Inverse Autoconvolution Problems with an Application in Laser Physics

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Inverse Autoconvolution Problems with an

Application in Laser Physics

Von der Fakultät für Mathematik der

Technischen Universität Chemnitz

genehmigte

Dissertation

zur Erlangung des akademischen Grades

doctor rerum naturalium

(Dr. rer. nat.)

vorgelegt von

Dipl.-Math. Steven Bürger

geboren am 16.3.1989 in Torgau

Tag der Einreichung: 2.5.2016

Gutachter:

Prof. Dr. Bernd Hofmann

Dr. Peter Mathé

Prof. Dr. Lothar Reichel

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Preface

Convolution and, as a special case, autoconvolution of functions are important in many branches of mathe-matics and have found lots of applications, such as in physics, statistics, image processing and others. While it is a relatively easy task to determine the autoconvolution of a function (at least from the numerical point of view), the inverse problem, which consists in reconstructing a function from its autoconvolution is an ill-posed problem. Hence there is no possibility to solve such an inverse autoconvolution problem with a simple algebraic operation. Instead the problem has to be regularized, which means that it is replaced by a well-posed problem, which is close to the original problem in a certain sense.

The outline of this thesis is as follows: In the ﬁrst chapter we give an introduction to the type of inverse problems we consider, including some basic deﬁnitions and some important examples of regularization methods for these problems. At the end of the introduction we shortly present some general results about the convergence theory of Tikhonov-regularization.

The second chapter is concerned with the autoconvolution of square integrable functions deﬁned on the interval[0, 1]. This will lead us to the classical autoconvolution problems, where the term “classical” means

that no kernel function is involved in the autoconvolution operator. For the data situation we distinguish two cases, namely data on[0, 1] and data on [0, 2]. We present some well-known properties of the classical

autoconvolution operators. Moreover, we investigate nonlinearity conditions, which are required to show applicability of certain regularization approaches or which lead convergence rates for the Tikhonov regular-ization. For the inverse autoconvolution problem with data on the interval[0, 1] we show that a convergence

rate cannot be shown using the standard convergence rate theory. If the data are given on the interval[0, 2],

we can show a convergence rate for Tikhonov regularization if the exact solution satisﬁes a sparsity assump-tion. After these theoretical investigations we present various approaches to solve inverse autoconvolution problems. Here we focus on a discretized Lavrentiev regularization approach, for which even a convergence rate can be shown. Finally, we present numerical examples for the regularization methods we presented.

In the third chapter we describe a physical measurement technique, the so-called SD-Spider, which leads to an inverse problem of autoconvolution type. The SD-Spider method is an approach to measure ultrashort laser pulses (laser pulses with time duration in the range of femtoseconds). Therefor we ﬁrst present some very basic concepts of nonlinear optics and after that we describe the method in detail. Then we show how this approach, starting from the wave equation, leads to a kernel-based equation of autoconvolution type.

The aim of chapter four is to investigate the equation and the corresponding problem, which we derived in chapter three. As a generalization of the classical autoconvolution we define the kernel-based autoconvo-lution operator and show that many properties of the classical autoconvoautoconvo-lution operator can also be shown in this new situation. Moreover, we will consider inverse problems with kernel-based autoconvolution op-erator, which reflect the data situation of the physical problem. It turns out that these inverse problems may be locally well-posed, if all possible data are taken into account and they are locally ill-posed if one special part of the data is not available. Finally, we introduce reconstruction approaches for solving these inverse problems numerically and test them on real and artificial data.

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Introduction

1.1 General theory for nonlinear inverse problems in Hilbert spaces

1.1.1 Problem setting

The setting for the inverse problems we will consider consists of an operator

𝐹 ∶ 𝑋 ⊃(𝐹 ) → 𝑌 (1.1)

mapping between Hilbert spaces𝑋 and 𝑌 and the corresponding equation

𝐹 (𝑥) = 𝑦, (1.2)

which we want to solve for a given right hand side𝑦 ∈ 𝑌 . In practice one usually has only noisy data 𝑦𝛿 _∈_𝑌

at hand, where the norm diﬀerence to the exact data𝑦†_∈_{𝑌 is controlled by an error bound 𝛿 > 0:}

‖𝑦𝛿₋_𝑦†_‖

𝑌 ≤ 𝛿.

We say that (1.2) is a linear inverse problem, if the forward operator𝐹 is a linear operator, otherwise it is

a nonlinear inverse problem. In general, nonlinear inverse problems are much harder to handle than linear ones, since there are only few general results, therefore regularization methods for nonlinear problems often are very adapted. An important concept in inverse problems is ill-posedness. Roughly speaking, a problem of type (1.2) is called ill-posed, if it cannot be solved directly in a reasonable way, for example by applying the inverse of𝐹 (if it exists) to the data. A commonly used deﬁnition for ill-posedness is according to

Hadamard (cf. [15]), which says that a problem is well-posed if for all admissible data • a solution exist (existence),

• the solution is unique (uniqueness),

• the solution depends continuously on the data (stability).

If at least one of these conditions is violated, the problem is called ill-posed. Of course this is just a heuristic deﬁnition, but since existence and uniqueness do not hold for autoconvolution problems in general, we will not go into details here, we rather are particularly interested in the stability condition. Therefor we introduce the concept of local ill-posedness:

Definition 1. An operator equation of type (1.2) is called locally well-posed at𝑥₀ ∈𝑋, if there exists an open neighbourhood𝑈 of 𝑥₀, such that for every sequence(𝑥𝑛)𝑛∈ℕ⊂ 𝑈 ∩(𝐹 ) the convergence condition

𝐹 (𝑥𝑛) → 𝐹 (𝑥0) implies𝑥𝑛→ 𝑥0. If an open neighbourhood with this property does not exist, the equation is called locally ill-posed at𝑥0.

While for a linear operator equation ill-posedness at one point implies ill-posedness on the whole domain (and the same with posedness), this is not the case for nonlinear operator equations, for which local ill-posedness is indeed a local property in general.

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1.1.2 Regularization

It is characteristic for ill-posed problems of type (1.2), that they cannot be solved by applying the inverse of the forward operator𝐹 to the data 𝑦𝛿_{, which is clear if a solution does not exist (this corresponds to}

𝑦 ∉(𝐹 )) or the solution is not unique (in this case 𝐹 is not injective). But even if a unique solution exists,

it is not reasonable to apply the inverse of𝐹 to the data in the presence of noise, since arbitrary small errors

in the data can lead to high reconstruction errors. For this reason one has to regularize the problem, that means to replace the original ill-posed problem by a well-posed one (the regularized problem), such that the solutions of these problems are somehow close to each other. There are many possibilities to replace an ill-posed problem by a regularized one, depending on which regularization method is applied. Regularization methods usually contain one or more regularization parameters which have inﬂuence on the stability of the problem and the connection between original and regularized problem. Such regularization parameters can for example be positive weights of penalty terms in a minimizing functional or the number of iterations for an iterative regularization method. To obtain a reasonable reconstruction it is necessary to choose these parameters appropriately. For this purpose there is a variety of parameter choice strategies. Furthermore many regularization methods allow to bring in a-priori information about the true solution in form of a reference element𝑥∗ _∈ _{𝑋, for which one assumes that the true solution is somehow close to 𝑥}∗_{. For a}

regularization method to be appropriate, it should have certain properties which we want to discuss in the following.

Properties of regularization methods

Existence of a regularized solution We say that a regularized solution exists for the data𝑦,

regulariza-tion parameter𝛼 (and possibly a reference element 𝑥∗_{), if the regularized problem has a solution for these}

parameters. This is not to be confused with the existence of a solution of (1.2). For an iteration method this could mean that the iteration terminates at some point. For a regularization method which consists in minimizing a functional, existence of a solution is equivalent to existence of a minimizer of the functional. Note that we do not demand that the regularized solution is unique.

Stability Since exact data are not available in practice, it is important that the reconstructions, obtained by a regularization method, are not signiﬁcantly aﬀected by small perturbations of the data. To formalize this, we consider a regularization method with regularization parameter𝛼 > 0, perhaps depending on a reference

element𝑥∗_{, for which solutions exist for all}_{𝑦 ∈ 𝑌 , as a function}

𝑅_𝛼 ∶𝑌 → 𝑋

which maps the data𝑦 to a corresponding regularized solution (if the regularized solution is not unique, 𝑅_𝛼

shall map to one of them). This notation leads us to the following deﬁnition of stability

Definition 2. Let𝛼 > 0, Ω ⊂ 𝑌 closed and 𝑅_𝛼 a regularization method, s.t solutions of𝑅_𝛼 exist for the parameters𝛼 and 𝑦_𝑛for all𝑛 ∈ ℕ. 𝑅_𝛼is stable inΩ, if for every sequence (𝑦_𝑛)_𝑛∈ℕ⊂ Ω with 𝑦_𝑛→ 𝑦 ∈ Ω

the sequence(𝑅_𝛼(𝑦_𝑛))_𝑛∈ℕhas a convergent subsequence(𝑅_𝛼(𝑦_𝑛_𝑘))_𝑘∈ℕand each convergent subsequence converges to a regularized solution for the data𝑦.

