Regularization of an Inverse Problem by Controlling the Stiffness of the Graphs of Approximate Solutions ? ?? Michal Cialkowski1 , Nikolai Botkin2[0000−0003−2724−4817] , Jan Kolodziej3 , Andrzej Fraçkowiak1 , and Wiktor Hoffmann1[0000−0001−7446−1369] 1 Institute of Thermal Engineering, Poznan University of Technology, Poznan, Poland {Michal.Cialkowski,andrzej.frackowiak}@put.poznan.pl, j.hoffmann@doctorate.put.poznan.pl 2 Mathematical Faculty, Technical University of Munich, Garching bei München, Germany botkin@ma.tum.de 3 Institute of Applied Mechanics, Poznan University of Technology, Poznan, Poland jan.kolodziej@put.poznan.pl Abstract. In this paper, an one-dimensional heat conductivity equa- tion is considered. Such an equation describes, for example, a wall which exhibits temperature changes across the thickness, whereas the temper- ature 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 flux are mea- sured at the right end point of the domain. Using the Laplace transform, the problem is reduced to an integral equation defining the unknown temperature at the left end point as function of time. An approximation of the integral equation yields a linear system defining 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 fitting 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 defines the stiffness of the solution graph. Appropriate values of the regularization parameter are being chosen as local minimizers of a discrepancy. Numerical experi- ments 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%). Keywords: Heat transfer · Inverse problems · Cauchy problems · Laplace transform · Regularization. ? 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 Com- mons License Attribution 4.0 International (CC BY 4.0). 42 M. Cialkowski et al. 1 Introduction Inverse problems, in contrast to stationary and non-stationary direct boundary value problems, are characterized by unknown boundary conditions on unreach- able 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 exam- ple in this connection (see Fig. 1). Measurements give the temperature and the heat flux 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 flux on the outer casing boundary. Such problems named after Cauchy have been first 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 distur- bances 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 flux are always affected by errors. For example, the presence of temperature sensors can essentially disturb the measurement of the heat flux. To suppress instabilities typical for inverse problems, Tikhonov’s reg- ularization techniques based on the minimization of fitting functionals are used. The idea of Tikhonov’s method consists in including a regularization term into the fitting functional. This provides the uniqueness and physically stipulated regularity of solutions. Cauchy problems are being intensively studied because of their practical sig- nificance, see works [3, 4, 6, 8–11] for an exemplarily overview of stable approxi- mation 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 finite differ- ence method supplemented by Fourier transform techniques is applied. Legen- dre polynomials are used in paper [11] for the solution of an one-dimensional Cauchy problem. Wavelet-Galerkin method supplemented by the Fourier trans- form 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 flux 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 com- mon points of neighboring segments/parabolas, which controls the rigidity of the graphs of approximate solutions. Numerical simulations show a very exact Regularization of an inverse problem 43 reconstruction of solutions even in the case of large measurement errors (up to 10%). 