Reconstruction of flow and temperature from boundary data

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Link¨

oping Studies in Science and Technology. Dissertations

No. 832

Reconstruction of Flow

and Temperature from

Boundary Data

Tomas Johansson

Γ0

Department of Science and Technology,

Campus Norrk¨

oping, Link¨

oping University,

S-601 74 Norrk¨

oping, Sweden

Norrk¨

oping 2003

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Link¨

oping Studies in Science and Technology. Dissertations

No. 832

Reconstruction of Flow

and Temperature from

Boundary Data

Tomas Johansson

Department of Science and Technology,

Campus Norrk¨

oping, Link¨

oping University,

S-601 74 Norrk¨

oping, Sweden

Norrk¨

oping 2003

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Preface

In this thesis, we study Cauchy problems for elliptic and parabolic equations. These include the stationary Stokes system and the heat equation. Data are given on a part of the boundary of a bounded domain. The aim is to reconstruct the solution from these data. These problems are ill-posed in the sense of J. Hadamard.

We propose iterative regularization methods, which require solving of a se-quence of well-posed boundary value problems for the same operator. Methods based on this idea were first proposed by V. A. Kozlov and V. G. Maz’ya for a certain class of equations which do not include the above problems. Regularizing character is proved and stopping rules are proposed.

The regularizing character for the heat equation is proved in a certain weighted L2_{space. In each iteration the Zaremba problem for the heat equation}

is solved. We also prove well-posedness of this problem in a weighted Sobolev space. This result is of independent interest and is presented as a separate paper.

Acknowledgement :

I wish to express my gratitude to my supervisor Professor Vladimir Kozlov for his invaluable advice during this work and for sharing his great mathematical knowledge in a nice and friendly way.

I would like to thank my co-supervisor George Bastay for fruitful discussions and for his stimulating enthusiasm and experience of ill-posed problems.

I also thank the following: Mark Bender for reading the manuscript of this thesis, Lionel Elliot, Derek B. Ingham, and Daniel Lesnic for sharing their knowledge about computational mathematics and for their hospitality, Stan Miklavcic for support in the beginning of this work, Bengt Ove Turesson for helping me with LA_{TEX, the Department}

of Mathematics at Link¨oping University for giving interesting and stimulating courses, and the Department of Science and Technology at Campus Norrk¨oping for providing an excellent research environment.

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vi Preface

AMS Mathematics Subject Classification (2000 ). Primary: 35R25.

Secondary: 47A52, 35M10.

Key words and phrases: Cauchy problem, heat equation, ill-posed problem, inverse problem, iterative regularization method, mixed problem, Stokes system, weighted Sobolev space.

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Principle notation

Throughout this thesis, Rn _{denotes Euclidian n-space equipped with the norm}

|x| = (Pn

i=1x 2

i)1/2, where x = (x1, . . . , xn) ∈ Rn. We let Ω be a bounded

domain in Rn_{, i.e., an open, bounded, and connected subset of R}n_{. The closure}

of Ω is denoted by Ω. The support of a function u defined on Ω is the closure of all points x∈ Ω with u(x) 6= 0 and is denoted as supp u. Let 0 ≤ k ≤ ∞. The elements of Ck_{(Ω) are k times continuously differentiable functions on Ω.}

By C₀k(Ω), we mean functions in Ck(Ω) having compact support in Ω.

The terms “integrable” and “locally integrable” will always refer to the Lebesgue measure dx = dx1· · · dxn. We denote by |Ω| the Lebesgue measure

of Ω.

Let α = (α1, . . . , αn) be a multi-index, i.e., αi is a non-negative integer

for i = 1, . . . , n. We write ∂_xα= ∂ α1 ∂xα1 1 . . . ∂ αn ∂xαn n , xα= xα1 1 · . . . · x αn n , |α| = α1+· · · + αn,

where x = (x1, . . . , xn)∈ Rn. This notation of|α| is inconsistent with the norm

of an element x∈ Rn _{defined above, but the meaning will always be clear from}

the context. If no confusion arises, we shall also write ∂α _{instead of ∂}α x. The

gradient of order k≥ 1 of a function u is ∇ku = (∂αxu)|α|=k and

|∇ku|2=

X

|α|=k

|∂xαu| 2_.

For k = 1, we omit the subscript k and use the notation∇u.