Convergence Since we are seeking for an approximate solution of (1.2), it is desirable that the

recon-structions are close to an exact solution𝑥†_{of (1.2). Especially a sequence of regularized solutions should}

somehow converge to an exact solution if the corresponding sequence of noise levels converges to zero. As we do not assume that the Forward operator is injective, a solution of (1.2) need not be unique. For this reason we use the concept of̄𝑥-minimum norm solutions (cf. [15, Chapter 10.1]).

Definition 3. Let𝑥∗_∈_{𝑋. Then 𝑥}†_∈_{𝑋 is called an 𝑥}∗_{-minimum-norm solution of (1.2) if}

𝐹 (𝑥†_{) =}_𝑦 and

‖𝑥†₋_𝑥∗_{‖ = min}{_{‖𝑥 − 𝑥}∗_{‖ | 𝑥 ∈ 𝑋, 𝐹 (𝑥) = 𝑦}}

In some books the term convergence is not generally defined for regularization methods for nonlinear operator equations, instead convergence Theorems are proven for different methods (cf. [15] and [37]). Since this definition requires some reference element𝑥∗_{, we assume that our regularization procedure also uses}

such a reference element. To obtain convergence, it is necessary to choose the regularization parameter𝛼

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Definition 4. Let𝑥†_∈_{(𝐹 ), 𝑦}†₌_{𝐹 (𝑥}†_),_𝑥∗_∈_{𝑋 a reference element and assume that the regularization} method𝑅_𝛼is stable in𝐵_𝑟(𝑦†_{) for all}_𝛼≤ 𝛼

0with𝛼0, 𝑟 > 0. Let 𝑆 be defined as the set of all 𝑥∗ minimum-norm solutions of

𝐹 (𝑥) = 𝑦†. (1.3)

𝑅𝛼together with the parameter choice rule𝛼 = 𝛼(𝛿, 𝑦𝛿) is convergent for𝑥†, if

inf 𝛿>0_‖𝑦 sup 𝛿−𝑦†‖≤𝛿 { dist(𝑅𝛼(𝑦𝛿) −𝑆 )} = 0. (1.4)

For𝑢 ∈ 𝑋 dist(𝑢, 𝑆) is the point-to-set distance defined as

dist(𝑢, 𝑆) ∶= inf

𝑥∈𝑆‖𝑢 − 𝑥‖ (1.5)

For the case𝑆 = {𝑥†_{} which means that}_𝑥†_{is the only}_𝑥∗_{-minimum-norm solution of} _{(1.3), the condition}

(1.4) is equivalent to inf 𝛿>0_‖𝑦_𝛿sup₋_𝑦_‖≤𝛿 {‖ ‖‖𝑅𝛼(𝑦𝛿) −𝑥†‖‖_‖ } = 0.

In the following we will present a number of regularization approaches which are frequently applied within nonlinear ill-posed problems.

Tikhonov-type regularization

The idea of Tikhonov-type regularization is to deﬁne a functional𝑇_𝛼𝑦∶𝑋 ⊃(𝑇_𝛼𝑦) → [0, ∞),

𝑇_𝛼𝑦(𝑥) = 𝑆(𝐹 (𝑥), 𝑦) + 𝛼Ω(𝑥), (1.6)

consisting of a ﬁtting functional𝑆 ∶ 𝑌 × 𝑌 → [0, ∞] and a regularization functional Ω ∶ 𝑋 → [0, ∞). Both

terms are coupled by a regularization parameter𝛼. The ﬁtting functional typically measures the discrepancy

between𝐹 (𝑥) and the given data 𝑦, whereas an appropriate regularization functional Ω is required to make

the minimization problem

𝑇_𝛼𝑦(𝑥) → min

𝑥∈𝑋 (1.7)

well-posed. Consequently, ﬁnding the minimizers of (1.7) is the regularization approach. For an introduc-tion to Tikhonov-regularizaintroduc-tion in a very general setting see [18]. In this book, suﬃcient condiintroduc-tions on𝐹 , 𝑆 and Ω to show existence, stability and convergence of minimizers are presented.

The classical Tikhonov-functional for operators (1.1) mapping between Hilbert spaces is given by

𝑆(𝑦₁, 𝑦₂) =‖𝑦₁−𝑦₂‖2

𝑌 and Ω(𝑥) =‖𝑥 − 𝑥∗‖2𝑋

for some reference element𝑥∗_∈_{𝑋 and thus}

𝑇_𝛼𝑦(𝑥) =‖𝐹 (𝑥) − 𝑦‖2_𝑌 +𝛼‖𝑥 − 𝑥∗‖2_𝑋 (1.8) An introduction to classical Tikhonov regularization for nonlinear inverse problems can be found in [15, chapter 10]. Existence, Stability and Convergence of classical Tikhonov-regularization can be shown if the forward operator𝐹 is continuous and weakly closed (see [15]). In the case that we have given noisy data 𝑦𝛿

we denote the minimizers of𝑇_𝛼𝑦𝛿 by𝑥𝛿

𝛼.

Remark 5. For linear operators𝐴 = 𝐹 the minimization problem (1.8) has a unique solution 𝑥𝛿

𝛼which can

be explicitly computed as

𝑥𝛿_𝛼= (𝐴∗𝐴 + 𝛼 I)−1(𝐴∗𝑦𝛿+𝑥∗)

For nonlinear operators the situation is completely different. The minimization problem need not have a unique solution and there is no explicit formula for the minimizers. Instead iteration methods are used to find an approximation of a minimizer. Since the functional (1.8) is not necessarily convex, the iteration process may end up in local minimizers, thus the result may depend on the starting point.

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Lavrent’ev regularization

Lavrentiev regularization is a method, which is frequently used to regularize ill-posed operator equations with monotone forward operator𝐹 . An operator (1.1) is monotone if

⟨𝐹 (𝑥1) −𝐹 (𝑥2), 𝑥1−𝑥2⟩ ≥ 0

for all𝑥₁, 𝑥₂ ∈ (𝐹 ). A prerequisite for applying this method is that the forward operator (1.1) maps between equal spaces, that is𝑋 = 𝑌 (here 𝑋 is assumed to be a Hilbert space). Then instead of the original

equation (1.2) one solves the regularized equation

𝛼𝑥 + 𝐹 (𝑥) = 𝑦𝛿+𝛼𝑥∗ (1.9)

for some reference element𝑥∗ _∈ _{𝑋. If 𝐹 is monotone and Fréchet diﬀerentiable, it can be shown that a}

unique solution of (1.9) exists (see [47, Theorem 1.1]). Moreover, it is easy to show that solution depends continuously on the data in the sense of Def. 2. Furthermore one can show convergence of the method for appropriate parameter choices, if𝐹 is monotone and weak-to-norm closed (see [29, Chapter 2]). For

an introduction to Lavrentiev regularization for monotone operators in a more general context see also [1]. If the nonlinear forward operator is not (at least locally) monotone, to our knowledge there are no general results concerning uniqueness, stability and convergence.

Iterative regularization methods for nonlinear problems

Since we will focus on Lavrentiev- and Tikhonov regularization for the deautoconvolution problem, we only mention a few prominent iterative regularization methods with the corresponding assumptions on the forward operator. All of the regularization schemes we present require Fréchet diﬀerentiability of𝐹 , which

means that for all𝑥 ∈(𝐹 ) it exist a linear operator 𝐴 ∶ 𝑋 → 𝑌 , such that

𝐹 (𝑥 + ℎ) − 𝐹 (𝑥) − 𝐴ℎ = 𝑜 (ℎ) , ℎ ∈ 𝑋.

𝐴 is called the Fréchet derivative at 𝑥 and we use the standard notation 𝐹′(𝑥) ∶= 𝐴. In this class of

regularization methods the number of iterations, denoted by𝑛, usually serves as regularization parameter.

For a survey on iterative regularization methods for nonlinear operator equations we refer to [37]. The material presented in this subsection is taken from that book.

