1490 đồng hóa số liệu trong truyền nhiệt luận văn tốt nghiệp

MINISTRY OF VIETNAM ACADEMY OF EDUCATION AND TRAINING SCIENCE AND TECHNOLOGY INSTITUTE OF MATHEMATICS NGUYEN THI NGOC OANH DATA ASSIMILATION IN HEAT CONDUCTION THESIS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY IN MATHEMATICS HANOI 2017 MINISTRY OF VIETNAM ACADEMY OF EDUCATION AND TRAINING SCIENCE AND TECHNOLOGY INSTITUTE OF MATHEMATICS NGUYEN THI NGOC OANH DATA ASSIMILATION IN HEAT CONDUCTION Spe iality: Di erential and Integral Equations Spe iality Code: 62 46 01 03 THESIS FOR THE DEGREE OF DOCTOR OF PHYLOSOPHY IN MATHEMATICS Supervisor: PROF DR HABIL ĐINH NHO HÀO HANOI 2017 BỘ GIÁO DỤC VÀ ĐÀO TẠO VIỆN HÀN LÂM KHOA HỌC VÀ CƠNG NGHỆ VIỆT NAM VIỆN TỐN HỌC NGUYỄN THỊ NGỌC OANH ĐỒNG HÓA SỐ LIỆU TRONG TRUYỀN NHIỆT Chun ngành: Phương trình Vi phân Tích phân Mã số: 62 46 01 03 LUẬN ÁN TIẾN SĨ TOÁN HỌC Người hướng dẫn khoa học: GS TSKH ĐINH NHO HÀO HÀ NỘI – 2017 A knowledgments I first learned about inverse and ill-posed problems when I met Professor Đinh Nho Hào in 2007, my final year of bachelor’s study I have been extremely fortunate to have a chance to study under his guidance since then I am deeply indebted to him not only for his supervision, patience, encouragement and support in my research, but also for his precious advices in life I would like to express my special appreciation to Professor Hà Ti ến Ngo ạn, Professor Nguyễn Minh Trí, Doctor Nguyễn Anh Tú, the other members of the seminar at Department of Differential Equations and all friends in Professor Đinh Nho Hào’s group seminar for their valuable comments and suggestions to my thesis I am very grateful to Doctor Nguyễn Trung Thành (Iowa State University) for his kind help on MATLAB programming I would like to thank the Institute of Mathematics for providing me with such an excellent study environment Furthermore, I would like to thank the leaders of College of Sciences, Thai Nguyen University, the Dean board as well as to all of my colleagues at the Faculty of Mathematics and Informatics for their encouragement and support throughout my PhD study Last but not least, I could not have finished this work without the constant love and unconditional support from my parents, my parents-in-law, my husband, my little children and my dearest aunt I would like to express my sincere gratitude to all of them Abstra t The problems of re onstru ting the initial ondition in paraboli equations from the observation at the nal time, from interior integral observations, and from boundary observations are studied We reformulate these inverse problems as variational problems of minimizing appropriate mis t fun tionals We prove that these fun tionals are Fré het di erentiable and derive a formula for their gradient via adjoint problems The dire t problems are rst dis retized in spa e variables by the nite di eren e method and the variational problems are orrespondingly dis retized The onvergen e of the solution of the dis retized varia- tional problems to the solution of the ontinuous ones is proved To solve the problems numeri ally, we further dis retize them in time by the splitting method It is proved that the ompletely dis retized fun tionals are Fré het di erentiable and the formulas for their gradient are derived via dis rete adjoint problems The problems are then solved by the onjugate gradient method and the numeri al algorithms are tested on omputer As a byprodu t of the variational method based on Lan zos' algorithm, we suggest a simple method to demonstrate the ill-posedness i Tóm tắt Các tốn xác định điều kiện ban đầu phương trình parabolic từ quan sát thời điểm cuối, từ quan sát tích phân bên trong, từ quan sát biên đ ược nghiên c ứu Chúng sử dụng phương pháp biến phân nghiên cứu toán ng ược b ằng cách c ực tiểu hóa phiếm hàm chỉnh Chúng chứng minh r ằng phi ếm hàm kh ả vi Fréchet đưa công thức gradient chúng thơng qua tốn liên h ợp Tr ước tiên, sử dụng phương pháp sai phân hữu hạn để r ời rạc hóa tốn thu ận toán liên hợp tương ứng theo biến không gian Chúng chứng minh s ự h ội t ụ c nghiệm