Luận án Một số bài toán ngược cho phương trình elliptic với quan sát biên (some inverse problems for elliptic equations with boundary observations) thesis for

ves the problem −∆u + au = g in Ω,∇u · n = 0 on ∂Ω. (3.41) Set φ¯(x) = φ(x)− u˜(x, 0) and denote φ¯k′ = ∫ Ω φ¯(x)ϕk′(x)dx. Hence, f (x) = ∞∑ |k′|=0 λk′φ¯kϕk′(x). (3.42) Since λk′ = a + ( k1π 2 )2 + ( k2π 2 )2 + · · · + ( kn−1π 2 )2 , tends to infinity as |k′| tends to infinity, we see from (3.42) that the problem of reconstructing f from φ is ill-posed, and we will use truncated Fourier series method for regularizing it. 3.2.3 Regularization by the truncated Fourier series Suppose that instead of φ we have only its approximate data φϵ ∈ L2(Ω) which satisfies (3.34). Then we see that the series (3.42) may not converge for this data. To avoid it, we shall truncate this series. Namely, we take fN,ϵ(x) = N∑ |k′|=0 λk′(φ ϵ k′ − u˜(x, 0))ϕk′(x) = N∑ |k′|=0 λk′φ¯ ϵ k′ϕk′(x), (3.43) fN (x) = N∑ |k′|=0 λk′(φk′ − u˜(x, 0))ϕk′(x) = N∑ |k′|=0 λk′φ¯k′ϕk′(x). (3.44) The purpose of this regularization method is to determine an appropriate N = N(ϵ) ∈ N such that ∥fN,ϵ − f∥L2(Ω) → 0 as ϵ→ 0. 87 Theorem 3.2.1. Let α be a positive given number, f a function in Hα(Ω). Further- more, suppose that there is a positive constant E such that ∥f∥Hα(Ω) ≤ E. Then with N = N∗ = [(E ϵ ) 1 2+2α (4(n− 1) π2 ) α 2+2α (a + π2 4 ) − 1(2+2α) ] , there exists a positive C3 = C3(E,α, n) independent of ϵ such that ∥f − fN,ϵ∥L2(Ω) ≤ C3ϵ α 1+α , which tends to zero as ϵ tends to zero. Proof. For N ∈ N, we have, ∥f − fN,ϵ∥L2(Ω) ≤ ∥f − fN∥L2(Ω) + ∥fN − fN,ϵ∥L2(Ω) := A +B. (3.45) We have A2 = ∥∥∥ ∞∑ |k′|=N+1 fk′ϕk′(·) ∥∥∥2 L2(Ω) = ∞∑ |k′|=N+1 f2k′ = ∞∑ |k′|=N+1 λ2αk′ f 2 k′λ −2α k′ ≤ λ−2αN+1 ∞∑ |k′|=N+1 λ2αk′ f 2 k′ ≤ λ−2αN+1∥f∥2Hα(Ω) ≤ λ−2αN+1E2. Using the Cauchy-Bunyakovsky inequality, we have λ−αk′ = ( a + ( k1π 2 )2 + ( k2π 2 )2 + . . . + ( kn−1π 2 )2)−α ≤ ( a + (k1 + k2 + · · · + kn−1)2 π2 4(n− 1) )−α = ( a + |k′|2π2 4(n− 1) )−α ≤ ( a + (N + 1)2π2 4(n− 1) )−α < ( N2π2 4(n− 1) )−α . (3.46) Thus, A < E ( 4(n− 1) π2 )α N−2α. (3.47) 88 On the other hand, we have λN = a + ( k1π 2 )2 + ( k2π 2 )2 + · · · + ( kn−1π 2 )2 , with k1 + k2 + · · · + kn−1 = N . So, λN ≤ a + (k1 + k2 + · · · + kn−1) 2π2 4 = a + N2π2 4 ≤ ( a + π2 4 ) N2. Since λ0 ≤ λ1 ≤ · · · ≤ λN , we have B2 = ∥ N∑ |k′|=0 λk′(φ¯ ϵ k′ − φ¯k′)ϕk′(x′)∥2L2(Ω) = N∑ |k′|=0 λ2k′ ((φk′ − u˜k(., 0))− (φϵk′ − u˜k(., 0)))2 = N∑ |k′|=0 λ2k′ (φk′ − φϵk′)2 ≤ λ2Nϵ2 ≤ ( a + π2 4 )2 N4ϵ2. (3.48) Therefore, B ≤ ( a + π2 4 ) N2ϵ. (3.49) From estimates (3.45), (3.47) and (3.49), we have ∥f − fN,ϵ∥L2(Ω) ≤ E ( 4(n− 1) π2 )α N−2α + ( a + π2 4 ) N2ϵ. (3.50) By taking N = N∗ = [(E ϵ ) 1 2+2α (4(n− 1) π2 ) α 1+2α (a + π2 4 ) − 1(2+2α) ] , and C3 = 3E 1 1+α (4(n− 1) π2 ) α 1+α ( a + π2 4 ) α 1+α , for example, we have ∥f − fN,ϵ∥L2(Ω) ≤ C3ϵ α 1+α , (3.51) which tends to zero as ϵ tends to zero. 89 3.3 Numerical examples In this section we apply the proposed method to some concrete examples for illustrating its efficiency. We wish to determine f = f (x) in the problem∆u + 2u = f (x) + g(x, y), (x, y) in Ω = (0, 2)× (0, 2),∇u · n = 0, (x, y) on ∂Ω (3.52) from the noisy observation on the boundary: u(x, 0) ≈ φϵ(x) = cos πx 2 ( 1 + p ∗ rand(−1, 1) ) . (3.53) Here, rand(−1, 1) generates a random number in (−1, 1), p is the percentage of the error. So, the noise level is ϵ = p.