Landweber Iteration For linear operator equations Landweber iteration is a simple to implement, but

slowly converging iterative regularization method. Using the Fréchet derivative it can be generalized to nonlinear problems by

𝑥𝑘+1=𝑥𝑘+𝐹′(𝑥𝑘)∗

(

𝑦𝛿−𝐹 (𝑥𝑘)

)

This recursion is obviously stable. A crucial assumption for applying this method is

‖𝐹′₍_𝑥)_{‖ ≤ 1} _(1.10)

for all𝑥 ∈ 𝐵₂_𝑟(𝑥∗_{) with some}_{𝑟 > 0. If (1.10) doesn’t hold it is possible to scale the forward operator in}

order to obtain this property. If (1.2) has a solution in the Ball𝐵_𝑟(𝑥∗_{), it is possible to show convergence of}

the method to an𝑥∗_{-minimum-norm solution in}_𝐵

𝑟(𝑥∗) if the following condition holds:

‖𝐹 (𝑥) − 𝐹 ( ̄𝑥) − 𝐹′₍_𝑥∗₎₍_{𝑥 − ̄𝑥)}_{‖ ≤ 𝜂‖𝐹 (𝑥) − 𝐹 ( ̄𝑥)‖}

for all𝑥, ̄𝑥 ∈ 𝐵₂𝑟(𝑥∗) with some𝜂 < 1₂. For details see [37, Theorems 2.4 and 2.6].

Levenberg-Marquardt method This method has the iteration rule

𝑥_𝑘+1=𝑥_𝑘+(𝐹′(𝑥_𝑘)∗𝐹′(𝑥_𝑘) +𝛼_𝑘I)−1𝐹′(𝑥_𝑘)∗(𝑦𝛿−𝐹 (𝑥_𝑘))

where the sequence of regularization parameters𝛼_𝑘has to be chosen appropriately. To show convergence the nonlinearity condition

‖𝐹 (𝑥) − 𝐹 ( ̄𝑥) − 𝐹′₍_𝑥∗₎₍_{𝑥 − ̄𝑥)}_{‖ ≤ 𝜂‖𝑥 − ̄𝑥‖ ⋅ ‖𝐹 (𝑥) − 𝐹 ( ̄𝑥)‖} _(1.11)

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Iteratively regularized Gauss-Newton method This method is deﬁned by 𝑥𝑘+1=𝑥𝑘+ ( 𝐹′(𝑥𝑘)∗𝐹′(𝑥𝑘) +𝛼𝑘I )−1 𝐹′(𝑥𝑘)∗ ( 𝑦𝛿−𝐹 (𝑥𝑘) +𝛼𝑘(𝑥∗−𝑥𝑘) )

where𝑥∗ _{is an initial guess for a solution}_𝑥†_{. Obviously this method is quite similar to the}

Levenberg-Marquardt iteration. Among other assumptions a source condition

𝑥†−𝑥∗=(𝐹′(𝑥†)∗𝐹′(𝑥†))𝜇𝜔 together with 𝐹′₍_{𝑥) = 𝑅(𝑥, ̄𝑥)𝐹}′₍_{̄𝑥) + 𝑄(𝑥, ̄𝑥)} ‖ I −𝑅(𝑥, ̄𝑥)‖ ≤ 𝑐𝑅 ‖𝑄(𝑥, ̄𝑥)‖ ≤ 𝑐𝑄‖𝐹′(𝑥†)(𝑥 − ̄𝑥)‖ ∀𝑥, ̄𝑥 ∈ 𝐵2𝑟(𝑥∗), 𝑐𝑅, 𝑐𝑄> 0 (1.12) if𝜇 < 1₂and ‖𝐹′₍_{𝑥) − 𝐹}′₍_̄𝑥)_{‖ ≤ 𝐿‖𝑥 − ̄𝑥‖} _{𝐿 > 0}

for𝜇≥ 1₂is required to show convergence (see [37, Theorem 4.12]).

1.1.3 Convergence rate theory of Tikhonov-regularization for nonlinear inverse

prob-lems

Most of the content, presented in this section, can be found in similar form in [9]. If one can show con-vergence for a regularization approach of a specific inverse problem, one is also interested in the speed of convergence, that is, in this case we want to find an upper bound for the discrepancy between exact and regularized solution in terms of the noise level𝛿. In other words, we want to find a function 𝜂, such that

‖𝑥𝛿

𝛼−𝑥†‖ ≤ 𝜂(𝛿) ∀𝛿 ∶ 0 < 𝛿≤ 𝛿0

for some𝛿₀ > 0 and a parameter choice rule 𝛼 = 𝛼(𝛿) (a priori) or 𝛼 = 𝛼(𝛿, 𝑦𝛿_{) (a posteriori). However,}

also lower bounds are of interest. We are mainly interested in the asymptotic behavior of𝜂 for 𝛿₀hence it is very common to write

‖𝑥𝛿

𝛼−𝑥†‖ ≤ (𝛿𝑝) as𝛿 → 0

if𝜂(𝑡) = 𝑐 ⋅ 𝑡𝑝_{for some constants}_{𝑐, 𝑝 > 0, without further specifying the constant 𝑐.}

In the convergence rate theory for nonlinear ill-posed problems there are only a few general results and most of them have assumptions which are not easy to verify in general. Therefore it is often necessary to develop problem-speciﬁc theory to show convergence rates for nonlinear inverse problems. Convergence theory has also been developed for other regularization methods than Tikhonov-regularization (see for ex-ample [29] for Lavrentiev regularization and [37] for iterative regularization methods), but, as we will see later, not even convergence can be shown in the autoconvolution case for the other regularization methods presented so far, hence we focus on Tikhonov-regularization here.

As we have seen before, Tikhonov regularization (1.8) guarantees convergence to𝑥∗_{-minimum-norm}

solutions also for nonlinear ill-posed problems. If the forward operator𝐹 is Fréchet-diﬀerentiable at 𝑥†

with Fréchet derivative𝐹′₍_𝑥†_{), a common way to obtain also convergence rates is by source conditions.}

Their most simple form is an equation

𝑥†₋_𝑥∗₌ 1

2𝐹

′₍_𝑥†₎∗_𝜔, _(1.13)

the so-called benchmark source condition. Here𝑥∗ _{is the reference element and} _𝐹′₍_𝑥†₎∗ _{is the adjoint}

operator to𝐹′(𝑥†). Moreover, this approach requires that there is a constant𝐾 > 0, such that the local

Lipschitz condition

‖𝐹 (𝑥) − 𝐹 (𝑥†_{) −}_𝐹′₍_𝑥†₎₍_{𝑥 − 𝑥}†₎_{‖ ≤ 𝐾‖𝑥 − 𝑥}†_‖2 _(1.14)

holds for all𝑥 ∈ 𝐵𝑟(𝑥†) for some𝑟 > 0 and the smallness condition

𝐾‖𝜔‖ < 1 (1.15)

is satisﬁed. In this situation a convergence rate ‖𝑥𝛿

𝛼−𝑥†‖ ≤ 

(√

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can be shown (see e.g. [15]).

If there is more than one𝑥∗_{-minimum-norm solution to (1.2), an immediate consequence of the result}

(1.16) is that only one such solution𝑥† _∈_{(𝐹 ) to (1.2) can satisfy the three conditions (1.14), (1.13) and}

(1.15), simultaneously.

In the case that the benchmark source condition holds, but a smallness condition cannot be veriﬁed, an alternative approach uses the tangential cone condition

‖𝐹 (𝑥) − 𝐹 (𝑥†_{) −}_𝐹′₍_𝑥†₎₍_{𝑥 − 𝑥}†₎_{‖ ≤ 𝑐‖𝐹 (𝑥) − 𝐹 (𝑥}†₎_‖, _(1.17)

where𝑐 > 0 is a constant. A generalization of this inequality is the nonlinearity condition

‖𝐹 (𝑥) − 𝐹 (𝑥†_{) −}_𝐹′₍_𝑥†₎₍_{𝑥 − 𝑥}†₎_{‖ ≤ 𝜑}(_{‖𝐹 (𝑥) − 𝐹 (𝑥}†₎_‖) _(1.18)

with a concave index function𝜑. Since the term index index function will also appear in the following, we

give a deﬁnition for it.

Definition 6. An index function𝜑 is a continuous, strictly increasing function

𝜑 ∶ [0, ∞) → [0, ∞) with𝜑(0) = 0.