2 Model Equation The governing equation and the initial and boundary conditions are the follow- ing:   ∂T ∂ ∂T ρc · = λ , x ∈ (0, δ) , t > 0, (1) ∂t ∂x ∂x T (x, 0) = T0 (x) , (2) T (δ, t) = H (t) , (3) ∂T −λ (δ, t) = Q (t) , (4) ∂x T (0, t) = F (t) . (5) Here, ρ denotes the density, c the relative heat, and λ is the heat conductivity coefficient. The temperature F (t) is unknown. Q Fig. 1. Fragment of a gas turbine casing with a one-dimensional wall permitting the heat transfer across the thickness only (left); calculation area (right). 44 M. Cialkowski et al. Obviously, the problem described by (1)–(4) is a Cauchy problem because the boundary conditions (3) and (4) are imposed at the same point x = δ. For the next considerations, it is convenient to introduce non-dimensional variables T x λ t ϑ= , ξ= , τ= · , where Tmax = max T (x, t), (6) Tmax δ ρc δ 2 x ∈ (0, δ) t≥0 to obtain the following non-dimensional formulation of the problem: ∂ϑ ∂2ϑ = , ξ ∈ (0, 1), τ > 0, (7) ∂τ ∂ξ 2 ϑ(ξ, 0) = ϑ0 (ξ) := T0 (x)/Tmax , ξ ∈ (0, 1), (8) ϑ (1, τ ) = h (τ ) := H(t)/Tmax , τ > 0, (9) ∂ϑ δ − (1, τ ) = q (τ ) := · Q(t), τ > 0. (10) ∂ξ λ · Tmax Finally, the unknown boundary temperature at ξ = 0 reads ϑ (0, τ ) = χ (τ ) := F (t)/Tmax , τ > 0. (11) 3 Analytical Solution 3.1 Laplace Transform In view of linearity of equations (7)–(10), the Laplace transform can be applied. Denote Z∞ Lϑ (ξ, τ ) = ϑ̄ (ξ, s) := ϑ (ξ, τ ) · e−sτ dτ (12) 0 and observe that the system of equations (7)–(10) is transformed to the form d2 ϑ̄ s · ϑ̄ (ξ, s) − ϑ (ξ, 0) = , (13) dξ 2 ϑ̄ (ξ, 0) = ϑ̄0 (ξ) , (14) ϑ̄ (1, s) = h̄ (s) , (15) ∂ ϑ̄ − (1, s) = q̄ (s) . (16) ∂ξ The unknown boundary condition at ξ = 0 reads ϑ̄ (0, s) = χ̄ (s) . (17) Regularization of an inverse problem 45 For simplicity, assume that ϑ0 (ξ) = ϑ0 = const. Then the solution of the direct problem given by (13), (14), (16), and (17) has the form √ cosh s (1 − ξ) ϑ̄ (ξ, s) =χ̄ (s) · √ − cosh s √  √  (18) sinh sξ ϑ cosh s (1 − ξ) q̄ (s) · √ √ + 0 · 1− √ . s · cosh s s cosh s Rewrite the previous equation as √ cosh s (1 − ξ) ϑ̄ (ξ, s) =s · χ̄ (s) · √ − s · cosh s √  √  (19) 1 sinh sξ 1 cosh s (1 − ξ) s · q̄ (s) · · √ √ + ϑ0 − √ s s · cosh s s s cosh s and observe that the poles of the right-hand side of (19) are given by the relations √ s = 0 and cosh( s) = 0. √ Setting s = iµ and √ observing that cosh(iµ) = cos(µ) yield the following roots of the equation cosh( s) = 0: π sn = −µ2n , where µn = (2n − 1) · , n = 1, 2, . . . . 