tóm tắt luận án tiến sĩ tiếng anh phương trình parabolic ngược thời gian

Parabolic equations backward in time appear frequently in the heat transfertheory, geophysics, groundwater problems, materials science, hydrodynamics, im-age processing ...This is the pr

MINISTRY OF EDUCATION AND TRAINING

Subject: Mathematical Analysis

Code: 62 46 01 01

PhD THESIS SUMMARY

VINH - 2011

Supervisors: 1 Prof Dr Sc Dinh Nho H`ao

2 Assoc Prof Dr Dinh Huy Hoang

Referee 1: Prof Dr Nguyen Huu Du

Hanoi University of Science, VNU

Referee 2: Assoc Prof Dr Ha Tien Ngoan

Institute of Mathematics - Vietnam Academy of Science and Technology

Referee 3: Assoc Prof Dr Nguyen Xuan Thao

Hanoi University of Science and Technology

The thesis will be defended at the exam Committee at

.

time date month 2011

The thesis is available at:

- National Library of Vietnam

- Library of Vinh University

Parabolic equations backward in time appear frequently in the heat transfertheory, geophysics, groundwater problems, materials science, hydrodynamics, im-age processing This is the problem, when the initial condition is not knownand we must determine it from the final condition These problems have beenintensively studied, but only for some special classes Moreover, finding efficientnumerical methods for them is always desired

Parabolic equations backward in time are ill-posed in sense Hadamard Aproblem is called well-posed if it fulfills the following properties:

a) For all admissible data, a solution exists

b) If a solution exists, it is unique

c) The solution depends continuously on the data

If one of the above properties is not satisfied, then the problem is called

ill-posed Hadamard supposed that ill-posed problems have no physical meaning.

However, many practical problems of science and technology have led to ill-posedproblems Therefore, since 1950 many papers concerned ill-posed problems havebeen published Mathematicians such as A N Tikhonov, M M Lavrent’ev, F.John, C Pucci, V K Ivanov are pioneers in this field

In 1955, John reported some results about a method to numerically solve theCauchy problem for the heat equation backward in time Then, Krein and cowork-ers also published some results on stability estimates and backward uniqueness forparabolic equations backward in time In 1963, Tikhonov proposed a regular-ization method which is applicable for almost all inverse and ill-posed problems.Especially, this method was applied successfully to the backward heat equation in

1974 by Franklin Addition to these, many authors also use another methods suchas: QR method, SQR method, the backward beam equation method, the method

1

of non-local boundary value problems, iterative methods, finite difference ods, mollification method for parabolic equations backward in time However,

meth-no method is universal for all problems For example, Tikhometh-nov method or QRmethod require to solve a equation of double of that of the original equation andchoosing regularization parameters is not easy Further, it is very difficult to useTikhonov method in Banach spaces

Until now, hundreds of papers devoted to parabolic equations backward in timethere have been published which focused mainly on

1) backward uniqueness,

2) stability estimates,

3) regularization methods, stable and efficient numerical methods

In this thesis, we focus on obtaining stability estimates and regularization ods for parabolic equations backward in time We regularize the problem

a priori and a posteriori parameter choice rules in order to yield order-optimalregularization methods Furthermore, the method were tested on the computerand the results are very encouraging

To our knowledge, Vabishchevich is one of the first using this method for theparabolic equations backward in time in 1981 He proposed an a priori methodfor (0.1) but without giving the convergence rates as we do in this thesis Further,

he suggested the following a posteriori method for (0.1)

Solve the well-posed problem

and take v α ((a − 1)T + t) as an approximation to u(t) We suggest a priori and

a posteriori strategies for choosing the parameter α and prove that these yield

order-optimal regularization methods for (0.1) The a priori method is given inTheorems 1.2.1, 1.2.3, 1.2.5, 1.3.1, the a posteriori method is as follows Suppose

ε < kf k and let τ > 1 satisfy τ ε < kf k Choose α > 0 such that

kv α (aT ) − f k = τ ε.

We note that, since f − v α (aT ) = αv α(0), the above discrepancy principle has the

very simple form: Choose α > 0 such that

Our results in case a = 1 are better than theirs We note that Denche and Bessila

approximated the problem (0.1) by the problem

establish stability estimates which are comparable to theirs

The problem becomes much more difficult if the operator A depends on timeand there are very few results in this case In this thesis, we improve the related

results by Krein, Agmon and Nirenberg Furthermore, we also suggest a larization method Our regularization method with a priori and a posteriori pa-rameter choice yields error estimates of H¨older type This is the only result when

regu-a regulregu-arizregu-ation method for bregu-ackwregu-ard pregu-arregu-abolic equregu-ations with time-dependentcoefficients provides a convergence rate

In the last part of the thesis, we use the mollification method to regularize for

the heat equation backward in time in Banach space L p (R) 1 < p < ∞ Namely,

we study the following problem: Let p ∈ (1, ∞), ϕ ∈ L p (R) and ε, E be given constants such that 0 < ε < E < ∞ Consider the heat equation backward in

u t = u xx , x ∈ R, t ∈ (0, T ), ku(·, T ) − ϕ(·)k L p(R) 6 ε, (0.5)

subject to the constraint

ku(·, 0)k L p(R) 6 E. (0.6)

We note that the case p 6= 2 is much more difficult, since we do not have the Parseval equality and in general the Fourier transform of a function in L p(R) with

p > 2 is a distribution.

This problem has been considered by the first author He gave a stability

estimate of H¨older type for the case p ∈ (1, ∞]: if u1 and u2 are two solutions of

the problem, there is a constant c ∗ such that

(0.5)–(0.6) However, instead of using the de la Vall´ee Poussin kernel for mollifying

the Cauchy data ϕ, we use the Dirichlet kernel and thus work with mollified data generated by the convolution of this kernel with ϕ The mollified data belong to

the space of band-limited functions, in which the Cauchy problem is well-posed,and with appropriate choices of mollification parameter we obtain error estimates

of H¨older type Stability estimates for the solutions of the problem (0.5)–(0.6) isthe direct consequence of these error estimates and the triangle inequality

In this thesis, supplementally to the result of Dinh Nho H`ao for p = 2, we establish stability estimates of H¨older type for all derivatives with respect to x and t of the solutions It is worth to note that such estimates are very seldom in

the literature of ill-posed problems

It is well known that with only the condition (0.6), we cannot expect any

con-tinuous dependence of the solution at t = 0 This can be recovered if an additional condition on the smoothness of u(x, 0) is available (see Theorem 3.2.7) To this

purpose, in the literature the regularization parameters are chosen dependently

on the parameters of this ”source condition” which are in general not known Toovercome this shortcoming, in Theorems 3.2.6 and 3.2.10 we propose a choice ofmollification parameters using only the condition (0.6) which guarantees error es-

timates of H¨older type in (0, T ] and a continuous dependence at t = 0 when a

source condition is available but without knowing its parameters This choice ofmollification parameters seems to be quite interesting for the numerical treatment

of the problem (0.5)–(0.6)

For p = 2, since the Fourier transform of mollified data has compact support,

one has at least two equivalent forms of the mollification method: one in its originalform, another uses the frequency cut-off technique These two forms lead to twodifferent numerical schemes which can be easily implemented numerically using

the fast Fourier transform technique (FFT) For p 6= 2, these schemes do not work

and we propose a stable marching difference scheme for (0.5) We test the methodsfor different numerical examples and see that they are very stable and fast

The thesis consists of an introduction, three chapters, conclusion and references.Chapter 1 presents results on the regularization of parabolic equations backward

in time with time-independent coefficients in Hilbert Spaces Theorems 1.2.1, 1.2.3

and 1.2.5 provide the results on an a priori parameter choice rule in case a = 1.

Theorems 1.3.1, 1.3.3 provide the results on a priori and a posteriori parameter

choice rules in case a > 1 At the end of Chapter 1, numerical results are presented

and discussed to confirm the theory

Chapter 2 presents results on stability estimates and regularization of parabolicequations backward in time with time-dependent coefficients in Hilbert spaces.Chapter 3 presents stability results for the heat equation backward in time in

Hilbert and Banach spaces, namely, in L p (R) for p ∈ (1, +∞) Theorems 3.2.1,

3.2.3 present stability and regularization results in Banach spaces with the same

convergence rate as in case L2(R) A slightly modifying the choice of ν in Theorem 3.2.1 gives a stability estimate of H¨older type for t ∈ (0, T ] which guarantees a continuous dependence of logarithmic type at t = 0 without explicitly knowing

˜

E and γ is the main result of Theorem 3.2.6 In case p = 2, Theorem 3.2.7

provides error estimates of H¨older type for all derivatives with respect to x and t

of the solutions.Theorem 3.2.10 shows that a slightly modifying the choice of ν in Theorem 3.2.7 guarantees a continuous dependence of logarithmic type at t = 0

without explicitly knowing ˜E and γ At the end of Chapter 3, a stable marching

difference scheme and numerical examples are presented and discussed

CHAPTER 1PARABOLIC EQUATIONS BACKWARD IN TIME WITH

with the positive self-adjoint unbounded operator A that admits an orthonormal

eigenbasis {φ i } i>1 in Hilbert space H with norm k · k, associated with the values {λ i } i>1 such that 0 < λ1 6 λ2 6 , and lim

eigen-i→+∞ λ i = +∞ In order to regularize the problem, we suppose that there is a positive constant E > ε > 0

with a > 1 being given and α > 0, the regularization parameter A priori and a

posteriori parameter choice rules are suggested which yield order-optimal

regular-ization methods The results of this chapter are published in Journal of

Mathe-matical Analysis and Applications and IMA Journal of Applied Mathematics.

Definition 1.1.1 Let H be a Hilbert space with the inner product h·, ·i and the

norm k·k, a and T are positive number The space C([0, aT ]; H) consist of all tinuous functions u : [0, aT ] → H with the norm kuk C([0,aT ];H) = max

con-06t6aT ku(t)k <

∞.

7

The space C1((0, aT ); H) consist of all continuously differentiable functions

u : (0, aT ) → H D(A) ⊂ H is domain of the operator A : D(A) ⊂ H → H.

Definition 1.1.2 A function v α : [0, aT ] → H is called a solution of (2.19) if

v α ∈ C1((0, aT ), H) ∩ C([0, aT ], H), v α (t) ∈ D(A), ∀t ∈ (0, aT ), and satisfies the

equation v αt + Av α = 0 in the interval (0, aT ) and the boundary value condition

1.2 Regularization of parabolic equations backward in time

by a non-local boundary value problem in case a=1

In this section, we regularize the problem (1.1), (1.2) by the non-local boundary

v αt + Av α = 0, 0 < t < aT,

We denote the solution of (1.1) by u(t), and the solution of (1.6) by v(t).

Theorem 1.2.1 The following inequality holds

K(t) := (t/T ) t/T (1 − t/T ) 1−t/T ∈ (0, 1), ∀t ∈ (0, T ),

K(0) = K(T ) = 1.

Theorem 1.2.1 does not give any information about the continuous dependence

of the solution of (1.1),(1.2) at t = 0 on the data, as the condition (1.2) is too

weak To establish this, we suppose that

for some positive constants β, γ, E1 and E2

Theorem 1.2.3 Suppose that instead of (1.2), we have (1.8) Then for all t ∈

Ã

e λ1T

λ β(t)1 C(t)

!1− t T

Remark 1.2.4 From final estimate of Theorem 1.2.3, at t = 0 we have an error estimate of logarithmic type Particularly, for β = 1, our error estimate at t = 0

is comparable to that of Denche and Bessila

Theorem 1.2.5 Suppose that instead of (1.2), we have (1.9) Then for all t ∈ [0, T )

ku(t) − v(t)k 6

½

Q(t, α)α t/T −1 ε + α (t+β)/T E2, if 0 < β < T − t, Q(t, α)α t/T −1 ε + αE2, if β > T − t.

1.3 Regularization of parabolic equation backward in time

by a non-local boundary value problem in case a > 1

We denote the solution of (1.1) by u(t), and the solution of (1.3) by v α (t).

1.3.1 A priori parameter choice rule

Theorem 1.3.1 (i) If u(t) satisfies (1.2), then with α = ¡E ε¢a we have

ku(t) − v α ((a − 1)T + t)k 6

Ã

H

µ1

a

¶+ 1

#−β(T −t)/T

(1 + o(1)),

a , β

¶¶t/T Ã

H

µ1

1.3.2 A posteriori parameter choice rule

Theorem 1.3.3 Suppose that ε < kf k Choose τ > 1 such that 0 < τ ε < kf k.

Then there exists a unique number α ε > 0 such that

τ − 1 C

µ1

a , β

¶

H

µ1

a

¶

+ C (0, β)

¶1− t T

.

(iii) if u(t) is a solution of the problem (1.1) and (1.9), then, for t ∈ [0, T ],

Remark 1.3.5 (a) In the first and second cases, our method is order-optimal

(b) For the third case, our method is of optimal order for γ ∈ (0, (a − 1)T ].

We tested on the computer for the a posteriori parameter choice rule in §1.3 with

two examples and find that that the method is stable and efficient

subject to the constraint ku(0)k 6 E (E > ε > 0) is regularized by the well-posed

non-local boundary value problem

or H¨older type at t = 0 if an additional condition on the smoothness of u(x, 0) is

available If only with the condition (2.2), an a posteriori parameter choice rule is

also suggested which yields order optimal regularization methods for all t ∈ (0; T ].

- In case a > 1, A priori and a posteriori parameter choice rules are suggested

which yield order optimal regularization methods

- Numerical results based on the boundary element method are presented anddiscussed to confirm the theory

CHAPTER 2PARABOLIC EQUATIONS BACKWARD IN TIME WITH

Here, H is a Hilbert space with the inner product h·, ·i and the norm k · k, f ∈ H,

A(t) (0 6 t 6 T ) is positive self-adjoint unbounded operators from D(A(t)) ⊂ H

to H, and E > ε > 0 are known The results of this chapter are published in

Inverse Problems.

2.1 Stability estimates

2.1.1 Agmon and Nirenberg’s results

For the reader’s convenience, we summarize Agmon and Nirenberg’s results

Suppose that:

(i) A(t) : D(A(t)) ⊂ H → H is a closed densely defined operator for each

t ∈ [0, T ], and u(t) belongs to the domain of A ∗ (t) as well as to that of A(t) (ii) In addition, A(t) is smooth in its dependence on t and A is ”almost self- adjoint” These hypotheses are best expressed in a single condition: if u(t) is the

solution of the equation

Lu = du

13

then for some positive constants k, c

−< d

dt hA(t)u(t), u(t)i ≥

1

2k(A(t) + A

∗ (t))u(t)k2 − c< h(A(t) + k)u(t), u(t)i

Theorem 2.1.1 (Agmon and Nirenberg) Let conditions (i)–(ii) be satisfied Then

the function log |e −kt u(t)| is a convex function of the variable s = e ct

The following results are direct consequences of Theorem 2.1.1

Proposition 2.1.2 (Stability estimates) Let conditions (i)–(ii) be satisfied Then,

In case A(t) is a self-adjoint, we have the following results.

Proposition 2.1.3 Suppose that

(iii) A(t) is a self-adjoint operator for each t, and u(t) belongs to the domain

of A(t).

(iv) if u(t) is the solution of the equation (2.3), then for some positive constants

k, c,

− d

dt hA(t)u(t), u(t)i ≥ 2kA(t)uk

2− c h(A(t) + k)u(t), u(t)i (2.6)

Then, for all t ∈ [0, T ],

ku(t)k ≤ e kt−kT µ(t) ku(T )k µ(t) ku(0)k 1−µ(t) (2.7)

Remark 2.1.4 We always have µ(t) < t

T , ∀t ∈ (0, T ) Thus, the order of the stability estimates by Agmon and Nirenberg is not higher than that in case time-independent coefficients.

2.1.2 An improvement of Agmon and Nirenberg’s results

Theorem 2.1.5 Suppose that

(i) A(t) is a self-adjoint operator for each t, and u(t) belongs to the domain of A(t).