∥u(., 0)∥. We test our method for three cases: f is a smooth function, f is continuous function but not smooth and f is a discontinuous function. Example 3.3.1. f is a smooth function g(x, y) = (1.25π2 + 2) cos πx 2 cos πy 2 − 3 sin(πx) + 1). The exact solution with p = 0 is (u, f ) = (cos πx 2 cos πy 2 , 3 sin(πx) + 1). Table 3.1: Example 3.3.1. The L2-norm of relative errors for smooth function p 5% 7 % 10% M = 10 0.0573 0.1427 0.3978 M=15 0.0477 0.0536 0.1992 90 Figure 3.1: Example 1. Solutions with different noise levels for smooth function, M = 15 Figure 3.2: Example 3.3.1. Errors with different perturbations for smooth func- tion, M = 15 Figure 3.3: Example 3.3.1. Solutions with different number of Fourier coefficients, smooth function, p = 7% Figure 3.4: Example 3.3.1. Errors with different number of Fourier coefficients for smooth function, p = 7% 91 Example 3.3.2. f is a continuous but non-smooth function g(x, y) = (1.25π2 + 2) cos πx 2 cos πy 2 − |x− 1|. The exact solution with p = 0 is (u, f ) = (cos πx 2 cos πy 2 , |x− 1|). Figure 3.5: Example 3.3.2. Solutions with different noise levels for continuous, non- smooth function, M = 15 Figure 3.6: Example 3.3.2. Errors with different noise levels, for continuous, non- smooth function, M = 15 Table 3.2: Example 3.3.2. The L2-norm of relative errors - non-smooth, continuous function. p 5% 7 % 10% M = 10 0.1095 0.2077 0.3626 M=15 0.0868 0.0987 0.2077 92 Figure 3.7: Example 3.3.2. Solutions with different number of Fourier coefficients, p = 7% Figure 3.8: Example 3.3.2. Errors with different number of Fourier coefficients, p = 7% Example 3.3.3. f is a discontinuous function. g(x, y) = (1.25π2 + 2) cos πx2 cos πy2 , (x, y) ∈ (0, 12)× (0, 2) ∪ (32 , 2)× (0, 2),(1.25π2 + 2) cos πx2 cos πy2 − 1, (x, y) ∈ [12 , 32 ]× 2. The exact solution is f (x) =  0, 0 ≤ x ≤ 13 and 23 ≤ x ≤ 1,1, 13 < x < 23 . (3.54) Table 3.3: Example 3.3.3. The L2-norm of relative errors for the discontinuous function p 5% 7 % 10% M = 15 0.1730 0.1776 0.3894 M=20 0.1520 0.1494 0.2522 In all examples, we use the Finite Difference Method with 80 × 80 nodes the domain and boundary. We compare the accuracy of the Finite Difference Method under differ- ent conditions: using the same number of Fourier coefficients with varying noise levels (5%, 7%, and 10%) and using different numbers of coefficients (M = 7, 10, 15 for Exam- ples 3.3.1 and 3.3.2 and M = 10, 15, 20 for Example 3.3.3) with a fixed noise level. 93 Figure 3.9: Example 3.3.3. Solutions with different noise levels for discontinu- ous function, M = 20 Figure 3.10: Example 3.3.3. Errors with different noise levels,for discontinu- ous function, M = 20 Figure 3.11: Example 3.3.3. Solutions with different number of Fourier coeffi- cients, p = 5% Figure 3.12: Example 3.3.3. Errors with different number of Fourier coefficients for discontinuous function, p = 5% 94 The results of Example 3.3.1 are presented in Figures 3.1–3.4 and Table 3.1 while results of Example 3.3.2 and Examples 3.3.3 are shown in Figures 3.5–3.8, Table 3.2 and Figures 3.9– 3.12, Table 3.3, respectively. From these figures and tables, we can see the decline in errors with decreasing noise: The errors in the computational solutions consistently decrease as the noise level drops from 10% to 5%, given a fixed number of Fourier coefficients. This pattern is evident across all examples. In Example 3.3.1, using 15 Fourier coefficients, the relative error drops from 0.1992 at a noise level of 10% to 0.0477 at a noise level of 5%. Increasing the number of Fourier coefficients also reduces errors: When employing the same noise level of 5% and increasing the number of Fourier coefficients from 7 to 15, the relative errors in Example 3.3.1 decrease from 0.0903 to 0.0477. This suggests that incorporating more coefficients can further enhance accuracy. Examples 3.3.2 and 3.3.1 exhibit similar trends, though their errors are generally larger than those in Example 3.3.1. This can be attributed to the underlying complexity of their exact solutions, which are non-smooth functions compared to the smooth solution in Example 3.3.1. 