Luận văn Phương pháp Kozlov-Maz''ya giải bài toán Cauchy cho phương trình Elliptic

MINISTRY OF EDUCATION AND TRAINING VIETNAM ACADEMY OF SCIENCE AND TECHNOLOGY GRADUATE UNIVERSITY OF SCIENCE AND TECHNOLOGY - Nguyen Vu Trung Quan KOZLOV-MAZ’YA’S METHOD FOR SOLVING THE CAUCHY PROBLEM FOR ELLIPTIC EQUATIONS Major: Mathematical Analysis Code: 46 01 02 MATHEMATICAL MASTER THESIS SUPERVISOR: Prof Dr Sc Dinh Nho Hao Hanoi – 2020 BỘ GIÁO DỤC VÀ ĐÀO TẠO VIỆN HÀN LÂM KHOA HỌC VÀ CÔNG NGHỆ VIỆT NAM HỌC VIỆN KHOA HỌC VÀ CÔNG NGHỆ - Nguyễn Vũ Trung Quân PHƯƠNG PHÁP KOZLOV-MAZ’YA GIẢI BÀI TỐN CAUCHY CHO PHƯƠNG TRÌNH ELLIPTIC Chun ngành: Tốn giải tích Mã số: 46 01 02 LUẬN VĂN THẠC SĨ TOÁN HỌC NGƯỜI HƯỚNG DẪN KHOA HỌC: GS TSKH Đinh Nho Hào Hà Nội - 2020 COMMITMENT I assure that the Thesis is my own exploration and study, under the supervision of Prof Dr Sc Dinh Nho Hao The results as well as the ideas of other authors are all specifically cited Up to now, this thesis topic has not been protected in any master thesis defense council and has not been published on any media I take responsibility for these guarantees Hanoi, October 2020 Student Nguyen Vu Trung Quan ACKNOWLEDGEMENTS Firstly, I am extremely grateful for my supervisor - Prof Dr Sc Dinh Nho Hao who devotedly guided me to learn some interesting fields in Mathematics and taught me to enjoy Ill-posed Problems and Inverse Problems with Machine Learning He took care and shared his experience in research career opportunities for me and help me to find a way for my research plan I want to share my appreciation with Dr Hoang The Tuan He took his time for talking and encouraging me a lot for a long time not only in the mathematical study, but also in many aspects of life In the time I study here, I sincerely thank all of my lecturers for teaching and helping me; and to the Institute of Mathematics Hanoi for offering me facilitation in a professional working environment I would like to say thanks for the help of the Graduate University of Science and Technology, Vietnam Academy of Science and Technology in the time of my Master program Especially, I really appreciate my family, my friends and my teachers at Thang Long Gifted High School - MSc Nguyen Van Hai and Dr Dang Van Doat for their supporting in my whole life Hanoi, October 2020 Student Nguyen Vu Trung Quan Contents Commitment Acknowledgement Contents Table of Figures Preface 1 Kozlov-Maz’ya’s Algorithm 1.1 A quick tour 1.1.1 Inverse Problems and Ill-posed Problems 1.1.2 The Cauchy problem for elliptic equations 1.2 Kozlov-Maz’ya’s Algorithm 1.2.1 Notations 1.2.2 Descrition of the Algorithm (the case of the Laplace equation) 1.2.3 Well-posedness 1.2.4 Convergence and Regularizing properties 10 1.2.5 Represent the algorithm in the form of an operator equation 16 1.3 Facts about the KMF Algorithm 22 1.4 Examples 24 Practicals and Developments 27 2.1 Relaxed KMF Algorithms 27 2.1.1 Relaxation Algorithms 28 2.1.2 The choice of Relaxation factors 30 2.1.3 Observations from numerical tests 36 2.2 Recasting KMF Algorithm as a form of Landweber Algorithm 37 2.2.1 Mathematical formulation 37 2.2.2 Landweber iteration for initial Neumann data 39 2.2.3 Introduction to Mann-Maz’ya Algorithm 45 2.3 Some related concepts 47 2.3.1 Minimizing an energy-like functional 48 2.3.2 In a point of view of Interface problems 51 Additional Topics 55 A Green’s Formulas 56 B Sobolev Spaces W k,p 56 C Well-posedness of mixed boundary value problems 57 D Equivalence of Norms 59 E Landweber Iteration 60 F Methods for solving elliptic