Operators will be denoted by capital letters, e.g., T . The symbol is used to mark the end of a proof. Finally, we point out that any positive constant whose value is irrelevant is denoted by the symbol C.

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Chapter 0 Introduction

In this chapter, we provide some background to ill-posed problems. A brief review about iterative regularization methods for partial differential equations and an outline of this thesis are also given.

0.1. Ill-posed problems

Following J. Hadamard, by a well-posed problem we mean a problem satisfying the following conditions:

1. A solution exists. 2. The solution is unique.

3. The solution depends continuously on the data of the problem.

This definition is made precise by defining the function spaces in which the data and solution lie and the notation of continuity. If one or more of these conditions are not fulfilled, then the problem is denoted as ill-posed . If we cannot guarantee the uniqueness of a solution, i.e., if condition two is not satisfied, one can possibly add some extra requirements, which bring uniqueness to the problem. Condition one is not so restrictive. Even if we do not know whether a solution to a given set of data exists, it can still be possible to construct a procedure which finds the solution. If the third condition above is not satisfied, then the problem is difficult to handle. In practice, one cannot in general avoid introducing some error when measuring the data. If the third condition is not satisfied, this means that one can get an unlimited amount of error in the calculated solution.

It is worth mentioning that ill-posed problems are of practical importance. Many people, including Hadamard, were convinced that ill-posed problems had

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2 0. Introduction

no physical application and therefore should be ignored. This became the gen-eral attitude but in the 1960’s this attitude began to change. This change was due to an increasing number of problems in the applied sciences that in fact were ill-posed and the need for theoretical understanding of them. Some areas where problems of this kind arise are in heat conduction, image reconstruc-tion, radar technology, signal processing, and X-ray tomography. For examples in these fields, see Chapter 1 in Engl, Hanke and Neubauer [7], Chapter 2 in Groetsch [10], and Tikhonov and Goncharski˘ı [35].

Ill-posed problems can arise as variations of direct problems. An example of a direct problem could be a problem where one wants to predict the future state of a physical situation. To predict the past state of the same physical problem given current information is a so-called inverse problem which can be ill-posed. An example of an inverse problem of this kind is the backward heat equation. A review of this problem can be found in Chapter 3 in Isakov [15]. For further examples and discussions about inverse problems, see Allison [1] and Keller [18]. We now give three examples of ill-posed problems. The first one is the simplest. The second is of the same kind as was given in Hadamard [11]. The third is an example of a general situation where ill-posed problems occur. Example 0.1.1. Let f ∈ C1_{[0, 1] and let δ}

∈ (0, 1). For n = 2, 3, . . . , define fn,δ(x) = f (x) + δ sin nx δ , x∈ [0, 1]. The derivative of fn,δ is f_n,δ0 (x) = f0(x) + n cosnx δ , x∈ [0, 1]. We then have sup x∈[0,1]|f(x) − f n,δ(x)| = δ, while sup x_∈[0,1] |f0(x)− fn,δ0 (x)| = n.

If we consider f as given exact data and fn,δ as a bounded perturbation of f ,

then the error in the result, the derivative, can be arbitrarily large. Hence, the derivative does not depend continuously on the data (in the supremum norm) and we have an ill-posed problem.

Example 0.1.2. Let us consider the problem of finding the two-dimensional steady temperature field inside a (thin) homogenous and square plate with side length equal to one. Suppose that the temperature is equal to zero at one of the edges of the plate. Moreover, assume that the heat flow across the same edge is equal to ϕn, where ϕn(x) = (nπ)−1sin nπx and n is a positive integer.

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0.2. Regularization methods 3

Then the temperature field un is a solution to a boundary value problem of the

following form    ∆un = 0 (x, y)∈ (0, 1) × (0, 1), un(x, 0) = 0 x∈ (0, 1), ∂yun(x, 0) = ϕn(x) x∈ (0, 1). The solution is

un(x, y) = (nπ)−2sin nπx sinh nπy.

As n tends to infinity, the data sequence ϕn tends uniformly to zero. However,

for y > 0, we see that un(x, y) tends to infinity. Hence, the third condition in

the definition of well-posedness above is not satisfied.

Example 0.1.3. Let T be an injective operator from one Hilbert space to an-other and let y be a given element in the range of T . If the inverse of T is not bounded, then the problem of finding x such that

T x = y, (0.1.1)

is ill-posed.