95 Conclusion This thesis is devoted to the Cauchy problem and to the source identification problem for second-order linear elliptic equations with boundary observation. These problems are ill-posed. The Cauchy problem for elliptic equations is studied in Chapter 1. There a new concept of very weak solution to the Cauchy problem is introduced. The Cauchy problem is regularized by a non-local boundary value problem, a solution to which is understood in the very weak sense. The last problem is discretized using the finite difference method, for which the stability and convergence are proved. Numerical examples are presented for illustrating the efficiency of the method. Chapter 2 is devoted to the source identification problem. The inverse problem is refor- mulated as an operator equation, which is regularized by Tikhonov regularization method. The regularized problem is discretized following Hinze’s variational discretization concept. A rule for choosing the regularization parameter depending on noise level and meshsize is suggested, that ensures optimal convergence rates. This abstract result is applied to the finite element method and the efficiency of the method has been showcased through several numerical examples. Chapter 3 addresses the inverse problem of identifying a term in the right-hand side of some special elliptic equations in a cylinder. Based on the special form of the considered equation in a cylinder, the solution of the direct and inverse problems can be represented by the Fourier series and standard truncated method is used to regularize the inverse problem. Error estimates of the method have been proved and some numerical examples are presented for showing its efficiency. 96 List of Author’s Related Papers 1. Ha`o D.N., Giang L.T.T., Kabanikhin S., and Shishlenin M. (2018). A fnite differ- ence method for the very weak solution to a Cauchy problem for an elliptic equation. Journal of Inverse and Ill-Posed Problems 26(6), 835–857. 2. Ha`o D.N., Giang L.T.T., and Oanh N.T.N.(2024). Determination of the right-hand side in elliptic equations. Optimization, 73(4), 1195–1227. 3. Giang, L.T.T. (2024), Determining a source term in an elliptic equation in a cylinder from boundary observations, Preprint IMH20240202, Institute of Mathematics, VAST.. 97 Bibliography [1] Abdelaziz B., El Badia A., and El Hajj A. (2013). Reconstruction of extended sources with small supports in the elliptic equation ∆u + µu = f from a single Cauchy data. Comptes Rendus Mathematique 351(21-22), 797–801. [2] Abdelaziz B., El Badia A. and El Hajj A. (2015). Direct algorithm for multipolar sources reconstruction. Journal of Mathematical Analysis and Applications 428(1), 306–336. [3] Amirov A.K. (1986). Solvability of the inverse problems for second-order equations. Functional Analysis and Its Applications 20(3), 236–237. [4] Amirov A.K. (1987). On a problem on the solvability of inverse problems. Sibirskii Matematicheskii Zhurnal 28(6), 3–11. [5] Anastasio M. A,, Zhang J., Modgil D., and La Rivie`re P.J. (2007). Application of inverse source concepts to photoacoustic tomography. Inverse Problems 23(6), S21. [6] Arridge S.R. (1999). Optical tomography in medical imaging. Inverse Problems 15(2), R41. [7] Badia A.E. and Hajj A.E. (2013). Identifcation of dislocations in materials from bound- ary measurements. SIAM Journal on Applied Mathematics 73(1), 84–103. [8] Baumeister J. (1987). Stable Solution of Inverse Problems. Vieweg & Sohn, Braun- schweig. [9] Berntsson F. and Elde´n L. (2001). Numerical solution of a Cauchy problem for the Laplace equation. Inverse Problems 17(4), 839–853. [10] Bidsadze A.V. and Samarskii A.A. (1969). On some simple generalization of linear el- liptic boundary value problems. Soviet Mathematics Doklady 10, 398–400. [11] Bourgeois L. (2005). A mixed formulation of quasi-reversibility to solve the Cauchy problem for Laplace’s equation. Inverse Problems 21(3), 1087–1104. 