Cauchy problems 61 G A numerical test 62 Conclusion 68 Bibliography 69 Table of Figures Figure Name Page(s) 1.1 Some Well-posed and Ill-posed in Partial Differential Equations 1.2 The Cauchy problem (1.1) 1.3 Step 1.4 Step (i) 1.5 Step (ii) 1.6 Comparison of the Kozlov-Maz’ya’s Algorithm with other methods 24 2.1 A numerical test for Relaxation algorithm 36 2.2 Variation of the Number of iterations as a function of Relaxation Parameter 67 2.3 Variation of the Number of iterations as a function of Relaxation Parameter 67 Preface The Cauchy problem for elliptic equations attracts many scientists and mathematicians since they have many applications such as (according to many papers, for e.g., [1], [2], [3], [4], etc.) the theory of potential, the interpretation of geophysical measurements, the bioelectric field application (electroencephalography (EEG), electrocardiography (ECG)) Because of its ill-posedness, there are difficulties in solving this problem Even in the case the (exact) solutions exist uniquely, it is still hard in approximating the solutions since their stability is not guaranteed Throughout the years, many regularizing methods to solve this problem are proposed and developed In this thesis, an alternating iterative method for solving Cauchy problem for elliptic equations, namely Kozlov-Maz’ya’s algorithm is considered This iterative procedure first introduced in 1990 by V.A Kozlov and V.G Maz’ya [5] In 1991, V.A Kozlov, V.G Maz’ya and A.V Fomin [6] proved the convergence of the method and its regularizing properties Shortly, this method regularizes the (ill-posed) Cauchy problems by constructing a sequence of (well-posed) boundary problems in order to approximate the exact solutions The thesis has two main chapters In Chapter 1, the author introduces in brief the Inverse Problems and the Ill-posed Problems; the convergence and regularizing properties of the Kozlov-Maz’ya method are represented; next, the author discusses some specific examples, advantages and disadvantages of the method, comparisons with other methods In Chapter 2, some related developments of the method by many researchers are reviewed Other concepts are shown in Additional Topics Chapter Kozlov-Maz’ya’s Algorithm 1.1 A quick tour 1.1.1 Inverse Problems and Ill-posed Problems A physical process can be described via a mathematical model I nput −→ Sy t em par amet er s −→ Out put In the most cases the description of the system is given in terms of a set of equations (for instance, ordinary and/or partial differential equations, integral equations), which contains certain parameters One can classfy into three distinct types of problems (A) The direct problem Given the input and the system parameters, find out the output of the model (B) The reconstruction problem Given the system parameters and the output, find out which input has led to this output (C) The identification problem Given the input and the output, determine the system parameters which are in agreement with the relation between the input and the output One calls a problem of type (A) is a direct problem and a problem of type (B) or type (C) is an inverse problem Now, let X and Y be normed spaces and K : X → Y be a (linear or nonlinear) mapping Consider the problem K x = y, where x ∈ X and y ∈ Y One has the following definition Definition 1.1 (Well-posedness) The equation K x = y is called properly-posed or well-posed (in the sense of Hadamard [7]) if the following conditions hold i) Existence For every y ∈ Y , there exists a solution x ∈ X to the equation K x = y, i.e R(K ) = Y where R(K ) is the range of K ii) Uniqueness For every y ∈ Y , the solution x ∈ X to the equation K x = y is unique, i.e the inverse mapping K −1 : Y → X exists iii) Stability The solution x ∈ X depends continuously on y, i.e the inverse mapping K −1 : Y → X is continuous Equations for which (at least) one of these properties does not hold are called improperly-posed or ill-posed Some examples of inverse problems and ill-posed problems can be found in J Baumeister [8], A Kirsch [9], L.E Payne [10] and S.I Kabanikhin [11] Some classifications of these fields can be found in S.I Kabanikhin [11] 59 unique weak solution Furthermore, by the continuity of ω, X , the coerciveness of A and the Cauchy-Schwarz’s inequality we have w C2 h ≤ H1 (Ω,S) φ H1 (Ω) ≤ C A (h, h) ≤ C h H1 (Ω,S) H1/2 (S) , w H1 (Ω) + h H1 (Ω,S) τ H−1/2 (L) , which lead to the estimate u H1 (Ω) ≤ w H1 (Ω) + h H1 (Ω,S) ≤ C φ H1/2 (S) + τ H−1/2 (L) (2.38) where C ,C ,C ,C are positive constants D Equivalence of Norms In this section we prove the equivalence between the usual Sobolev norm with the Sobolev norm which are used in Section 1.2.5 and for the proof of Lemma 2.9, see also [1] or [13] We denote by |||.|||H−1/2 (S) the Sobolev norm used in Section 1.2.5 and for the proof of Lemma 2.9, and we use H−1/2 (S) to denote the usual Sobolev norm Let ψ ∈ H−1/2 (S) and let u be the solution of     −∆u = i n Ω,            So ψ H −1/2 (S) = ∇u L (Ω) ∂u ∂ν = ψ on S, u = on L Let any v ∈ H1 (Ω) and pick φ ∈ H1/2 (∂Ω) such 60 that v = φ on ∂Ω, by Green’s formula we have ∂u φ= ∂Ω ∂ν Ω ∇u∇v From this identity we have ∂u ∂ν H−1/2 (∂Ω) ≤ ∇u L (Ω) But ∂u ∂ν H−1/2 (S) ≤ ∂u ∂ν , H−1/2 (∂Ω) then we have ψ H−1/2 (S) ≤ ∇u L (Ω) ≤ u H1 (Ω) Besides, we imply from (2.38) that u H1 (Ω) ≤ C ψ H−1/2 (S) Therefore ψ E H−1/2 (S) ≤ ψ H−1/2 (S) ≤ C ψ H−1/2 (S) Landweber Algorithm To solve the equation K x = y, one can transform it into a fixed point equa- tion such as x = x + K ∗ (y − K x) 61 The Landweber algorithm is given as follows, starting with an initial guess x , then one constructs a sequence x k+1 = x k + K ∗ y − K x k with a given assumption that K ≤ If the assumption does not hold, one introduced a relaxed version for the algorithm x k+1 = x k + aK ∗ y − K x k , where ≤ a ≤ K The Landweber algorithm can be seen as a case of the Steepest Descent algorithm to minimize the functional J (x) = Kx−y 2 whose gradient at x is −K ∗ y − K x The study of the Landweber algorithm including the explainations about the selection criterion for the norm operator K and the relaxation factor a can be find in [9], [27] F Methods for solving Cauchy problems Here are some methods as the writer knows 1) Energy Error-based method 2) Tikhonov regularization 62 3) Lavrentiev regularization 4) Logarithmic Convexity method 5) Variational approaches based on Steklov-Poincare operators 6) Quasi-reversibility 7) Conjugate Gradient Method 8) Fourth-order modified method 9) Moment method associated to Backus-Gilbert algorithm 10) Singular Value Decomposition method (SVD) 11) Fixed point algorithms and their variants [1], [2], [3], [4],[5], [6], [12], [13], [14], [15], [16], [24],[28], [29], [30] For the study of these methods, one can find in the references therein of the cited materials above, and also in [8], [9], [10], [27] G A numerical test In this section the author performs a numerical experiment for Relax- ation algorithm One can compare it with the one in Sub-section 2.1.3 """ A numerical test for the Cauchy problem for the Poisson equation with relaxed Kozlov - Maz ’ ya ’s algorithm , by using Finite Element method with the help of FEniCS Project - Delta ( u ) = f in \ Omega , \ frac { partial u } { \ partial \ nu } = g_1 on \ Gamma_1 , u = v_1 on \ Gamma_1 Follow the link below for documentation , including the book : 63 Solving PDEs in Python by Hans Petter Langtangen and Anders Logg https :// fenicsproject org / tutorial / """ # Libraries from fenics import * from mshr import * import matplotlib pyplot as plt import math as m import numpy as np from time import time # Define Function Space and Mesh , domain_vertices = Rectangle ( Point ( 0 , ) , Point ( , ) ) mesh = generate_mesh ( domain_vertices , 64 ) V = FunctionSpace ( mesh , ’P ’ , ) # Lagrange element of degree # Define datum of the Cauchy problem v_1 = Expression ( ’1 + * x [ ]* x [ ]+ * x [ ]* x [ ] ’ , degree = 10 ) g_1 = Expression ( ’4 * x [ ] ’ , degree = 10 ) f = Constant ( - ) epsilon = 1E - tol = 1E - 14 """ Use for defining the boundaries , since the exact comparison == is not good in programming pratice """ dGamma_0 = FacetNormal ( mesh ) # Define noram vector to the entire Gamma u_exact = Expression ( ’1 + * x [ ]* x [ ]+ * x [ ]* x [ ] ’ , degree = 10 ) # Define boundaries boundary_markers = MeshFunction ( " size_t " , mesh , ) Gamma_0 = Compiled SubDomai n ( ’ on_boundary & & ( near ( x [ ] ,0 , tol ) | | near ( x [ ] ,1 , tol ) ) ’ , tol = tol ) Gamma_0 mark ( boundary_markers , ) Gamma_1 = Compiled SubDomai n ( ’ on_boundary & & ( near ( x [ ] ,0 , tol ) | | near ( x [ ] ,1 , tol ) ) ’ , tol = tol ) Gamma_1 mark ( boundary_markers , ) # Define the PDE solvers def solver1 ( u_D0 , u_N1 ) : 64 # Define boundary condition bc1 = DirichletBC (V , u_D0 , Gamma_0 ) # Define variational problem usolver1 = TrialFunction ( V ) v = TestFunction ( V ) a_1 = dot ( grad ( usolver1 ) , grad ( v ) ) * dx L_1 = f * v * dx - u_N1 * v * ds ( ) # Compute solution usolver1 = Function ( V ) solve ( a_1 = = L_1 , usolver1 , bc1 ) return usolver1 def solver2 ( u_N0 , u_D1 ) : # Define boundary condition bc2 = DirichletBC (V , u_D1 , Gamma_1 ) # Define variational problem usolver2 = TrialFunction ( V ) v = TestFunction ( V ) a_2 = dot ( grad ( usolver2 ) , grad ( v ) ) * dx L_2 = f * v * dx - u_N0 * v * ds ( ) # Compute solution usolver2 = Function ( V ) solve ( a_2 = = L_2 , usolver2 , bc2 ) return usolver2 # Perform Kozlov - Maz ’ ya Algorithm # theta = # Define the stopping criterion def error (x ,y ,z , t ) : return abs ( errornorm (x ,y , ’ L2 ’) - errornorm (z ,t , ’ L2 ’) ) # Define the number of iterations as a function of relaxation parameter def iteration ( theta ) : # Step Specify an initial guess a_0 a_0 = Expression ( ’x [ ] ’ , degree = 10 ) # Step Find u_0 and obtain b_0 , find u_1 and obtain a_1 u_0 = solver1 ( a_0 , g_1 ) 65 b_0 = dot ( grad ( u_0 ) , dGamma_0 ) u_1 = solver2 ( b_0 , v_1 ) a_1 = theta * u_1 + ( - theta ) * a_0 num_iterations = u_2 = solver1 ( a_1 , g_1 ) b_1 = dot ( grad ( u_2 ) , dGamma_0 ) u_3 = solver2 ( b_1 , v_1 ) a_0 = theta * u_3 + ( - theta ) * a_1 num_iterations = # Step if error ( u_0 , u_1 , u_2 , u_3 ) < epsilon : print ( ’ Stop at iteration ’ , num_iterations ) else : while error ( u_0 , u_1 , u_2 , u_3 ) > = epsilon : u_0 = solver1 ( a_0 , g_1 ) b_0 = dot ( grad ( u_0 ) , dGamma_0 ) u_1 = solver2 ( b_0 , v_1 ) """ Since the test function v vanishes on the Dirichlet boundary Gamma_1 , I use the solver2 function with the normal derivative b_0 , which is defined on the entire boudary Gamma See the reference , page 85 for explanation """ a_1 = theta * u_1 + ( - theta ) * a_0 num_iterations + = u_2 = solver1 ( a_1 , g_1 ) b_1 = dot ( grad ( u_2 ) , dGamma_0 ) u_3 = solver2 ( b_1 , v_1 ) a_0 = theta * u_3 + ( - theta ) * a_1 num_iterations + = print ( error ( u_0 , u_1 , u_2 , u_3 ) ) if num_iterations > 300 : break """ To limit the number of iterations in case the method does not converge """ if error ( u_0 , u_1 , u_2 , u_3 ) > = : num_iterations = 300 66 break """ Another condition to limit the number of iterations in case the method does not converge """ return num_iterations # Plot function with theta from to , I take 20 values theta = [ ] iters = [ ] for i in range ( 20 ) : theta append ( + i / 10 ) iters append ( iteration ( theta [ i ] ) ) plt figure () plt plot ( theta , iters , ’b ’) plt title ( ’ Variation of the Number of iterations as a function of Relaxation Parameter ’) plt xlabel ( ’ Relaxation parameter ’) plt ylabel ( ’ Number of iterations ’) plt grid ( True ) plt savefig ( ’ relaxation algorithm png ’ , dpi = 300 , bbox_inches = ’ tight ’ ) plt savefig ( ’ relaxation algorithm pdf ’ , dpi = 300 , bbox_inches = ’ tight ’ ) plt show () After running this code, we have the Fig 2.2 Note that if we replace L_1 and L_2 in the functions solver1 and solver2, respectively, by L_1 = f * v * dx - u_N1 * v * ds L_2 = f * v * dx - u_N0 * v * ds (i.e without using boundary marker), we have the Fig 2.3 In this code above the author treats each stage (i) and (ii) of Step in Relaxation KMF algorithm as one iteration So the exact number of iterations is equal to num_iterations divided by 67 Figure 2.2: Figure 2.3: 68 Conclusion In this Thesis I have introduced a method for solving the Cauchy problem for elliptic equation, named Kozlov-Maz’ya’s algorithm, and have rewritten it in my own way in order to make it more understandable In Section 1.2, I have clarified the calculations, estimations, arguments, etc in the original paper by Kozlov et al [6] I have used the relation u (2k+2) = NL ∂u (2k) ∂ν + D S u ∗ = NL (Ψk ) + D S u ∗ , L to obtain the estimate R 2k+2 u ∗ ,p ∗ ,p (0) H1 (Ω) ≤c M (k+2) u ∗ +(k+2) H1/2 (S) p∗ H−1/2 (S) + p (0) H−1/2 (L) , so I not need to use the operators Φk , F and A which are presented in Subsection 1.2.5 In Section 1.3 and 1.4 and Chapter 2, based on papers which cited the paper Kozlov et al [6] in Mathscinet.org (up to now, this paper has 111 citations), I have collected some relations around the method such as information about KMF algorithm in Section 1.3, examples from various PDEs in Section 1.4, and some selected topics (developments, related methods) in Chapter In Additional Topics, to perform a numerical test, I have presented my code with detailed explanation in Python language, and the method I have used is Finite Element based on FEniCS library 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1998 [33] Haim Brezis Functional Analysis, Sobolev Spaces and Partial Differential Equations Universitext, Springer-Verlag New York, 2010 ... - Nguyễn Vũ Trung Quân PHƯƠNG PHÁP KOZLOV-MAZ’YA GIẢI BÀI TOÁN CAUCHY CHO PHƯƠNG TRÌNH ELLIPTIC Chun ngành: Tốn giải tích Mã số: 46 01 02 LUẬN VĂN THẠC SĨ TOÁN HỌC NGƯỜI HƯỚNG DẪN KHOA... fields can be found in S.I Kabanikhin [11] 4 1.1.2 The Cauchy problem for elliptic equations Instead of only introducing the Cauchy problem for elliptic equations, Figure 1.1, which is captured from... the famous example for the Cauchy problem for elliptic equations which is proposed by Hadamard [7], see also [8] or [11] This example says that the solution of the Cauchy problem for the Laplace