2 3.2 Inverse Laplace Transform Let L−1 f and Resx f denote the inverse Laplace transform of f and the residue of f at x, respectively. The following auxiliary calculations are true:  √  cosh s (1 − ξ) L−1 √ = s · cosh s √ ∞ √ cosh s (1 − ξ) X cosh s (1 − ξ) s·τ Res √ + Res √ ·e = s=0 s · cosh s n=1 s=sn s · cosh s ∞ √ X (s − sn ) · cosh s (1 − ξ) sτ 1+ lim √ ·e = (20) n=1 s→sn s · cosh s ∞ X cos µn (1 − ξ) 2 1−2 · e−µn ·τ = n=1 µn · sin µn ∞ 4 X sin µn ξ −µ2n ·τ 1− ·e , π n=1 2n − 1  √  ∞ 1 sinh sξ X n−1 sin µn ξ 2 L−1 ·√ √ =ξ−2 (−1) · 2 · e−µn τ , (21) s s · cosh s n=1 µ n 46 M. Cialkowski et al. L−1 [s q̄ (s)] =q 0 (τ ) + q0 · δ (τ ) , L−1 [s χ̄ (s)] = χ0 (τ ) + χ0 · δ (τ ) , (22) cf. e.g. L [q 0 (τ )] = s · q̄ (s) − q0 ,    1 L−1 = η(τ ), (23) s where δ (τ ) is Dirac’s delta function, η(τ ) is the unit step function, q0 = q(0), and χ0 = χ(0). Application of the above auxiliary calculations yields the following results: " ∞ # −1   4 X sin (2n − 1) π2 ξ −µ2n τ ϑ (ξ, τ ) = L ϑ̄ (ξ, s) = ϑ0 ·e + π n=1 2n − 1 ∞ " # −1 4 X sin (2n − 1) π2 ξ −µ2n τ L [s χ̄ (s)] ∗ 1 − ·e − π n=1 2n − 1 ∞ ! π 8 X n−1 sin (2n − 1) ξ 2 L−1 [s q̄ (s)] ∗ ξ − 2 (−1) 2 2 · e−µn τ = π n−1 (2n − 1) " ∞ # 4 X sin (2n − 1) π2 ξ −µ2n τ = ϑ0 ·e + [χ0 (τ ) + χ0 · δ (τ )] ∗ η (τ ) − π n=1 2n − 1 ∞ Zτ 4 X sin (2n − 1) π2 ξ −µ2n τ 2 ·e · [χ0 (p) + χ0 · δ (p)] · eµn p · dp − π n=1 2n − 1 0 [q 0 (τ ) + q0 · δ (τ )] ∗ η (τ ) · ξ + ∞ π Zτ 8 X n−1 sin (2n − 1) 2 ξ −µ2n τ 2 (−1) 2 ·e · [q 0 (p) + q0 · δ (p)] eµn p · dp = π 2 n=1 (2n − 1) 0 " ∞ # " ∞ # π 4 X sin (2n − 1) 2 ξ −µ2n τ 4 X sin (2n − 1) π2 ξ = ϑ0 ·e + χ (τ ) · 1 − + π n=1 2n − 1 π n=1 2n − 1 ∞ Zτ −µ2n τ 2 X 2· µn · sin µn ξ · e · χ (p) · eµn p · dp − n=1 0 ∞ Zτ X π 2 2 sin (2n − 1) · ξ · e−µn τ · q (p) · eµn p · dp. (24) n=1 2 0 Since ∞ 4 X sin (2n − 1) π2 ξ =1 π n=1 2n − 1 for ξ > 0, the square bracket following χ(τ ) vanishes. Regularization of an inverse problem 47 4 Inverse Problem Henceforth, the case where q(τ ) ≡ 0 and ϑ0 = 0 will be considered. Then equation (24) assumes the form Zτ ϑ (ξ, τ ) = χ (p) · ψ (ξ, τ, p) dp, ξ ∈ (0, 1), τ ≥ 0, (25) 0 where ∞ X 2 ψ (ξ, τ, p) = 2 µn · sin µn ξ · e−µn (τ −p) . (26) n=1 The boundary condition (9) and formula (25) yield the integral equation Zτ χ (p) · ψ (1, τ, p) · dp = h (τ ) (27) 0 for the determination of χ(τ ). 4.1 Approximation of the Integral Equation Assume that the temperature h(t) is measured with the sampling time ∆τ so that the values hk = h(τk ), τk = k · ∆τ, k = 0, 1, 2, . . ., are available. Then equation (27) assumes the form Zτk χ (p) · ψk (p) · dp = hk , with ψk (p) = ψ (1, τk , p) . (28) 0 Using the sampling χj = χ(τj ), j = 0, . . . , k, and choosing a mixing coeffi- cient Θ ∈ (0, 1), we have Zτk τj k Z X χ (p) · ψk (p)dp = χ (p) · ψk (p) dp ≈ 0 j=1τ j−1 τj k Z X [Θ · χj−1 + (1 − Θ) χj ] · ψk (p) dp = j=1τ j−1 k X [Θ · χj−1 · rkj + (1 − Θ) · χj · rkj ] = j=1 k X k X Θ · χ0 · rk1 + (1 − Θ)χk · rkk + χj [Θ · rkj+1 + (1 − Θ) · rkj ] =: χj ψkj . j=1 j=0 48 M. Cialkowski et al. Here, for any k ≥ 1, Zτj rkj = ψk (p) dp, j = 1, . . . k, τj−1 (29) ψkj = Θ · rkj+1 + (1 − Θ) rkj , j = 1, . . . k − 1, ψk0 = Θ · rk1 , ψkk = (1 − Θ) rkk . Thus, the approximation of (28) can be written as: k X χj ψkj = hk , k = 1, . . . M, j=0 where M defines the time horizon. The matrix form of this system reads [ψ] {χ} = {h} , dim [ψ] = M × (M + 1), dim {h} = M. (30) Remark. Use (26) to explicitly obtain Zτj ∞ Zτj X 2 rkj = ψ(1, τk , p)dp = 2 µn · sin µn · e−µn (τk −p) dp = τj−1 n=1 τj−1 ∞ X sin µn  −µ2n ∆τ ·(k−j) 2  =2 · e − e−µn ∆τ ·(k−j+1) . (31) n=1 µn Note that rkj = rk+l,j+l , and hence ψkj = ψk+l,j+l for any l > 0, which saves the computation efforts. 