The papers [26] and [5] have discussed consequences of nonlinearity conditions of the form (1.18) for Banach space regularization, but they also apply to the Hilbert space situation of Tikhonov regularization (1.8) under consideration here. In this situation, we obtain for a choice𝛼 = 𝛼(𝛿, 𝑦𝛿_{) of the regularization}

parameter by the sequential discrepancy principle (cf. [3, 30]) convergence rates ‖𝑥𝛿

𝛼(𝛿,𝑦𝛿₎−𝑥†‖ = 

(√

𝜑(𝛿)) as 𝛿 → 0 (1.19)

whenever (1.18) is satisﬁed for some concave index function𝜑 together with the benchmark source condition

(1.13), and no smallness condition is required. If the benchmark source condition fails, but the derivative

𝐹′₍_𝑥†_{) ∶}_{𝑋 → 𝑌 is an injective and bounded linear operator, then under (1.18) the method of approximate} source conditionsdeveloped in [27] can be used together with variational inequalities combining solution smoothness and nonlinearity structure in one tool (cf. [28], [44, Chapt. 3], [18, Chapt. 12] and [24]). This yields convergence rates

‖𝑥𝛿

𝛼(𝛿,𝑦𝛿₎−𝑥†‖2 =  (𝜓(𝛿)) as 𝛿 → 0, (1.20)

which are lower than the rates in (1.19). Taking into account [5, Theorem 5.2] and [30, Theorem 2] it can be seen that the rate function𝜓 in (1.20) is an index function of the form

𝜓(𝛿) = 𝑑(Ψ−1(𝜑(𝛿))) with Ψ(𝑅) ∶= 𝑑(𝑅)

2

𝑅 ,

essentially based on the decay rate of the concave decreasing and strictly positive distance function

𝑑(𝑅) ∶= min{‖𝑥†−𝑥∗−1 2𝐹

′₍_𝑥†₎∗_𝑤_{‖ ∶ 𝑤 ∈ 𝑌 , ‖𝑤‖ ≤ 𝑅},} _{𝑅 > 0,}

which indicates for𝑥†_{the degree of violation with respect to (1.13). Here we have to assume that}_{𝑑(𝑅) → 0}

as𝑅 → ∞, The rate (1.20) can be arbitrarily slow if 𝑥†_{misses the benchmark source condition signiﬁcantly,}

which goes hand in hand with a very low decay of𝑑(𝑅) → 0 as 𝑅 → ∞.

If the benchmark source condition (1.13) fails, but the Fréchet derivative 𝐹′₍_{𝑥) exists for all 𝑥 ∈} 𝐵_𝑟(𝑥†₎ _⊂ _{(𝐹 ) and some 𝑟 > 0, by extending the ideas of [25, 43, 48] two further alternatives for}

ob-taining convergence rates to (1.8) have been presented in the paper [36] with focus on low order Hölder source conditions (see also [32, 48]):

𝑥†=𝑥∗+ (𝐹′(𝑥†)∗𝐹′(𝑥†))𝜈𝑤, 𝑤 ∈ 𝑋, 0< 𝜈 < 1

2, and logarithmic source conditions (cf. [33])

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To show convergence rates under these source conditions, as ﬁrst option the nonlinearity condition

𝐹′(𝑥) = 𝑅(𝑥, 𝑥†)𝐹′(𝑥†), ‖𝑅(𝑥, 𝑥†) −𝐼‖_{𝑌 →𝑌} ≤ 𝐶_𝑅‖𝑥 − 𝑥†‖𝜅, 0 < 𝜅 ≤ 1, (1.21) for some constant0 < 𝐶_𝑅 < ∞ and all 𝑥 ∈ 𝐵_𝑟(𝑥†_{) is recommended. Then the mean value theorem in}

integral form yields (cf. [25, p.28])

‖𝐹 (𝑥) − 𝐹 (𝑥†_{) −}_𝐹′₍_𝑥†₎₍_{𝑥 − 𝑥}†₎ ‖ = ‖ ∫₀1[𝐹′₍_𝑥†₊_{𝑡(𝑥 − 𝑥}†_{)) −}_𝐹′₍_𝑥†_)](_{𝑥 − 𝑥}†_{) d}_𝑡_‖ =‖ ∫ 1 0 [𝑅(𝑥†+𝑡(𝑥 − 𝑥†), 𝑥†) −𝐼] 𝐹′(𝑥†)(𝑥 − 𝑥†) d𝑡‖ ≤ 𝐶𝑅 ( ∫ 1 0 𝑡𝜅 d𝑡 ) ‖𝐹′₍_𝑥†₎₍_{𝑥 − 𝑥}†₎_{‖ ‖𝑥 − 𝑥}†_‖𝜅 and hence ‖𝐹 (𝑥) − 𝐹 (𝑥†_{) −}_𝐹′₍_𝑥†₎₍_{𝑥 − 𝑥}†₎_{‖ ≤} 𝐶𝑅 1 +𝜅‖𝐹 ′₍_𝑥†₎₍_{𝑥 − 𝑥}†₎_{‖ ‖𝑥 − 𝑥}†_‖𝜅 (1.22) Now the inequality (1.22) implies on the one hand that

‖𝐹 (𝑥) − 𝐹 (𝑥†_{) −}_𝐹′₍_𝑥†₎₍_{𝑥 − 𝑥}†₎_{‖ ≤ ̃}_𝐶_‖𝐹′₍_𝑥†₎₍_{𝑥 − 𝑥}†₎_‖ _(1.23)

holds for some constant0< ̃𝐶 < ∞ and all 𝑥 ∈ 𝐵𝑟(𝑥†). On the other hand, by using the triangle inequality,

from (1.22) we even derive the tangential cone condition (1.17) in the case of suﬃciently small𝑟 > 0, which

is then also a consequence of (1.21).

As second option the nonlinearity condition

𝐹′(𝑥) = 𝐹′(𝑥†)𝑅(𝑥, 𝑥†), ‖𝑅(𝑥, 𝑥†) −𝐼‖_𝑋→𝑋≤ 𝐶𝑅‖𝑥 − 𝑥†‖𝜅, 0 < 𝜅≤ 1, (1.24)

for some constant0 < 𝐶_𝑅 < ∞ and all 𝑥 ∈ 𝐵_𝑟(𝑥†_{) has been suggested, which is very diﬀerent from}

the tangential cone condition but can be veriﬁed for inverse problems with boundary measurements (cf., e.g., [8]). For Hölder and logarithmic rates under (1.24) we refer to [36, Theorem 2.1] and should mention in this context that for the proof of those convergence rates a condition of form (1.24) must be valid with a uniform constant𝐶𝑅for all𝑥 and 𝑥†lying in a small ball.

All results we presented up to this point relied on some kind of condition that controls the nonlinearity of the forward operator. A modern approach to show convergence rates for Tikhonov regularization, which avoids such nonlinearity conditions, are variational smoothness assumptions. An introduction to variational smoothness assumptions in the more general Banach space situation can be found in [18]. Such a variational smoothness assumption reads as

𝛽𝐸_𝑥†(𝑥)≤ Ω(𝑥) − Ω(𝑥†) +𝜑

(

𝑆(𝐹 (𝑥), 𝐹 (𝑥†₎₎) _∀_{𝑥 ∈ 𝑋} _(1.25)

with a constant𝛽, an error functional 𝐸𝑥† and the functionalsΩ and𝑆 introduced in (1.6). In the Hilbert

space situation, we usually have

𝑆(𝑦₁, 𝑦₂) =‖𝑦₁−𝑦₂‖2 𝑌 𝑦1, 𝑦2∈𝑌 and Ω(𝑥) =‖𝑥‖2 𝑋 𝑥 ∈ 𝑋. Now we deﬁne 𝐸𝑥†(𝑥) ∶= dist(𝑥, 𝑆),

where dist is the point-to-set distance deﬁned in (1.5) and𝑆 denotes the set of minimum-norm solutions.

Then (1.25) becomes

𝛽 dist(𝑥, 𝑆)≤ ‖𝑥‖2−‖𝑥†‖2+𝜑(‖𝐹 (𝑥) − 𝐹 (𝑥†)‖2) ∀𝑥 ∈ 𝑋. (1.26)

From this inequality one obtains

𝛽 dist(𝑥𝛿_𝛼, 𝑆)≤ 2 ( 𝛿2 𝛼 + (−𝜑) ∗(₋1 2𝛼 )) .

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Here we denote by𝑓∗_{the conjugate function of}_{𝑓 ∶ [0, ∞) → ℝ, which is deﬁned as} 𝑓∗(𝜉) ∶= sup

𝑡∈[0,∞)

(𝜉𝑡 − 𝑓 (𝑡)) .

If a variational smoothness assumption cannot be shown and the benchmark source condition (1.13) fails or the source element𝑣 ∈ 𝑌 in (1.13) violates the smallness condition (1.15) and if moreover neither a

con-dition (1.18) with any concave index function𝜑 nor the condition (1.24) are satisﬁed, but only a nonlinearity

condition (1.14) holds, then to our knowledge the literature provides no convergence rate result. Hence, this situation of low solution smoothness in combination with a poor structure of nonlinearity describes an un-explored area with respect to convergence rates for the Tikhonov regularization. In the next section we will show that this situation may arise for the real-valued autoconvolution problem on the unit interval.

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Classical autoconvolution problems

For real- or complex-valued functions deﬁned on ℝ the most simple kind of autoconvolution operator one can imagine is

𝐹 ∶ 𝑋 → 𝑌 , [_{𝐹 (𝑥)](𝑠) = ∫ℝ}𝑥(𝑠 − 𝑡)𝑥(𝑡) d𝑡. (2.1)

To ensure that[𝐹 (𝑥)](𝑠) is well-deﬁned, the elements of 𝑋 have to be at least square integrable functions.