For example, it is well known that if T is a linear, injective, and compact operator on a Hilbert space with infinite dimension, then it has no bounded inverse.

To recover some information about the solution to an ill-posed problem of the form (0.1.1) in a stable way, there are so-called regularization methods. We give a precise definition of such methods in the next section.

0.2. Regularization methods

Let H1 and H2 be two Hilbert spaces and let T : H1 → H2 be a bounded

linear operator such that the kernel of T contains only zero, i.e., ker(T ) =_{0}. Consider the following problem. For a given element y in the range of T , i.e., y ∈ R(T ), find x ∈ H1 such that equality (0.1.1) holds. Since ker(T ) = {0},

there exists an inverse mapping from R(T ) to H1. As mentioned above, if this

mapping is unbounded, then (0.1.1) is an ill-posed problem.

Consider now the case when there is some error in y. That is, instead of y, we have ye, where

ky − yekH2 ≤ δ, (0.2.1)

with δ > 0. Let the inverse of T be unbounded. Then we cannot guarantee that the inverse mapping of ye(if ye∈ R(T )) will approximate x, since this mapping

is not continuous. Instead, one shall attempt to find a family of continuous operators that in some sense approximate this inverse mapping.

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4 0. Introduction

Definition 0.2.1. Let H1 and H2 be two Hilbert spaces and let T : H1 → H2

be a bounded linear operator such that ker(T ) ={0}. Also, let {Rk}∞k=1 be a

family of continuous operators (not necessarily linear), where Rk : H2→ H1, for k = 1, 2, . . . .

This family is a regularization method for equation (0.1.1) at x∈ H1, if there

exists a positive number δ0and functions k(δ) and ε(δ) defined on (0, δ0), such

that ε(δ) → 0, as δ → 0 and if the inequality kv − T xkH2 ≤ δ implies the

estimatekRk(δ)v− xkH1≤ ε(δ).

The family{Rk}∞k=1 is also called a regularization or a regularization

oper-ator for the inverse mapping of T . The function k(δ) in the above definition is often denoted as a stopping criterion or a stopping rule. From Definition 0.2.1, it follows that limδ_→0Rk(δ)(T x) = x. On the other hand, it is well-known that

a family of continuous operators{Rk}∞k=1 is a regularization method for

equa-tion (0.1.1) at a point x∈ H1, if Rk(T x) → x, as k → ∞, see Engl et al. [7,

p. 52].

A family_{Rk}∞k=1of operators is a regularization method for equation (0.1.1)

on a set M , if it is a regularization method for the equation at every element in M and the functions k(δ) and ε(δ) can be chosen independent of x∈ M.

In the equations to be studied in this thesis, ker(T ) ={0}. If ker T 6= {0}, then one can instead consider the so-called generalized inverse and obtain similar results.

There are many ways of constructing regularization methods. In some cases one can modify the operator T in such a way that the modified oper-ator is bounded, injective, and close to T . One such method for modifying the operator is called Tikhonov regularization or Tikhonov-Phillips regularization, see Phillips [29] and Tikhonov [33] and [34]. In Tikhonov’s papers [33] and [34], the foundations of the theory of ill-posed problems and regularization meth-ods are laid. Important contributions to the theory of ill-posed problems and methods for solving them have also been given by, among others, V. K. Ivanov, F. John, M. M. Lavrente’v, and J. L. Lions. For a brief survey, see the intro-duction to the monograph by V. A. Morozov [28].

Another group of regularizing methods includes iterative procedures. In the next two sections, we present some basic facts about two iterative procedures for solving equation (0.1.1). Iterative methods for solving well-posed problems have been studied during a long time. We focus on properties of these methods which show that they are regularization methods for (0.1.1).

0.2.1. The Landweber iteration

The Landweber iteration is an example of an iterative method for solving prob-lem (0.1.1). This method has been investigated by, among others, Fridman [9]

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0.2. Regularization methods 5

and Landweber [24]. In this procedure, we find elements xk = xk(y) from the

scheme

xk= xk−1+ ωT∗(y− T xk−1) (0.2.2)

for k ≥ 1. Here, x0 is equal to some initial guess, which starts the procedure,

T∗ is the adjoint operator, y∈ R(T ), and ω is a real-valued parameter. The Landweber iteration is a regularization method for ω chosen from a cer-tain interval. A proof of this and the other results in the following Theorem can for example be found in Engl et al. [7, pp. 155–158].