4.2 A Known Particular Solution In order to validate the solution of integral equation (27) via approximation (30), use a known particular solution of equation (7) with the initial value ϑ(ξ, 0) = 0 and the boundary conditions ∂ϑ ϑ (0, τ ) = Tb · 1 − e−βτ ,  − (1, τ ) − Bi · ϑ (1, τ ) = 0. (32) ∂ξ Here, Bi is the Biot number. The solution has the form   Bi · ξ 1 − e−βτ +  ϑ (ξ, τ ) =Tb · 1 − Bi + 1 ∞ X 1 2Tb · β · e−βτ · wn (ξ) · 2 − (33) n=1 pn − β ∞ X 1 2 2Tb · β · wn (ξ) − · e−pn τ , n=1 p2n − β Regularization of an inverse problem 49 where   sin pn ξ Bi wn (ξ) = − · 1− 2 , pn Bi + Bi + p2n and pn are the roots of the equation pn  π  tan pn = − , n = 1, 2, . . . , lim pn − (2n − 1) = 0, uniformly in n. Bi Bi→0 2 Thus, if Bi = 0, we can set r(τ ) = ∂ϑ(1, τ )/∂ξ = 0, h(τ ) = ϑ(1, τ ), and χ(τ ) = Tb · 1 − e−βτ . Therefore, the solution given by approximation (30) can be compared with χ(τ ). Computer experiments show that system (30) is numerically unstable (see Fig. 2). The instability occurs near to the time hori- zon M · ∆. Thus, the regularization of solutions of (30) is necessary. Fig. 2. Oscillations of a solution given by non-regularized system (30). 4.3 Regularization of solutions Remember that approximate solutions are searched as grid functions assuming values χj at grid points τj , 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 first, second, third, or fourth derivatives. Regularization with the 2nd derivative. Consider two adjacent intervals [τi−1 , τi ] and [τi , τi+1 ] (see Fig. 3) and denote the finite difference approximations of the ˜ respectively. Impose the condition left and right first derivatives at τi by χ̃ and χ̃, ˜ = χi − χi−1 − χi+1 − χi = − 1 (χi+1 − 2χi + χi−1 ) ≈ 0 δi = χ̃ − χ̃ (34) h h h that expresses a small jump of the approximate first derivative at τi . 50 M. Cialkowski et al. Fig. 3. Fitting of the approximate first derivatives. Note that χi+1 − 2χi + χi−1 00 2 = χ (τi ) + 0 (h) , i = 1, 2, . . . , M − 1, h and therefore M −1  2 Z t X χi+1 − 2χi + χi−1 00 2 ≈ χ (τ ) dt =: J1 . i=1 h2 0 The matrix corresponding to the relation (34) has the following form:   1 −2 1  1 −2 1   , dim[w] = (M − 2) · M, [w] =  (35)  ···  1 −2 1 or   −1 1  1 −2 1    [w] =   1 − 2 1 ,  dim[w] = (M − 1) · M,  ···  1 −2 1 if the value of χ0 (0) is available. Regularization with the 3rd derivative. Consider now three adjacent intervals ˜ the finite difference [τi−1 , τi ], [τi , τi+1 ], and [τi+1 , τi+2 ] and denote now by χ̃ and χ̃ approximations of the second derivative at τi and τi+1 , respectively. Impose the condition ˜ = χi+1 − 2χi + χi−1 − χi+2 − 2χi+1 + χi = δi = χ̃ − χ̃ h2 h2 (36) 1 (χi−1 − 3χi + 3χi+1 − χi+2 ) ≈ 0 h2 Regularization of an inverse problem 51 that guarantees the closeness of the parabolas shown in Fig. 4 on the common interval [τi , τi+1 ]. Fig. 4. Fitting of the approximate second derivatives. Note that χi−1 − 3χi + 3χi+1 − χi+2 000 = −χ (τi ) + 0 (h) , i = 1, 2, . . . , M − 2, h3 and therefore M −2  2 Z t X χi−1 − 3χi + 3χi+1 − χi+2 000 2 ≈ χ (τ ) dt =: J2 . i=1 h3 0 The matrix corresponding to the relation (36) has the following form:   1 −3 3 −1  1 −3 3 −1  [w] =  , dim[w] = (M − 2) · M. (37)  ···············  1 −3 3 −1 Regularization with the 4rth derivative. Consider fourth adjacent intervals formed by the points τi−2 , τi−1 , τi , τi+1 , τi+2 , and two cubic parabolas corresponding to the points {τi−2 , τi−1 , τi , τi+1 } and {τi−1 , τi , τi+1 , τi+2 }, respectively (see Fig. 5). Impose the condition χi−2 − 4χi−1 + 6χi − 4χi+1 + χi+2 ≈ 0 (38) that guarantees the closeness of the cubic parabolas on the common intervals [τi−1 , τi ] and [τi , τi+1 ]. 52 M. Cialkowski et al. Fig. 5. Fitting of the approximate third derivatives. It easily to see that χi−2 − 4χi−1 + 6χi − 4χi+1 + χi+2 0000 = χ (τi ) + 0 (h) , i = 2, 3, . . . , M − 2, h4 and therefore M −2  2 Z t  X (χi−2 − 4χi−1 + 6χi − 4χi+1 + χi+2 ) 0000 2 ≈ χ (τ ) dt =: J3 . i=2 h4 0 The matrix corresponding to the relation (38) has the following form:   1 −4 6 −4 1  1 −4 6 −4 1  [w] =   , dim[w] = (M − 4) · M. (39) ···············  1 −4 6 −4 1 When solving equation (30), conditions (34), (36), and (38) can be accounted for by minimizing the following functional (the upper index ς denotes the noise level in data measured with error): 2 2 J ({χ}) = k[ψ]{χ} − {h}ς k + α2 k[w]{χ}k , which can be interpreted as the application of minimum square method to the system    ς [ψ] {h} {χ} = . (40) α [w] {0} The optimality condition (the zero first variation of the functional J) reads 1 δJ ({χ}) = [ψ]T [ψ]{χ} + α2 [w]T [w]{χ} − [ψ]T {h}ς = 0, 2 and therefore [ψ]T [ψ] + α2 [w]T [w] {χ} = [ψ]T {h}ς , rank [ψ]T [ψ] + α2 [w]T [w] = M.   Regularization of an inverse problem 53 Finally  −1 {χ}ςα = [ψ]T [ψ] + α2 [w]T [w] [ψ]T {h}ς . (41) Let [ψ]+ := lim→0 ([ψ]T [ψ]+I)−1 [ψ]T be the Moore-Penrose-Inverse matrix, and {χ}+ := [ψ]+ {h} a unique Moore-Penrose solution of (30). Estimate the difference −1 {χ}+ − {χ}ςα = [ψ]+ {h} − [ψ]T [ψ] + α2 [w]T [w] [ψ]T {h}ς = −1 T [ψ]+ {h} − [ψ]T [ψ] + α2 [w]T [w] [ψ] {h}+ T 2 T −1 T [ψ] [ψ] + α [w] [w] [ψ] · ({h} − {h}ς ) = (42)  −1 T  [ψ]+ − [ψ]T [ψ] + α2 [w]T [w] [ψ] · {h}+ −1 [ψ]T [ψ] + α2 [w]T [w] · ({h} − {h}ς ) .  Therefore, −1 T {χ}+ − {χ}ςα 2 ≤ [ψ]+ − [ψ]T [ψ] + α2 [w]T [w] [ψ] · k{h}k2 + 2 −1 T [ψ]T [ψ] + α2 [w]T [w] [ψ] · k{h} − {h}ς k2 ≤ 2 −1 T [ψ]+ − [ψ]T [ψ] + α2 [w]T [w] [ψ] · k{h}k2 + 2 (43) −1 T [ψ]T [ψ] + α2 [w]T [w] [ψ] ·ς = 2 [ψ]+ − E (α) 2 · k{h}k2 + kE (α)k2 · ς, where −1 T E (α) = [ψ]T [ψ] + α2 [w]T [w] [ψ] , rank E = M. In numerical computations, the distance k{χ}+ − {χ}ςα k2 will be used for finding optimalvalues of α(ς) for given noise levels  ς. Moreover, if an exact stable solution χ0 of (30) would be known, say χ0 = {χ}+ , its rigidity can be compared with that of the solution searched. This motivates the following modification of the objective functional: 2  2 J {χ} , χ0 = k[ψ] {χ} − {h}k + α2 [w] {χ} − χ0    (44) which corresponds to the system         [ψ] {h} {h} [0]  0 {χ} =  0 = + χ . (45) α [w] α [w] χ {0} α [w] Figures 6–9 show simulation results. 54 M. Cialkowski et al. k{χ}+ − {χ}ςα k2 α Fig. 6. Dependency of the distance between {χ}+ and {χ}ςα on α at fixed error levels ς. Appropriate values of α are local minimizers of this function. The case of matrix (35) is considered. Fig. 7. Comparison of the exact temperature at x = 1 with the temperature recon- structed from the inverse problem for several appropriate values (local minimizers, see Fig. 6) of the regularization parameter α. The random noise level ς equals 5.0%. The case of matrix (35) is considered. Regularization of an inverse problem 55 Fig. 8. Comparison of the exact temperature