98 [12] Bourgeois L (2006). Convergence rates for the quasi-reversibility method to solve the Cauchy problem for Laplace’s equation. Inverse Problems 22(2), 413–430. [13] Budak B.M., Samarskii A.A., Tikhonov A.N. (1964). A Collection of Problems in Mathematical Physics., Pergamon Press, Oxford. [14] Bukhge˘ım, A.L. (1988). Introduction to the Theory of Inverse Problemsz. Nauka, Norosibirsk (in Russian). [15] Caldero´n, A.P. (1958). Uniqueness in the Cauchy problem for partial differential equa- tions. American Journal of Mathematics 80(1), 16–36. [16] Cannon J.R. and Ewing R.E. (1975). The locations and strengths of point sources. Improperly Posed Boundary Value Problems (Research Notes in Mathematics) 1, 39–53. [17] Cannon J.R. and Rundell W. (1987). An inverse problem for an elliptic partial differential equation. Journal of Mathematical Analysis and Applications 126(2), 329–340. [18] Carleman T. (1939). Sur un proble`me d’unicite´ pour les syste`mes d’e´quations aux de´rive´es partielles a` deux variables inde´pendantes. Arkiv for Matematik Astronomioch Fysik 26(17), 9pp. [19] Colli-Franzone P., Guerri L., Tentoni S., Viganotti C., Baruf S., Spaggiari S., and Tac- cardi B. (1985). A mathematical procedure for solving the inverse potential problem of electrocardiography. Analysis of the time-space accuracy from in vitro experimental data. Mathematical Biosciences 77(1-2), 353–396. [20] Daniel T. (1995). Unique continuation for solutions to pde’s; between Ho¨rmander’s the- orem and Holmgren’s theorem. Communications in Partial Differential Equations 20(5-6), 855–884. [21] Denisov A.M. and Solov’eva S.I. (1993). A problem for determining a coefcient of a nonlinear stationary heat equations’. Computational Mathematics and Mathematical Physics 33, 1294–1304. [22] El Badia A. and Ha-Duong T. (2002). On an inverse source problem for the heat equa- tion. Application to a pollution detection problem. Journal of Inverse and Illposed Problems 10(6), 585–599. 99 [23] Engl H.W. and Leitao A.(2001). A Mann iterative regularization method for elliptic Cauchy problems. Numerical Functional Analysis and Optimization 22(7-8), 861– 884. [24] Falk R.S. and Monk P.B. (1986). Logarithmic convexity for discrete harmonic functions and the approximation of the Cauchy problem for Poisson’s equation. Mathematics of Computation 47(175), 135–149. [25] Fang W. and Zeng S. (2009). Numerical recovery of Robin boundary from boundary mea- surements for the Laplace equation. Journal of Computational and Applied Mathe- matics 224(2), 573–580. [26] Farcas A., Elliott L., Ingham D.B., Lesnic D., and Mera N.S.(2003). A dual reciprocity boundary element method for the regularized numerical solution of the inverse source problem associated to the Poisson equation. Inverse Problems in Engineering 11(2), 123–139. [27] Franzone P.C., Guerri L., Taccardi B., and Viganotti C.(1985). Finite element approx- imation of regularized solutions of the inverse potential problem of electrocardiography and applications to experimental data. Calcolo 22(1), 91–186. [28] Franzone P.C. and Magenes E. (1979). On the inverse potential problem of electrocar- diology. Calcolo 16(4), 459–538. [29] Friedman A. and Gustafsson B. (1987). Identifcation of the conductivity coeficient in an elliptic equation. SIAM Journal on Mathematical Analysis 18(3), 777–787. [30] Fursikov A. V.