Thông tin tóm tắt về những đóng góp mới của luận văn thạc sĩ: Phương pháp Kozlov-Maz'ya giải bài toán Cauchy cho phương trình Elliptic.

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 introdu[r]

AND TRAINING 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: 8 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 TOÁN CAUCHY

CHO PHƯƠNG TRÌNH ELLIPTIC

Chuyên ngành: Toán giải tích

Mã số: 8 46 01 02

LUẬN VĂN THẠC SĨ TOÁN HỌC

NGƯỜI HƯỚNG DẪN KHOA HỌC:

GS TSKH Đinh Nho Hào

Hà Nội - 2020

I assure that the Thesis is my own exploration and study, under the vision of Prof Dr Sc Dinh Nho Hao The results as well as the ideas of otherauthors are all specifically cited Up to now, this thesis topic has not beenprotected in any master thesis defense council and has not been published

super-on any media I take respsuper-onsibility for these guarantees

Hanoi, October 2020

Student

Nguyen Vu Trung Quan

Firstly, I am extremely grateful for my supervisor - Prof Dr Sc DinhNho Hao who devotedly guided me to learn some interesting fields in Math-ematics and taught me to enjoy Ill-posed Problems and Inverse Problemswith Machine Learning He took care and shared his experience in researchcareer 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 histime for talking and encouraging me a lot for a long time not only in themathematical study, but also in many aspects of life

In the time I study here, I sincerely thank all of my lecturers for teachingand helping me; and to the Institute of Mathematics Hanoi for offering mefacilitation in a professional working environment

I would like to say thanks for the help of the Graduate University of ence and Technology, Vietnam Academy of Science and Technology in thetime of my Master program

Sci-Especially, I really appreciate my family, my friends and my teachers atThang Long Gifted High School - MSc Nguyen Van Hai and Dr Dang VanDoat for their supporting in my whole life

Hanoi, October 2020

Student

Nguyen Vu Trung Quan

Acknowledgement

Contents

Table of Figures

1.1 A quick tour 2

1.1.1 Inverse Problems and Ill-posed Problems 2

1.1.2 The Cauchy problem for elliptic equations 4

1.2 Kozlov-Maz’ya’s Algorithm 5

1.2.1 Notations 6

1.2.2 Descrition of the Algorithm (the case of the Laplace equa-tion) 6

1.2.3 Well-posedness 9

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

2 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

Figure Name Page(s)1.1 Some Well-posed and Ill-posed in Partial Differential Equations 4

The Cauchy problem for elliptic equations attracts many scientists andmathematicians since they have many applications such as (according tomany papers, for e.g., [1], [2], [3], [4], etc.) the theory of potential, the in-terpretation of geophysical measurements, the bioelectric field application(electroencephalography (EEG), electrocardiography (ECG)) Because of itsill-posedness, there are difficulties in solving this problem Even in the casethe (exact) solutions exist uniquely, it is still hard in approximating the so-lutions since their stability is not guaranteed Throughout the years, manyregularizing methods to solve this problem are proposed and developed

In this thesis, an alternating iterative method for solving Cauchy problemfor elliptic equations, namely Kozlov-Maz’ya’s algorithm is considered Thisiterative 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 ofthe method and its regularizing properties Shortly, this method regularizesthe (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 inbrief the Inverse Problems and the Ill-posed Problems; the convergence andregularizing 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 relateddevelopments of the method by many researchers are reviewed Other con-cepts are shown in Additional Topics

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 tem par ameter s −→ Out put

In the most cases the description of the system is given in terms of a set ofequations (for instance, ordinary and/or partial differential equations, in-tegral equations), which contains certain parameters One can classfy intothree 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

out-put, find out which input has led to this output

(C) The identification problem Given the input and the output, determine

2

the system parameters which are in agreement with the relation betweenthe 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

nonlin-ear) 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]

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 S.I Kabanikhin [11], shows more prob-lems in Partial Differential Equations (PDEs)

Figure 1.1: Some Well-posed (left columm) and Ill-posed (right column) in PDEs.

What is given below is the famous example for the Cauchy problem for liptic equations which is proposed by Hadamard [7], see also [8] or [11] Thisexample says that the solution of the Cauchy problem for the Laplace equa-tion does not depends continuously on the given data For the similar exam-ples in the "hyperbolic" and "parabolic" cases, one can find in J Baumeisterand A Leitao [12].

el-Example 1.2 Consider the problem

For any fixed x > 0 and sufficiently large n, we see that u n (x, y) is also large, while the given data f n (y) tends to zero as n tends to infinity.

1.2 Kozlov-Maz’ya’s Algorithm

From now on, Kozlov-Maz’ya’s algorithm is abbreviated as the KMF rithm This section is totally based on the original work of [6] Notice that in

algo-[6], any numerical experiments were not considered.

1.2.1 Notations

In this chapter we let

• Ω ∈ Rn be an open, bounded and connected domain

• S and L are nonempty open subsets ofΓ sharing a common boundary

Π and Γ = S ∪ L ∪ Π is a Lipschitz dissection.

• U∗, u∗∈ H1/2(S): Exact and approximate Dirichlet conditions on S.

• P∗, p∗∈ H−1/2(S): Exact and approximate Neumann conditions on S.

1.2.2 Descrition of the Algorithm (the case of the Laplace equation)

Let U be the exact solution of the Cauchy problem

where U∗∈ H1/2(S) and P∗∈ H−1/2(S) are given Dirichlet data and Neumann

data, respectively (Fig 1.2)

Figure 1.2: The Cauchy problem (1.1)

Assume that u∗∈ H1/2(S) and p∗∈ H−1/2(S) be "good" specified

approxima-tions of Dirichlet data and Neumann data, respectively Now we shall studythe algorithm

Step 1 Specify an initial guess p(0)∈ H−1/2(L).

the mixed problem (Fig 1.4)

Figure 1.4: Step 3 (i)

(ii) Having constructed u (2k+1) , we can obtain u (2k+2) by solving themixed problem (Fig 1.5)

Step 4 Repeat Step 3 for k ≥ 0 until a prescribed stopping criterion is

satis-fied

1.2.3 Well-posedness

Letξ ∈ H−1/2(L), ϕ ∈ H1/2(S), ψ ∈ H1/2(L), and η ∈ H−1/2(S) One can prove

that the following problems are well-posed (in the sense of Sobolev space

For the proof, see Section C in Additional Topics

To show that the problems (1.2)–(1.4) are well-posed, Kozlov et al [6]stated and proved the following proposition

Proposition 1.3 Let u be a harmonic function belonging toH1(Ω) Assume that ∂u/∂ν| L∈ H−1/2(L), then ∂u/∂ν| S∈ H−1/2(S).

(v1 − v2)| L ∈ H001/2(L) Since u is harmonic, it follows by Green’s formula that

With the help of Proposition 1.3, since u∗ ∈ H1/2(S), p∗ ∈ H−1/2(S) and

p(0) ∈ H−1/2(L), we have ∂u (2k+2)/∂ν¯¯ L ∈ H−1/2(L) Therefore, each of

prob-lems (1.2)-(1.4) is well-posed (inH1(Ω))

1.2.4 Convergence and Regularizing properties

For each k, one can write u (k) as u (k) = R k ¡u∗, p∗, p(0)¢ ∈ H1(Ω), where

R k :H1/2(S) × H−1/2(S) × H−1/2(L) → H1(Ω)

The definition of a regularizing family of operators for problem (1.1) reads asfollows (see [6] or [5])

Definition 1.4 A family of operators R k ¡·,·, p(0)¢ : H1/2(S)×H−1/2(S) → H1(Ω),

k = 0,1, , regularizes the problem (1.1) on the exact solution U if there exist

a positive numberδ0 and functions k( δ) and ²(δ) defined on (0,δ0) such that

²(δ) → 0 as δ → 0,

and the inequality

°

°u∗−U∗°°H 1/2(S)+°°p∗− P∗°°H −1/2(S) ≤ δ (1.8)

implies the estimate

°

°R k( δ) âu∗, p∗, p(0) đ −U°

°

H 1 ( Ω)≤ Ề(δ).

Here the initial approximation p(0)∈ H−1/2(L) plays the role of a parameter

for the family of operators

In what follows the convergence of the algorithm is presented

Theorem 1.5 (V A Kozlov, V G Maz’ya and A V Fomin [6], 1991) Let U∗ ∈

H1/2(S) and P∗ ∈ H−1/2(S) Let U be the solution of problem (1.1) belonging

toH1(Ω) Then, for every p(0)

∈ H−1/2(L), the sequence ẪR k âU∗, P∗, p(0)đà verges to U inH1(Ω).

con-Proof We can divide the proof into three steps.

∂U/∂ν| L ∈ H−1/2(L) Hence, U can be written as U = R k (U∗, P∗,∂U/∂ν| L).Therefore

So it suffices to prove that for everyη ∈ H−1/2(L), the sequence

does not increase as k increases Since

Ω£2|∇r 2k+2(ξ) − ∇r 2k+1(ξ)|2

+ 2 |∇r 2k+1(ξ) − ∇r2k(ξ)|2¤ d x

= 2Z

thatẪr 2kâ

ηđà k≥0 tends to zero as k tends to infinity Therefore, η ∈M

Now, we denote byM⊥ is the orthogonal set of MinH−1/2(L) We have

M⊥ is non-empty since 0 ∈M⊥ Pick anyϕ ∈M⊥, then we have

for allξ ∈ H−1/2(L) We shall prove ϕ ≡ 0 to obtain thatM⊥= {0}

Let q be a function (inH1(Ω)) such that

Let w be a function (inH1(Ω)) such that

∂ν = ξ on L and ξ is an arbitrary function, we obtain w = ϕ on L Thus

har-monic function, then w − q = 0 in Ω Moreover, w = ∂q ∂ν = 0 on S Hence,

w = q = 0 and so that ϕ = 0 We complete the proof.

At a consequence of the Theorem 1.5, the following assertion shows theregularizing property (in the sense of the Definition 1.4) of the algorithm

Theorem 1.6 (V A Kozlov, V G Maz’ya and A V Fomin [6], 1991) Let p(0)∈

H−1/2(L) Then the family of operators ẪR k â·,·, p(0)đà regularizes problem (1.1)

on the exact solution U

For each k ≥ 0 we denote by ρ k the norm of the operator

ε1(δ) = min

½

δ,inf k

Then k( δ) and ²(δ) satisfy the conditions of the Definition 1.4.

1.2.5 Represent the algorithm in the form of an operator equation

We shall study how to represent the KMF algorithm in the form of an erator equation Many generalizations of the KMF algorithm were proposed

op-by using this idea (and we shall explore them more in and Chapter 2)

Let us come back to the problems (1.5) and (1.6) in Section 1.2.3

We define the operator D L = D L(ψ) that assign the solution of problem (1.6)

forη = 0 and for any ψ ∈ H1/2(L), i.e.

Next, we define the operator N L = N L(ξ) that assign the solution of problem

(1.5) forϕ = 0 and for any ξ ∈ H−1/2(L), i.e.

tinuous Then, the KMF algorithm (1.2)-(1.4) can be written as follows

Now, our goal is to estimate the norm of the operator R k by using the

rep-resentation of R k in terms of A and B

First, we prove the inequality

d x (1.16)

which helps us to estimate the norm of B Indeed, by Green’s formula and the definition of N L we have

RΩ

Then, by using the Cauchy-Schwarz’s inequality, we obtain (1.16)

Next, we estimate the norm of B To do this, the authors equip the space

H−1/2(L) with the norm

kξkH−1/2(L)=

µZ

Ω|∇N L ξ|2d x

¶1/2,

notice that one can prove this norm is equivalent to the usual Sobolev norm

of this space (see Section D in Additional Topics, or also [1], [13]) With thehelp of (1.16), Green’s formula and Cauchy-Schwarz’s inequality, we have

¯

¯∇D L ¡ N L ψ¯¯ L¢¯

¯2

d x∇

´1/2³RΩ

which leads us to the estimate

H −1/2(S)

´

Remark 1.7 Obtaining the estimations for°°R 2k+1 ¡u∗, p∗, p(0)¢°

°

H 1 ( Ω)and for

°R 2k+2 ¡u∗, p∗, p(0)¢°°H 1 ( Ω), one can prove Theorem 1.6 without using rem 1.5

Theo-1.3 Facts about the KMF Algorithm

Here are some remarks and observations has been collected The items 1and 2 are originally taken from [6]

1 The alternating process can begin with the problem

S alternate as before.

3 In caseΩ ⊂ R2, by using a similar technique to the one in Section 1.2.5,another proof for the convergence of the KMF algorithm via an analytical-

functional approach for elliptic operators was proposed, see A Leitao[13].

4 For numerical study of the algorithm, the author mentions the works

by Daniel Lesnic and his co-authors (for instance, see [14], [15], [16]) inwhich the algorithm is numerically implemented using the boundaryelement method (BEM)

5 The algorithm may not converge if its corresponding elliptic operator(in the problems (1.2)-(1.4), this operator is the Laplacian) is not coer-cive For e.g., Amen Ben Abda et al [17] said that the algorithm applied

to Helmholtz equations diverges linearly; or see numerical tests in [16])

So B Tomas Johansson and V A Kozlov [18] proposed a modified KMFalgorithm and gave a convergence proof, in the case the elliptic opera-tor is self-adjoint and non-coercive

6 According to [2], [3], [4], [14], [15], the algorithm has some advantagessuch as the simplicity of the computational schemes, the similarity ofthe schemes for problems with linear and nonlinear operators; one pos-sible disadvantage is the large number of iterations that may be required

to achieve the convergence (i.e the algorithm may converge slowly)

7 Some numerical comparisons of the KMF algorithm with other rithms include Conjugate Gradient Method, Tikhonov Regularization,Singular Value Decomposition can be found in [19] The Table 1.6 be-low, which is captured from [19], shows us the comparison betweenKMF algorithm with the other first three algorithms for a numerical test

algo-of Helmholtz equation, where e T , eΦ indicate the accuracy errors, and

p(%) indicates the noise level in the data One also observes in [20], [21],

[22] that, the Energy Functional Minimization Methods outperformedthe KMF algorithm in the number of iterations

Figure 1.6: Comparison of the KMF algorithm, Tikhonov regularization, CGM and SVD.

1.4 Examples

In addition to the Laplace equation, one can find the implementations ofthe KMF algorithm in Poisson equation (see [3], in which the Poisson equa-tion is reduced into the Laplace one by using the sequence of error func-tions); Helmholtz equation [16], [18], [19]; hyperbolic and parabolic equa-tions [5], [12]; biparabolic equation [23]; isotropic and anisotropic linear elas-ticity [6], [24]; etc

In this section we introduce an example – the Cauchy problem for Stokessystem ([25]) to illustrate for the algorithm, the convergence analysis is omit-ted

Consider the problem

Step 1 Specify an initial guess ξ0∈¡H1/2

(i) Having constructed¡u (2k) , p (2k)¢ , then solve the problem

to obtain¡u (2k+2) , p (2k+2)¢ andη k+1 = u (2k+2)¯¯Γ 1

Step 4 Repeat Step 3 for k ≥ 0 until a prescribed stopping criterion is

satis-fied

Practicals and Developments

2.1 Relaxed KMF Algorithms

One of efforts to improve the rate of convergence of the classical KMFalgorithm is using the relaxation procedures The author studied these algo-rithms in M Jourhmane and A Nachaoui [2], [3], M Jourhmane et al [4].Consider the Cauchy problem

bound-is the external unit vector normal toΓ

27

2.1.1 Relaxation Algorithms

Let us rewrite the classical KMF algorithm for the above problem

Classical KMF algorithm.

Step 1 Specify an initial guess a0∈ H1/2(Γ0)

Step 2 Solve the mixed problem below

(ii) Solve the problem

whereγ n,δ n , n ≥ 1 are positive relaxation factors.

2.1.2 The choice of Relaxation factors

In the case of constant relaxation factors, it is easy to see that

Remark 2.1. • Ifβ n= 1 for all n ≥ 1, Relaxation algorithm 1 reduces to theclassical KMF algorithm

• If α n = 1 for all n ≥ 1, Relaxation algorithm 2 reduces to the classicalKMF algorithm

• Ifγ n= 1 for all n ≥ 1, Relaxation algorithm 3 reduces to Relaxation rithm 1

algo-• Ifδ n= 1 for all n ≥ 1, Relaxation algorithm 3 reduces to Relaxation rithm 2

algo-Talking about the selection criteria for relaxation factors of the Relaxationalgorithm 2, one has some interesting results:

Theorem 2.2 (M Jourhmane and A Nachaoui [2], 1999) There exists some

a > 0 and there are some α0,α00 with 0 < α0 < α00 < min 2, a2/¡a2

− 1¢ such that, for all α n ∈ £α0,α00¤, the sequence u (n) converges to the exact solution independently of the initial value a0.

Theorem 2.3 (M Jourhmane and A Nachaoui [3], 2002) There exists a

con-stant α∗ ∈ (1, 2] such that, for all α n ∈ (0, α∗], the sequence u (n) converges to the exact solution independently of the initial value a0.

Next, for the classical KMF algorithm we denote a set of four¡a n , b n , u (2n−1),

u (2n)¢ By analogy, Relaxation algorithm 1, says ¡ ˜a n, ˜b n, ˜u (2n−1), ˜u (2n)¢; ation algorithm 2¡ ¯a n, ¯b n, ¯u (2n−1), ¯u (2n)¢ Then we have the following result

Relax-Theorem 2.4 (M Jourhmane, D Lesnic and N.S Mera [4], 2004) If α n = β n , ∀n ≥

may be written as the sum of the solutions of the following two problems

known from the previous iteration, the first problem at the n-th iteration is

Remark 2.5 Moreover, from Theorem 2.4 (more precisely, from the proof of

Theorem 2.4) one can deduce that ifα n = β n , ∀n ≥ 1, then for all n ≥ 1 the