Luận án: Bài toán ngược phương trình elliptic với quan sát biên - Lê Thị Thu Giang

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 BOUNDARY OBSERVATIONS) Speciality: Applied Mathematics Speciality Code: 9 46 01 12 THESIS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY IN MATHEMATICS Supervisor: Prof. DINH NHO HAO HA NOI – 2024 Abstract This thesis is devoted to the Cauchy problem and the problem of determining the right- hand side for second-order linear elliptic equations with boundary observations. The Cauchy problem for elliptic equations is studied in Chapter 1. There we introduce a new concept of very weak solution to the Cauchy problem for elliptic equations and for a non-local boundary value problem which regularizes the Cauchy problem.

We discretize the non-local boundary problem by the finite difference method and prove the stability of the scheme 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 ob- servation on the boundary. We formulate the inverse problem as an operator equation and regularize it by the Tikhonov regularization method.

To discretize the problem, we follow Hinze’s idea on variational discretization and suggest a rule of choosing the regularization parameter depending on the noise level in the observation data and the discretization mesh size which yields the optimal convergence rate. This abstract result is applied to the finite element method for numerical solving the source identification problem. Some numerical examples 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 elliptic equation 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 direct and inverse problems can be represented by the Fourier series. Since the source problem is ill-posed, we regularize it using the truncated Fourier series method and propose a method for 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 present several numerical examples. i Tóm tắt Luận án này nghiên cứu về bài toán Cauchy và bài toán xác định vế phải của phương trình eliptic tuyến tính bậc hai với quan sát trên biên.

Chương 1 nghiên cứu về bài toán Cauchy cho các phương trình elliptic. Trong chương này, chúng tôi giới thiệu một khái niệm mới về nghiệm rất yếu cho bài toán Cauchy và bài toán hiệu chỉnh của nó - bài toán biên không địa phương cho phương trình elliptic. Sau đó, chúng tôi rời rạc hóa bài toán biên không địa phương bằng phương pháp sai phân hữu hạn và chứng minh sự ổn định của phương pháp, từ đó suy ra sự hội tụ của phương pháp. 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. ii Declaration This work has been completed at Institute of Mathematics, Vietnam Academy of Science and Technology under the supervision of Prof.

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 iii Acknowledgments First and foremost, I want to express my deepest gratitude to my advisor, Pro- 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. 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 Mathe- 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. 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 parents- 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. List of Figures 1.

Numerical solution with p=0. Approximation error with p=0. Approximation error with p=0. Comparison of the exact solution with numerical solutions for various p, a priori method.

Comparison of the approximation errors of the numerical so- lutions for various p, a priori method. Exact and approximate solutions at T = 1 for various p, a posteriori method. Approximation errors at T = 1 for various p, a posteriori method. Exact and approximate solutions at T = 1 for various p in case u(x, T ) is a hat function.

Approximation errors at T = 1 for various p in case u(x, T ) is a hat function. Exact and approximate solutions at T = 1 for various p, x ∈ [0, π ]. Approximation errors at T = 1 for various p with x ∈ [0, π ]. Exact and approximate solutions at T = 1 for various p, x ∈ [0, 5π ].

Approximation errors at T = 1 for various p, x ∈ [0, 5π ]. Reconstruction of the smooth source function f1 (x1 ). Reconstruction of the continuous and nonsmooth source func- tion f2 (x1 ). Reconstruction of the discontinuous source function f3 (x1 ).

Exact and approximation solutions with f ∗ = 0, 10 and 20. f 20x1 +x2 and its numerical solution with f ∗ close to f 20x1 +x2. f 0 and the numerical solution with f ∗ close to f 0. f −5 sin πx2 and the numerical solution with f ∗ close to f −5 sin πx2.

Numerical results for f satisfying the source condition (2. Numerical results for f satisfying the source condition (2. Solutions with different noise levels for smooth function, M = 15 91 3. Errors with different perturbations for smooth function, M = 15 91 3.

Solutions with different number of Fourier coefficients, smooth function, p = 7%. Errors with different number of Fourier coefficients for smooth function, p = 7%. Solutions with different noise levels for continuous, non-smooth function, M = 15. Errors with different noise levels, for continuous, non-smooth function, M = 15.

Solutions with different number of Fourier coefficients, p = 7% 93 3. Errors with different number of Fourier coefficients, p = 7%. Solutions with different noise levels for discontinuous function, M = 20. Errors with different noise levels,for discontinuous function, M = 20.

Solutions with different number of Fourier coefficients, p = 5% 94 vi 3. Errors with different number of Fourier coefficients for discon- tinuous function, p = 5%. 94 vii List of Tables 1.1 Regularization parameters chosen due the a posterior method for various p. Regularization parameters with the exact solutions at T being a hat function.

Regularization parameters with the exact solutions at T being a step function. The L2 -norm of errors. L2 -error in the smooth case. L2 -error in the non-smooth but continuous case with α = 10−5.

L2 -error in the discontinuous case with different regularization parameters. L2 -error between the exact solution and numerical ones with relative noise 0. L2 -errors with f ∗ close to exact solutions. L2 -error behavior - the source condition 1.

L2 -error behavior - the source condition 2. The L2 -norm of relative errors for smooth function. The L2 -norm of relative errors - non-smooth, continuous func- tion. The L2 -norm of relative errors for the discontinuous function 93 viii List of Notations Rn n− dimensional Euclidean space; Ω open set in Rn ; ∂Ω boundary of Ω; Γ subset of ∂Ω; Q = Ω × (0, T ) S = ∂Ω × (0, T ); C(Q) space of continuous functions on Q; C k (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 (resp.

S, Q); ∂u H 1 (Ω) Sobolev space= {u(x) ∈ L2 (Ω) : ∂x i ∈ L2 (Ω)}; H01 (Ω) the closure of Cc∞ (Ω) in the norm of H 1 (Ω) ; ∇J gradient of J; ⟨, ⟩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.