(1989). The Cauchy problem for a second-order elliptic equation in a condi- tionally well-posed formulation.Trudy Moskovskogo Matematicheskogo Obshchestva 52, 138–174. [31] Giang, L. T. T. (2024), Determining a source term in an elliptic equation in a cylinder from boundary observations, Preprint IMH20240202, Institute of Mathematics, VAST. [32] Glasko V., Mudretsova E., Strakhov V., Tikhonov A., and Goncharsky A. (1987). In- verse problems in gravimetry and magnetometry. Ill-posed Problems in the Natural Sciences: MIR, 115–129. [33] Grisvard P.(1985). Elliptic Problems in Nonsmooth Domains. Classics in Applied Mathematics, Pitman, Boston. 100 [34] Hadamard J.(1902). Sur les proble`mes aux de´rive´es partielles et leur signification physique Princeton university bulletin, 49–52. [35] Hadamard J. (1923). Lectures on Cauchy’s Problem in Linear Partial Differential Equations. Yale University Press. [36] Ha¨ma¨la¨inen M., Hari R, Ilmoniemi R. J., Knuutila J., and Lounasmaa O.V. (1993). Mag- netoencephalography—theory, instrumentation, and applications to noninvasive studies of the working human brain. Reviews of Modern Physics 65(2), 414–497. [37] Han H. and Reinhardt H.-J. (1997). Some stability estimates for Cauchy problems for elliptic equations. Journal of Inverse and Ill-posed Problems 5(5), 437–454. [38] Han H.D. (1982). The finite element method in the family of improperly posed problems. Mathematics of Computation 38(157), 55–65. [39] Ha`o D.N. (1994). A mollifcation method for ill-posed problems. Numerische Mathe- matik 68(4), 469–506. [40] Ha`o D.N., Duc N.V., and Lesnic D. (2009). A non-local boundary value problem method for the Cauchy problem for elliptic equations. Inverse Problems 25(5), 055002. [41] Ha`o D.N., Giang L.T.T., Kabanikhin S., and Shishlenin M. (2018). A fnite difference method for the very weak solution to a Cauchy problem for an elliptic equation. Journal of Inverse and Ill-Posed Problems 26(6), 835–857. [42] Ha`o D.N., Giang L.T.T., and Oanh N.T.N.(2024). Determination of the right-hand side in elliptic equations. Optimization, 73(4), 1195–1227. [43] Ha`o D.N. and Lesnic D. (2000). The Cauchy problem for Laplace’s equation via the conjugate gradient method. IMA Journal of Applied Mathematics 65(2), 199–217. [44] Ha`o D.N., Van T.D., and Gorenflo R. (1992). Towards the Cauchy problem for the Laplace equation. Banach Center Publications 27(1), 111–128. [45] Hinze, M. (2005). A variational discretization concept in control constrained optimiza- tion: the linear-quadratic case. Computational Optimization and Applications 30, 45–61. [46] Hon, Y. C. and Wei T (2001). Backus-Gilbert algorithm for the Cauchy problem of the Laplace equation. Inverse Problems 17(2), 261–271. 101 [47] Ho¨rmander L. (1983). Uniqueness theorems for 2nd-order elliptic diffrential equations. Communications in Partial Differential Equations 8(1), 21–64. [48] Ho¨rmander L. (2003). The Analysis of Linear Partial Differential Operators I:Distribution Theory and Fourier Analysis. Springer, Berlin, Heidelberg. [49] Hu, S.J. and Yu W.H. (1983). Identifcation of floated surface temperature in floated gyroscope. IFAC Proceedings Volumes 16(10), 347–352. [50] Il’in, V.A. (1960). The solvability of mixed problems for hyperbolic and parabolic equa- tions. Russian Mathematical Surveys 15(2), 85–142. [51] Il’in, V.A. and Shishmarev I.A. (1959). On the connection between the classical and the generalizied solution to Dirichet problem and to the problem of eigenvalues. Doklady Akademii Nauk SSSR 126(6), 