,.k...\\
.1 f

 

L :_
Jan.

3‘ 24.).»
4...; x ti.

1.
$9131.
:1 ixz|
1.9.1:. .
i.

(5

$1

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

. r2“ . ‘ .
.. . if. u. A ton
5‘21; I

 

 

.
.. Lav.

ESE; .EEEEE EEM

, ‘ A;n.d..x...4: , . ........y.£?,. .... Law...“ .25. ..: _. r z. ‘ . E . ‘ . V
I s as . £4 . EEEEE E gig fig E . , , . WE

 

LIBRARIES
MICHIGAN STATE UNIVERSITY
EAST LANSING, MICH 48824-1048

This is to certify that the
dissertation entitled

Local Regularization for the Autoconvolution Problem

presented by

Zhewei Dai

has been accepted towards fulfillment
of the requirements for the

PhD. degree in Applied Mathematics

 

 

(Paws; KW

 

Major Professor’s Signature
7 I l 5 / o 5'

Date

MSU is an Affirmative Action/Equal Opportunity Institution

 

PLACE IN RETURN BOX to remove this checkout from your record.
TO AVOID FINES return on or before date due.
MAY BE RECALLED with earlier due date if requested.

 

DATE DUE

DATE DUE

DATE DUE

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

2/05 cm.mms

 

Local Regularization for the Autoconvolution

Problem
By

Zhewei Dai

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements

for the degree of
DOCTOR OF PHILOSOPHY
Department of Mathematics

2005

ABSTRACT

Local Regularization for the Autoconvolution Problem

By

Zhewei Dai

We develop a local regularization theory for the nonlinear autoconvolution prob-
lem. Unlike the classic regularization techniques such as Tikhonov regularization,
this theory provides regularization methods that preserve the causal nature of the
autoconvolution problem, allowing for fast sequential numerical solution. We prove
the convergence of the regularized solutions to the true solution as the noise level in
the data shrinks to zero, with a certain convergence rate. We propose several regu-
larization methods and provide theoretic basis for their convergence. Our numerical
results confirm effectiveness of the methods, suggesting superiority of our methods

over the existing ones, especially in recovering sharp features in the solution.

To Michael

iii

ACKNOWLEDGMENTS

First and foremost, I would like to thank my advisor and mentor, Professor Patri-
cia Lamm. Thank you for your constant encouragement, your generosity with your
time, your tireless proofreading and commenting on this thesis and your contagious
enthusiasm about mathematical research. The great deal I have learned from you are
all very invaluable, and I simply could not have asked for a better advisor.

I would also like to thank other members of my committee, Professor Michael
Frazier, Chang Yi Wang, Baisheng Yan and Zhengfang Zhou. They all have been
my teachers at some point of my graduate life, and have impressed me with their
dedication to mathematical research and education.

I could not have made this far without the love and support of my family and
friends. While there are too many to name, I would like to express my deepest
appreciation for my parents, for raising me into what I am today, for always trusting
me in every endeavor I pursue in life.

Finally, thanks to Michael Hamlin, my husband and best friend, who has been
through this all. Thank you for helping with the numerical work of this thesis, thank

you for your love, your patience, your confidence in me and in us.

iv

TABLE OF CONTENTS

Introduction

1 Properties of the Autoconvolution Equation
1.1 Continuity and Compactness .......................
1.2 Weak Closedness and Injectivity .....................
1.3 Local Ill-posedness ............................

2 Convergence of Local Regularization
2.1 Formulation of the Regularized Equation ................
2.2 Convergence of the Regularized Equation ................
2.3 Alternate Methods on [0, R] .......................

3 Discretization and Numerical Results
3.1 Discretization of the Regularized Equation ...............
3.2 Numerical Results .............................
3.2.1 Example with continuous 1‘: function ..............
3.2.2 Example with discontinuous a": function .............

4 Summary and Future Work

BIBLIOGRAPHY

1

8
9
10
12

15
15
30
42

59
60
64
65
71

76

85

Introduction

Linear and nonlinear Volterra integral equations arise in various applications, for
example, in recovering a space curve from its curvature and torsion, in the theory of
industrial inventory problems, and in nuclear reactor kinetics [23].

In this paper, we will study the solution of a nonlinear Volterra problem. We

consider the autoconvolution problem of finding :16 E L2(0, T) solving
G(;1:)(t) _-_—. f(t), a.e. t E [0,T], (1)
where G is the nonlinear Volterra Operator given by

G(.1:)(t) 22/0 17(t -- s):r(s) ds, a.e. t E [0,T], (2)

and f E Range(G) Q L2(0,T).
Before turning to this problem, we first give some background information on the
linear counterpart to this problem. Let us consider a linear first-kind Volterra integral

equation of the form

An = f, (3)

where A is a bounded linear operator on L2(0, T) defined by

Au(t) = [0‘ k(t — s)u(s) ds, a.e. t E [0,T], (4)

with the kernel function k E L2(0, T). Here, f is in the range of A and our objective
is to find 11 E L2(0, T) that satisfies equation (4).

A classic example of such equation is the Inverse Heat Conduction Problem
(IHCP). If we apply heat at the end of a semi-infinite bar where we call location
a: = 0, and measure the temperature f (t) as a function of time t somewhere away
from the heat source at location a: = 1, then the problem of recovering the tempera-

ture u(t) at the heat source is to solve equation (4), with the kernel given by

k<t> = é—fifl— exp(—,—,;)- (5)

Unfortunately, such problems are ill-posed because solutions do not have a con-
tinuous dependence on data: very small errors in the measured data can lead to large
errors in the solution. Since the available data in practice always contain uncertainty,
regularization methods should be employed to stabilize the problem.

One of the most well-known regularization methods is Tikhonov regularization.
Instead of solving for u that satisfies Au 2 f5, we solve the following constrained

minimization problem

min llAu- f6||2+a||LUI|2, (6)

where f5 is the measured noisy data, and a > 0 is the regularization parameter.
Here L is a closed linear operator often picked as either the identity operator or the
derivative operator. The role of L then is to penalize highly oscillatory solutions, thus
stabilize the problem.

The Tikhonov theory states that there is a choice of a = 0(6) such that as the

noise level 6 —+ 0,
0 (1(6) ——) 0,

o and the corresponding Tikhonov solution it? to (6) converges to the true 17..

Classical regularization methods such as Tikhonov regularization have inherent
disadvantages in solving Volterra problems. Volterra problems are causal in the sense
that the solution it at a given time t does not affect the data f on the interval
[0, t). Therefore, it makes sense to use future data f on the interval [t,T] only in
reconstructing u(t). Tikhonov regularization however replaces the original causal
problem by a “full domain” problem. It uses all values of data in the whole domain
[0, T] in reconstructing the solution u at any given time t. This becomes apparent

when we consider the necessary condition for the minimization problem (6):
(A*.A + aL*L)u = A*f6,

where A*.A is a non-causal operator even with a Volterra operator A.

In the late 1960’s, J. V. Beck developed a regularization scheme for the discretized
IHCP which retains the causal nature of the problem [2]. The numerical implemen-
tation of the Beck method is also more efficient than those of classical regularization
methods, because of its sequential nature. It was not until mid-1990’s that P. K.
Lamm established the theoretical basis for the convergence of the sequential local
regularization method, and Beck’s approach was generalized to a wide class of linear
first-kind Volterra problems [13][14][15]. While Beck’s method was an approach devel-
oped to handle a finite dimensional problem, the current theory of local regularization
methods can be placed in both finite and infinite dimensional settings.

To motivate the sequential local regularization method for linear Volterra prob-
lems, we let R > 0 be a small fixed constant and assume that equation (4) holds on

an extended interval [0, T + R] for 0 < R < R. Then {f(t) solves

[0 pk(t +p— S)u(s)ds = f(t + p), t E [0, T], p 6 [QB].

Split the integral at t, then change the variable of integration, we get

[0k(t+p-—s)u(s)ds+[opk(p—s)u(t+s)ds=f(t+p) t€[0,T],pE[O,R].

We integrate both sides of the equation with respect to a suitable Borel measure

7) = nR(p) (which we will clarify later) on [0, R], then

[O/RM ()t+p—sd773( u(s)ds+/OR/ok( p)p—su(t+s)dsdna()

=/OR f(t+p)dmz(p), t€l0»Tl- (7)

Note that we still have an equation that 2] satisfies exactly.

In reality, we often only have access to some perturbed f ‘5 E C [0, T+ R] such that
||f6 — flloo g 6 for some 6 > 0. (8)

Some regularization method needs to be employed.

We motivate the regularization method by considering what would happen if, for
fixed t, we momentarily held it constant on a small local interval [t,t + R]. This
motivates us to replace u(t + s) by u(t) in the second term of equation (7). Here, the
length R of the local interval becomes the regularization parameter. We then obtain

the regularized equation in u valid for all t E [0, T],

013 u(t) +/0 1}};(t— s) 11(3) ds 2 fat), (9)

where

~ R

k..(t)=f k(t+p)dmz(p), (10)

fiz(t‘)=/0 f5(t+p)dnn(p), (11)
R 9

0F] / ktp—sMsantp). (12)

The existing theory for the local regularization of linear Volterra problems requires

the assumption that the convolution kernel k in equation (4) is u—smoothing, i.e.,

k E C”[0,T] such that k(0) = k’(0) = = ICU—2(0) = O and [CV—1(0) 75 0.

It is well-known that the degree of ill-posedness of problem (4) is characterized by the
the degree of smoothness of the kernel k and the behavior of k at 0: the ill-posedness
increases as u increases. For example, the Inverse Heat Conduction Problem is
severely ill-posed since the heat kernel (5) is infinitely smoothing, i.e., k(P)(0) = 0
forp=0,1,---.

Let the Borel measure 7);,(p) on [0, R] satisfy the following three conditions:

1. For 2' = 0,1,...,1/, there is some a 6 IR and c,- = c,(,u) E R independent of R

such that
R
/ pi an(p) 2 R”" (c,- + (9(R)), as R ——> 0, (13)
o
with Cu 75 0.
2. The parameters c,-, 2' = 0,1,...,z/, satisfy the condition that all roots of the

polynomial pu(/\) defined by

Cu- u_ c c
1,/\‘+...+—1/\+—°-

CV V
p”(’\)—fi’\+(u—1) 1! 0!

have negative real parts.

3. There exists a C 2 0 independent of R such that

R
[0 9(pldnn(p) scnguw,

 

 

for all g E C [0, R] and all R > 0 sufficiently small.

It is worth noting that the Borel measures 173 satisfying these conditions are not
necessarily positive. Therefore, a signed Borel measure is allowed. It was shown in
[18] that under the above three conditions on 771200), we have (13 aé O for all R > O
sufficiently small and all u-smoothing k, where V = 1,2, 3,.... Therefore (9) is a
well-posed second-kind Volterra equation, with solutions depending continuously on
data f 5. We summarize the convergence theory loosely in the following Theorem, for

a more precise version and information about convergence rate, we refer to [18].

Theorem 0.1 Let 2) denote the solution of equation (4) given ‘true’ data f E
C[0,T + R]. Assume k is V—smoothing and that m; is a family of signed Borel mea-
sures satisfying hypotheses (1)-(3) for all R E [0, R]. Then for allt E [0, T] and
f6 E C[0,T + R] satisfying (8), there is a choice ofR : R(6) such that R(6) —> 0 as
6 —> 0 and

u‘]2(t) —> a(t,) as 6 —-> 0,

where u‘]z is the solution to the regularized equation {9)

Even though the full convergence of the sequential local regularization method
for linear first-kind Volterra equations is obtained by allowing signed Borel measures,
we would like to point out that convergence was also obtained using positive Borel
measures for u-smoothing Volterra problems, where 1/ = 1, 2, 3, 4. There is to date no

convergence theory for positive Borel measures except in those cases. For details, see

[13], [15] and [25].

Different variations of the local regularization methods for linear ill-posed prob-
lems were developed over the last decade [3] [4] [5][14][16][17] [20] [21][22]. For example,
in [3] and [4], the motivation for the regularization equation is to consider using a
polynomial function to fit the data on the local interval instead of a constant function.
In [16], [21] and [22], a variable regularization parameter R(t) is used instead of a sin-
gle constant R. This technique allows for more or less smoothing at different parts of
the domain, and it enforces another advantage of local regularization over Tikhonov
regularization in the ability of recovering sharp features of the solution. The methods
were even extended to Fredholm (non-Volterra) problems in [5] and [17], where the
solution is obtained by iteratively solving many small localized problems.

While the theory for the local regularization methods of linear Volterra problems
is rather complete, the nonlinear theory is largely absent. It was only recently that

local regularization theory was extended to the nonlinear Hammerstein equation:

[0 k(t,s)g(u(s),s)ds = f(t) for t E [0,T].

Notice the Hammerstein equation composes the desired solution u with another arbi-
trary function g. Based on the linear convergence theory, the key issue here becomes
how we can stably recover u from inverting the function g. We have shown in [19]
that local regularization of this problem is successful under certain conditions on this
external function g. It is not surprising that the proof utilizes the linear theory.

In this paper, we develop a local regularization theory for the nonlinear autocon-
volution problem of finding a: E L2(0, 1) solving equation (1) with G given by (2). We
will use the underlying ideas of local regularization to formulate the regularization
equation for the autoconvolution problem. However, due to its nonlinearity, we ex-
pect the convergence theory to be fundamentally different from what is in the linear

C358.

CHAPTER 1

Properties of the Autoconvolution

Equation

Autoconvolution has been of interest to scientists for decades because of its appli-
cations in various fields. It arises in stochastics where the density function of a
continuous random variable V is reconstructed after observing the density function
of the random variable 5 2 VI +V2, where V1 and V2 are identically and independently
distributed random variables of V. For more details, see [10]. Another application
of autoconvolution occurs in spectrosc0py. Baumeister presents in [1] a reference list
of physically motivated papers concerning this class of problems. He also discusses
in detail the mathematical model of deconvolution of “appearance potential”(AP)
spectra to investigate electronic properties of solids in their surface region. In this
context, the density of unoccupied states in the surface region of a solid is recovered
from the measured AP-spectrum data.

In this chapter, we will summarize the properties of the autoconvolution equation.
For details of these results, we refer to [7] [8] [9] [10]. We will also mention some

existing regularization methods towards the end the chapter.

1.1 Continuity and Compactness

In the linear case, the inverse problem (3) is ill-posed when the operator A is compact.
As we will Show in this section, the autoconvolution operator is continuous in L2(0, 1),
but fails to be compact in L2(0, 1).

Recall the autoconvolution Operator G : L2(O, 1) —+ L2(O, 1) defined as

G(:r)(t) z/o :r(t — s)x(s)ds = f(t), a.e. t 6 [0,1]. (1.1)

More generally, G : D(G) g B1 -—> 82 where B1 and 82 are Banach spaces containing

real functions on [0, 1], with properties to be specified below.

Proposition 1.1 [10] If a: E L2(0,1), then C(33) E C[0, 1] and [G(a:)](0) = 0.

Moreover, G : L2(0,1) —> C[0,1] is a continuous nonlinear integral operator with

||G(37)ll1.2(0,1) S llG(I)llcto,u £ lll‘lligton

Note that the continuity of G : D(G) _C_ B1 —> 32 remains true if B1 has a stronger

norm than L2(O, 1) or if the norm of 82 is weaker compared to that of C[0,1].

Definition 1.1 We call a linear or nonlinear operator A : D(A) Q Bl —> B2 compact
if the range RA(S) E {y 6 B2, y = .4(1:),:r E S} is a relatively compact subset of B2
whenever S is a Bl-bounded subset of D(A).

Proposition 1.2 [10] The autoconvolution operator G : L2(O,1) ——> L2(0,1) is not
a compact operator. 0n the other hand, the Fre’chet derivative G’(:1:) : L2(0, 1) ——>

L2(O, 1) defined by
G'(;I:)(h)(t) = 2/t:r(t — s)h(s) ds, t 6 [0,1], h E L2(0,1),

is a compact bounded linear operator for all .7: E L2(0, 1).

9

As an example of noncompactness of G, we consider an infinite sequence

17,,(t) =sin(nt), OStS 1,n=1,2,...,

 

then,
_ _ _t cos(nt) sin(nt)
yn(t) “' [C(xnllU) "‘ 2 2,” a
for O S t S 1 and n = 1,2,---. Note that 2:0(t) = 0 andy0(t) = [G(:ro)](t) = 0

for t 6 [0,1]. Evidently, the sequence {sin} is bounded in L2(0,1), but we don’t

J6
12
the autoconvolution operator G doesn’t take every bounded subset in L2(O, 1) into a

have strong convergence of yn -> yo = 0 since llynllL2(0,1) ——> 76 0. Therefore,
relatively compact subset in L2(0,1).

Even though the operator G is not compact in the general setting where G :
L2(O, 1) ——+ L2(0, 1), compactness of G can be obtained by restricting its domain D(G)
to a relatively compact subset of L2(O, 1) due to the continuity of G. For example,
if D(G) only contains equibounded monotone functions, then G : D(G) —+ L2(0, 1)

becomes a compact operator.

1.2 Weak Closedness and Injectivity

In the following, we restrict the domain Of the autoconvolution problem to
D(G) E {:L‘ E L2(O,1), .r.(t) _>_ 0 for a.e. t 6 [0,1]}. (1.2)

To apply the classical Tikhonov regularization theory to nonlinear inverse
problems[6], it is required that the nonlinear Operator be weakly closed. We can

ensure the weak-Closedness of the autoconvolution operator G if using the restricted

domain D(G) as defined in (1.2).

10

Definition 1.2 A linear or nonlinear operator A : D(A) (_Z 81 —> B2 (where 31 and
82 are Hilbert spaces) is weakly sequentially closed if, for any sequence {:cn}$,:1 C
D(A), weak convergence xn —\ $0 in BI and A(:z:,,) -—3 yo imply $0 6 D(A) and

A(3170) = 310-

Proposition 1.3 The autoconvolution operator G : D(G) C L2(0,1) —) L2(0,1) is
weakly continuous, D(G) is weakly closed in L2(0,1), therefore G is weakly sequen-

tially closed.

We can also prove the injectivity of the autoconvolution operator using the Titch-

marsh’s Lemma.

Lemma 1.1 For f,g E L2(0,1), let there exist a value V (0 < V < 1) such that

[0f<t—s>.q(s)ds=o (031:9).

Then there exist numbers a,,8 6 [0,1] with a + R Z V, f(t) = 0 a.e. in t 6 [0,0] and

g(t) = 0 a.e. in t E [0,/3]

Then it follows:

Theorem 1.1 If we define for any a: E L2(0,1),
5(517) E sup{0 g 8 $1: :1:(t)= 0 a.e. on [0,5]},

then the autoconvolution equation (1.1) subject to the domain (1.2) has a unique
solution if and only if f(t) Z 0 a.e. t 6 [0,1] and 5(f) = 0. If 35* is the uniquely

determined solution, then it fulfills the condition e(:1:*) = 0.

11

1.3 Local Ill-posedness

Let us first consider a general operator equation
F (.13) = y, (1.3)

where the operator F : D(F) (_Z X —> Y maps between Hilbert spaces X and Y. We
denote the norms in X and Y by [I - “x and I] - ”y. If F is nonlinear, we will focus
our attention to a solution point 23* E D(F) of equation (1.3) and a family of closed

balls centered at :1:* with radius r, i.e., B(a:*,r) E {x E X, Ila: — x*||X g r}.

Definition 1.3 We call the equation (1.3) locally ill-posed in at" if, for arbitrary small

r > 0, there is an infinite sequence {17"} E D(F) fl B($*,r) with
||F(:L‘,,) — F(at*)]|y ——> 0, but “an" — a:*||X 4+ 0 as n —> oo.

Otherwise the equation is called locally well-posed in 17*.

In linear inverse problems, ill-posedness typically occurs when the operator F is
compact; in particular, if F is linear and compact, then the problem is unstable if and
only if the range of F is infinitely dimensional. However, ill-posedness can also occur
when F fails to be compact, and the autoconvolution problem represents a nonlinear

example of this case.

Proposition 1.4 For the D(G) defined in (1.2), the inverse autoconvolution operator
G'—1 is discontinuous at every point y 2 C(17) E L2(0,1), :1: E D(G) C L2(0,1), i.e.,

the autoconvolution equation is locally ill-posed at every point a: E D(G) C L2(0,1).

Various degrees of ill-posedness for the autoconvolution equation are discussed in

[8] and [10]. In short, we expect a correlation between the degree of ill-posedness and

12

boththe smoothness of the solution a: and the behavior of a: at 0. It is not surprising
that the same kind of dependence exists in the linear problems except that the role
of the kernel function It is now taken over by the solution 2:: the global smoothness
of k is replaced by the local smoothness of 3:. Furthermore, the main difficulty of
the autoconvolution problem is associated with values of the solution x with small t,
particularly at 33(0).

Various regularization methods have been studied for the autoconvolution equa-
tion. We can utilize Tikhonov regularization theory for nonlinear inverse problems
since the autoconvolution operator G : D(G) g L2(0, 1) —> L2(0, 1) is continuous and
weakly closed, and G has a compact Fréchet derivative at any a: E L2(0, 1) that satis-
fies the assumptions to guarantee stability in the Tikhonov theory [6] [10]. However,
the drawbacks of Tikhonov regularization (e.g., loss of causality Of the Volterra prob-
lem) still exist here, and because Of the nonlinearity of the problem, the numerical
implementation becomes even more expensive.

The ill-posed autoconvolution problem (1.1) can be changed into a well-posed one
by imposing appropriate a prior restrictions 1' E Q. If Q is a relatively compact
subset of L2(0,1) and G is injective on Q, then the inverse operator G‘1 exists
and is continuous by Tikhonov lemma. In [9], the domain of the autoconvolution
equation is restricted to a subset B of D(G) (as defined in (1.2)), where B contains
solutions a: that are uniformly bounded below and above by positive constants, and
with a prescribed upper bound c for the total variation. Here, c is the regularization
parameter.

The most recent approach of which we are aware, to regularize the autoconvolution
problem, was studied by Janno in [12]. Lavrent’ev regularization method was applied
to the autoconvolution equation and convergence was obtained. The advantage of
Lavrent’ev method is that it preserves the causal nature of Volterra problems and

therefore leads to a fast sequential method. The major drawback for the method is

13

that it requires and depends on an initial guess of the true solution. When the initial
guess is far away from the true solution, the method appears less able to recover the
true solution.

In the next chapter, we develop a local regularization method for the autocon-
volution equation. A convergence theorem is proved, and we provide an effective
regularization method which preserves the casual nature of the problem for the most

part without having to introduce an initial guess.

14

CHAPTER 2

Convergence of Local

Regularization

2.1 Formulation of the Regularized Equation

Recall that we are considering the problem of finding a: E L2(0, 1) satisfying the

autoconvolution equation
G(.r)(t) : f(t), a.e. t 6 [0,1], (2.1)
where G is the nonlinear Volterra Operator given by

G(.r)(t) :/0 37(t — s):1:(s) ds a.e. t 6 [0,1], (2.2)

and f E Range(G) g L2(0,1). Note that without loss of generality, we have assumed
that T = 1.

Suppose 0 < :i: E C1[0, 1] is the true solution of equation (2.1). As in the local
regularization approach described in the Introduction for the linear Volterra problems,

we extend equation (2.1) slightly into the future, i.e., let R > 0 be a small fixed

15

constant such that R < 1 and assume that equation (2.1) holds on an extended

interval [0,1 + R] for 0 < R < R. Then f(t) solves

Aft-pm“ + p —" 3) 17(8) d8 = f(t + p), t 6 [0,1], p 6 [0,12]

Split the integral at p and t, then change the variable of integration, we get

2/0px(t+p—s):r(s)ds+/ :1:(t+p—s):r(s)ds = f(t+p), t 6 [0,1], p 6 [0, R]. (2.3)

In order to consolidate the local future information introduced by the variable p, we
integrate both sides of the equation (2.3) with respect to a suitable Borel measure

77 2: 71(p) > 0 (which we will clarify later) on [0, R], then

2/R/Opa z‘(.)1:t+p—s s)dsdn(p )+/0R/ta:( a:().rt+p-—s r(s)dsdn(p)

=/OR f(t+p)dn(p). t€€l0,1l- (24)

Note that i: still satisfies equation (2.4) exactly.
In reality, instead of having the exact data f, we always only have access to some

approximation f6 E C [0,1 + R], such that
[If6 — fl]CO S 5 for some 6 > 0. (2.5)

So, instead of solving G2: = f, we need to solve G3: 2 f5. As discussed in Chapter
1, this latter problem is locally ill-posed in L2(0, 1) in the sense that 6 ——> 0 does not
guarantee the convergence of the solution to 17:, i.e., the solution of the equation does
not depend continuously on the right hand side. Thus a regularization method is
needed.

As in the linear case, we motivate the regularization method by momentarily

16

holding :1: constant on a small local interval [t,t + R]. Therefore, we replace :1:(t + p)
by $(t) for p E [0, R] in the first term of equation (2.4). Here, the length of this local
interval R serves as the regularization parameter. We then obtain the regularization

equation

0R($)$ + FR($) : IR? (26)

where for t E [0, 1],

03(23) E 2/0R/0x( s)(pdsdn
new) a f f Haw—s) (s)dsdn(p),
R P
rim 2 / f"(t+p)dn(p). (2.7)

We will first study the class of the Borel measures we use. If 77(p) > 0 is a

continuous Borel measure on [0, R] of the form

R R
/0 9006100)): /0 s(p)w(p)dp,

where w(p) E Loo[0, R], such that 0 < col 3 w(p) S (.02 < oo,p E [0, R], for constants

w1,w2 > 0, then for any real numbers m and n,

foR pm d77(p) = foR me(P) dp < 602 I: pm dp : w2(n + 1) Rm’"
If p" dn(p) 15R W102) dp — an I: p" dp Wm + 1)

 

 

If n(p) > 0 is a discrete Borel measure on [0, R] Of the form

R k
[0 go) (177(0) = Zgoaa,

where for i = 1,--- ,k, 0 < (2113 if), < 02 < oo,p,- 2 CR for G,- E [0, 1] and there

17

exists at least some C.- for 1 g i S k such that C,- 95 0, then

 

 

1:
GR m E),- _
f: p’" (17700) _ i; ) wzk m-"
R — k — - ' n
n d ~ w mm C,-
f0 P ”(W §(CiR)" wi 13%,:

Therefore, a reasonable general assumption for our generic Borel measure 1] is that

for any real numbers m and n, there exists a constant C (m, n, n) > 0, such that

If p'" Cir/(p)
If p" dn(p)

 

S C(m, n, 17)Rm_". (2.8)

Our main convergence result is stated below. It follows as a corollary to Theorem

2.2 of the next section.

Theorem 2.1 Assume the perturbed data f6 satisfies |f(t) — f6(t)| g 6 fort 6
[0,1 + R], and that the Borel measure 77 > 0 satisfies (2.8). Then there exists 5' >
0 and k1 > 0 independent of R such that if the true solution it E C'1[0, 1] 0f the

autoconvolution equation is positive and satisfies

then for R = R((S) > 0 selected satisfying

 

R((S) —) 0 and

as 6 —+ 0, it follows that the solution rim) of the regularization equation (2.6) asso-

18

ciated with data f6 satisfies

“th - illews) = 0(51/2)

as 6—)0.

The proof of this result will take the remainder of this section and involves intro-
ducing some additional spaces and norms. To begin, we rewrite equation (2.4) using

similar notation as in equation (2.6), then :7: satisfies
OR(j)j+FR(3—3) = fR+€R, (2-9)

where for t E [0, 1],

013(517) : 2/0R/Opx(. s)dsd7)(p

FR(1>)(t) = [OR/p1 (1t+p—s) )1‘8/1()dsdn()

Mt) E /R f(t+p)dn(p),

6R

2/R/pae )—;r(t+p—s)) (s)dsdr}(p.) (2.10)

Let us define a new R-dependent topology on L2(0,1):

1
(17, y>a,R C(R) (£13, y)Lg(o,R) + (113, y)Lg(o,1)

 

8 a II) t l 1‘ (it 8 0 IL' t t dt,

where 0 > 0 and C(R) > 0, with conditions on 0 and C(R) to be specified later.

19

Note that the weighted inner product

T
(:r,y)Lg(0,T) =/ e—2U‘x(t)y(t)dt for T > 0
o

induces a weighted norm

1/2

T
”mum = (/ esteem) ,
0

which is equivalent to the L2 norm since

8”W -|

 

L2(0.T) S H ° ”Lamm S H ° ||L2(o,r)

for T > 0.
Throughout this chapter, we will couple the L2(0, 1) space with the newly defined
R-dependent topology and denote it as Lg’R (0,1). We then denote a closed ball

centered at 1'0 with radius r under this new topology by
B(z0,r) = {z E LZ’R(0,1),||2 — rollmg S r}, (2.11)

where

1 1/2
H - ”as = {5627” - Him + n - Him} .

Note that if :1: E LS’R(O, 1), then a? E L‘2’(O, 1) and I][0,R] E Lg(0, R) and thus the

norms I] - ”1,ng) and H ~ llL-‘2’(0,R) still have meaning for such IE.

Lemma 2.1 The Operator FR : Lg’R(O, 1) —> Lg’R(O, 1) as defined in (2 7) is Fréchet
differentiable and F h is uniformly Lipschitz in Lg’R(0, 1), i.e., there exists some con-

stant k > 0, such that fora 21/2 and R S 1,

lthUTI) “ Fh($2)ll Skill-1‘1 - $2lla.e

20

for 221,32 6 Lg’R(O, 1), where I] - || denotes the usual £(Lg’R(O, 1)) operator norm.

Proof: For any 3:, h E Lg’R(0, 1) and t E [0, 1], we calculate

FR (:1: + h)( —FR(:r)(t)

= [OR/:)h(t)+p—sx)(sdsd17H/OR/$(t+p—S)hs()d3d7l(Pl
+f0R [p h(t + p — s)h(S) 181770))

= 2/0R/ptzr(t+p—s)h(s)dsdr}(p)+/0R/:h(t+p—s)h(s)dsdn(p).

For fixed t 6 [0,1], p E [0, R], let T = t + p — s, then

t 1/2 t 1/2
3 (/ h2(t + p — 3) ds) (/ h2(s) d3)
P P
t 1/2 t 1/2
S (/ 62016—2arh2(7_) d7“) (/ 62036—2ash2(8) d8)
p p

2 2
S e a llhllLf2’(0,1)°

 

 

/th(t+p—s)h(s)ds

Therefore,

[OR/ptMHp—S)h(8)dsdn(p)] S [OR

R
r0 2
$62 llhllrg(o,1)/O (177(9),

dn(p)

 

 

fth(t + p — s) h(s)ds

 

and

R
h( + p - 8) h(S) d8 d71(1)) S 62" llhlligm,” [0 (171(0) ”lllL‘2’(0,1)

 

 

       

L.‘_,’(0,1)

R
s llhlligm,” / duo) (2.12)

21

for a Z 1 / 2. Here we have used the fact that

1 ”2 1_e—2a 1/2
1 ., = "2"‘dt = <1
|l||L2(0.1) (f e ) ( ,0 ) _

 

for a 21/2.

Similarly, we can show for 0 Z 1 / 2,

R
h(- + p - 8) h(8) ds(177(p) S 62" llhlligm) [0 (117(1)) (2-13)

       

 

 

Lg(o,n)

Combining inequalities (2.12) and (2.13), we get

 

 

 

 

       

 

 

           

 

 

 

 

R - 2
f / h(-+p—s>h<s>dsdn(p)
0 p a,R
2
= h(° + p - 8) h(S) ds d77(p)
Lg(o,1)
2
h(- + p - 8) h(8)1817200)
Laws)
- 2 2 R 2 1 2 2 R 2
g e" h] a / d7](p)] +—[e" h a / dnp]
_ H lL2(0,1) 0 C(R) H llL2(0,R) 0 ( )
.. R 12 1
3 e20/ (177(2) llhlngmi) [lllllllg(01)+C(R) llhlngm In]
. o .12
F R T
s / d72(p) thifinhnifi
Therefore,

h(- + p — s) h(s) ds d1)(p)

       

 

 

R
s / dn(p) “hut...
R 0

Hence, FR is Fréchet differentiable at :L' E Lg’R(0, 1) with Fk(:c) E £(Lg’R(O, 1))

0a

given by

F'(.r h)(t)=2/0R /tr J(t+p— s) hs()dsdr)(p) (2.14)

22

for any h E Lg’R(O, 1) and t 6 [0,1].
To show the Lipschitz condition, we will first consider ||F,'{(:z:1) — Fk(x2)||1,g(o,1),

for 131,1:2 E Lg’R(0, 1). For fixed t E [0,1],p E [0, R], let T = t + p — s, we have

(x1(t+ p - S) — 2:2(t + p — 3)) h(8)618

S (fpt(:c1(t+ p — .3) _ $2“ + p _ 3))2ds)1/2 (A, h2(3) (13)”?
(f; 62076—2”’(x1(r) — x2('r))2 d7) ”2 (f: 62.,e_203h2(8) d3)

62" ”$1 " $2lng(o,1)llh“L‘§(0I1)’

 

 

N

1/2

I/\

therefore,

($105 + p - 8) - 222(t + p - 8)) h(8) d8 dn(p)

0
R
s f
0

R
S 620 ”1'1 _ 332lng(0,1)llh‘lngwIl) / 6177(9)-
0

     

 

 

/ <qu + p — s) — szt + p — 3)) h(s) ds anp)

 

 

 

 

               

Hence,
lle'z(:v1)( h)“ F'R($2)(h)lllg(0, 1)
= 2//p(a~( +p—s) —a~2(+p—s>)h(s)dsdn(m
Lg(0,1)
s 2e20IIwI — aerIIII IIIhIILI; III) / d7I(P)H1HLg(0.1)
0
R
s 2e2°IIrI—I~2IILI ,gI,/Im anp)
0
for021/2.

23

Similarly, we can get

lth(-'171)(h)’ Flz($2)(h)lng(o,R)

R
g 2e2°llx1-$2||Lg(o,R)||hHLg(o,R)/O d’llpl

R
s Mum.-ngILgIIIIIIhIILng / WI
0

for a 21/2.

Therefore, for any h E Lg’R(0, 1),

IIFIIIxIIIh) — qu(132)(h)||:,n
= lth($1)(h) - Fk($2)(h)lllg(o,1) + 6711}; lle’z($1)(h) - Fk(I2)(h)”iz(o,1)

. R 2
a 1
2e2 / anpIIIxI-szILgIIIJ [IIhIIigIo,I,+—C(R)IIhIIigI0,RI]

|/\

|/\

 

_ R 2
2&0 / d77(P)ll$1—$2lla,n] IIhIIZII.
_ 0

Hence,

R
IIFIIIIII — Fawn s 2e” / d77(p)ll~’v1-172.,,Rll

o
for any 171,172 6 Lg’R(0,1),i.e., F}2 is uniformly Lipschitz in Lg’R(0, 1) with Lipschitz

constant k = 262” ff d7)(p). D

The following Lemma follows immediately from Lemma 2.1.

Lemma 2.2 Let 21,111,222 6 3(0, 7‘) g Lg’R(0, 1) and a: E Lg’R(O, 1) , then the remain-
der

RR($, v) E FR(IL' + v) — FR(LE) — F};(r)v (2.15)

24

of the Fréchet derivative F,’2(:1:) satisfies

R
Imam”... s f anp) IIvIIiIII, (2.16)

R
Hence, an) — RRIx. v11“... 3 2e?“ f dn(p) max{||v1l|a,zz, Ilvzlla.n}||v1 — «and.
0

(2.17)

Proof: We can write
1
723(22, v) =/ (FMJ: + tv) — F};(:c)) vdt, (2.18)
0
then

1
IIRIIIx, v>II.,R s / “(FM-”v +tv1— Fax» 'vllmdt

l
s / (IIFI'I($+t'v)-F}I(r)llllvlla,a) dt

1 R
S/ (2620/ M")“tullmrzllvllag dt
0

0
I R 1 -.
S 262”] 6177(1)) llvlliflf W
0 0

R
= / d71(p)||v||3,n-
0

Similarly, we can write

1
RR(III,U1) — RR(1',’U2):/ (17;?(1' +tU1+(1—t)2)2)— Fk($))(1’1- U2) dt,
0

25

then,

llRR(x?v1) - 723(22, Uzlllafi
1
S / ”(FACE + “II + (1 — t)‘Uzl — F},(2:))(v1— v2)”o,R dt
0
l
_<. / (”Fax +tvI +11 —t)v2> — F}I(w)|lllv1 — 2221012) dt
0
l R
s f (M / dn(p)lltv1+(1-t)v2lla,nllv1 — 22210.3) dt
0 0

R 1
_<. 2&0] anp) Ilvl-v2lla,R max{||v1||a,R,|lv1||a,R} f Idt
0 0

R
= 2&0 f anp) 1221 - vIIIIII max{||v1|la,n,llvlllo,n}-
0

C]
For h E Lg’R(0, 1), t 6 [0,1], we define
R t
BR(T)(h)(t) E 2/ / f(t+p— s)h(s)dsd7)(p), (2.19)
o o
R Io
DR(i)(h)(f) 2 2 / / :w + p — s) h(s) dsd-nIpI, (2.20)
o 0
then
mix/w.) = BR(:z><h)<t) — DR(i)(h)(t)- (2.21)
Note that both 33(1) and DR(§:) are bounded linear operators in LZ’R(O, 1).
Expand FR(:1:) in equation (2.6) using the Fréchet derivative FMi), we get
aR(:r);17 + PRC?) + Fk(i)(.r —— i) + 723(17, .1: — :E) 2 fig. (2.22)

After combining equation (2.9), (2.21) with equation (2.22) and some simple al-

26

gebra, we obtain

(012073)] + Bah—Ema” ‘ 51'7) = fist — fR - 6R — Rafi-ax — i)

+ DR(T)(.’L‘ — 11-3) + (03(2) - OR($))IL‘, (2.23)

where I is the identity operator on Lg’R(O, 1).

Let us further denote for v E Lg’R(0, 1),

E365, v) = Dn(i)(v) - 0112(1)) :73, (2-24)

where we note that v —+ ER(:E, v) is linear. Then equation(2.23) becomes

(03(T)I + 33(2))“: — 53) 1: f2 — f3 - ER - RR(.’TP,$ — 3—3)

+ ER(;2‘:,:1: — i2) + (aR(:z:) — aR(i))(:f: — :12). (2.25)

In order to be able to invert (aR(;i:)I + BR(§:)) E £(LS‘R(O,1)), we need the

following lemma.

Lemma 2.3 If f(t) > O for t 6 [0,1 + R] and :7: E li"2’°°(0,1 + R), then for positive
Borel measure 77(p), there exists 00 > 0 independent ofR > 0, such that the operator

BR(1‘7) is accretive in Lg’R(0, 1) for o 2 0‘0; i.e.,

(BR(i‘)v,v)o‘R Z 0 for any v E LS’R(0, 1).

27

Proof: Recall that for h e Lg*R(o,1), t 6 [0,1],

8;; (5:)( h)(t)=2/OR/Ot5: a:()t+p—s h(s)dsdr)(p)

= [0 (2 / x(t+p— S)dn(p)) hoods.

Let us denote for t E [0, 1],

R
ago) a 2 f w +p)dn(p),

then

BR(:E)(h)(t) 2/0 aR(t - s)h(s) ds. (2.26)

'7

We will first show 33(1) is accretive in Lf’(0,1),i.

[01 (3‘2“ (Alum — as)l)(8)d8) o(t)dt
=//Ote“’( H(t—s)e"”o (s)dse'U‘o (t)dt_>_0

for any 1) E Lg(0, 1). This is equivalent to the following condition

i/OAtftsaRt—g)()d91()dt20

for any o E L§(0,1), i.e., the operator Bfli‘) is accretive under (., -)0=0 (the regular
L2 norm) where 8%(57) is defined as in (2.26) except that the kernel aR(t) for BR is
replaced by aR[o](t) = “.taR( ). In what follows, we will show accretivity of B"(.r )
in L2(0, 1) for all a sufficiently large.

Let us define

._ ' —. , __ —’ _ ._ .-." _
A0 -t€[13}llilml(t) > 0, ‘41 - “17 ”Loc[0,1+R]a ‘42 -— H-l' llLoo[o,1+R}a

28

then

R
' t > 2A (1* ,
trerféglafl ) _ 0 f0 00))

R
lla'RllemA] _<. 2A1/ 6177(0),
0

R
”a’l’illemJ] S 2A2/0 (177(0)-

Consider aR[o](t) = e‘U‘aR(t), we have for a.e. t 6 [0,1],

R
anl0](t) Z 6“" (2 A0 [0 dn(p)),
a'R[o](t) = e_"t(—oa3(t) + a}{(t)) _<_ 6“” 2 (—vo + A1)/0 dn(p),

R
a)}[o](t) = e—”t(o2aR(t) — 2oa'R(t) + a','{(t)) Z 6“” 2 (02140 — 20/11 - 242)]; d7)(p).

Since fOR d77(p) > 0, we can take

 

> 111+ A? +A0.42

00 _ 40 (independent of R)

then for o 2 00, we have
aRlal(t) 2 01 aR[o]'(t) S 0’ aRlalfl(t) Z O

for a.e. t 6 [0,1].

Therefore, the kernel a R[o] is nonnegative, nonincreasing and convex. According
to Lemma 2 of [12], for o 2 00, we have Bfli) accretive on L2(0,1), from which
it follows that BR(J':) is accretive in Lg(0,1). Similarly, we can verify that BR(;1’:) is

accretive in Lg(0, R) for a Z 00, where the 00 is the same as in L§(0, 1) case.

29

Hence, for any o E LZ’R(0,1),0 2 00,

1
(312077)”, v)a.R = WWWE)”, 700240.12) + (Bsfilv, ’U)Lg(0,1)
1
> —0 0 =
- C(R) + 0’
we then have BR(§:) accretive with respect to (-, -),,,R for o 2 00. [:1

Consequently, (aR(;‘1‘:)I + BR(.7‘:))‘l E £(Lg’R(0, 1)) and we have the following

estimates (see [24]):

Il(aR(i:)I + BR(i:))“II s

 

(112(27)

”(01453)! + BR(i))"lBR(f)I| S 1

for a 2 00, where 00 > O is independent of R.

We are now ready to formularize our regularized equation as
:1: = H R :r, (2.27)

where

H33: : (aR(;'1§)I + BR(.ir))‘1[ffg — fR — €R — Rid-73,113 — :72)

+ ER(.1“:, :17 — it) + (03(33) — a,{(:1‘7))(:1’: — 1:)] + it. (2.28)

2.2 Convergence of the Regularized Equation

In this section, we are going to prove that the regularized equation (2.27) has a unique

solution, and this solution converges to the true solution :7: as the noise level in the

30

data 6 ——> 0. In the following lemmas, we bound the right hand side of equation (2.28)

term by term.

Lemma 2.4 If |f5(t) — f(t)| S 6 for t E [0,1+ R] and 0 21/2, then

R 2
us. — all... s 6/ 6177(0) \/1+ W (2.29)

 

Proof: Since

1
6 6 6
”fR - fallin = mllfk - fRHLngZ) + “fR - falligm),

we first consider

 

 

R
llfh—fRHLgmfi) S /0 |f6('+P)—f('+P))ld77(/)) L( R)
‘2"),

R R
/ M770?) S 6/ (172(1)) ||1||Lg(o.n>
0 1.3mm 0

R R 1/2 R 1 _ 6—2012
S 6 / dn(p) (/ Ema) = 5 / d77(p) V -———-
0 o 0 20

~20R

|/\

 

 

 

 

Taylor expansion of 6 around R = 0 gives

6-2.3 =1— 20R + 0(R2).

Thus,

R
”first — lell.f§(0,R) S 5/ d77(p) R+ CUP).
0

31

Similarly, we consider

R 1
“fit " fRHLg(0,1) S 5/0. dTI(P) (f0 8—2atdt)
R _ ~20 R
=6/0 dr/(p)\/1 2: S 6/0 (177(1))

for a _>_ 1 / 2. The lemma then follows. C]

1/2

 

Lemma 2.5 If f(t) E C1[0,1] and a 2 1/2, then 63 defined in (2.10) satisfies

 

R R 2
116311.,Rs]2fu-'umz<o> / p2dn(p)+ 3—[nx'nio / p3dn(p>]\/1+5—E%§—l
(2.30)

Proof: For fixed t E [O,1],p E [0, R], we have

 

/0p(i7(t) — :I:(t + p — 8)) 2(3) ds

3 (foo) — at + p — 3W3) ”2 (fro) ds) U2.

Since f(t) E C1[0,1], there exist {1,62 6 [0,1], such that for 0 < s < p < R,

 

f(t+p- 5') -i‘( ) =16 ('51)(p- 8) 17(3) = f(O) +i"(52)8,

32

then

 

A2270) — f(t + p — 3)) 55(3) ds

Ili'lloo ([(p— Was)”2 (2 [op 5(0)2ds + 2/0" (3%) 3)2ds)1/2

—I (23/2 — 2 —I 2 p3 1/2
s llxlloo 7342 (2(0) p + Inn... —3—) .

 

I/\

|/\

Now consider

 

 

II€RIIL301):H2/ /p()i‘(-)-fr( +p-s>) WHdenH

       

 

 

 

Lam)
S -i‘(-+p—8))i(8)d8 d77(1))
Laws)
p3 1/2
S 2/255II 'Iloo\/:pi)3/2( (0) p+ll$’II§o 1:7) dn(p)II1IILgIo,1)
\/;p3/1/2 p3/2
g :r 00 —— 0 + 17 00*— d 1 a
2]: II II 22:(()p II II )3) n(p)|| new»
R
2f
S V553“ IIooJTIOI/ pzd'(r/)||1||L001)+——|l1.||2 /R 10 dUIPlIIlIIL;(0,1)-
2f 0 o

Similarly,we have

2f R 2
IIeRIILgIom 7— IIi-Imo )/ panpIIIIIILgIO,R,
0

+2f _,
——II Ilio / panpIIIIIILgIo.R).

As seen in the proof of the previous lemma,

1_2o

 

“I'll/310,1): <1 fOI' 0' 21/2,

33

and

II1IILgI0.R) = 3+ C(32)-

Therefore, the lemma follows. E]

Lemma 2. 6 If :1: (t)€ Cl[0, 1] andforo >1/2, then ER(CL‘, :1: — :1:) defined in (2. 24)

satisfies

 

R 2
IIERIM— mums]. fun: as'Ilooe””II:v-illa,n / p3/2anpI\/1+Eié—(—iff—). (231)

Proof: For fixed t E [0, 1], p E [0, R], we have

I Edi, 2: - i)(t) )I = IDIiIII - 57W) — (022(1) - an(i))i(t)l

<2] / |rt+p—:>)—:v()llr()-:v()|dsdn()

s2“ “00/ ijp —sI <sz -i(s))|dsdn(p)-

Consider, for fixed p > 0,

£90 — 8) |(rr(s) — stIsIII ds

S 00”“) — 8):! ‘15)”2 (fope2me_2"s(rr(s) — 2(3))2 d3)1/2

p3/2
S 1736"” III — iIILgIOJI-

Therefore, for fixed t and p,

_ _ 2 a _ R
IER(:E,I- III/SIM 7H1? I'lloo'e RIII-HSIILgImI/ 93/261002)-
0

34

It is now not hard to see that

_ _ 2 _ _ R
HEM-13$ — $IIIL3I0,1) S — IIx'IIooBUR IIJI " JIIILgIOJI / 03/261779) II1IILgIo.1I,
0

J3

and
2 R R 3 2
._ _ —I _
IIER(~’E,$ — $lIILg(0.R) S 7—3 II33 IIoo 6” II?” — $IILg(o.1) [0 P/ dUIP) II1IILg(0.R)-
Combine the above two inequalities with the fact that

IISF - iIILgIOJ) S ”17 — iIIpRs

we obtain the lemma. L__I

Lemma 2.7 Assume the Borel measure 77 > 0 satisfies (2.8), if further f(t) E

C1[0, 1], then for R > 0 sufficiently small,

 

 

_ S R . 2.32
03(13) it(0)f0pd77(/)l ( )

Proof: We can write for some {(9) E [0, R],

)2/0R/p1-(zspwsdm
2//p(f(0 I+sp'( (tsIIIIdsdepI

=2a;(0)/OR pd7](p )[1+g(R)ls

35

where

 

 

 

 

 

1 R P _,
9(R)—i(0)fo,spdn(p) f f ssItIsIIdsanI.

Since

1 is R g: IIi'IIoo C(2a 1: 77)

I9(R)|_ (off/)d()” Ilse/0 2depIs mo) 12.

then

g(R)=O(R)—>0 as R—>0.
Thus,

1 _ 1 . 1
ORG?) _ 2i(0)f0den(p) 1+9IR)
1

- [1+ 9(R) + 0(R2)I,

 

: 2 33(0) If pdn(p)

where |g(R) + 0(R2)| 3 CR for R sufficiently small and some constant C. Therefore,

 

 

 

1 1 1
_, S _ R -(1+0(R))S_ R
“8(1) 217(0) f0 panp) 17(0) f0 pdn(p)
for R > 0 sufficiently small. CI

Lemma 2.8 For 23,5: 6 Lg‘R(O, 1), we have

R
IastI—aRIpIIsz/ p1/2dn(p)-x/C(R)-II1?-i‘lls.n. (2.33)

0

36

Proof: For fixed p E [0, R], we have

S (fop($(3) — 57(3))2ds)1/2 , ([0” 1d8>1/2

1/2.

 

[0p(s(s> — s<s)) ds

 

S eaRllx - i‘lngwfl) '10

Therefore,

lam) —as(s s)I=l2 ffptsts updsdnnl

32/0 /0(s<s>:j( s(s)>ds ss(p)
R
£2/eaRllw-illsgonp1/2d0(p)

 

 

R
= 2 803 III — ilng(0,R)/ 101/2d72(;0)-
0

R
3 2s“ / p”? ss(p) - MR) - Hs — sum.
0

We are now ready to state our main convergence theorem.

Theorem 2.2 Assume the autoconvolution problem (2.4) has a positive solution :7: E
C1[O, 1 + R] satisfying
:E(O) > 9 b2 e20 - ”sum (2.34)

for o = max {00, %} and l) = max {C(O, 1,11), C(2,1,77)}. Assume further that the
Borel measure 71(p) > 0 satisfies (2.8). Then there exist constants k1 > 0, kg 5:9 0,
and C’ > 0, all independent of R, such that if

Wt) — f(t)! s. 6 s kle, t e [0,1+ R], (2.35)

37

then the regularized equation (2.27) has a unique solution 23% satisfying

1

s _ 2 6 — 2 ‘2 2
films. _ xlngsz) + ”51712 _ $HLg(o,1) S C R -

Proof: We will apply the contraction mapping principle to the regularized equation
(2.27) in the ball Bet, C'R).
Let C(R) 2 k3 R, we have

 

 

 

R+O(R2) _ R+O(R2) _ leg-+1.
\/1+———C—(-R—)——\/1+'—F22—R'—— kg \/1+O(R).

k§+1
s3

 

Let k3 = , then k3 >1 and

 

R+O(R‘2)_ _
VHF—m— _ k3(1+0(R)) _k3 +O(R).

38

Combining the results of the previous lemmas in this chapter, we have

IIHRII: - film

1 _
S —.-l|fR- fallen +—- Ilénllan +-— ”Rat”? -x)|lo,a

01(113) “(1573 l “(153)
1 (Ex—_ |OR(xx)'-alt(i)|$_i.
+—a(:1':)”ER( )llan+ (2) II ”0,12
6 -1
< _—(—0)C(0,1,n)RC (k3+0(R))
+23|( zfili 2f_lls'll

0(0 1 n)R‘1||x—:i||3fl

 

——°—° C(3, 1, me?) a, + one»

 

530()

2 llx’lloo em 3

_ ”2 113—113 R
\/§:i:(0) C( 1,77)R (k3+O(R))H “a,

2,
2e“it C(R) 1

C— 1, 12-1/2 -'. 2
(f(O) (27 77) ”I ‘Tllo,R

 

1
for o 2 max{oo, 2}

Since ”17 — fllafl g C'R and assumptions (2.35), we have

”HR-T — j:“ofl

k1(k3il0()9(R)) C(O, 1, n) R + 3—\/§||:1‘3’lloo C(2, 1, h)(ks + 0(3)”?

\/3

_2_____'____\/§”T’Hoo 2 620
+————— 357(0) C(3,1n) (k3+O(R))R “52(0)

2||:7:’||ooe"R 3 , § 2e0Rk20(;,1,n)
——————-—C—,1,7 A:;+(’)R 0122+ _

 

 

C(O, 1, n) (1‘2 R

C12 R2

 

Therefore, for sufficiently small R, to have ”HRIE — illafl 3 OR for some C’ > 0, a

sufficient condition is

1 2\/2 62"
——bk k. + it' 00 ' _

          

< C. (2.36)

 

39

Let us pick k1 such that

_ 2J2 _
k1 — (2—75) llxlloosw),

equation (2.36) then becomes

e2ob A

2___ A -I
s(0)C C+2bk3|lx|loo<0. (2.37)

 

L(é) E

It is not hard to see L(C') = O has two distinct positive solutions by assumption
(2.34) when 19:, E (1,9/8]. Let us denote these two solutions by C; and (:‘2 such
that 0 < C”, < Cg. Then for C’ satisfying C} < C' < 6'2, we have L(C‘) < 0, thus
“ng — illmg 3 CR for R sufficiently small.

To further demonstrate that H R is a contraction on B(E,C'R), we let 21,272 6

3(2, CR), then

”HR 1'1 — HR I2llo,R
= ||(ch(:i7)I + B;g(.i:))‘l{RR(:it,:172 — :ir) — RR(:T:,271— Li?) + ER(.1‘:,2:1 — 2:2)

“ [WM-Tl) “ 03(2))(1, “ é?) "' (03(132) — 03(2))(22 — illHlmR

 

_ _ _ _ 1 _
‘ IIRR(113,172 — 37) — Ri«‘:(17,5171— $)||a,n + ——_—HER(17,I131— $2)l|a,R
1 ant”)

 

+ , ,_ ||(0'R(131)— 0R(f))(171— 17) — (0120132) — 03(f))(12 — illlafi-
03(13)

40

Since

__1_

012(5)

: 1_ ”(03(111) — OR($2))(I1— j) + (03(32) _ 03(j))($1— x2)”0sR
(13(3))

”(012031) — C112(53))(5131 — 5'3) - (012(32) ~ 012073))(332 — 2:)||,,,R

 

laa($1)- 03(32ll 103(352) - 0M5?“

 

 

 

 

 

 

 

 

_ 0(53) ”$1 - 15”an + at?) “$1 — $2”O‘,R
26"” Rpidv p)- CR)
3 {0 3‘ ( -{|lx1—xslla,allx1—illa,s+llxs-illa,nllw1—sv2|la,n}
3(0)fo pdn(p)
4eankgC(%,l,n)C'Rll II
_ j(O) 1:1 1:2 (LR)
then
2e2aC 0,1, .
“Haiti — Hiflzllafi S [ 55((0) 7?) C] “171— $2Ha,R
2||:1‘:’||ooe"R 3 1 4eaRk2C(.l,1,n)CR
+————-C—,1, k1+OR R2+ _2 x—xa.
I: fiiw) (2 77)( 3 ( )) 15(0) H 1 2” ,1?
Thus,
. 33(0) . 52(0)
2.
C<2€20b2>0<282a (0,1777), ( 38)

and for R > O sufficiently small this leads to

“HR-1‘1 - HR $2Ha.a _<_ (1' ”$1 — $2Ha,R

for any 131,232 E B(i:,C'R) with some q < 1.

 

 

C‘ (3‘ - 0
Notice further that —1—%——3 _-= 2:1:(2 )b’ therefore our regularized equation (2.27)
e a
. ~ - - ’ 0
has a unique solution in B(.i:, CR) for C that satisfies C1 < C < 2:1:(2 )b' C]
e 0

Remark 2.2.1 The usual convergence result is stated using the regularization para-

meter as a function of the noise level 6 in the the data. In this case, the results of

41

Theorem 2.2 indicate that we need R = R((S) so that

R((S) —) 0 (17161 < k1

_£‘_
32(5) _
as 6 —> 0.

Remark 2.2.2 The local convergence rate obtained by Theorem 2.2 is 0 (61/2), i. e.,

“1732— in”; ~ (9 (61/2) as 6 —> 0.

2.3 Alternate Methods on [0, R]

As we will show in the next chapter, if we use a piecewise function to approximate the
solution 2:, and recover :1: using the standard collocation on the interval [0,1] of the
regularization equation (2.6), we have a sequential numerical method for recovering
:r(t) for R < t S 1. Unfortunately, for the :1: values on the interval 0 S t g R, we
have to solve a nonlinear system of equations which can be numerically expensive.
This disadvantage motivates us to look for cheaper alternatives to recover a: on the
interval 0 S t S R. In this section, we will propose some alternative methods and
give theoretical basis for them.

In what follows we will show that for any function 2:); sufficiently close to i on

0, R , we may find a unique 25 E L2(0, 1) for which
R

(t) = 13W), t6 [0,52],

:61

A
11'
I

and such that if; satisfies equation (2.6) on the restricted interval (R, 1]. That is,

2‘}; E L2(0, 1) satisfies

aR(:r);r(t) + FR(;r)(t) = ffz(t), t E (R, 1]. (2.39)

42

Further, we will see that under suitable conditions on the true solution :1: and the
choice of R =-- R((S), the function 53‘}; is a good approximation of 5: for 6 small. The
advantage to this new approach is that we are free to find easier ways of determining
an approximation :13); to 51’: on [0, R] than that obtained by solving equation (2.6) on

the interval [0, R].

Theorem 2.3 Assume the autoconvolution problem (2.4) has a positive solution :7: E
Cl[0, 1 + R] satisfying
2(0) > 9 b2 e20 - ”sum (2.40)

for o = max {00, 2} and b = max {C(O,1,n), C(2, 1,17)}. Assume further that the
Borel measure 77(p) > 0 satisfies (2 8), and that {$R}Re(0,fz] is any family of Loo(0, 1)
functions satisfying

sup (23(1) — f(tll S CRP, (2.41)
te[0,R]

for some 6' > O andp > 1. Then there exist constants In > 0, k2 > 0, and C > 0, all

independent of R, such that if
lf‘5(t) — f(t)l s 6 .<_ Islet t 6 [01+ R], (2.42)

there exists an unique if}, E L2(0, 1) with 2‘}, = xR(t), t E (0, R], such that 2;, satisfies

equation (2.6) fort E (R, 1] and for which

1 as — 2
fill-TR — $|ng(0,R) +

 

.5 _ N
11712 — $l|2Lg(0,1) 3 CZRZ-

43

53(0)
2 e20 b

 

Proof: To begin, let C‘ < and define a new ball around :7: by

A

83(2) 5 {x e Lgfim, 1) 3$l102Rl = .2.) [0,R], le — in... 5 CR}. (2.43)

In what follows we will use the fact that 83(23) Q B(:E,C'R), where B(§:,CR) was
fundamental to the proof of Theorem 2.2. We claim that the ball BARCE) is not empty.
Indeed, if 5:3 E Lg’R(0, 1) is defined as

xR(t) ift E [0, R]

A

(ER:

f(t) ift e (R, 1],
then, for C(R) = 12312,

1
.. - 2 _ ‘2 2 A " 2
“IR _ xllofl — mllxs _ llngsz) + “55R — 113|ng(0,1)

1 ~ R ~ R
g _02122 / .22....222 / .22..
CW) 0 0
1

< ~2 2? _
(Hm).
62R2p

_ —k§

(1 + 12.312) (2.44)

3 C2122

for R sufficiently small and p > 1, i.e., 2:}; E BR(:7‘).

Let us further define an new operator RR such that

_ x(t) ift E [0, R]
HR(1E)(t) =-
HRx(t) ift e (R, 1].

We will Show that the operator RR(x) has a fixed point in the ball 33(2) by contrac-

44

tion mapping theorem.

We let 231,222 E 133(2), then

Ill-112031) - Ililz(:r2)||§,n
1
5(7)

= o + f (fauna) — HR($2)(t))2 dt

llHR($1)— HR(372)Hi.g(0,R) + llHR($1) — HR($2)Hi.g(0,1)
1 ._ _
= f (HR($1)(t) — 2122222))th

R

S ”1112051) ‘ HR($2)lli.g(o,1)

g ||HR(x1) - HR(~772)“3,R'

Therefore the same condition (2.38) as in Theorem 2.2 is needed for HR to be a

contraction in the ball 83(7)

We now consider [IRR(x) — xllafl for x E BR(:77). Note that

_ _ , 1 _ ~ , - _
“HM-'1’) — Illgn :- E—(—R_)HHR(1) _ fruigmfi) + ”HRU') — $l|ig(0,1)s

where

R

“1512(2) — 2:11.21... = / 2 (22(2) — 2(2))222
0
R
S CV2 RZp / -2ot dt
0
C2 R2,) 1 _ 6—20}?
20 ’

45

and

III—112(3) " illigmn]

R _ 1 _
= / 2?"? (H.(2)(t) — 2(2))? dt + l. (H.(s)(t) — 2(2))?22

R
= f 22?"? (22(2) — 2(2))? dt + / 2?“? (H.(.2)(t) - 2(2))? st
0 R R
2p 1 -— 6‘2”

S 02 R T + ”HR($) " illigma)

We therefore have

— _ _ ~ 1___ e—20’R 1
HH.(2) — 2113,. s “212(2) — xllisss) + 02 32” T. (1+ 6(2))

< ”112(2) — 2n? ,. + e“? R2” L153 (1+ _L
7 ”’ 22 6(2) ’

or,

 

_ ~ 1__ l—2aR 1
IIHR(1?)- 277110.12 SIIHR(17)— ills}; + CRp\/——2:— (1+ 5‘05)

If we let M = 3, then
k2

(———-(—~—)((——)

= ~11; + 0(R) S 11!
k2

 

 

for R sufficiently small. Thus

((HR(.1:) — 2“,. g ((11,.(2) — 2H... + CM RP. (2.45)

Notice that if p > 1, the corollary now follows from the proof of Theorem 2.2 under

the exact same conditions.

Cl

46

This theorem suggests that as long as we have a higher than C(72) approximation
for :7: on the interval [0, R] and still use equation (2.6) to recover a new solution on
the interval (R, 1], convergence is guaranteed under the same conditions of Theorem

2.2. For example, if we have access to 57(0) and i’(0), then the function

23(1) = 2(0) + 2’(0)t (2.46)

for t E [0, R] satisfies (2.41) for p = 2. Therefore, it is theoretically justified that we
can utilize this linear approximation in recovering the solution on the interval [0, R].

Unfortunately, we do not always have access to 52(0), or to 2(0) for that matter,
despite the fact that a full convergence theory for other prominent regularization
methods cannot be established unless the value of 17(0) is actually made an explicit
part of these methods. Indeed, convergence rates for Tikhonov regularization [6] and
the Levrent’ev regularization method [12] (another method preserving the causality
of the Volterra problem) cannot be obtained unless an auxiliary function x0 is used

in an essential way in these methods; here, x0 is assumed to satisfy

(1:0 2 :7: + G'(;7:)w

for suitable w. From the form of G’(:7:) we see that 170(0) 2 572(0), so that :7:(0) actually
must be used as part of these methods. Thus, if we too make the assumption that
17(0) is known, then by letting

2,.(2) : 2(0) (2.47)

for t E [0, R], we only have an 0(R) approximation of f(t) for t E [0, R]. We will
show in the following theorem that the convergence can still be obtained in the case

of p = 1 under a slightly tighter condition on the true solution .7: and a restriction on

47

C‘ in (2.41).

Theorem 2.4 Assume the autoconvolution problem (2.4) has a positive solution :7: E
C'1[0, 1 + R] satisfying

2(0) > 13 b? e20 - ”2'“,o (2.48)

for o = max{oo, %} and b = max {C(O, 1,11), C(2,1,77)}. Assume further that the
Borel measure r](p) > 0 satisfies (2.8), and that {xR}R€(0,R] is any family of Loo(0,1)

functions satisfying

4-

sup IxR(t) — 2":(t)| < CR”, (2.49)

tE[0,R] _
where p = 1 and some 6' > 0. Then there exist constants k1 > 0, k2 > 0, and C" > 0,

all independent of R, such that iffor V [,t > 0 fixed,

 

~ Ck.
C < . 2.50
" 1+ [1 ( )
and
MW) — f(t)l S 6 s kle, t e [0,1+ R]. (2.51)

then there exists an unique 2% E L2(0, 1) with it]. = xR(t), t E (0,R], such that 2:53

satisfies equation {2.6) fort E (R, 1] and for which

1 25 _ 25 _ . .
ig—Rllia — $]]2g(0,n) + H1711 — 37]]i.g(0,1) S CZRQ-
2

We are not going to repeat the proof since it follows the same procedure as the proof

of Corollary 2.3. The following changes need to be made to adapt the smaller p

48

To guarantee the ball 83(2) is nonempty, we need the extra condition (2.50).

This can be easily seen from inequality (2.44).

We cannot immediately use Theorem 2.2 after (2.45) is obtained. Instead, we
argue that for sufficiently small R, a sufficient condition for l]ER$“i]]o-,R 5 CR

is that

~ 1 2 2 2"
MC + :—bk1k3 + illi'llmbkg +_e
2(0)

12(0) J3 00 < C. (2.52)

 

As in the proof of Theorem 2.2, we may still pick k1 such that

_ N2 _, _
1., -— (2— 73—) Hz II..2(0).

and it is also convenient to add a second condition on C, namely

22.1121)... < 2222211211..

 

 

- < .
C _ M _ 2 (2 53)
Then inequality (2.52) is true if
,. ,. 820’) -2 .. 4
L(C) 2 2(0) 0 -— C + 3012. Ha: H... < 0. (2.54)

It is not hard to see £(C) = O has two distinct positive solutions by assumption
(2.48) when k3 E (1, 13/ 12]. Let us denote these two solutions by C1 and C2
such that. 0 < C1 < C2. Then for C satisfying C1 < C < C2, we have £(C) < 0,

thus IIHRx — :7:||0,R 3 CR for R sufficiently small.

Notice that we have imposed two conditions on C so far, and we need to Show

that they are compatible, i.e., if C satisfies (2.53), it also satisfies (2.50). This

49

implies that we need to show

51.. am)... < (:2.
2 _ 1+u’

 

01'

2 bk 7' 00

 

We note that the C’s allowed by the corollary satisfy

 

 

Cl < C < = (2.56)

where 0 < C1 < C2 are the two solutions of £(C) = 0. Combining (2.55) and

(2.56), to ensure the existence of C, all we need is that

bks ||~”T7'|los(1 + u) 13(0)
2 2e20b’

 

 

which is equivalent to

2(0) > (1+ 11)?)262” k3||:1:’||oo. (2.57)

We point out further that ,u could be taken small when R is sufficiently small,
therefore the condition (2.57) is ensured by the assumption (2.48) of the corol-

lary.

Remark 2.3.1 The results of Corollary 2.3 and Corollary 2.4 indicate that we need
R = R(6) so that
R((i) —> 0 and

 

as (5 —> 0 to obtain convergence.

Remark 2.3.2 The local convergence rates obtained by Corollary 2. 3 and Corollary

50

2.4 are 0 (61/2), the same as Theorem 2.2; i.e.,
||x‘]Z — :7:]|0,R ~ (9 (61/2) as 6 -—> 0.

As mentioned earlier, Theorem 2.3 and 2.4 provide us the freedom of finding easier
ways of determining the approximation x R to :7 on [0, R] than that obtained by solving
equation (2.6) on the interval [0, R]. We have seen so far two possible xR’s we can
use: a constant function on [0, R] as defined in (2.47) or a linear function on [0, R] as
defined in (2.46), both of which require the value :7:(0), which is not always available.
We also notice a waste of data when using those two xR’s since the collected data
values f6(t) for t E [0, R] are not utilized at all in recovering the solution.

After investigating the numerical solution of the autoconvolution equation (2.1)
(with f replaced by f6) after a standard collocation on the interval [0,1], we notice
that the calculated solution recovers :7: quite well for small it, even though it does
substantially worse as t increases. Note that no special regularization method is used
other than changing an infinite—dimensional problem into a finite dimensional problem
in the course of discretization. The increasingly worse solution on the interval [0, 1] is
not surprising since the error in the earlier part of the solution can propagate through
the rest of the interval and due to the ill-posedness of the problem. But it does not
prevent us from hoping that if we only solve the unregularized equation (2.1) on the
small interval [0, R], this solution x); will be close to :7: on the interval [0, R] for R
sufficiently small.