Since we are only interested in functions with compact support, we will restrict ourselves to functions𝑥

deﬁned on the interval[0, 1]. Such functions can be considered as functions on 𝑅 with support in [0, 1].

Then due to (2.1) the support of𝐹 (𝑥) is contained in [0, 2]. However, in the Literature also the case where 𝐹 (𝑥) is treated as a function on [0, 1] can often be found and allows special regularization techniques, as

we will see later. Hence we will also be concerned with this situation. The mostly considered preimage space is the Hilbert space𝐿2(0, 1), which contains the square integrable functions on [0, 1]. Then there

are two natural choices for the image space, namely𝐿2(0, 1) or 𝐿2(0, 2). The following two subsections

are concerned with these cases. In the following we will often make use of the standard norms and inner products on𝐿₂(0, 1) and 𝐿₂(0, 2) respectively without distinguishing them by a diﬀerent notation. For 𝑥, 𝑥₁, 𝑥₂∈𝐿₂(0, 1) let ‖𝑥‖ ∶= √ ∫ 1 0 |𝑥(𝑡)| 2_d_𝑡, _⟨𝑥 1, 𝑥2⟩ ∶= ∫ 1 0 𝑥₁(𝑡)𝑥₂(𝑡) d𝑡

whereas for𝑦, 𝑦₁, 𝑦₂∈𝐿₂(0, 2) we introduce

‖𝑦‖ ∶= √ ∫ 2 0 |𝑦(𝑠)| 2_d_𝑠, _⟨𝑦 1, 𝑦2⟩ ∶= ∫ 2 0 𝑦₁(𝑠)𝑦₂(𝑠) d𝑠.

2.1 Autoconvolution on the unit interval

We deﬁne the autoconvolution operator as

𝐹 ∶ 𝐿2(0, 1) → 𝐿2(0, 1), [𝐹 (𝑥)](𝑠) = ∫

𝑠

0

𝑥(𝑠 − 𝑡)𝑥(𝑡) d𝑡, (2.2)

where𝐿₂(0, 1) = 𝐿ℝ

2(0, 1) contains only real-valued functions. We want to ﬁnd approximate solutions of

the equation

𝐹 (𝑥) = 𝑦 (2.3)

for given noisy data𝑦𝛿 with‖𝑦 − 𝑦𝛿‖ ≤ 𝛿. We will also refer to the operator (2.2) as autoconvolution in𝐿2(0, 1), since it maps this function space to itself. We denote the convolution of two not necessarily

identical functions𝑥₁, 𝑥₂∈𝐿₂(0, 1) by 𝐵(𝑥₁, 𝑥₂). That means we introduce the bilinear operator𝐵 by 𝐵 ∶ 𝐿₂(0, 1) × 𝐿₂(0_{, 1) → 𝐿2}(0, 1), [𝐵(𝑥₁, 𝑥₂)](_{𝑠) = ∫}

𝑠

0

𝑥₁(𝑠 − 𝑡)𝑥₂(𝑡) d𝑡 (2.4)

Note that𝐵 is symmetric, that is 𝐵(𝑥₁, 𝑥₂) =𝐵(𝑥₂, 𝑥₁) for all𝑥₁, 𝑥₂∈𝐿₂(0, 1).

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The properties of the forward operator𝐹 have been studied extensively in [23] for the case of real-valued

functions. At first we show that the operator𝐹 , defined in (2.2) is well-defined. Therefor let 𝑥 ∈ 𝐿₂(0, 1).

Then we obtain with the Hölder inequality ‖𝐹 (𝑥)‖2 = ∫ 1 0 ( ∫ 𝑠 0 𝑥(𝑠 − 𝑡)𝑥(𝑡) d𝑡 )2 d𝑠≤ ∫ 1 0 ( ∫ 𝑠 0 𝑥(𝑠 − 𝑡)2d𝑡 ) ( ∫ 𝑠 0 𝑥(𝑡)2d𝑡 ) d𝑠 = ∫ 1 0 ( ∫ 1 0 𝑥(𝑡)2d𝑡 )2 d𝑠 =‖𝑥‖4,

which implies that𝐹 (𝑥) ∈ 𝐿₂(0, 1). In [23, Lemma 1] it has been shown that even 𝐹 (𝑥) ∈ 𝐶(0, 1), or in

other words

(𝐹 ) ⊂ 𝐶(0, 1). (2.5)

The inequality

‖𝐹 (𝑥2) −𝐹 (𝑥1)‖ ≤ ‖𝑥2−𝑥1‖ ⋅ ‖𝑥2+𝑥1‖ (2.6)

for𝑥1, 𝑥2∈𝐿2(0, 1) has also been shown in [23, Lemma 1]. Since it is frequently used in proofs, we derive

it here: ‖𝐹 (𝑥2) −𝐹 (𝑥1)‖2= ∫ 1 0 ( ∫ 𝑠 0 ( 𝑥₂(𝑠 − 𝑡)𝑥₂(𝑡) − 𝑥₁(𝑠 − 𝑡)𝑥₁(𝑡))d𝑡 )2 d𝑠 = ∫ 1 0 ( ∫ 𝑠 0 ( 𝑥₂(𝑠 − 𝑡) − 𝑥₁(𝑠 − 𝑡))(𝑥₂(𝑡) + 𝑥₁(𝑡))d𝑡 )2 d𝑠 ≤ ∫₀1(∫ 𝑠 0 ( 𝑥₂(𝑠 − 𝑡) − 𝑥₁(𝑠 − 𝑡))2d𝑡 ) ( ∫ 𝑠 0 ( 𝑥₂(𝑡) + 𝑥₁(𝑡))2d𝑡 ) d𝑠 ≤ ‖𝑥2−𝑥1‖ ⋅ ‖𝑥2+𝑥1‖

The continuity of𝐹 has also been shown in [23] as a consequence of (2.6), but it also follows directly

from the (well-known) fact that𝐹 is Fréchet diﬀerentiable.

Proposition 7. The autoconvolution operator (2.16) is Fréchet differentiable everywhere and its Fréchet derivative at𝑥 ∈ 𝐿₂(0, 1) is given by 𝐹′₍_{𝑥) ∶ 𝐿} 2(0, 1) → 𝐿1(0, 2), [𝐹′(𝑥)𝑣](𝑠) = 2 ∫ 𝑠 0 𝑥(𝑠 − 𝑡)𝑣(𝑡) d𝑡 For its adjoint we have

𝐹′₍_𝑥)∗_∶_𝐿

2(0, 2) → 𝐿2(0, 1), [𝐹′(𝑥)∗𝑤](𝑡) = 2 ∫ 1

𝑡

𝑥(𝑠 − 𝑡)𝑤(𝑠) d𝑠 Proof. For𝑥, ℎ ∈ 𝐿₂(0, 1) and 𝐺 deﬁned as

𝐺 ∶ 𝐿₂(0, 1) → 𝐿2(0_{, 1), [𝐺𝑣](𝑠) = 2 ∫} 𝑠 0 𝑥(𝑠 − 𝑡)𝑣(𝑡) d𝑡 we have ‖𝐹 (𝑥 + ℎ)−𝐹 (𝑥) − 𝐺ℎ‖ = ( ∫ 1 0 ( ∫ 𝑠 0 ( (𝑥(𝑠 − 𝑡) + ℎ(𝑠 − 𝑡))(𝑥(𝑡) + ℎ(𝑡)) − 𝑥(𝑠 − 𝑡)𝑥(𝑡) − 2𝑥(𝑠 − 𝑡)ℎ(𝑡))d𝑡 )2 d𝑠 )1 2 = ( ∫ 1 0 ( ∫ 𝑠 0 ℎ(𝑠 − 𝑡)ℎ(𝑡) d𝑡 )2 d𝑠 )1 2 =‖𝐹 (ℎ)‖ ≤ ‖ℎ‖2,

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hence𝐺 is the Fréchet derivative of 𝐹 at 𝑥. To compute the adjoint operator, let 𝑥, 𝑣 ∈ 𝐿₂(0, 1) and 𝑤 ∈ 𝐿₂(0, 2). Then ⟨𝐹′₍_{𝑥)𝑣, 𝑤} ⟩ = 2 ∫₀1∫ 𝑠 0 𝑥(𝑠 − 𝑡)𝑣(𝑡) d𝑡 𝑤(𝑠) d𝑠 = 2 ∫ 1 0 𝑣(𝑡) ∫ 1 𝑡 𝑥(𝑠 − 𝑡) 𝑤(𝑠) d𝑠 d𝑡 =⟨𝑣, 𝐹′(𝑥)∗𝑤⟩

In [23, Theorem 2] it was shown that the restriction of𝐹 to non-negative functions is weakly continuous.

We will show weak continuity for the whole operator𝐹 .

Proposition 8. The operator𝐹 is weakly continuous.