1176–1179. [52] Il’in, V.A. and Shishmarev I.A. (1960). On the equivalence of the systems of generalized and classical eigenfunctions. Izvestiya Rossiiskoi Akademii Nauk. Seriya Matem- aticheskaya 24(5), 757–774. [53] Isakov, V. (1990). Inverse Source Problems 34. American Mathematical Society, Prov- idence, Rhode Island. [54] Isakov, V. (2006). Inverse Problems For Partial Differential Equations. Springer, New York. [55] Ivanov, V.K. (1963). On ill-posed problems.Matematicheskii sbornik 103(2), 211–223. [56] Ivanov V K., Vasin V.V., and Tanana V.P. (2002). Theory of Linear Ill-posed Prob- lems and Its Applications. De Gruyter, Berlin, Boston. [57] Johansson T (2004). An iterative procedure for solving a Cauchy problem for second order elliptic equations. Mathematische Nachrichten 272(1), 46–54. [58] John F. (1955). A note on “improper” problems in partial differential equations. Com- munications on Pure and Applied Mathematics 8(4), 591–594. [59] John F. (1960). Continuous dependence on data for solutions of partial differential equa- tions with a prescribed bound. Communication on Pure and Applied Mathematics 13, 551–585. [60] Johnson C. (1987). Numerical Solution of Partial Differential Equations by the Finite Element Method., Cambridge University Press, Cambridge. 102 [61] Johnson C.R. (1997). Computational and numerical methods for bioelectric feld prob- lems. Critical Reviews™ in Biomedical Engineering 25(1), 1–81. [62] Khaidarov A. (1984). A class of inverse problems for elliptic equations. Doklady Akademii Nauk 277(6), 1335–1337. [63] Khaidarov A. (1987). On estimates for and the existence of solutions of a class of inverse problems for elliptic equations. Doklady Akademii Nauk 294(1), 41–43. [64] Khaidarov A. (1990). A class of inverse problems for elliptic equations. Siberian Math- ematical Journal 31(4), 657–666. [65] Kirsch A. (2011). An Introduction to the Mathematical Theory of Inverse Problems. Springer, New York. [66] Klibanov M.V. and Santosa F. (1991). A computational quasi-reversibility method for Cauchy problems for Laplace’s equation. SIAM Journal on Applied Mathematics 51(6), 1653–1675. [67] Kohn, R.V. and Vogelius M. (1985). Determining conductivity by boundary measure- ments II. interior results. Communications on Pure and Applied Mathematics 38(5), 643–667. [68] Ladyzenskaya O.A. (1954). On solvability of the fundamental boundary problems for equations of parabolic and hyperbolic type. Doklady Akademii nauk SSSR 97(3), 395–398. [69] Ladyzhenskaya O. A. (1958). On non-stationary operator equations and their applica- tions to linear problems of mathematical physics. Matematicheskii Sbornik 45(87), 123–158. [70] Ladyzhenskaya O.A. (1985). The Boundary Value Problems of Mathematical Physics. Springer, New York. [71] Landis E.M. (1956). Certain properties of equations of elliptic type. Doklady Akademii Nauk SSSR 107, 640–643. [72] Landis E.M. (1959). Some questions in the qualitative theory of elliptic and parabolic equations. Uspekhi Matematicheskikh Nauk 14(1), 21–85. [73] Landis E.M. (1963). Some problems of the qualitative theory of second order ellip- tic equations (case of several independent variables). Russian Mathematical Surveys 18(1),1–62. 103 [74] Latte`s R., Lions J.L. and Bellman R.E. (1969). The Method of Quasi- reversibility:Applications to Partial Differential Equations. Elsevier, New York. [75] Lavrent’ev M.M. (1957). On Cauchy problem for linear elliptical equations of the second order.Doklady Akademii Nauk 112, 195–197. Russian Academy of Sciences. [76] Lavrent’ev M.M. (1962). Some ill-posed problems of mathematical physics (in Russian) Izdat. Sibirsk. Otdel. Akad. Nauk SSSR. Novosibirsk. [77] Lavrent’ev, M.M. (1967). Some Improperly Posed Problems of Mathematical Physics. Springer, New York. [78] Lavrent’ev M.M., Romanov V.G., and Shishatski S.P. (1986). Ill-posed Problems of Mathematical Physics and Analysis. American Mathematical Society Providence, Rhode Island. [79] Lavrent’ev M.M.