Theorem 0.2.2. Let T : H1→ H2be a bounded linear operator between Hilbert

spaces with ker(T ) ={0}. Assume that T x = y and let xk = xk(y) be the k-th

iterate in the Landweber iteration. If 0 < ω <kT k−2_{, then :}

(i) xk→ x, as k → ∞, in the Landweber iteration.

(ii) The Landweber iteration is a regularization method for equation (0.1.1) in the sense of Definition 0.2.1. Moreover, there exists a stopping rule k(δ) such that k(δ) = O(δ−2).

0.2.2. The minimal error method

Another iterative procedure for solving (0.1.1) is the minimal error method. This method was investigated by Craig [5] and Shamanskii [30] and is one of the so-called conjugate gradient methods, see Hestenes and Stiefel [14]. Assume, as before, that T : H1→ H2 is a bounded linear operator between Hilbert spaces.

The minimal error method can then be described as: First, calculate

r0= y− T x0 and d0= T∗r0,

where x0 ∈ H1 is some initial guess and T∗ is the adjoint operator. Then

compute further approximations xk+1= xk+1(y) according to:

αk = kr kk2 kdkk2 , xk+1= xk+ αkdk, rk+1= rk− αkT dk, βk = kr k+1k2 krkk2 , dk+1= T∗rk+1+ βkdk.

It is easy to check that rk = y− T xk for k≥ 0 in this procedure.

Note that the minimal error method does not involve any parameter as the Landweber method did. This is one advantage. Moreover, the convergence

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6 0. Introduction

rate of the minimal error procedure is in general much better than that of the Landweber method. A proof of the following results can be found in Chapter 4 in Hanke [12].

Theorem 0.2.3. Let T : H1→ H2be a bounded linear operator between Hilbert

spaces with ker(T ) =_{{0}. Assume that T x = y and let x}k = xk(y) be the k-th

iterate in the minimal error method. Then we have: (i) xk→ x, as k → ∞, in the minimal error method.

(ii) The minimal error method is a regularization method for equation (0.1.1) in the sense of Definition 0.2.1.

0.3. Iterative methods for ill-posed partial

differ-ential equations

Regularization methods for ill-posed partial differential equations is a vast area, for an introduction see Isakov [15]. A common regularization method for such problems is the method of quasi-reversibility, see Latt`es and Lions [25]. This method uses the above idea that one can obtain a well-posed problem by mod-ifying the operator in an appropriate way.

Some difficulties are incurred when one uses methods of quasi-reversibility. The way in which to modify the operator is not always unique. The new prob-lem that arises might be non-standard, which implies that there is no stan-dard numerical way to handle the perturbed problem. These observations led V. A. Kozlov and V. G. Maz’ya to study iterative methods for ill-posed partial differential equations which preserve the operator of the problem. Kozlov and Maz’ya proposed iterative methods for solving some boundary value problems for elliptic, parabolic, and hyperbolic equations, see [22]. One of the advantages with these methods is that they preserve the original operator and that the reg-ularizing character is achieved by appropriate change of boundary conditions. Most of the methods were suggested for differential operators which are self-adjoint. In Kozlov, Maz’ya and Fomin [23], an alternating iterative method is applied to Cauchy problems for equations of anisotropic elasticity.

Bastay [2] extended some of the results in [22] to problems which are not formally self-adjoint. Bastay also proposed accelerated methods and presented some numerical results.

Numerical investigations and results for alternating iterative procedures can for example be found in Baumeister and Leit˜ao [4], Engl and Leit˜ao [8], Jourhmane and Nachaoui [17], and Mera, Elliot, Ingham, and Lesnic [27].

Lundgren [26] investigated an alternating iterative procedure for the bihar-monic equation with incomplete Cauchy data.

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0.4. Outline of the thesis 7

An iterative procedure for Cauchy problems for parabolic and elliptic equa-tions with not necessarily self-adjoint operators, was proposed by Bastay, Kozlov and Turesson [3]. Numerical results were also presented (performed by the au-thor of this thesis). A similar method for the Laplace equation is investigated in H`ao and Lesnic [13].