Proof. Let(𝑥𝑛)𝑛∈ℕ ⊂ 𝐿2(0, 1) and 𝑥𝑛 ⇀ 𝑥 ∈ 𝐿2(0, 1). We will show that 𝐹 (𝑥𝑛) ⇀ 𝐹 (𝑥). Therefor let

𝑦 ∈ 𝐿₂(0, 1). It is enough to show that⟨𝐹 (𝑥𝑛) −𝐹 (𝑥), 𝑦⟩ → 0 as 𝑛 → ∞. We have

⟨𝐹 (𝑥𝑛) −𝐹 (𝑥), 𝑦⟩ = ∫ 1 0 ∫ 𝑠 0 (𝑥𝑛(𝑠 − 𝑡)𝑥𝑛(𝑡) − 𝑥(𝑠 − 𝑡)𝑥(𝑡)) d𝑡 𝑦(𝑠) d𝑠 = ∫ 1 0 ∫ 𝑠 0 (𝑥𝑛(𝑠 − 𝑡) − 𝑥(𝑠 − 𝑡))(𝑥𝑛(𝑡) + 𝑥(𝑡)) d𝑡 𝑦(𝑠) d𝑠 = ∫ 1 0 (𝑥_𝑛(_{𝑡) + 𝑥(𝑡)) ∫} 1 𝑡 (𝑥_𝑛(𝑠 − 𝑡) − 𝑥(𝑠 − 𝑡))𝑦(𝑠) d𝑠 d𝑡 = ∫ 1 0 (𝑥_𝑛(_{𝑡) + 𝑥(𝑡)) ∫} 1−𝑡 0 (𝑥_𝑛(𝑠) − 𝑥(𝑠))𝑦(𝑠 + 𝑡) d𝑠 d𝑡 For𝑛 ∈ ℕ we deﬁne 𝑔𝑛(𝑡) ∶=∫1− 𝑡 0 (𝑥𝑛(𝑠) − 𝑥(𝑠))𝑦(𝑠 + 𝑡) d𝑠 and obtain ⟨𝐹 (𝑥𝑛) −𝐹 (𝑥), 𝑦⟩ = ⟨𝑥𝑛+𝑥, 𝑔𝑛⟩

Now we show that the set(𝑔_𝑛)_𝑛∈ℕis equicontinuous, so let𝜖 > 0 and 𝑡 ∈ [0, 1] be arbitrary. Since the

sequence(𝑥_𝑛)_𝑛∈ℕis weakly convergent, its norm is bounded. That is, it exists𝑅 ∈ ℝ, such that‖𝑥_𝑛‖ ≤ 𝑅

for all𝑛 ∈ ℕ. Now let ̃𝑡 ∈ [0, 1] and for 𝑢 ∈ [0, 1] we deﬁne 𝑦(. + 𝑢) ∈ 𝐿₂(0, 1) by

(𝑦(. + 𝑢))(𝑠) =

{

𝑦(𝑠 + 𝑢) for𝑠 + 𝑢≤ 1

0 else

With the Hölder inequality we get |𝑔𝑛(𝑡) − 𝑔𝑛(̃𝑡)| ≤ ∫ 1 0 ||| ( 𝑦(𝑠 + 𝑡) − 𝑦(𝑠 + ̃𝑡)) (𝑥𝑛(𝑠) − 𝑥(𝑠))||| d𝑠 ≤ ‖𝑦(. + 𝑡) − 𝑦(. + ̃𝑡)‖ ⋅ ‖𝑥𝑛−𝑥‖ ≤ ‖𝑦(. + 𝑡) − 𝑦(. + ̃𝑡)‖ ⋅(‖𝑥𝑛‖ + ‖𝑥‖ ) ≤ ‖𝑦(. + 𝑡) − 𝑦(. + ̃𝑡)‖ ⋅ (𝑅 + ‖𝑥‖)

Since𝑦 ∈ 𝐿₂(0, 1), we know that‖𝑦(. + 𝑡) − 𝑦(. + ̃𝑡)‖ → 0 as ̃𝑡 → 𝑡. Hence it exists 𝛿 > 0 such that

‖𝑦(. + 𝑡) − 𝑦(. + ̃𝑡)‖ ≤ _{𝑅 +}𝜖_‖𝑥‖ for|𝑡 − ̃𝑡| ≤ 𝛿. Together with the last inequality this yields

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for|𝑡 − ̃𝑡| ≤ 𝛿 and thus (𝑔_𝑛)_𝑛∈ℕis equicontinuous. Next we observe that(𝑔_𝑛)_𝑛∈ℕconverges pointwise to zero. For arbitrary𝑡 ∈ [0, 1] we have

𝑔𝑛(𝑡) = ∫

1−𝑡

0

(𝑥𝑛(𝑠) − 𝑥(𝑠))𝑦(𝑠 + 𝑡) d𝑠 =⟨𝑥𝑛−𝑥, 𝑦(. + 𝑡)⟩ → 0

as𝑛 → ∞. Since (𝑔𝑛)𝑛∈ℕis an equicontinuous sequence of functions on a compact interval, that converges

pointwise,(𝑔𝑛)𝑛∈ℕeven converges uniformly, that is

lim

𝑛→∞‖𝑔𝑛‖∞= 0

and Hölder’s inequality yields

lim

𝑛→∞‖𝑔𝑛‖ ≤ lim𝑛→∞

√

‖𝑔𝑛‖∞ = 0

and ﬁnally we obtain

||⟨𝐹(𝑥𝑛) −𝐹 (𝑥), 𝑦⟩|| = ||⟨𝑥𝑛+𝑥, 𝑔𝑛⟩|| ≤ ‖𝑥𝑛+𝑥‖ ⋅ ‖𝑔𝑛‖ ≤ (𝑅 + ‖𝑥‖)‖𝑔𝑛‖ → 0

as𝑛 → ∞. Hence 𝐹 is weakly continuous.

Corollary 9. The operator𝐹 is weakly sequentially closed.

Proof. This is a direct consequence of the weak continuity of the operator𝐹 since(𝐹 ) is closed.

An advantage of the setting𝑋 = 𝑌 = 𝐿₂(0, 1) is of course, that preimage space and image space are

equal. Unfortunately, for𝑥 ∈ 𝐿₂(0, 1) the image 𝐹 (𝑥) need not contain the whole information about the

autoconvolution performed on the corresponding functions, which are deﬁned on ℝ. This becomes clear if one considers the example of a function𝑥 ∈ 𝐿₂(0, 1) which is zero on [0,1₂]. It is easy to see that𝐹 (𝑥) = 0

on[0, 1] then and hence 𝑥 cannot be reconstructed on [1

2, 1] at all. On the other hand, the continuity of 𝐹 (𝑥) implies that a solution of (1.2) need not exist for arbitrary 𝑦 ∈ 𝐿₂(0, 1), thus (2.3) is ill-posed in the

sense of Hadamard. In [23, Chapter 3] the authors restricted the domain of𝐹 to nonnegative functions and

investigated the uniqueness of solutions. We cite the main result here:

Proposition 10 (Gorenﬂo, Hofmann 1994). Let ̃ 𝐹 ∶+_{→ 𝐿} 2(0, 1), [ ̃𝐹 (𝑥)](𝑠) = ∫ 𝑠 0 𝑥(𝑠 − 𝑡)𝑥(𝑡) d𝑡 be the restriction of𝐹 to the non-negative functions

+_{∶= {}_{𝑥 ∈ 𝐿}

2(0, 1) ∶ 𝑥(𝑡)≥ 0 a.e. in [0, 1]} and let𝑦 ∈( ̃𝐹). Then the operator equation

̃ 𝐹 (𝑥) = 𝑦

has a unique solution if and only if𝑦 ∈ 𝑅+₀, where we define for0≤ 𝜎 ≤ 1

𝑅+_𝜎 ∶= {𝑦 ∈ 𝐶(0, 1) ∶ 𝑦≥ 0, 𝜎 = max{𝑠 ∶ 𝑦 = 0 a.e. in [0, 𝑠]}} If for0≤ 𝜎 < 1, 𝑦 = ̃𝐹(𝑥) ∈ 𝑅+

𝜎 with𝑥 ∈+, then𝑥 possesses the form

𝑥(𝑡) = ⎧ ⎪ ⎨ ⎪ ⎩ 0 a.e. in𝑡 ∈ [0, 𝜎∕2]

uniquely determined from𝑦 a.e. in𝑡 ∈ [𝜎∕2, 1 − 𝜎∕2]

arbitrarily non-negative a.e. in𝑡 ∈ [1 − 𝜎∕2, 1]

The proof uses a theorem, which was proven by Titchmarsh (see [49])

Lemma 11 (Titchmarsh 1925). If𝜙 and 𝜓 are integrable functions, such that

∫

𝑠

0

𝜙(𝑠 − 𝑡)𝜓(𝑡) d𝑡 = 0

almost everywhere in the interval0< 𝑠 < 𝜅, then 𝜙(𝑡) = 0 a.e. in (0, 𝜆) and 𝜓(𝑡) = 0 a.e. in (0, 𝜇), where

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Since every square integrable function on(0, 1) is also integrable on (0, 1) and can be considered as a

function on ℝ by setting it zero outside[0, 1], the theorem is applicable in our situation. The more general

case of autoconvolution with domain𝐿₂(0, 1) can be treated almost analogously to Proposition 10.

Proposition 12. For𝜎≥ 0 we define

𝑅_𝜎 ∶= {𝑦 ∈ 𝐶(0, 1) ∶ 𝜎 = max{𝑠 ∶ 𝑦(𝜉) = 0 ∀𝜉 ∈ [0, 𝑠]}}.

Let𝑦 ∈(𝐹 ). Then it exists 𝜎 ∈ [0, 1] such that 𝑦 ∈ 𝑅_𝜎and the solutions of the operator equation possess the form 𝑥(𝑡) = ⎧ ⎪ ⎨ ⎪ ⎩ 0 a.e. in𝑡 ∈ [0, 𝜎∕2]

determined from𝑦 up to sign a.e. in𝑡 ∈ [𝜎∕2, 1 − 𝜎∕2]

arbitrary a.e. in𝑡 ∈ [1 − 𝜎∕2, 1]

Proof. The prove the existence of𝜎 ∈ [0, 1] with 𝑦 ∈ 𝑅𝜎we ﬁrst observe that𝑦(0) = 0, since there exists

𝑥 ∈ 𝐿₂(0, 1) with 𝑦 = 𝐹 (𝑥) and thus 𝑦(0) = [𝐹 (𝑥)](0) = 0. Moreover ̇

⋃

𝜌∈[0,1]

𝑅_𝜌= {̃𝑦 ∈ 𝐶(0, 1) ∶ ̃𝑦(0) = 0} =∶ 𝑀

by the deﬁnition of𝑅𝜌and with (2.5) we obtain that𝑦 ∈ 𝑀 and hence 𝑦 ∈ 𝑅𝜎for some𝜎 ∈ [0, 1]. From

Titchmarsh’s Theorem we conclude now that a solution𝑥 of (2.3) must satisfy 𝑥 = 0 a.e. on [0, 𝜎∕2].

Then it is clear that𝑥_|_[1−_{𝜎∕2, 1]} has no inﬂuence on𝐹 (𝑥). Now assume that 𝐹 (𝑥₁) = 𝐹 (𝑥₂) = 𝑦 with 𝑥₁, 𝑥₂∈𝐿₂(0, 1). This implies

0 =𝐹 (𝑥₁) −𝐹 (𝑥₂) =𝐵(𝑥₁−𝑥₂, 𝑥₁+𝑥₂)

Hence it exists𝜖 ≥ 0, such that 𝑥₁−𝑥₂= 0 on [0, 𝜖] and 𝑥₁+𝑥₂= 0 on [0, 1 − 𝜖] by Titchmarsh’s Theorem.

Now suppose that𝜖 ∈ (𝜎∕2, 1 − 𝜎∕2). Then 𝑥₁−𝑥₂=𝑥₁+𝑥₂= 0 on [0, 𝜏] with 𝜏 ∶= min{𝜖, 1 − 𝜖}, thus 𝜏 > 𝜎∕2. This is obviously equivalent to 𝑥₁=𝑥₂= 0 on [0, 𝜏], but it implies that 𝑦 = 𝐹 (𝑥₁) = 0 on [0, 2𝜏].

This is a contradiction, since𝑦 ∈ 𝑅_𝜎, but𝜎 < 2𝜏.

Consequently we have𝜖≤ 𝜎∕2 or 𝜖 ≥ 1 − 𝜎∕2 which implies 𝑥₁=𝑥₂on[0, 1 − 𝜎∕2] or 𝑥₁= −𝑥₂on [0, 1 − 𝜎∕2]. Hence a solution 𝑥 of (2.3) is uniquely determined on [𝜎∕2, 1 − 𝜎∕2] up to the sign

In the light of the last proposition we see that it is hopeless to try to reconstruct a function𝑥 with 𝑥 = 0 on

some interval[0, 𝜎] with 𝜎 > 0 from a measurement of 𝐹 (𝑥) arbitrarily well, since then we have 𝐹 (𝑥) ∈ 𝑅𝜌

for some𝜌≥ 2𝜎 and hence we have no information about 𝑥|_[1−_{𝜌, 1]}at all.

The following results concerning non-compactness and local ill-posedness of𝐹 can already be found

in [23].

Proposition 13. [23, Proposition 4] The operator𝐹 is not compact.

Idea of a proof. Deﬁne the weakly convergent sequence(𝑥𝑛)𝑛∈ℕby

𝑥𝑛(𝑡) = sin(𝑛𝑡)

and show that𝐹 (𝑥𝑛) ⇀ 0 but ‖𝐹 (𝑥𝑛)‖ =

√ 6 12 ≠ 0.

Proposition 14 (cf. [23, Example 2]). The operator equation (2.3) is locally ill-posed everywhere. Idea of a proof. For given𝑥₀∈𝐿₂(0, 1) and an open neighbourhood 𝑈 of 𝑥₀we can ﬁnd𝑟 > 0 s.t 𝐵_𝑟(𝑥₀)⊂ 𝑈 . Deﬁne the sequence (Δ_𝑛)_𝑛∈ℕby

Δ_𝑛(𝑡) ∶= { 0 for0≤ 𝑡 ≤ 1 −1_𝑛 √ 𝑛 for1 − 1 𝑛< 𝑡≤ 1

and set𝑧𝑛 ∶= 𝑥0+𝑟Δ𝑛. Obviously‖𝑧𝑛−𝑥0‖ = 𝑟 for all 𝑛 ∈ ℕ, thus (𝑧𝑛)𝑛∈ℕ ∈ 𝑈 . Now show that

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2.1.1 Nonlinearity conditions

The topic of nonlinearity conditions for autoconvolution on𝐿₂(0, 1) has been treated intensively in [9,

Chap-ter 2]. Most of the results presented here can already be found in this paper.

Proposition 15. For the autoconvolution operator𝐹 mapping in 𝐿₂(0, 1) (defined in (2.2)) and any element 𝑥†_∈_𝐿

2(0, 1) there is no index function 𝜂 in combination with a radius 𝑟 > 0 such that

‖𝐹 (𝑥) − 𝐹 (𝑥†₎_{‖ ≤ ̂𝐶 𝜂(‖𝐹}′₍_𝑥†₎₍_{𝑥 − 𝑥}†₎_‖) _(2.7) for some constant0< ̂𝐶 < ∞ and all 𝑥 ∈ 𝐵_𝑟(𝑥†_).

Proof. To construct a contradiction it is enough to ﬁnd a sequence{𝑥𝑛}∞_𝑛=1⊂ 𝐵𝑟(𝑥†) such that‖𝐹′(𝑥†)(𝑥𝑛−

𝑥†₎_{‖ → 0 as 𝑛 → ∞, but lim}

𝑛→∞‖𝐹 (𝑥𝑛) −𝐹 (𝑥†)‖ > 0. Along the lines of Example 4 from [23] we can

consider the sequence of functions𝑥𝑛=𝑥†+ Δ𝑛∈𝐵𝑟(𝑥†) with Δ𝑛(𝑡) =

√

2𝑟 sin(2𝜋𝑛𝑡) and‖Δ𝑛‖ = 𝑟 > 0.

Taking into account the weak convergence𝑥𝑛−𝑥† ⇀ 0 in 𝐿2(0, 1) we have‖𝐹′(𝑥†)(𝑥𝑛−𝑥†)‖ → 0

and for any index function𝜂 also 𝜂(‖𝐹′₍_𝑥†₎₍_𝑥

𝑛−𝑥†)‖) → 0 as 𝑛 → ∞, because 𝐹′(𝑥†) is a compact

operator. However, 𝐹 is not compact and lim_𝑛→∞‖𝐹 (𝑥_𝑛) − 𝐹 (𝑥†₎_{‖ = lim}

𝑛→∞‖𝐵(2𝑥†+ Δ𝑛, Δ𝑛)‖ =

lim_𝑛→∞‖𝐹 (Δ_𝑛)‖ =

√ 6 6 𝑟

2 _{> 0. This proves the proposition. Note that we have used in this context the}

limitlim_𝑛→∞‖𝐵(𝑥†, Δ_𝑛)‖ = 0, which is again a consequence of the compactness of linear convolution operators.

Now the following corollary of Proposition 15 is valid.

Corollary 16. For the autoconvolution operator (2.2) a condition (1.23) and consequently a nonlinearity condition (1.21) cannot hold. Moreover, also the tangential cone condition (1.17) cannot hold with a small constant0< 𝐶 < 1.