(1956). On the Cauchy problem for the Laplace equation. Izvestiya Akademii Nauk SSSR. Seriya Matematicheskaya(in Russian) 20, 819–842. [80] Leitao, A.(1998). Applications of the Backus-Gilbert method to linear and some nonlin- ear equations.Inverse Problems 14(5), 1285–1297. [81] Levine H.A. and Vessella S. (1985). Estimates and regularization for solutions of some ill-posed problems of elliptic and parabolic type.Rendiconti del Circolo Matematico di Palermo 34(1), 141–160. [82] Lions, J.L. (1971). Optimal Control of Systems Governed by Partial Differential Equations. Springer-Verlag, Heidelberg. [83] Lions J.L. and Magenes E. (2012). Non-homogeneous Boundary Value Problemsand Applications: Vol. 1. Springer Berlin, Heidelberg. [84] Medeiros L.A.(1986). Remarks on a non-well posed problem. Proceedings of the Royal Society of Edinburgh Section A: Mathematics 102(1-2), 131–140. [85] Melnikova I.V. and Filinkov A.(2001).Abstract Cauchy Problems: Three Approaches. CRC Press. [86] Mikhailov V.P.(1978). Partial Differential Equations. Mir Publishers, Moscow. [87] Miller K. (1962)Three circle theorems in partial differential equations and applica- tions to improperly posed problems. Ph.D. Thesis, Rice University. 104 [88] Miller K. (1970). Least squares methods for ill-posed problems with a prescribed bound. SIAM Journal on Mathematical Analysis 1(1), 52–74. [89] Mu¨ller C. (1954). On the behavior of the solutions of the differential equation δu = f (x, u) in the neighborhood of a point. Communications on Pure and Applied Math- ematics 7(3), 505–515. [90] Nikolaevich T. and Goncharsky A.V. (1987). Ill-posed Problems in The Natural Sci- ences. Mir Publishers, Moscow. [91] Orlovsky D.G.(1989). An inverse problem for a second-order differential equation in a Banach space. Differentsial’nye Uravneniya 25(6), 1000–1009. [92] Payne L. E. (1960). Bounds in the Cauchy problem for the Laplace equation. Archive for Rational Mechanics and Analysis 5(1), 35–45. [93] Payne L.E. (1970). On a priori bounds in the Cauchy problem for elliptic equations. SIAM Journal on Mathematical Analysis 1(1), 82–89. [94] Payne L.E. (1975). Improperly Posed Problems in Partial Differential Equations. SIAM, Philadelphia. [95] Payne L.E. and Sather D. (1967). On some improperly posed problems for the Chaplygin equation. Journal of Mathematical Analysis and Applications 19(1), 67–77. [96] Prilepko A.I (1967). Inverse problems of potential theory.Differentsial’nye Uravneniya 3(1), 30–44. [97] Prilepko A.I (1973). Inverse problems of potential theory (elliptic, parabolic, hyperbolic, and transport equations). Matematicheskie Zametki 14(5), 755–767. [98] Prilepko A.I., Orlovsky D.G., Vasin I.A., et al. (2000). Methods for Solving Inverse Problems in Mathematical Physics. CRC Press, Boca Raton. [99] Pucci C. (1955). “Sui problemi di Cauchy non ben posti,”(“on a non-well-posed problem of Cauchy”). Atti della Accademia Nazionale dei Lincei 18, 473–477. [100] Pucci C. (1982). Problemi non ben posti per l’equazione delle onde. Rendiconti del Seminario Matematico e Fisico di Milano 52, 473–484. [101] Samarskii A. and Nikolaev E. (1989). Numerical Methods for Grid Equations. Birkha¨user, Verlag. 105 [102] Samarskii A.A.