0.4. Outline of the thesis

The purpose of this thesis is to develop iterative regularization methods for reconstruction of the solution to elliptic and parabolic equations from Cauchy data given on a part of the boundary of a bounded domain. These equations include the stationary Stokes system and the heat equation.

A problem which sometimes occurs in fluid mechanics is that of reconstruc-tion of a stareconstruc-tionary flow from boundary data. An approximareconstruc-tion to the fluid velocity can be found by solving the so-called generalized stationary Stokes sys-tem

Lu − ∇p = 0 in Ω,

divu = 0 in Ω, (0.4.1)

supplied with boundary conditions

u = ϕ on Γ0,

pν_{− Nu = ψ} on Γ0.

(0.4.2) Here, we suppose that Ω is a bounded domain in Rn, where n≥ 2, of class C2_{, Γ}

0

is a closed part the boundary Γ, and L is an elliptic operator of second order. The function u = (u1, . . . , un) is the velocity of the fluid and the scalar function p

is the pressure. We assume that the distance between Γ0 and Γ1 = Γ\ Γ0

is greater then zero. The data ϕ and ψ belong to the space L2_(Γ

0)n. This

problem is investigated in Paper 1. Neither of the methods in [3] and [23] can be applied, since the problem is not formally self-adjoint and not strongly elliptic but elliptic in the sense of Douglis and Nirenberg [6]. We propose an iterative procedure. At each iteration step, we solve well-posed problems for (0.4.1) and its adjoint, obtained by changing the boundary conditions, see Section 1.2 for a complete description of this method. Well-posedness of the problems used in this procedure are proved in the space L2_(Ω)n _{× (H}1_(Ω))∗_{, see Lemma 1.3.4.}

Convergence of the procedure to the solution of (0.4.1) and (0.4.2) is proved in Theorem 1.4.6.

In Paper 2, we consider problem (0.4.1) in the case when L is the Laplace operator, i.e., the stationary Stokes system, with boundary conditions (0.4.2). The domain Ω of this problem is bounded and has a Lipschitz boundary. The data are given at a Lipschitz domain of the boundary. The alternating method in [23] cannot be directly applied since the system is not strongly elliptic. In Sec-tion 2.3, we propose an alternating iterative procedure where, at each iteraSec-tion,

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8 0. Introduction

we solve well-posed problems for (0.4.1) obtained by changing the boundary conditions. The convergence of this procedure is demonstrated in the space H1(Ω)n× L2_{(Ω), see Theorem 2.6.1. In the case of noisy data, we give a}

stop-ping rule which brings stability to the reconstruction of the velocity and pressure from boundary measurements.

The remaining part of the thesis is then devoted to elliptic and parabolic equations which have applications to heat transfer problems. We start in Paper 3 by considering the problem

   Lu = 0 in Ω, u = ϕ on Γ0, N u = ψ on Γ0, (0.4.3)

in a bounded two dimensional domain Ω. Here, L is an elliptic operator of second order with smooth coefficients, N is the co-normal derivative, and Γ0is

the union of a finite number of open and connected parts of the boundary. One of the applications of problem (0.4.3) is reconstruction of the temperature in a non-homogenous and non-isotropic medium from boundary measurements. We propose an iterative method where in each step, we solve mixed boundary value problems for the operator L and its adjoint L∗with Dirichlet data on Γ1= Γ\Γ0

and Neumann data on Γ0. Solvability results for elliptic equations in domains

with conical points was investigated in Kondrat’ev [19]. Using such results, we show well-posedness of the mixed problems in a weighted L2space. The weight is of the form rβ−2, where the function r is the distance to the endpoints of Γ0,

and β is a real number in the interval (1/2, 3/2), see Lemma 3.3.4. Convergence of the method is also proved in the space mentioned above, see Theorem 3.4.1. It is also demonstrated that the method works with inexact data.

In the procedure that we propose, the factors rβ−3/2 _{and r}3/2−β _occur.

Formally, if β = 3/2, then the procedure is the same as in Bastay et al. [3]. As is indicated in Section 3.5, it is not possible to take β = 3/2 in order to have convergence. Thus, for the case that we consider in Paper 3, the usual L2_space

cannot be used as in Bastay et al. [3].