at x = 0 with the temperature obtained from the inverse problem for several appropriate values (local minimizers, compare with Fig. 6) of the regularization parameter α. The random noise level ς equals 5.0%. The case of matrix (37) is considered. Fig. 9. Comparison of the exact temperature at x = 0 with the temperature obtained from the inverse problem for several appropriate values (local minimizers, compare with Fig. 6) of the regularization parameter α. The random noise level ς equals 5.0%. The case of matrix (39) is considered. 56 M. Cialkowski et al. 5 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 parabo- las, and the condition of close values of the derivatives of neighboring seg- ments/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 solu- tions. Numerical experiments show that appropriate values of the regularization parameters correspond to local minimizers of k{χ}+ − {χ}ςα k2 . Simulations show a good agreement of reconstructed functions with exact solutions even for the high level (up to 10%) of random disturbances in mea- surements. References 1. Cialkowski, M.J., Fraçkowiak, A., von Wolfersdorf, J.: Numerical solution of a two- dimensional inverse heat transfer problem in gas turbine blade cooling. Archives of Thermodynamics 27(4), 1–8 (2006) 2. Cialkowski, M.J., Fraçkowiak, A., Grysa, K.: Physical regularization for inverse problems for stationary heat conduction. Journal of Inverse and Ill-Posed Problems 15, 1–18 (2007). https://doi.org/10.1515/jiip.2007.019 3. De Lillo, S., Lupo, G., Sanchini, G.: A Cauchy problem in nonlinear heat conduc- tion. Journal of Physics A: Mathematical and General 39, 7299–7303 (2006) 4. Eldén, L.: Numerical solution of the sideways heat equation by difference approx- imation in time. Inverse Problems 11(4), 913–923, (1995) 5. Fraçkowiak, A., Botkin, N.D., Cialkowski, M., Hoffmann, K.-H.: A fit- ting algorithm for solving inverse problems of heat conduction. Inter- national Journal of Heat and Mass Transfer 53, 2123–2127 (2010). https://doi.org/10.1016/j.ijheatmasstransfer.2009.12.039 6. Fu, C.-L.: Simplified Tikhonov and Fourier regularization methods on a general sideways parabolic equation. Journal of Computational and Applied Mathematics 167(2), 449–463 (2004). https://doi.org/10.1016/j.cam.2003.10.011 7. Hansen, C., Prost O’Leary, D.: The use of the L-curve in the regularization of discrete ill-posed problems. SIAM Journal on Scientific Computing 14(6), 1487– 1503 (1993). https://doi.org/10.1137/0914086 8. Haò, D.N.: A noncharacteristic Cauchy problem for linear parabolic equa- tions I: Solvability. Mathematische Nachrichten 171(1), 177–206 (1995). https://doi.org/10.1002/mana.19951710112 9. Marin, L., Lesnic, D.: The method of fundamental solutions for the Cauchy prob- lem associated with two-dimensional Helmholtz-type equations. Computers and Structures 83 267–278 (2005). https://doi.org/10.1016/j.compstruc.2004.10.0050 10. Reginska, T., Eldén, L.: Solving the sideways heat equation by a wavelet-Galerkin method. Inverse Problems 13(4), 1093–1106 (1997) 11. Shidfar, A., Pourgholi, R.: Numerical approximation of solution of an inverse heat conduction problem based on Legendre polynomi- als. Applied Mathematics and Computation 175(2), 1366–1374 (2006). https://doi.org/10.1016/j.amc.2005.08.040