(2001). The Theory of Difference Schemes. CRC Press, Boca Raton. [103] Samarskii A.A. and Vabishchevich P.N. (2007). Numerical Methods for Solving In- verse Problems of Mathematical Physics. Walter de Gruyter, Berlin. [104] Samarskij A.A. and Nikolaev E.S. (2012).Numerical Methods for Grid Equa- tions:Volume II Iterative Methods. Birkha¨user, Basel. [105] Stoer J., Bulirsch R., Bartels R., Gautschi W., and Witzgall C. (1980). Introductionto Numerical Analysis. Springer, Heidelberg. [106] Sun Z. (1989). On an inverse boundary value problem in two dimensions. Communi- cations in Partial Differential Equations 14(8-9), 1101–1113. [107] Sylvester J. and Uhlmann G. (1988). Inverse boundary value problems at the boundary- continuous dependence. Communications on Pure and Applied Mathematics 41(2), 197–219. [108] Tataru, D. (1996). Carleman estimates and unique continuation for solutions to bound- ary value problems. Journal de Mathe´matiques Pures et Applique´es 75(4), 367–408. [109] Tikhonov, A.N. (1963). On the solution of ill-posed problems and the method of regu- larization. Doklady Akademii Nauk 151, 501–504. [110] Tikhonov A.N. et al. (1943). On the stability of inverse problems. Doklady Akademii Nauk 39, 195–198. [111] Tro¨ltzsch F (2010). American Mathematical Society Optimal Control Of Partial Dif- ferential Equations: Theory, Methods, and Applications., Providence, Rhode Island. [112] Vabishchevich P. (1981). Nonlocal parabolic problems and the inverse heatconduction problem. Differensial’nye Uravneniya Equations 17(7), 1193–1199. [113] Vabishchevich, P.N. (1979). Numerical solution of the Cauchy problem for elliptic equa- tions and systems. Vestnik Moskovskogo Universiteta. Seriya XV. Vychislitelnaya Matematika i Kibernetika, 3–10. [114] Vabishchevich, P.N. (1982). Uniqueness of the solution of an inverse problem for deter- mining the right-hand side of an elliptic equation.Differentsial’nye Uravneniya 18(8), 1450–1453. [115] Vabishchevich, P. N. (1983). Numerical solution of nonlocal elliptic problems.Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika(5), 13–19. 106 [116] Vabishchevich, P. N. (1985). Inverse problem of fding the 2nd member of an elliptic equation and its numerical solution. Differential Equations 21(2), 201–207. [117] Vabishchevich P. N. (1988). Uniqueness of some inverse problems for elliptic equations. Differentsial’nye Uravneniya 24(12), 2125–2129. [118] Vabishchevich P.N. (2007). Reconstruction of the right-hand side of the elliptical equa- tion from observation data obtained at the boundary. Kragujev J Math 30, 45–58. [119] Vabishchevich P.N. and Denisenko A.Y. (1993). Regularization of nonstationary prob- lems for elliptic equations. Journal of Engineering Physics and Thermophysics 65(6), 1195–1199. [120] Vabishchevich, P.N. Glasko V.B., and Kriksin Y.A. (1979). Solution of the Hadamard problem by a Tikhonov-regularizing algorithm. USSR Computational Mathematics and Mathematical Physics 19(6), 103–112. [121] Vabishchevich, P.N. and Pulatov P.A. (1984). A method of numerical solution of the Cauchy problem for elliptic equations. Vestnik Moskovskogo Universiteta. SeriyaXV. Vychislitelnaya Matematika i Kibernetika 38. [122] Wang G., Li Y. and Jiang M. (2004). Uniqueness theorems in bioluminescence tomog- raphy. Medical Physics 31(8), 2289–2299. [123] Yu W.(1993). On determination of source terms in the 2nd order linear partial differ- ential equations. Acta Mathematica Scientia 13(1), 23–32. [124] Yu W. (1994). Well-posedness of determining the source term of an elliptic equation. Bulletin of the Australian Mathematical Society 50(3), 383–398. [125] Yu Emanuilov, O. (1994). A class of inverse problems for semilinear elliptic and parabolic equations. Transactions of the Moscow Mathematical Society 55, 221–238. 107