toán biến phân rời rạc tới nghiệm toán bi ến phân liên t ục Đ ể giải số toán, chúng tơi tiếp tục rời rạc tốn theo biến thời gian phương pháp sai phân phân rã (phương pháp splitting) Chúng chứng minh phiếm hàm rời rạc khả vi Fréchet đưa công thức gradient c chúng thông qua tốn liên hợp rời rạc Sau chúng tơi sử dụng phương pháp gradient liên hợp để giải thuật tốn số thử nghiệm máy tính Ngoài ra, nh s ản phẩm phụ phương pháp biến phân, dựa thuật toán Lanczos, đề xu ất phương pháp đơn giản để minh họa tính đặt khơng chỉnh tốn ii De laration 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: Nguyen Thi Ngoc Oanh ii List of Figures 2.1 Example 1: Singular values 52 2.2 Example 2: Re onstru tion results: (a) exa t fun tion v; (b) estimated one; ( ) point-wise error; (d) the omparison of v|x1=1/2 and its re onstru tion (the dashed urve: the exa t fun tion, the solid urve: the estimated fun tion) 53 2.3 Example 3: Re onstru tion result: (a) exa t fun tion v; (b) estimated one; ( ) point-wise error; (d) the omparison of v|x1=1/2 and its re onstru tion (the dashed urve: the exa t fun tion, the solid urve: the estimated fun tion) 54 2.4 Example 4: Re onstru tion result: (a) exa t fun tion v; (b) estimated one; ( ) point-wise error; (d) the omparison of v|x1=1/2 and its re onstru tion (the dashed urve: the exa t fun tion, the solid urve: the estimated fun tion) 55 2.5 Example 5: Re onstru tion result: (a) exa t fun tion v; (b) estimated one; ( ) point-wise error; (d) the omparison of v|x1=1/2 and its re onstru tion (the dashed urve: the exa t fun tion, the solid urve: the estimated fun tion) 56 3.1 Example Singular values: three observations and various time intervals of observations 68 3.2 Example 2: Re onstru tion results for (a) uniform observation points in (0, 0.5), error in L2norm = 0.006116; (b) uniform observation points in (0.5, 1), error in L2-norm = 0.006133; ( ) uniform observation points in (0.25, 0.75), the error in L2-norm = 0.0060894; (d) uniform observation points in Ω, the error in L2-norm = 0.0057764 69 3.3 Re onstru tion result of Example 3: (a) τ = 0.01 ; (b) τ = 0.05; ( ) τ = 0.1; (d) τ = 0.3 70 3.4 Re onstru tion result of Example 4: (a) τ = 0.01 ; (b) τ = 0.05; ( ) τ = 0.1; (d) τ = 0.3 72 3.5 Re onstru tion result of Example 5: (a) τ = 0.01 ; (b) τ = 0.05; ( ) τ = 0.1; (d) τ = 0.3 73 3.6 Example Re onstru tion results: (a) Exa t initial ondition v; (b) re onstru tion of v; ( ) 3.7 Example Re onstru tion results: (a) Exa t initial ondition v; (b) re onstru tion of v; ( ) 3.8 Example Re onstru tion results: (a) Exa t initial ondition v; (b) re onstru tion of v; ( ) point-wise error; (d) the omparison of v|x1=1/2| and its re onstru tion 74 point-wise error; (d) the omparison of v|x1=1/2| and its re onstru tion 76 point-wise error; (d) the omparison of v|x1=1/2| and its re onstru tion 77 i 4.1 Example 1: Singular Values for 1D Problem 87 4.2 Example 2, 3, 4: 1D Problem: Re onstru tion results for smooth, ontinuous and dis ontinuous 4.3 Example 5: Exa t initial ondition (left) and its re onstru tion (right) 89 4.4 Example ( ontinue): Error (left) and the verti al sli e of the exa t initial ondition and its 4.5 Example 6: Exa t initial ondition (left) and its re onstru tion (right) 90 4.6 Example ( ontinue): Error (left) and the sli e of the exa t initial ondition and its re onstru - 4.7 Example 7: Exa t initial ondition (left) and its re onstru tion (right) 90 4.8 Example ( ontinue): Error (left) and the sli e of the exa t initial ondition and its re onstru - initial onditions 88 re onstru tion along the interval [(0.5, 0), (0.5, 1)] (right) 89 tion along the interval [(0.5, 0), (0.5, 1)] (right) 90 tion along the interval [(0.5, 0), (0.5, 1)] (right) 91 v List of Tables 3.1 Example 3: Behavior of the algorithm with di erent starting points of observation τ 71 3.2 Example 6: Behavior of the algorithm when the number of observations and the positions of 3.3 Example 6: Behavior of the algorithm when the number of observations and the positions of observations vary (N = 4) 75 observations vary (N = 9) 75 v