 0 (h) ; i = 1; 2; : : : ; M 1; i+1 2 i + i 1 h2 2
Z t 0 00 ( ) 2 dt =: J1: The matrix corresponding to the relation (34) has the following form:
Regularization with the 3rd derivative. Consider now three adjacent intervals [ i 1; i], [ i; i+1], and [ i+1; i+2] and denote now by ~ and ~~ the nite di erence approximations of the second derivative at i and i+1, respectively. Impose the condition i = ~ that guarantees the closeness of the parabolas shown in Fig. 4 on the common interval [ i; i+1]. and therefore
M 2 X i=1
The matrix corresponding to the relation (36) has the following form: Regularization with the 4rth derivative. Consider fourth adjacent intervals formed by the points i 2; i 1; i; i+1; i+2, and two cubic parabolas corresponding to the points f i 2; i 1; i; i+1g and f i 1; i; i+1; i+2g, respectively (see Fig. 5). Impose the condition i 2 4 i 1 + 6 i 4 i+1 + i+2 0 (38) that guarantees the closeness of the cubic parabolas on the common intervals [ i 1; i] and [ i; i+1]. and therefore 2; ( i 2 The matrix corresponding to the relation (38) has the following form:
When solving equation (30), conditions (34), (36), and (38) can be accounted for by minimizing the following functional (the upper index & denotes the noise level in data measured with error):
J (f g) = k[ ]f g fhg& k2 + 2[w]T [w] f g = [ ]T fhg& ; rank [ ]T [ ] + 2[w]T [w] = M:
f g& = [ ]T [ ] + 2[w]T [w] 1
Let [ +]+:=:=[ lim]+f!h0g([a ]uTn[iq]u+e IM)oo1[re-]PTebnerotshee sMolouotrieo-nPeonfr(o3s0e-)I.nEvesrtsimemataetrtihxe, and f g di erence
+ f g Therefore, f g +
& f g 2 [ ]T [ ] + 2[w]T [w] E ( ) = [ ]T [ ] + 2[w]T [w] [ ]T ; rank E = M:
1 [ ]T 2 kfhgk2 + (41) (42) (43)
In numerical computations, the distance kf g+ f g& k2 will be used for nding optimal values of (&) for given noise levels &. Moreover, if an exact stable solution 0 of (30) would be known, say 0 = f g+, its rigidity can be compared with that of the solution searched. This motivates the following modi cation of the objective functional:
J f g ; 0 = k[ ] f g
fhgk2 + 2 [w] f g which corresponds to the system [ ] [w] f g =
fhg [w] 0 = fhg f0g +
Concluding Remarks The paper proposes a practical method of regularization of inverse problems where a function (a set of functions) of time has to be reconstructed. Such a function can be approximated by a sequence of segments or overlapping parabolas, and the condition of close values of the derivatives of neighboring segments/parabolas at common points can be imposed. This introduces a rigidity of the graph of the searched function. Corresponding matrix weighted penalty terms multiplied by a regularization parameter provide the stabilization of solutions. Numerical experiments show that appropriate values of the regularization parameters correspond to local minimizers of kf g+ f g& k2.
Simulations show a good agreement of reconstructed functions with exact solutions even for the high level (up to 10%) of random disturbances in measurements.