In Paper 4, we consider the following mixed boundary value problem        ∂tu− Lu = f in Ω× (0, T ), u = η on Γ1× (0, T ), Bu = ψ on Γ0× (0, T ), u(x, 0) = ϕ0 for x∈ Ω, (0.4.4)

where L is an elliptic operator of second order with coefficients which do not depend on t, B is an operator of first order, and Ω is bounded domain in Rn, where n≥ 3. Both the coefficients and the domain are assumed to be sufficiently smooth. We state this assumption more precisely in Section 4.2.

Solvability of problem (0.4.4) when n = 2 and L is equal to the Laplace operator was proved in Kozlov and Maz’ya [20]. In the case when Ω is a di-hedral angle and L is the Laplace operator, the above problem was studied

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0.4. Outline of the thesis 9

in Solonnikov [31] and [32]. For general parabolic operators and bounded two-dimensional domains such problems was studied in Kozlov [21].

In paper 4, we prove solvability results for (0.4.4) when n≥ 3. The boundary of Ω is the union of three non-empty and disjoint pieces, Γ0, Γ1, and M , where

M = Γ0∩ Γ1 is a smooth (n− 2)-dimensional manifold without boundary of

class C2` with ` ≥ 1. This in particular implies that Γ0 and Γ1 are (n−

1)-dimensional manifolds of class C2`with boundary equal to M . We study (0.4.4) in a certain weighted Sobolev space denoted by W_β2`,`(Ω× (0, T )). Functions in this space have generalized derivatives of order `≥ 1 with respect to t and of order 2` with respect to x. Each derivative is square summable with a weight. This weight is a certain power, involving the real number β, of the distance to the common part of Γ0 and Γ1.

The data satisfy the so-called consistency condition, see Section 4.2.3 for a definition of this condition. The main result that we shall prove is that for β in a certain interval, a unique solution to the above problem exists in the space W_β2`,`(Ω× (0, T )), see Theorem 4.2.3. Moreover, the solution can be estimated by the data of the problem. A proof of this theorem is given in Section 4.3.

In paper 5, we consider the Cauchy problem        ∂tu− Lu = 0 in Ω× (0, T ), u = ϕ on Γ0× (0, T ), N u = ψ on Γ0× (0, T ), u(x, 0) = 0 for x∈ Ω. (0.4.5)

Here, Ω is a bounded domain in Rn_{, where n > 2, of class C}2_{. We assume}

that the boundary of Ω is the union of three non-empty and disjoint pieces, Γ0, Γ1, and M , where M = Γ0∩ Γ1 is a smooth (n− 2)-dimensional manifold

without boundary of class C2_. _{The operator L is an elliptic operator with}

smooth coefficients which do not depend on the variable t. We generalize the method given in Paper 3 to this problem, see Section 5.3. In each step, we solve mixed boundary value problems for L and L∗with Dirichlet data on Γ1×(0, T ),

where Γ1= Γ\ Γ0, and Neumann data on Γ0× (0, T ). The weight is of the form

rβ−2, where the function r is the distance to the manifold M , and β is a real number in the interval (1/2, 3/2). Using the results in Paper 4, we prove that these problems are well-posed. Convergence of the method is also proved in the space mentioned above, see Theorem 5.7.1. It is also demonstrated that the method works with inexact data and a stopping rule is given in Section 5.7.2. The case when Ω is a bounded plain domain is dealt with at the end of Paper 5, see Section 5.8.

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Reconstruction of flow and temperature from boundary data

Link¨

oping Studies in Science and Technology. Dissertations

No. 832

Reconstruction of Flow

and Temperature from

Boundary Data

Tomas Johansson

Department of Science and Technology,

Campus Norrk¨

oping, Link¨

oping University,

S-601 74 Norrk¨

oping, Sweden

Norrk¨

oping 2003

Link¨

oping Studies in Science and Technology. Dissertations

No. 832

Reconstruction of Flow

and Temperature from

Boundary Data

Tomas Johansson

Department of Science and Technology,

Campus Norrk¨

oping, Link¨

oping University,

S-601 74 Norrk¨

oping, Sweden

Norrk¨

oping 2003

Contents

Preface

Principle notation

Chapter 0

Introduction

0.1. Ill-posed problems

0.2. Regularization methods

0.2.1. The Landweber iteration

0.2.2. The minimal error method

0.3. Iterative methods for ill-posed partial

differ-ential equations

0.4. Outline of the thesis

References

Papers

The articles associated with this thesis have been removed for copyright

reasons. For more details about these see: