Deautoconvolution in the two-dimensional case Yu DengBernd HofmannandFrank Werner

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Deautoconvolution in the two-dimensional

case

Yu Deng∗

,Bernd Hofmann†and Frank Werner‡

October 26, 2022

Abstract: There is extensive mathematical literature on the inverse problem of deau-

toconvolution for a function with support in the unit interval [0,1] ⊂R, but little is

known about the multidimensional situation. This article tries to ﬁll this gap with

analytical and numerical studies on the reconstruction of a real function of two real

variables over the unit square from observations of its autoconvolution on [0,2]2⊂R2

(full data case) or on [0,1]2(limited data case). In an L2-setting, twofoldness and

uniqueness assertions are proven for the deautoconvolution problem in 2D. Moreover,

its ill-posedness is characterized and illustrated. Extensive numerical case studies give

an overview of the behaviour of stable approximate solutions to the two-dimensional

deautoconvolution problem obtained by Tikhonov-type regularization with diﬀerent

penalties and the iteratively regularized Gauss-Newton method.

Keywords: deautoconvolution, inverse problem, ill-posedness, case studies in 2D, Tikhonov-

type regularization, iteratively regularized Gauss?Newton method

AMS-classiﬁcation (2010): 47J06, 65R32, 45Q05, 47A52, 65J20

1 Introduction

The object of research in this work is the problem of deautoconvolution, where our focus is on

the two-dimensional case, which means that a square integrable real function of two variables

x(t1, t2) (0 ≤t1, t2≤1) is to be identiﬁed from the function y=x∗xof its autoconvolution. If

we consider xas an element in the Hilbert space L2(R2) with support supp(x)⊆[0,1]2, then it

is well-known that x∗xalso lies in L2(R2) with support supp(x∗x)⊆[0,2]2. In this context,

the elements xand x∗xboth can be considered as tempered distributions with compact support,

where supp(·) is regarded as the essential support with respect to the Lebesgue measure λin R2.

Instead of yitself, noisy data yδ∈L2(R2) to ywith some noise level δ≥0 are available only. Since

the inverse problem of deautoconvolution tends to be ill-posed, the aim of the recovery process is

to ﬁnd stable approximate solutions of xbased on the data yδ. We are going to distinguish the

full data case, where noisy data are available for y(s1, s2) (0 ≤s1, s2≤2), and the limited data

case, where data are given for y(s1, s2) (0 ≤s1, s2≤1). Since the scope of the data in the limited

data case is only 25% compared to the full data case, the eﬀect of ill-posedness is stronger in that

∗Chemnitz University of Technology, Faculty of Mathematics, 09107 Chemnitz, Germany, e-mail:

yu.deng@math.tu-chemnitz.de

†Faculty of Mathematics, Chemnitz University of Technology, 09107 Chemnitz, Germany,

e-mail: hofmannb@mathematik.tu-chemnitz.de.

‡Institute for Mathematics, University of W¨urzburg, Emil-Fischer-Str. 30, 97074 W¨urzburg, Germany, e-mail:

frank.werner@mathematik.uni-wuerzburg.de

arXiv:2210.14093v1 [math.NA] 25 Oct 2022

case. As a consequence, also the chances for the accurate recovery of xare more restricted in the

limited data case.

The simplest application of our deautoconvolution problem in two dimensions is the recovery of

the density function xwith support in the unit square [0,1]2of a two-dimensional random variable

Xfrom observations of the density function yof the two-dimensional random variable Y:= ˆ

X+¯

where X,ˆ

Xand ¯

Xare assumed to be of i.i.d. type.

The deautoconvolution problem in one dimension has been considered extensively in the literature

motivated by physical applications in spectroscopy (see, e.g., [6, 29]). Its mathematical analysis

has been implemented comprehensively in the last decades with focus on properties of the speciﬁc

forward operator, ill-posedenss and regularization based on the seminal paper [20]. In this context,

we refer to [8, 10, 11, 14, 16, 17, 23] for investigations concerning the stable identiﬁcation of real

functions xon the unit interval [0,1] from noisy data of its autoconvolution x∗x. A new series of

interdisciplinary autoconvolution studies was unleashed by a cooperation started in 2010 between

a Research Group of the Max Born Institute for Nonlinear Optics and Short Pulse Spectroscopy,

Berlin, led by Prof. G¨unter Steinmeyer and the Chemnitz research Group on Regularization, and

we refer to the publications [3, 9, 18, 19] presenting the output of this cooperation. The goal of

this cooperation between mathematics and laser optics was the extension of the one-dimensional

deautoconvolution problem to complex-valued functions combining amplitude and phase functions

for characterizing ultrashort laser pulses.

In this article, in an L2-setting we are considering a series of numerical case studies for the nonlinear

Volterra-type integral equation

F(x) = y , with F(x) := x∗x , (1)

the solution of which solves the deautoconvolution problem in two dimensions. Equation (1) is a

special case of a nonlinear operator equation

F(x) = y, F :D(F)⊆X→Y , (2)

with forward operator Fmapping between real-valued Hilbert spaces Xand Ywith norms k·kX

and k·kY, respectively, and domain D(F).

In dependence of the data situation, we have to distinguish in the full data case the forward

operator F:X=L2([0,1]2)→Y=L2([0,2]2) deﬁned as

[F(x)](s1, s2) :=

min(s2,1)

max(s2−1,0)

min(s1,1)

max(s1−1,0)

x(s1−t1, s2−t2)x(t1, t2)dt1dt2(0 ≤s1, s2≤2) (3)

and in the limited data case the forward operator F:X=L2([0,1]2)→Y=L2([0,1]2) as

[F(x)](s1, s2) := Zs2

0Zs1

x(s1−t1, s2−t2)x(t1, t2)dt1dt2(0 ≤s1, s2≤1).(4)

In general we consider D(F) = X=L2([0,1]2), but for the limited data case we partially focus

on non-negative solutions expressed by the domain D(F) = D+with

D+:= {x∈X=L2([0,1]2) : x≥0 a.e. on [0,1]2}.(5)

For any function x∈L2([0,1]2) the autoconvolution products F(x) = x∗xand F(−x) = (−x)∗

(−x) coincide for both forward operator versions (3) and (4). However, it is of interest whether

for y=x∗xthe elements xand −xare the only solutions of equation (1) or not. Moreover it is of

interest whether in the limited data case the restriction of the domain D(F) to D+from (5) leads

to unique solutions. Some answers to those questions will be given in the subsequent Section 2.

The remainder of the paper is organized as follows: Section 2 is devoted to assertions on twofoldness

and uniqueness for the deautoconvolution problem in two dimensions, preceded by a subsection

with relevant lemmas and deﬁnitions. As an inverse problem, deautoconvolution tends to be ill-

posed in the setting of inﬁnite dimensional L2-spaces. After the presentation of two functions

deﬁned over the unit square as basis for later numerical case studies, in Section 3 the speciﬁc

ill-posedness character for the deautoconvolution of a real function of two real variables with

compact support is analyzed and illustrated. To suppress ill-posedness phenomena, variants of

variational and iterative regularization methods are used, which will be introduced in Section 4.

The numerical treatment, including discretization approaches of forward operator and penalty

functionals for the Tikhonov regularization as well as for the iterative regularization by using the

Fourier transform, is outlined in Section 5. Section 6 completes the article with comprehensive

numerical case studies.

2 Assertions on twofoldness and uniqueness for the

deautoconvolution problem in two dimensions

2.1 Preliminaries

Assertions on twofoldness and uniqueness for the deautoconvolution problem in one dimension

have been formulated in the articles [20] for the limited data case and [19] for the full data case.

The respective proofs are based on the Titchmarsh convolution theorem from [31], which was

formulated as Lemma 3 in [20] and will be recalled below in a slightly reformulated form as

Lemma 1.

Lemma 1. Let the functions f, g ∈L2(R)have compact supports supp(f)and supp(g). Then we

have for the convolution that f∗g∈L2(R)and that the equation

inf supp(f∗g) = inf supp(f) + inf supp(g) (6)

holds. In particular, for supp(f)and supp(g)covered by the unit interval [0,1], we conclude from

[f∗g](s) =

min(s,1)

max(s−1,0)

f(s−t)g(t)dt = 0 a.e. for s∈[0, γ] (γ≤2)

that there are numbers γ1, γ2∈[0,1] with γ1+γ2≥γsuch that

f(t) = 0 a.e. for t∈[0, γ1]and g(t) = 0 a.e. for t∈[0, γ2].

For an extension of the Titchmarsh convolution theorem to two dimensions, we mention the

following Lemma 2 (cf. [26, 27]).

Lemma 2. Let the functions f, g ∈L2(R2)have compact supports supp(f)and supp(g). Then

we have for the convolution that f∗g∈L2(R2)and that the equation

ch supp(f∗g) = ch supp(f) + ch supp(g) (7)

holds, where ch Mdenotes the convex hull of a set M⊆R2. In the special case that supp(f∗g) = ∅

we have that at least one of the supports supp(f)or supp(g)is the empty set.

Deﬁnition 1. For given y∈L2([0,2]2), we call x†∈L2([0,1]2)with supp(x†)⊆[0,1]2asolution

to the operator equation (1) in the full data case if it satisﬁes the condition

[x†∗x†](s1, s2) = y(s1, s2)a.e. for (s1, s2)∈[0,2]2.(8)

Deﬁnition 2. For given y∈L2([0,1]2), we call x†∈L2([0,1]2)with supp(x†)⊆[0,1]2asolution

to the operator equation (1) in the limited data case if it satisﬁes the condition

[x†∗x†](s1, s2) = y(s1, s2)a.e. for (s1, s2)∈[0,1]2.(9)

For x†∈ D+with D+from (5) we call it non-negative solution in the limited data case.

Deﬁnition 3. We call x∈L2([0,1]2)with supp(x)⊆[0,1]2satisfying (8) or (9) a factored

solution to equation (1) in the full data case or in the limited data case, respectively, if we have

the structure x(t1, t2) = x1(t1)x2(t2) (0 ≤t1, t2≤1) with xi∈L2([0,1]),supp(xi)⊆[0,1] for

i= 1 and i= 2. If moreover xi≥0a.e. on [0,1] for i= 1 and i= 2, then we call it non-negative

factored solution in the respective case.

2.2 Results for the full data case

Lemma 2 allows us to prove the following theorem for the forward autoconvolution operator

F:L2([0,1]2)→L2([0,2]2) from (3), which is an extension of [19, Theorem 4.2] to the two-

dimensional case of the deautoconvolution problem.

Theorem 1. If, for given y∈L2([0,2]2), the function x†∈L2([0,1]2)with supp(x†)⊆[0,1]2is

a solution to (1) with Ffrom (3), then x†and −x†are the only solutions of this equation in the

full data case.

Proof. Let x†∈L2([0,1]2) supposing supp(x†)⊆[0,1]2and h∈L2([0,1]2) supposing supp(h)⊆

[0,1]2. We assume that x†and x†+hsolve the equation (1), which means that [x†∗x†](s1, s2) =

[(x†+h)∗(x†+h)](s1, s2) a.e. for (s1, s2)∈[0,2]2. Then we have [(x†+h)∗(x†+h)−x†∗x†](s1, s2) =

[h∗(2x†+h)](s1, s2) = 0 a.e. for (s1, s2)∈[0,2]2. By setting f:= hand g:= 2x†+hwe now apply

Lemma 2. Taking into account that

supp(h∗(2x†+h)) ⊆[0,2]2we then have supp(h∗(2x†+h)) = ∅and consequently also

ch supp(h∗(2x†+h)) = ∅. This implies due to equation (7) that either supp(h) = ∅or

supp(2x†+h) = ∅. On the one hand, supp(h) = ∅leads to the solution x†itself, whereas on

the other hand supp(2x†+h) = ∅leads to [2x†+h](t1, t2) = 0 a.e. for (t1, t2)∈[0,1]2and

consequently with h=−2x†to the second solution −x†. Alternative solutions are thus excluded,

which proves the theorem.

2.3 Results for the limited data case

For solutions x†∈L2([0,1]2) to equation (1) with supp(x†)⊆[0,1]2, the condition 0 ∈supp(x†)

plays a prominent role in the limited data case. This condition means that for any ball Br(0)

around the origin with arbitrary small radius r > 0 there exists a set Mr⊂Br(0) ∩[0,1]2with

Lebesgue measure λ(Mr)>0 such that x†(t1, t2)6= 0 a.e. for (t1, t2)∈Mr. Vice versa, for

0/∈supp(x†) we have some suﬃciently small radius r > 0 such that x†(t1, t2) = 0 a.e. for

(t1, t2)∈Br(0) ∩[0,1]2.

First, we generalize in Theorem 2 those aspects of [20, Theorem 1] that concern the strong non-

injectivity of the autoconvolution operator in the limited data case.

Theorem 2. If, for given y∈L2([0,1]2), the function x†∈L2([0,1]2)with supp(x†)⊆[0,1]2is

a solution to (1) with Ffrom (4) that fulﬁls the condition

0/∈supp(x†),(10)

then there exist inﬁnitely many other solutions ˆx†∈L2([0,1]2)to (1) with supp(ˆx†)⊆[0,1]2in

the limited data case.

Proof. If (10) holds, there is some 0 < ε < 1/2 such that x†(t1, t2) = 0 a.e. for (t1, t2)∈[0, ε]2.

Then we have, for all elements h∈L2([0,1]2) with supp(h)⊆[0,1]2satisfying the condition

h(t1, t2) = 0 a.e. for (t1, t2)∈[0,1]2\[1 −ε, 1]2,

that ˆx†=x†+hobeys the condition

[ˆx†∗ˆx†](s1, s2) = y(s1, s2) a.e. for (s1, s2)∈[0,1]2.

This is a consequence of the fact that [h∗(2x†+h)](s1, s2) = 0 a.e. for (s1, s2)∈[0,1]2is true for

each such element h.

To formulate uniqueness assertions for solutions x†to equation (1) in the limited data case, we

restrict our considerations now to non-negative solutions and the domain D(F) = D+from (5)

for the forward operator Ffrom (4). We present in Theorem 3 a result that extends to the two-

dimensional autoconvolution operator F:D+⊂L2([0,1]2)→L2([0,1]2) from (4) those aspects

of [20, Theorem 1] which concern the solution uniqueness. Precisely, we are able to handle the

special case of factored non-negative solutions in the sense of Deﬁnition 3, occurring for example

when x†is a density function for the two-dimensional random variable X= (X1,X2), where X1

and X2are uncorrelated one-dimensional random variables.

Theorem 3. Let, for given y∈L2([0,1]2),x†be a non-negative factored solution to equation (1)

in the limited data case, which satisﬁes the condition

0∈supp(x†).(11)

Then there are no other non-negative factored solutions in this case.

Proof. For the factored situation, we have that the right-hand side yis also factored with

y(s1, s2) = y1(s1)y2(s2) (0 ≤s1, s2≤1) and y1=x†

1∗x†

1, y2=x†

2∗x†

Moreover, the condition (11) implies that

inf supp(x†

1) = inf supp(x†

2)=0.(12)

Otherwise, there would be a square [0, ε]2with ε= max{inf supp(x†

1),inf supp(x†

2)}>0 on which

x†vanishes almost everywhere such that 0 /∈supp(x†) would then apply. Now we suppose that

for i= 1 and i= 2 quadratically integrable perturbations hi(ti) (0 ≤ti≤1) exist such that

x†

i+hi≥0 a.e. on [0,1] and

(x†

1+h1)∗(x†

1+h1) = y1and (x†

2+h2)∗(x†

2+h2) = y2.(13)

To complete the proof of the theorem we still show that h1and h2have to vanish almost ev-

erywhere on [0,1]. This can be done with the help of Titchmarsh’s convolution theorem in the

one-dimensional case (cf. Lemma 1). From (13) we derive for i= 1 and i= 2 that

[hi∗(2x†

i+hi)](si) = 0 a.e. for si∈[0,1],

where x†

i+hi≥0 implies that 2x†

i+hi≥x†

iand inf supp(2x†

i+hi) = 0 as a consequence of (12).

Then it follows from Lemma 1 that

inf supp(hi) + inf supp(2x†

i+hi) = inf supp(hi∗(2x†

i+hi)) ≥1

and hence inf supp(hi)≥1 for both i= 1,2, which gives hi= 0 a.e. on [0,1] and completes the

proof.

3 Examples and ill-posedness phenomena of deautoconvolution

in 2D

3.1 Two Examples

For the numerical case studies of deautoconvolution in 2D, we have selected two examples of

solutions x†to the autoconvolution equation in 2D. The ﬁrst example refers to the function

x†(t1, t2) = −3t2

1+ 3t1+1

4(sin(1.5πt2) + 1) (0 ≤t1, t2≤1) (14)

to be reconstructed from its own autoconvolution F(x†) = x†∗x†. This smooth and non-negative

factored function x†is illustrated in Figure 1, in a line with the F(x†)-images for the limited and

full data case, respectively.

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Deautoconvolutioninthetwo-dimensionalcaseYuDeng*,BerndHofmannandFrankWernerOctober26,2022Abstract:Thereisextensivemathematicalliteratureontheinverseproblemofdeau-toconvolutionforafunctionwithsupportintheunitinterval[0;1]R,butlittleisknownaboutthemultidimensionalsituation.Thisarticletriestollthis...

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Deautoconvolution in the two-dimensional case Yu DengBernd HofmannandFrank Werner

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