Proof. From (1.23) we would obtain by using the triangle inequality

‖𝐹 (𝑥) − 𝐹 (𝑥†₎_{‖ ≤ ‖𝐹 (𝑥) − 𝐹 (𝑥}†_{) −}_𝐹′₍_𝑥†₎₍_{𝑥 − 𝑥}†₎_{‖ + ‖𝐹}′₍_𝑥†₎₍_{𝑥 − 𝑥}†₎_‖

≤ ( ̃𝐶 + 1)‖𝐹′₍_𝑥†₎₍_{𝑥 − 𝑥}†₎_{‖ and hence (2.7) which contradicts Proposition 15. By taking into account that}

(1.23) is a consequence of the nonlinearity condition (1.21) we see that also (1.21) cannot hold. Moreover, a tangential cone condition (1.17) would yield

‖𝐹 (𝑥) − 𝐹 (𝑥†₎_{‖ ≤ ‖𝐹 (𝑥) − 𝐹 (𝑥}†_{) −}_𝐹′₍_𝑥†₎₍_{𝑥 − 𝑥}†₎_{‖ + ‖𝐹}′₍_𝑥†₎₍_{𝑥 − 𝑥}†₎_‖

≤ 𝐶 ‖𝐹 (𝑥) − 𝐹 (𝑥†₎_{‖ + ‖𝐹}′₍_𝑥†₎₍_{𝑥 − 𝑥}†₎_{‖, and in particular with 0 < 𝐶 < 1}

‖𝐹 (𝑥) − 𝐹 (𝑥†₎_{‖ ≤} 1

1 −𝐶 ‖𝐹

′₍_𝑥†₎₍_{𝑥 − 𝑥}†₎_‖,

which is also incompatible with Proposition 15. For the nonlinearity condition (1.11) we have

Proposition 17. For the autoconvolution operator (2.2) the nonlinearity condition (1.11) cannot hold for

any𝑥∗_∈_𝐿

2(0, 1) and 𝑟, 𝜂 > 0.

Proof. Let𝜂, 𝑟 > 0 and 𝑥∗∈𝐿2(0, 1) be given and assume that (1.11) holds with these parameters. Consider

the sequence(ℎ𝑛)𝑛∈ℕ⊂ 𝐿2(0, 1) deﬁned by ℎ𝑛(𝑡) =

√ 2

𝑛 sin(2𝜋𝑛𝑡). Then obviously‖ℎ𝑛‖ =

1

𝑛 for all𝑛 ∈ ℕ.

Hence it exists𝑁 > 0, such that ℎ_𝑛 ∈ 𝐵₂_𝑟(0) for all 𝑛 ≥ 𝑁 and if we deﬁne the sequence (𝑥_𝑛)_𝑛∈ℕ by

𝑥_𝑛=𝑥∗₊_ℎ

𝑛, this yields𝑥𝑛∈𝐵2𝑟(𝑥∗) for all𝑛≥ 𝑁 An easy calculation shows that

[𝐹 (ℎ_𝑛)](𝑠) = 1 𝑛2 ( −𝑠 cos(2𝜋𝑛𝑠) + 1 2𝜋𝑛sin(2𝜋𝑛𝑠) ) . (2.8)

Setting𝑥 ∶= 𝑥_𝑛and ̄𝑥 ∶= 𝑥∗_{in condition (1.11) yields}

‖𝐹 (ℎ𝑛)‖ ≤ ‖ℎ𝑛‖‖𝐹 (𝑥∗+ℎ𝑛) −𝐹 (𝑥∗)‖ = 1 𝑛‖𝐹 ′₍_𝑥∗₎_ℎ 𝑛+𝐹 (ℎ𝑛)‖ ≤ 1 𝑛 ( ‖𝐹′₍_𝑥∗₎_ℎ 𝑛‖ + ‖𝐹 (ℎ𝑛)‖ )

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for all𝑛≥ 𝑁. After Multiplication with 𝑛2_{we obtain} 𝑛2‖𝐹 (ℎ_𝑛)‖ ≤(‖𝑛𝐹′(𝑥∗)ℎ_𝑛)‖ + 𝑛‖𝐹 (ℎ_𝑛)‖). (2.9) Since𝑛 ⋅ ℎ𝑛⇀ 0 we have lim 𝑛→∞𝑛𝐹 ′₍_𝑥∗₎_ℎ 𝑛= 0

due to the compactness of𝐹′₍_𝑥∗_{). From (2.8) we see that}

lim 𝑛→∞𝑛 2_{‖𝐹 (ℎ} 𝑛)‖ = √ 6 6 and𝑛→∞lim𝑛‖𝐹 (ℎ𝑛)‖ = 0. (2.10)

Taking the limes for𝑛 → ∞ of (2.9) then yields

√ 6 6 ≤ 0

which is a contradiction. Thus our assumption was false and consequently the assertion of Proposition 17 holds.

Proposition 18. A range invariance condition of type (1.12) cannot hold for the autoconvolution operator

(2.3).

Proof. The proof is indirect and it is based on deriving a nonlinearity condition similar to (1.23). Assume that (1.12) holds for some𝑐_𝑅, 𝑐_𝑄> 0 and 𝑥∗_∈_𝐿

2(0, 1). Then we have for 𝑥 ∈ 𝐵2𝑟(𝑥∗)

‖𝐹 (𝑥 − 𝑥∗₎_‖ =‖𝐹 (𝑥) − 𝐹 (𝑥∗) −𝐹′(𝑥∗)(𝑥 − 𝑥∗)‖ =‖ ∫ 1 0 [𝐹′₍_𝑥∗₊_{𝑡(𝑥 − 𝑥}∗_{)) −}_𝐹′₍_𝑥∗_)](_{𝑥 − 𝑥}∗_{) d}_𝑡_‖ =‖ ∫ 1 0 [ [𝑅(𝑥∗+𝑡(𝑥 − 𝑥∗), 𝑥∗) −𝐼]𝐹′(𝑥∗) +𝑄(𝑥∗+𝑡(𝑥 − 𝑥∗), 𝑥∗)](𝑥 − 𝑥∗) d𝑡‖ ≤ ∫01 ( ‖𝑅(𝑥∗₊_{𝑡(𝑥 − 𝑥}∗₎_{, 𝑥}∗_{) −}_𝐼_‖‖𝐹′₍_𝑥∗₎₍_{𝑥 − 𝑥}∗₎_{‖ + ‖𝑄(𝑥}∗₊_{𝑡(𝑥 − 𝑥}∗₎_{, 𝑥}∗₎_{‖‖(𝑥 − 𝑥}∗₎_‖) _d_𝑡 ≤ ∫01 ( 𝑐_𝑅‖𝐹′(𝑥∗)(𝑥 − 𝑥∗)‖ + 𝑡‖𝐹′(𝑥†)(𝑥 − 𝑥∗)‖ ⋅ 2𝑟) d𝑡‖ ≤ 𝑐𝑅‖𝐹′(𝑥∗)(𝑥 − 𝑥∗)‖ + 𝑐𝑄𝑟‖ 𝐹′(𝑥†)(𝑥 − 𝑥∗)‖ (2.11)

For the sequence similar to the one in the last proposition, namely(𝑥_𝑛)_𝑛∈ℕ ⊂ 𝐵₂_𝑟(𝑥∗_{) deﬁned by}_𝑥

𝑛(𝑡) ∶= 𝑥∗₍_{𝑡) +}√₂_{𝑟 sin(2𝜋𝑡) we have 𝑥} 𝑛⇀ 𝑥∗and consequently lim 𝑛→∞‖𝐹 ′₍_𝑥∗₎₍_{𝑥 − 𝑥}∗₎_{‖ = lim} 𝑛→∞‖𝐹 ′₍_𝑥†₎₍_{𝑥 − 𝑥}∗₎_{‖ = 0.} Moreover lim 𝑛→∞‖𝐹 (𝑥𝑛−𝑥 ∗₎_{‖ =} √ 6 6

(cf. (2.10)), thus taking the limes for𝑛 → ∞ in (2.11) yields a contradiction.

Proposition 19. For the autoconvolution operator (2.2) a nonlinearity condition (1.24) cannot hold. Proof. For𝑥†_{= 0 the assertion is obviously true since}_𝐹′₍_𝑥†_{) is the zero-operator in this case, but there are}

non-zero operators𝐹′₍_{𝑥) for elements 𝑥 in any ball 𝐵}

𝑟(0). Hence we can restrict our proof to the case that

𝑥†_{≠ 0. Now let us assume that condition (1.24) is satisﬁed. From (1.24) we have that, for all 𝑥 ∈ 𝐵}

𝑟(𝑥†),

𝑅(𝑥, 𝑥†_{) ∶}_{𝑋 → 𝑋 denotes bounded linear operators with a uniform norm bound and}

‖𝑅(𝑥, 𝑥†₎∗₋_𝐼_‖

Inverse Autoconvolution Problems with an Application in Laser Physics