một số bài toán ngược cho phương trình elliptic với quan sát biên some inverse problems for elliptic equations with boundary observations

Some numericalexamples are presented for showing the efficiency of the method.ob-Chapter 3 is devoted to the problem of determining a term in the right-hand side of ellipticequation with

VIETNAM ACADEMY OF SCIENCE AND TECHNOLOGY

INSTITUTE OF MATHEMATICS

LE THI THU GIANG

MỘT SỐ BÀI TOÁN NGƯỢC CHO PHƯƠNG TRÌNH ELLIPTIC VỚI QUAN SÁT BIÊN

(SOME INVERSE PROBLEMS FOR ELLIPTIC EQUATIONS WITH

BOUNDARY OBSERVATIONS)

THESIS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY IN MATHEMATICS

HA NOI – 2024

VIETNAM ACADEMY OF SCIENCE AND TECHNOLOGY

INSTITUTE OF MATHEMATICS

LE THI THU GIANG

MỘT SỐ BÀI TOÁN NGƯỢC CHO PHƯƠNG TRÌNH ELLIPTIC VỚI QUAN SÁT BIÊN

(SOME INVERSE PROBLEMS FOR ELLIPTIC EQUATIONS WITH

HA NOI – 2024

This thesis is devoted to the Cauchy problem and the problem of determining the hand side for second-order linear elliptic equations with boundary observations

right-The Cauchy problem for elliptic equations is studied in Chapter 1 right-There we introduce

a new concept of very weak solution to the Cauchy problem for elliptic equations and for anon-local boundary value problem which regularizes the Cauchy problem We discretize thenon-local boundary problem by the finite difference method and prove the stability of thescheme and its convergence We present some numerical examples for showing the efficiency

of the regularization method

In Chapter 2, we study the source identification problem for elliptic equations with servation on the boundary We formulate the inverse problem as an operator equation andregularize it by the Tikhonov regularization method To discretize the problem, we followHinze’s idea on variational discretization and suggest a rule of choosing the regularizationparameter depending on the noise level in the observation data and the discretization meshsize which yields the optimal convergence rate This abstract result is applied to the finiteelement method for numerical solving the source identification problem Some numericalexamples are presented for showing the efficiency of the method

ob-Chapter 3 is devoted to the problem of determining a term in the right-hand side of ellipticequation with constant and variable coefficients in a cylinder from boundary observations.Based on the special form of the considered equation in a cylinder, the solution of the directand inverse problems can be represented by the Fourier series Since the source problem isill-posed, we regularize it using the truncated Fourier series method and propose a methodfor selecting the number of Fourier coefficients to ensure the convergence of the method and

to indicate the rate of convergence To demonstrate the efficiency of the method, we presentseveral numerical examples

Trong Chương 2, chúng tôi nghiên cứu bài toán xác định nguồn cho các phương trình elliptic với quan sát trên biên Bài toán này có thể viết dưới dạng phương trình toán tử Chúng tôi đã sử dụng phương pháp hiệu chỉnh Tikhonov để chỉnh hóa phương trình toán tử và rời rạc nó dựa trên ý tưởng rời rạc biến phân (variational discretization) của Hinze, rồi sau đó đề xuất một quy tắc chọn tham số hiệu chỉnh phụ thuộc vào mức độ nhiễu trong dữ liệu quan sát và bước lưới để nhận được tốc độ hội tụ tối ưu Kết quả đã này được áp dụng cho phương pháp phần tử hữu hạn để giải số bài toán xác định vế phải và được thực hiện bằng số trên máy tính.

Trong Chương 3, chúng tôi xét bài toán xác định vế phải của một phương trình eliptic với hệ số hằng và hệ số biến thiên trong hình trụ từ các quan sát trên biên Dựa vào cấu trúc đặc biệt của phương trình đang xét và miền hình trụ, lời giải của bài toán thuận và bài toán ngược có thể biểu diễn qua chuỗi Fourier Do bài toán đặt không chỉnh nên chúng tôi hiệu chỉnh nó bằng phương pháp chặt cụt chuỗi Fourier và đưa ra cách chọn số các hệ số Fourier để phương pháp hội tụ và chỉ ra tốc độ hội tụ Cuối mỗi chương, chúng tôi trình bày một số ví dụ số để thể hiện tính hiệu quả của các phương pháp đã đề xuất.

This work has been completed at Institute of Mathematics, Vietnam Academy of Science and Technology under the supervision of Prof Dr Habil Đinh Nho Hào I declare hereby that the results presented in it are new and have never been published elsewhere.

Author: Lê Thị Thu Giang

First and foremost, I want to express my deepest gratitude to my advisor, fessor Đinh Nho Hào for his invaluable help and support in my research He spent a lot of time and energy on my work as well as giving me many valuable opinions and comments, and he gradually guided me to get acquainted with my scientific research work For me, he is not only an extremely respected teacher but also my second fa- ther, who always gives me a lot of support and patience, encourages me to overcome difficulties in my study and my daily life.

Pro-I would like to express my special appreciation to Professor Hoàng Thế Tuấn, Dr Đào Quang Khải, Dr Lương Thái Hưng and other members of the weekly seminar at Department of Differential Equations for many interesting discussions Also, I would like to thank Professor Nguyễn Văn Đức (Vinh University), Professor Nguyễn Trung Thành (Rowan University, USA) and all of friends in Professor Đinh Nho Hào’s group seminar for their valuable comments and suggestions to my research papers.

Thanks go to my sincere friend, Dr Nguyễn Thị Ngọc Oanh (College of Sciences, Thai Nguyen University) for her listening, offering me advice, and supporting me through my entire study process I am very grateful to Dr Phan Xuân Thành (Hanoi University of Science and Technology) for his kind help on computer programming and giving useful comments for improving my PhD thesis.

My great appreciation is also expressed to the leaders of the Institute of matics, the Center of Postgraduate Training and the International Center of Research and Postgraduate Training in Mathematics for providing me with such an excellent study environment as well as financial support during my PhD study.

Mathe-I would like to thank the leaders of Thuong mai University, the Dean, as well as all of my colleagues at the Faculty of Mathematical Economics and my friends for their encouragement and support throughout my PhD study.

Last but not least, this journey would not have been possible without the support

of my family I would like to express my sincere gratitude to my parents, my in-law, my husband, my children, my brothers and sisters for their unconditional love and encouragement to me Especially, this thesis is dedicated to my beloved father who was my first teacher inspiring me to study Mathematics Although he passed away, his boundless love, trust and hope for me are my motivation to complete this work.

parents-List of Figures

1.1 Example 1.5.1 Exact solution 33

1.2 Example 1.5.1 Numerical solution with p=0.01, a priori method 33

1.3 Example 1.5.1 Approximation error with p=0.01, a priori method 34

1.4 Example 1.5.1 Approximation error with p=0.035, a priori method 34

1.5 Example 1.5.1 Comparison of the exact solution with numerical solutions for various p, a priori method 34

1.6 Example 1.5.1 Comparison of the approximation errors of the numerical so-lutions for various p, a priori method 34

1.7 Example 1.5.1 Exact and approximate solutions at T = 1 for various p, a posteriori method 35

1.8 Example 1.5.1 Approximation errors at T = 1 for various p, a posteriori method 35

1.9 Example 1.5.2 Exact and approximate solutions at T = 1 for various p in case u(x, T) is a hat function 36

1.10 Example1.5.2 Approximation errors at T = 1 for various pin caseu(x, T) is a hat function 36

1.11 Example 1.5.3 Exact and approximate solutions at T = 1 for various p, x ∈ [0, π] 37

1.12 Example 1.5.3 Approximation errors at T = 1 for various pwith x ∈[0, π] 37

1.13 Example 1.5.3 Exact and approximate solutions at T = 1 for various p, x ∈ [0,5π] 37

1.14 Example 1.5.3 Approximation errors at T = 1 for various p, x ∈[0,5π] 37

2.1 Example 2.4.1 Reconstruction of the smooth source function f1(x1) 652.2 Example 2.4.1 Reconstruction of the continuous and nonsmooth source func-tion f2(x1) 652.3 Example 2.4.1 Reconstruction of the discontinuous source function f3(x1) 662.4 Example 2.4.1 Exact and approximation solutions with f∗ = 0,10 and 20 672.5 Example 2.4.2 f20x1 +x 2 and its numerical solution with f∗ close tof20x1 +x 2 692.6 Example 2.4.2 f0 and the numerical solution with f∗ close to f0 702.7 Example 2.4.2 f−5 sin πx2and the numerical solution withf∗close tof−5 sin πx2.

712.8 Example 2.4.3 Numerical results for f satisfying the source condition (2.29)with θ = 1/2 Relative noise = 0.01%. 732.9 Example 2.4.4 Numerical results for f satisfying the source condition (2.29)with θ = 1 Relative noise = 0.01% 753.1 Example 1 Solutions with different noise levels for smooth function, M = 15 913.2 Example 3.3.1 Errors with different perturbations for smooth function,M = 15 913.3 Example 3.3.1 Solutions with different number of Fourier coefficients, smoothfunction, p= 7% 913.4 Example 3.3.1 Errors with different number of Fourier coefficients for smoothfunction, p= 7% 913.5 Example 3.3.2 Solutions with different noise levels for continuous, non-smoothfunction, M = 15 923.6 Example 3.3.2 Errors with different noise levels, for continuous, non-smoothfunction, M = 15 923.7 Example 3.3.2 Solutions with different number of Fourier coefficients, p= 7% 933.8 Example 3.3.2 Errors with different number of Fourier coefficients, p= 7% 933.9 Example 3.3.3 Solutions with different noise levels for discontinuous function,

M = 20 943.10 Example 3.3.3 Errors with different noise levels,for discontinuous function,

M = 20 943.11 Example 3.3.3 Solutions with different number of Fourier coefficients, p= 5% 94

3.12 Example 3.3.3 Errors with different number of Fourier coefficients for tinuous function, p= 5% 94

discon-List of Tables

1.1 Example 1.5.1 Regularization parameters chosen due the a posterior method

for various p 34

1.2 Example 1.5.2 Regularization parameters with the exact solutions atT being a hat function 36

1.3 Example 1.5.3 Regularization parameters with the exact solutions atT being a step function 37

1.4 Example 1.5.3 The L2-norm of errors 38

2.1 Example 2.4.1 L2-error in the smooth case 66

2.2 Example 2.4.1 L2-error in the non-smooth but continuous case with α = 10−5 66 2.3 Example 2.4.1 L2-error in the discontinuous case with different regularization parameters 66

2.4 Example 2.4.1 L2-error between the exact solution and numerical ones with relative noise 0.1%,1% for f∗ = 0,10, and 20 67

2.5 Example 2.4.2 L2-errors withf∗ close to exact solutions 68

2.6 Example 2.4.3 L2-error behavior - the source condition 1 72

2.7 Example 2.4.4 L2-error behavior - the source condition 2 75

3.1 Example 3.3.1 The L2-norm of relative errors for smooth function 90

3.2 Example 3.3.2 TheL2-norm of relative errors - non-smooth, continuous func-tion 92 3.3 Example 3.3.3 The L2-norm of relative errors for the discontinuous function 93

C(Q) space of continuous functions on Q;

Ck(Q) space of k-times continuously differentiable functions on Q;

C0∞(Q) space of infinitely differentiable functions

with compact support in Ω;

L2(Ω), L2(S), L2(Q) space of measurable, square-integrable functions in Ω

⟨, ⟩H inner product in Hilber space H;

∥.∥H norm in Hilber space H;

CG Conjugate Gradient Method;

FEM Finite Element Method;

FDM Finite Difference Method

1 The Cauchy problem for elliptic equations 11

1.1 Problem settings and the non-local boundary problem regularization method 11

1.2 The very weak solution to the Cauchy problem for elliptic equations 14

1.3 The well-posedness of the non-local boundary value problem for elliptic equations 18 1.3.1 The very weak solution 18

1.3.2 The classical solution 20

1.4 Finite difference scheme for the non-local boundary value problem 23

1.4.1 The three-dimensional problem 24

1.4.2 The two-dimensional problem 31

1.5 Numerical examples 32

2 The source term determination in the Robin boundary problem for elliptic equations 39 2.1 Problem setting 40

2.2 The inverse source problem and its abstract setting 40

2.2.1 The inverse problem as an operator equation 40

2.2.2 Example on the instability 42

2.2.3 The non-uniqueness of the solution 45

2.2.4 Discretization of linear ill-posed problems 45

2.2.5 The Tikhonov regularization parameter selection rule 49

2.3 The inverse source problem and its finite element approximation 53

2.3.1 The inverse problem 53

2.3.2 Conjugate Gradient Method 55

2.3.3 Finite element method for the direct problem 57

2.3.4 Regularized discretized variational problem 62

2.4 Numerical examples 62

3 Determining a source term in an elliptic equation in a cylinder from boundary observations 76 3.1 Problem setting 76

3.1.1 A representation of the solution 79

3.1.2 Regularization by the truncated Fourier series 81

3.2 A special case of equations with constant coefficients in a parallelepiped 83

3.2.1 Problem setting 83

3.2.2 A representation to the solution 84

3.2.3 Regularization by the truncated Fourier series 87

3.3 Numerical examples 90

• the domain in which the process is studied;

• the equation (including the parameters and the relations) that describes the process;

• the initial conditions (if the process is nonstationary);

• the conditions on the boundary of the domain

If all the inputs, parameters and relations between them are known in a process, and we want

to know how the process look like, we have a direct problem The difficulties of solving theseproblems cause by their ill-posedness Following Hadamard, [35], a problem is well-posed if

• there exists a solution,

• the solution is unique,

• the solution continuously depends on the data in an appropriate topology

Once, at least one of these conditions is violated we say that the problem is ill-posed In

1902 Hadamard [34] considered the Cauchy problem for the Laplace equation

He showed that ifφ and ψ are continuous, then there exists a solution to the problem (0.1)

ψ(ξ) log | x − ξ | dξ (0.2)

is analytic in a < x < b Thus, not for every Cauchy data φ and ψ there exists a solution

to the Cauchy problem for the Laplace equation, and if there exists one some compatibilitycondition between them has to be satisfied Further, in 1917 Hadamard [35] remarked that,with the Cauchy data φn(x) = 0, ψn(x) =ne−

sin(nx) sinh(ny).

It is clear that, fora= 0, b=π, although∥φn∥L∞ (0,π), ∥ψn∥L∞ (0,π) → 0,∥un(., y)∥L∞ (0,π) →

∞, asy >0, n → ∞ Thus, the solution does not continuously depend on the Cauchy data.Hence, the problem is ill-posed

At that time, Hadamard thought that ill-posed problems have no physical meaning ever, many problems in science, technology and practice lead to ill-posed problems (see[32, 34, 54, 78, 94]) To our best knowledge, A N Tikhonov, [90, 109, 110], M M Lau-vrent’ev, [75, 77–79], F John [59], C Pucci [99], V K Ivanov [55, 56] are the first whostudied ill-posed problems From their studies, one can see that while the first two condi-tions of the well-posedness conditions can be overcome by imposing an additional data orrestricting the set of admissible solutions, the third condition is very hard to overcome Sincethere always contain random noise in the data generated during the measurement process,the proposed problem cannot be satisfied exactly Even when the noise level in a measuredoutput data is small, many algorithms developed for well-posed problems do not work forunstable problems because of round-off errors Due to the instability of the problems, a smallperturbation in the data can cause very large errors in the numerical solutions Therefore,

How-if the stability condition does not hold, it is impossible to directly apply traditional tational methods for ill-posed problems This is the reason why regularization methods play

compu-a centrcompu-al role in the theory compu-and compu-appliccompu-ations of inverse compu-and ill-posed problems The purpose

of regularization methods is to approximate the ill-posed problems by a series of well-posedproblems such that the solutions of these problems converge to the exact solution

Two popular regularization methods are proposed by Lavrent’ev [76] and Tikhonov [109]

A typical inverse problem can be written as the operator equation Af = g, where f is theunknown which we have to seek, g is the data which is perturbed by noise (so, in practice,this datum is replaced by gϵ) and A is a compact operator mapping from a Hilbert space

X to a Hilbert space Y The idea of Lavrent’ev was that, instead of taking fϵ = A−1gϵ as

a solution, he approximated it by (A+αI)−1gϵ with α = α(ϵ) > 0 being a regularization

parameter such that whenϵ →0, then α →0 and fϵ → f.

In 1963, Tikhonov [109] published the well-known Tikhonov regularization method byminimizing the following functional with respect to f

• The regularization, least squares methods [81, 88, 109, 112, 119, 120]

Among inverse problems in PDEs, those for elliptic equations are of great interest to manyresearchers because of their crucial role in practice These problems have been studied fordecades, yielding a wealth of interesting results in both quantitative and qualitative aspects.For surveys on inverse problems for elliptic equations, we refer to [17, 21, 29, 67, 106, 107]for identifying coefficients, [49] for identifying boundary values, [62, 83, 91, 96, 97, 116, 124]for identifying source terms of elliptic equations and [25, 123] for computational methods.However, despite a vast literature for inverse problems for elliptic equations, those withobservations on the boundary are not many The practical situation is that the data of

a concerned physical process could be measured only on the boundary (or a part of theboundary) and we have to determine what the process is in the domains or at the otherparts of the boundary For example, in Medicine, to determine the electroencephalogram orelectrocardiogram, people use the voltage and current measured on the skull or chest andthen solve a Cauchy problem for the corresponding elliptic equations, [19, 27, 28, 58]

In Chapter 1 of this thesis, we are interested in the Cauchy problem for elliptic equations

with c >0 and a= a(x)≥ 0 a.e in Ω

The Cauchy problem for second-order linear elliptic equations occurs in many real-worldproblems For example, to figure out what is happening with a field (gravity, electricity,magnetism) in unseen areas, we often have to determine the potential of the field in anexternal part of the mass (charge, current) creating the field, from observations in a part ofthis domain This lead to a problem of extending an analytic vector field inR3or a curve inR2

to a harmonic vector field outside with determination of its singularities [44] Some literaturereviews of the Cauchy problems for elliptic equations can be found in [28, 44, 54, 61, 94].Generally, the problems are proposed as follows: Let Ω be a domain in Rn, n ≥ 2, withsmooth enough boundary Consider the problem of determining u=u(x) such that

(ν, xi) is the angle formed by the outer normal vector of∂Ω andxi It seems that Lavrent’evwas the first who studied the Cauchy problem for elliptic equations from the viewpoint ofill-posed problems [75, 77–79] and up to now there have been many papers devoted to thisproblems Lavrent’ev established stability estimates of H¨older type for Cauchy problemsfor the Laplace equations and then generalized it for second-order elliptic equations Similarresults were given by Landis in [71–73] by using different methods In 1958, Pucci studied theconditional stability of the problems with assuming that the solutions are non-negative Afterthat, Miller [87, 88], Payne [94], Falk and Monk [24], Han [37, 38], Fursikov [30], etc., usedlogarithm-convex method; Payne [92, 93] presented the weighted function method to achievestability estimates results for these problems The most popular and effective method forgetting stability estimates for ill-posed problems is the Carleman estimate method, see [14,

18, 20, 48, 54, 108] From these studies, the stability estimate has been partly proved for someclasses of the Cauchy problems However, in general, the solutions of the Cauchy problemsfor elliptic equations are unstable That means, a small perturbation in the Cauchy data canlead to an arbitrary error in the solution, so, it is very difficult to apply numerical methods

to solve them The reason is that besides perturbations in measuring, there are unadvoidableerrors due to the discretization process and the round-off errors of the computers Therefore,finding stable and effective algorithms for the problems desired There have been many papers

on regularization and approximation methods for Cauchy problems for elliptic equations

so far In 1955, C Pucci [100] suggested a method to approximate the solution of theCauchy problem for the Laplace equation in a stable way Later, in 1956, Lavrent’ev [79]proposed two methods to regularize the Cauchy problems for the Laplace equation The firstbased on the Fourier method and the regularization led to determining the coefficients of theFourier series However, these methods of Pucci and Lavrent’ev were not feasible in practice.The second was Lavrent’ev regularization that mentioned above In 1963, Tikhonov [109]created the so-called Tikhonov regularization method to stabilize ill-posed problems and sincethen the theory of ill-posed problems has been intensively developed Besides regularizationmethods mentioned above, many other methods are proposed including: Quasi-reversibilitymethod [11, 12, 66, 74], Iterative method [23, 43, 57]; Backus-Gilbert method [46, 80], Finitedifference method [9, 103, 113], Mollification method [39], Non-local boundary value problemmethod [85, 115, 119, 121], etc However, there is no universal method for all the Cauchyproblem for elliptic equations Therefore, developing a new method for special classes ofelliptic equations is desired

In [40], to solve the Cauchy problem

withγ >1 being given andα >0 the regularization parameter Furthermore, they suggested

a priori and a posteriori parameter choice rules to yield order-optimal convergence rates.These results are devoted for the abstract equations, hence, to apply them to the Cauchyproblem (0.3) we have to define the solution to it as well the solution to its regularizedequation

et al is that they suggested the priori and posteriori strategies of choosing regularizationparameter depending on the perturbation such that the numerical solutions tend to theexact solution with optimal convergence rate However, the exact solutions defined in [40]are classic solutions, which demand high smoothness of the data Furthermore, it seems thatnumerical methods for the non-local boundary value problem (0.8) are not developed Wetherefore introduce concept of very weak solution to the Cauchy problem for elliptic equationswith Cauchy data inL2(Ω) and very weak solution to the non-local boundary value problem(0.8) and then suggest a stable finite difference scheme for it We note that the notion of veryweak solution has been introduced by Ladyzhenskaya for mixed boundary value problems forparabolic and hyperbolic equations [68, 69] and by Lions [82, 83] for another equations

The next problem is studied in this thesis is the problem of determining a source termfor linear elliptic equations:

(Ω)∩ C( ¯Ω) However, if this term is independent of one of the spatialvariables, with some more conditions, the uniqueness of solution can be established, see

e.g [26, 97, 114, 124] In [97] Prilepko proved that the source term of the Poisson equationcan be uniquely determined if it is independent of one of the variables and is monotone.Vabishchevich [114] also proved that determining the source term is unique but it mustsatisfy some curious conditions In [124], W Yu considered the problem of determining(u, f) from

(0.14)

with observation at the final point ofy:

y|y=Y =φ4(x), x ∈ Ω, (0.15)where

1 h(x, y) is strictly positive inQ

2 h is monotonic with respect to y, that is ∂yh(x, y)≥ 0 and∂yh ̸= 0

W Yu [124] proved that results under the weaker assumptions Specifically, Yu proved that

if the problem satisfied one of following conditions:

1 h(x, y)>0 almost everywhere in Q and ∂yh(x, y)≥ 0 and ∂yh(x, y)̸= 0,

2 h(x, y)≥ α >0 and has a derivative bounded from below with respect to y,

3 f does not depend upon y, that is∂yf = 0 when f >0,

then the existence in the above-mentioned inverse problem in the same space as that inthe investigation of uniqueness is obtained, using the theory of solvable operators betweenBanach spaces Furthermore, he obtained also the continuous dependence of the solution tothe inverse problem on data Hence, problem (0.14)-(0.15) is well-posed in sense of Hadamard

He also noted that the results in [124] could be applied for the similar problems with Dirichlet

or Neumann boundaries

In general, however, the source determination problem for elliptic equations is an ill-posedproblem So, we need some appropriate definition to ensure the uniqueness of solution incertain sense and regularization methods to approximate it by a well-posed problem Since in-verse problem (0.9)-(0.10) may have many solutions, we introduce the so-calledf∗-minimumnorm solution which is nearest the a-priorif∗ among all the least squares solutions to it Wewill give a specific example to show that the solution to the above inverse source problem isunstable Then the problem is reformulated in an abstract setting and using Tikhonov reg-ularization for solving it This leads to an optimal control problem with control constraints.

To solve the problem numerically we discretize the regularized problem by finite-dimensionalproblems based on Hinze’s variational discretization concept in optimal control [45] This

is a new discretization concept for optimal control problems with control constraints Itskey feature is not to discretize the space of admissible controls but to implicitly utilize thefirst order optimality conditions and the discretization of the state and adjoint equationsfor the discretization of the control For discrete controls obtained in this way an optimalerror estimate is proved However, we go a little further than that for optimal control byHinze, namely, we suggest a choice of the regularization parameter depending on the noiselevel in the observation data and the discretization mesh size which yields the convergence ofthe solution to the discretized regularized problem to the solution of the continuous inverseproblem as these quantities tend to zero This is one of the main contributions of Chapter

2 Furthermore, with this choice a convergence rate is also established

As problem (0.14)-(0.15) has a special form, its solution can be represented by Fourierseries Therefore, instead of using Tikhonov regularization method for it, in Chapter 3 weshall apply the standard truncated Fourier series method There we prove error estimates ofthe method and present some numerical examples for showing its efficiency

This thesis is organized as follows:

Chapter 1 is devoted to the Cauchy problem for elliptic equations There we introduce

a new concept of very weak solution to the Cauchy problem for elliptic equations and for anon-local boundary value problem which regularizes the Cauchy problem We discretize thenon-local boundary problem by the finite difference method and prove the stability of thescheme and its convergence We present some numerical examples for showing the efficiency

of the regularization method

In Chapter 2, we study the source identification problem for elliptic equations with servation on the boundary We formulate the inverse problem as an operator equation andregularize it by the Tikhonov regularization method To discretize the problem, we followHinze’s idea on variational discretization and suggest a rule of choosing the regularization

ob-parameter depending on the noise level in the observation data and the discretization meshsize which yields the optimal convergence rate This abstract result is applied to the finiteelement method for numerical solving the source identification problem Some numericalexamples are presented for showing the efficiency of the method.

Chapter 3 is devoted to the problem of determining a term in the right-hand side of ellipticequation with constant and variable coefficients in a cylinder from boundary observations.Based on the special form of the considered equation in a cylinder, the solution of the directand inverse problems can be represented by the Fourier series As the problem is ill-posed,

we regularize it by truncating the Fourier series We prove error estimates of the methodand present some numerical examples for showing its efficiency

it by the finite difference method as well as prove the stability of the scheme and thus itsconvergence Some numerical examples are given for illustrating the efficiency of the method.This chapter is written on the basis of the paper [41].(1).

problem regularization method

Let Ω be a bounded domain inRnandT be a given positive number Letaij ∈ C(Ω), a ∈

withc > 0 anda =a(x)≥ 0 a.e in Ω In this chapter we consider following Cauchy problem

The exact solution to (1.4) is u(x, t) = 1nsin (nx) cosh(nt) We can see that, as n tends

to infinity, ∥φ(.)∥L∞ (0,π) → 0, while ∥u(., t)∥L∞ (0,π) → ∞ Therefore, the stability of thesolution to problem (1.3) does not hold, indicating that problem (1.3) is ill-posed So, it

is reasonable to apply a regularization method for it In [40], H`ao et al. (see also [115])proposed to regularize this problem by the non-local boundary problem (see [10]):

suggested a rule of choosing the regularization parameterα =α(ϵ) so that when ϵ →0 then

vα(t)→ u(t) with the optimal convergence rate as following results:

Denote by vα(x, t) ∈ C2((0, T), L2(Ω))∩ C([0, T]) the solution to (1.5) with the larization parameterα,u(., t) the solution to (1.2) and ∥ · ∥ :=∥ · ∥L2 (Ω)

regu-Theorem 1.1.1 (A priori parameter method choice) Suppose that γ > 1 If there exists a positive number E such that

∥u(·, T)∥ ≤ E, (1.6)

then with α= (Eϵ)γ we have

∥u(·, t)− vα(·, t)∥ ≤ Cϵ1−TtETt, ∀t ∈ [0, T]. (1.7)

Here, C is a finite constant depending only on γ and T

Theorem 1.1.2 (A posteriori parameter method choice) Let ϵ < ∥φ∥ and γ > 1 Taking τ∗ >1 such that 0< τ∗ϵ < ∥φ∥, if there exists a positive number E such that

Remark 1.1.1. Suppose that A admits an orthonormal system of eigenfunctions vk ∈

L2(Ω): Avk =λkvk such that 0< λ1 ≤ λ2 ≤ , lim

−βt/T

, t ∈[0, T] Remark 1.1.2. The question what does the constraint (1.10) mean remains open in [40]and in many other papers on ill-posed problems We now give a partial answer to this question

by using Lemma 1.3.3 below Due to this result, if Ω is a connected, bounded domain inRnand aij ∈ Cℓ

(Ω), a ∈ Cℓ−1(Ω) with a positive integer ℓ, then foru(x, T) = Φ such that

Aℓ+12 Φ
2dx ≤ E1 forℓ odd,

yields the constraint (1.10) forβ = ℓ −1

Remark 1.1.3. (i) The results of Theorem 1.1.1 are valid for γ = 1

(ii) The two above parameter choosing methods are of optimal order

(iii) The a posterior parameter choice in Theorem 1.1.2 is equivalent to the equation

for elliptic equations

Let D be a bounded domain in Rk with boundary ∂D consisting of two non-intersectedparts S0 and S1 Let the functions bij ∈ C(D), i, j = 1 , k, b ∈ L∞(D) satisfying thecondition of uniform ellipticity

with b >0 being a positive constant and b ≥0 a.e x ∈ D Consider the Cauchy problem

It is common that the solution to the first equation in the Cauchy problem (1.11) issought in the space H1(D) Therefore, if S0 is not too rough, φ must be in H1/2(S0).However, in practice this Cauchy datum should be inL2(S0) and so it is reasonable to find

u in L2(D) But, in this case there is no trace of u on S0 To give a precise definition forthe solution in L2(D), multiplying the both side of the first equation of (1.11) by a smoothfunctionψ defined on D then formally taking integration by parts, we get

As u and ∂u/∂N are given on S0 and not known on S1, to make the above expressiondefined, we suppose that ψ solves the Cauchy problem

0 =

Z

D

(u1− u2)pdx (1.17)for all p ∈ P Since P is dense in L2(D), it follows thatu1− u2 = 0 in L2(D)

Now we turn to the Cauchy problem:

Z

Ω

(utt − Au)ψdxdt=

Z T 0

Z

∂Ω

u ∂ψ

∂N dξ.

In this formal expression, letp ∈ L2(Ω×(0, T)) and ψ ∈ H1(Ω×(0, T)) be a solution tothe adjoint Cauchy problem

Z

Ω

f ψdx, (1.20)

for all p ∈ L2(Ω ×(0, T)) for which (1.19) has a unique solution Suppose that for any

p ∈ P, a dense set in L2(Ω×(0, T)), there exists a unique solution in H1(Ω×(0, T)) toproblem (1.19) Then, similarly to problem (1.11), we can conclude that all items in theboth sides of (1.20) have a meaning and the very weak solution to (1.18) is unique We nowpresent a representation to the solution In doing so, consider the eigenvalue problem







− Pn i,j=1(aijuxi)xj +au=λu, x ∈ Ω, u|∂Ω = 0.

with φk = RΩφ(x)vk(x)dx and φ1k = RΩφ1(x)vk(x)dx converges in L2(Ω) for almost

t ∈(0, T), it is the very weak solution to (1.18) It is clear that in this case φ1 and φ2 arevery smooth (see below), however, as we see from the condition (0.2), in general the Cauchydataφ and φ1 are not independently chosen Hence, a sufficient and necessary condition forthe solvability of the Cauchy problem (1.18) is not clear to us

When f = 0 andφ1 = 0, we have,

value problem for elliptic equations

In this section we analyze the solvability of problem (1.5) Since the parameter γ plays

no role in this part, we suppose that it equals to 1 As for the Cauchy problem (1.18), if thesolution to this problem is sought inH1(Ω×(0, T)), thenu(·,0) =φ(·) must be inH1/2(Ω),while this datum is given in L2(Ω) Therefore, we have to introduce a new definition for asolution to this problem

From the formal formula (1.14) we introduce the following notion to a solution to problem(1.5)

Definition 1.3.1. A function v ∈ C([0, T], L2(Ω)) is called the very weak solution toproblem (1.5) if it satisfies

for all ψ ∈ H1(Ω×(0, T)) such that ψtt− Aψ ∈ L2(Ω×(0, T)) andψ|∂Ω =ψ(x, T) = 0

Theorem 1.3.1 Let φ be a function in L2(Ω) Then, the exists a unique very weak solution to problem (1.5) which can be represented in the form

Here, vk are orthonormalized in L2(Ω) eigenvalues corresponding to the eigenvalues

λkT) uk(x). (1.30)Substitutingψk into the first equation of (1.28) and using ˜v(x,0) =−α v˜(x, T), we have

Z

Ω

˜

v(ψtt− Aψ)dxdt = 0, (1.32)for all ψ ∈ H1(Ω×(0, T)) such that ψtt− Aψ ∈ L2(Ω×(0, T)) andψ|∂Ω =ψ(x, T) = 0.For any p ∈ L2(Ω× (0, T)) there exists a unique solution in H01(Ω× (0, T)) to theDirichlet problem

ψ(x, T) = 0,

ψ(x,0) = 0.

(1.33)

Z T 0

Z

Ω

˜

vp(x, t)dxdt = 0, (1.34)for all p ∈ L2(Ω×(0, T)) It follows that ˜v = 0 The uniqueness has been proved

Moreover, the solution v(x, t) defined in (1.27) is stable Indeed,

1.3.2 The classical solution

A classical solution to problem (1.5) is a function u twice continuously differentiable in

Ω×(0, T) and satisfies the problem pointwise For guaranteeing the existence of a classicalsolution, some regularity conditions on the coefficients and data of problem (1.5) are required.Following [50] we suppose that Ω⊂Ω, an open bounded domain ine Rn and

aij ∈ C(1,µ)(Ω)e , a ∈ C(0,µ)(Ω)e , µ >0, (1.35)which means that aij are differentiable in eΩ and their first derivatives satisfy the H¨oldercondition with exponentµ in any closed subset ofΩ, the functione a ∈ C(eΩ) and satisfies theH¨older condition with exponentµin any closed subset ofΩ In [51, 52], Il’in and Shishmareve

proved that if the Dirichlet problem for the Laplace equation is solvable in Ω for arbitrarycontinuous boundary data on∂Ω and the condition (1.35) is satisfied, then the eigenfunctions

in H01(Ω) of problem (1.21) are classical eigenfunctions that means that the eigenfunctions

vk are twice continuously differentiable and satisfy (1.21) in the classical sense Furthermore,

we have the following results concerning the Fourier series generated by eigenfunctionsvk:

Lemma 1.3.2 [50, Lemma 1, p 123–124] Let Ω ⊂ Ωe , an open bounded domain in

Rn, and the conditions (1.35) be satisfied Then the series from the eigenfunctions of problem (1.21) of the form

(1.36)

is uniformly convergent in the closed domain Ω and the series from the first and second-order derivatives of derivatives of eigenfunctions of the form

(1.37)

are uniformly convergent in any closed subdomain of Ω.

Lemma 1.3.3 [50, Lemma 5, p 139] Let Ω be a connected, bounded domain in Rnand aij ∈ Cℓ(Ω), a ∈ Cℓ−1(Ω) with a positive integer ℓ Furthermore, suppose that the function Φ satisfies

• Φ ∈ Hℓ+1(Ω),

• Φ, AΦ, , A[2ℓ]Φ∈ H1

0(Ω) Then

i Aℓ2Φ
∂x∂

j Aℓ2Φ
+a Aℓ2Φ
2



dx for ℓ even, R

solv-φ, Asolv-φ, , A[n4 ]+1φ ∈ H01(Ω), (1.39)

the coefficients aij ∈ C[n2]+2(Ω) and the coefficient a ∈ C[n2 ]+1(Ω) Then the series

(1.27) is uniformly convergent in Ω×(0, T), its second order derivatives vtt and vxixjare uniformly convergent in any closed subdomain of Ω×(0, T) The series (1.27) is the classical solution to the problem (1.5).

Proof. 1) The uniform convergence of the series (1.27) in Ω×(0, T) Since

+1 k

1/2

.

Since the conditions of Lemmas 1.3.2, 1.3.3 are satisfied withℓ = [n2] the two factors in thelast product converge uniformly in Ω×(0, T) Hence this part is proved It follows furtherthat the series (1.27) satisfies the condition v(x,0) +αv(x, T) =φ(x) pointwise

2)The uniform convergence of the second derivatives of v(x, t). The formal secondderivatives with respect to tand to xi and xj of the series (1.27) have the form

In case Ω = (0, L) and Au = uxx, the eigenfunctions and eigenvalues of the

one-dimensional problem (1.21) correspondingly are

2

, k = 1,2, (1.45)Hence

boundary value problem

In this section, we introduce a finite difference scheme for problem (1.5) We restrictourselves for two and three-dimensional problems and for Ω being either the interval (0, L)

or the parallelepiped (0, L1)×(0, L2)

Following [102], we subdivide the domain [0, L]×[0, γT] into cells by the rectangularuniform grid specified by

The interior ofωh is denoted byωh and its boundary by Ξh We denote byL2(ωh) the space

of grid functions given in ωh with the scalar product

Let Ω = (0, L1) × (0, L2) Consider the three-dimensional non-local boundary valueproblem

u(x,0) +αu(x, γT) = φ(x), x ∈Ω,

ut(x,0) = 0, x ∈Ω.

(1.52)

Here, x= (x1, x2) and Au :=−(a1ux1)x1 −(a2ux2)x2 +au, γ > 1 The coefficients a1, a2

and c belong toC(Ω) and satisfy the condition of uniform ellipticity (1.1):

0< c1 ≤ a1(x), a2(x)≤ c2,0 ≤ c3 ≤ a(x)≤ c4 for all x ∈ Ω (1.53)for some given positive constantsc1, c2, c3 and c4

To discretize φ ∈ L2(Ω) we either take its average over a cell or we first mollify it by asmooth function and then take its values at nodes of ¯ωh Since problem (1.52) is well-posed,the mollification process does not magnify the error in φ Hence we take convention that

φ ∈ C(Ω) and discretize problem (1.52) by

We rewrite problem (1.55) into the operator equation Ahv =λv by defining the operator

Ahv = −(a1v˜x¯1)x1 − (a2v˜¯x2)x2 +a v˜, where v ∈ L2(ωh), ˜v ∈ L2h0 and v(x) = ˜v(x) for

x ∈ ωh

Define the discrete Laplace operator Lhv =− v˜x¯1x1− v˜x¯2x2, where v ∈ L2(ωh), ˜v ∈ L2h0

and v(x) = ˜v(x) for x ∈ ωh We have the following result:

Lemma 1.4.2 [104, Lemma 19, p 19] The operator Ah is self-adjoint in L2(ωh)

and the following estimates hold:

(c1+c3/∆)(Lhv, v)L2 (ωh) ≤(Ahv, v)L2 (ωh) ≤(c3+c4/δ)(Lhv, v)L2 (ωh), (1.56)(c1δ+c3)(v, v)L2 (ωh) ≤(Ahv, v)L2 (ωh) ≤(c2∆ +c4)(v, v)L2 (ωh), (1.57)

It follows that the eigenvalues λkℓ, k = 1, , M1 − 1; ℓ = 1, , M2 − 1, of lem (1.55) are positive and the corresponding eigenfunctions vkℓ which are supposed to beorthonormalized in L2(ωh).

prob-Now, we look for a solution y(t, x) to the finite difference system (1.54) in the form

detA=αdetAN + (−1)N +2(−1)N =αdetAN + 1.

We prove thatAN is symmetric and positive definite We have AN =A1N +A2N, with