We first discretize the autoconvolution equation

C(x)(t) =/0 x(t — s).r(s) ds = f6(t), t E [0, R], (2.58)

with |f5(t) —f(t)| g 6 fort E [0, R]. Let K = K(R) Z 1 be an integer (we will specify

later), we partition the interval [0, R] into K equal-length subintervals; i.e., let

At = R/K,

tiziAt, i=0,1,---,K.

For i = 2,3, - ~ , K, let h(t) be the indicator function on the interval (t,_1,t,-]. Let
h(t) be the indicator function on the interval [t0,t1]. We further denote an ap-
proximation space of piecewise constant functions on [0, R] as 8K 2 span{ xi, i =
1, 2, - - - , K}.

A standard discretization of equation (2.58) involves finding 2:}; E SK, i.e., x}; of

the form

K
22(2) = :2. 2(2). (2.59)
(:1

where the constants x) E R, l = 1, 2, - 2- ,K, are determined by requiring a: to solve

the collocation equations

i.e.,
ti
/ x(t, — s):1:(s)ds 2' f6(t,-), i = 1,2, - - - ,K. (2.60)
0
Therefore,
1' t, K K
2/ [Z$1X1(ti—S)] [ZTPXP(S)] d8:f6(ti)3 Z: 112)'°° ,K.
7:1 t7—1 (=1 p21

Note that for s E (t,_1,t,], X)(t, — s)=1ifft,-—t, =t1_1,i.e.,l=i— 7 + 1; and for

52

s E (t,_1, t,], xp(s) = 1 iff p = 7. Thus the collocation equations become

i t,
Z/ xi—7+l$7d3:f6(ti)a i21121'°°7K1
7-1

i.e.,

i f6(t2) -
£13.27.“ (137 = T, Z =1,2,' " ,K. (2.61)

The collocation equations (2.61) allow us to explicitly solve for 25,-, i = 1,2, - -- ,K,

provided that f6(t1) > 0. Namely,

 

 

1:601)
11: = , 2.62
1 A, ( )
and if x1, - -- ,x,_1 have been found already, x,- is determined by
f 6 t2)
"jg?" — (mi—1172 + ° ° ° + 11721754)
2.731
therefore, .172, 2:3, ~ -- ,xK can be found uniquely and sequentially.

So far, we have shown that there is a unique solution 2:}; to the discrete autocon-
volution equation (2.60), and 273(t) = 21:117le for t E [0, R] with constants xi’s
specifically given by (2.62) and (2.63). We will show in the following claim that this

x); is a close approximation of 7: on the interval [0, R] for R sufficiently small.

Corollary 2.1 Assume the autoconvolution problem (2.1) has a positive solution it E
C1[0,1]. Let x}; = £331,300) be the unique solution of the discrete autoconvolution
equation (2.60) on lthle interval [0,R], where the constants x,, i = 1,2,--- ,K, are
specified in (2.62) and (2.63) and K = K(R) Z 1 is an integer. Then if there exists

a constant 117 > 0, such that K = K(R) g 117 uniformly in R, and

If“(t) - f(t)| = W)! s 6 s (2112?, 12(01), (264)

53

convergence of :rR(t) to the true solution f(t) fort E [0,R] occurs at the collocation

points as R —) 0, i.e.,
l$R(ti) —.’i‘(ti)l ~O(R), fori=1,2,~- ,K, (2.65)

for R sufficiently small. Further, we have a constant 6' depending on :E but indepen-
dent ofR such that
|:1:R(t) — a”:(t)| < CR (2.66)

for R sufl‘iciently small and all t E [0, R]. Thus ifs": is such that 6' satisfies (2.50),
we obtain the conclusions of Theorem 2.4 if we use the family {lenemfi} where 113;;

is defined by (2.59), (2.62) and (2.63).

Proof: The true solution 5: solves the autoconvolution equation (2.1) for any t E [0, 1],

i‘ then solves more specifically the following collocation equations.

Of
:3]:( i(t-—s).i( (s)ds=f(t,-), i=1,2,-~,K.

Since If E C1[0,1], then for 0 < t,_1 << t < R, there exist C,- ( ) (,1) E [0, R]

such that

j301—3) = i(t,-,+1)+(t,~ — 3 t._,+s1) (Ci 1( l) = i(t,-_ 7+1) + GUM),

and

as) = at.) + (s —t.)r: m >> = u )+ 0W)

54

Therefore, we have

Eli‘s

i.e.,
At 2““

01'

where |g,-(At)| S hAt for i = 1,2, - --

The collocation equations (2.67) allow us to solve for (t(t ) for i =1, 2,

ti)_7+1)(jt(7)+O(At))d f(t), 2 21,2, ' ,K,
ti— 7+1) (t.7=)+O(At)) f(ti)$ 2 :112, ' 1K1
was) at.) = 1):—g) + amt), (2.67)

, K and To > 0 constant independent of K, R, 6.
,K. To

simplify the notation, we will denote :1“:(t,~) as it,- from now on. Then we have

and ifi1,--~

 

 

 

_ f (h)
= —— 2.

l... (.8)

“in--1 have been found already, :73,- is determined by

t: _ _ _ _

f: ) + g,(At)—(;r,-_1.T2 + - - - + arzzri_1)

T,“ Z x t 2’?) , (2.69)
- 1

therefore, i2, .73, - - -

,iK can be found using equation (2.69).

We will now prove by induction that

for R sufficiently small.

For i = 1, we have

r,—:1’7,-|~O(R), fori=1,2,-~-,K,
_ 6
371—33212" ’A—lt“ 91(At)

55

therefore

 

_ _ 61 1
$1 —1I1 _ (Kt ngl(At)) £131 +1731,

where 6,- : f6(t,~) — f(t,-) for i = 1,2, - -- ,N. Notice that

= 715—], (We) + 6.+ x/f(t1)+ Ammo) .

 

 

 

since |6,—| S 6 g k1R2 for i = 1,2,o-~ ,N, we have

Vf(t1) + 51: Vf(t1) (1 + 0(R))
\/f(t1) + Atg(At) = \/f(t1)(1+ C(12))-

 

Therefore, for At = R/ K and R sufficiently small, we have

__ “Ammo. 1
[2:1 SEIIS m 2 f(t1)(1+0(R))

S (lel/2 123/? _+_ RK~3/2 R3/2) .

 

 

1
f(t1)(1+O(R))’

obviously, I s K = K (R) S M guarantees ll‘1 — i1| to be at least 0(R). Let there
exist a constant q] such that 51:1 — 51?; = qu.

Assume that for 1 g j g i - 1 and R sufficiently small, [22,- —— 53] = Cj(R)R for
some constant Cj(R) > 0 uniformly bounded independent of R, then there exist a

constant q2 = q2(i, R) > 0 such that
Kiri—11% + ° ' ' + $2$i—1)—(ii-1J_32 +"'+4i'21i‘i—1)l S (12R,

where q2(i, R) depends 011 Cj’s forj = 1, . -- ,i — 1 and q2(i, R) uniformly bounded

56

‘11,. . ~

 

111....-.

 

 

independent of R. Let us denote

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

A,‘ E f6(ti) — (131-1132 + ' ' ' + $2.’L‘i_1),
- ti _ _ _ _
A,‘ E fét) + 91(At) — ($i_1.’l?2 + ' ’ ' + (13223-1),
then
- A. Ai IlAi ‘— INA:
1:. Sta-l — - _ = _
22:1 21:1 2231131
_ TIA: —' (531 + qu)Ai A1 — Ai QIRAi
— 211311-31 25131 21171521
Note that
~ (5.- _ _ _ _
IA,- — A,| = E—t— — g,(At) — [(x,_1:132 + - - ~ + 33213-4) — (417,-_1.I:2 + - - - + x2517,_1)]|
1‘;le — R
< — k— rR
- R/K + K + ‘12

k
_<.(k1K + K + (1'2)R,

and A, 2 fr, - 2&1 is a positive constant, then for 1 s K = K(R) S M uniformly in
R, we have

lift — it] ~ 0(3)

for R sufficiently small. Therefore, (2.65) is true.

It follows immediately that for t E [0, R],

~ 0(a)

 

k
23mm) — f(t)
(:1

 

for R sufficiently small, since for t,_1 g t 3 t1, f(t) = f(t,) + 0(R) for R sufficiently

small. [:1

We have shown in Corollary 2.1 that the unique solution :cR(t) 2 ix, X1(t) to the
discrete autoconvolution equation (2.60) for t E [0, R] is an 0(R) anroximation of
f(t) on the interval [0, R]. By Theorem 2.4, this could provide us another alternative
for constructing a convergent regularized solution 52%. Namely, we solve the discrete
autoconvolution equation (2.60) for t E [0, R], and then the regularized equation (2.6)
for t E (R, 1]. This approach allows us to fully utilize the measured data ( f 5(t) for all
t E [0, 1]) and the measured data only (no a": value required) in recovering the solution.
Note that the size of 6‘ depends on the true solution 5:, so that the verification of

the condition (2.50) depends on the problem. However, as we will show in the next

chapter, the numerical results suggest the success of this approach.

58

CHAPTER 3

Discretization and Numerical

Results

To solve the autoconvolution equation (2.1) on the interval [0, 1] numerically without
any special regularization, the same kind of discretization approach as described in
Section 2.3 can be utilized on the interval [0,1]. Due to the causal nature of the
problem, the discretized version of equation (2.1) can be sequentially, thus efficiently
solved. However, the inherent instability of the problem remains prominent in the
solution. In this chapter, we will consider a discretized version of the regularized
equation (2.6), which leads to a stable method 011 the interval [0, 1] that is sequential
for the majority of the interval. We will then summarize the approaches motivated
by Theorem 2.3 and 2.4, all of which are sequential on the interval [0,1]. We also

include the numerical results for all discussed methods.

 

3.1 Discretization of the Regularized Equation

Recall the regularized equation (2.6) of finding a: E L2(0, 1) satisfying

Z/OR/opd s)(pdsdv )N()+/OR/pt$(t+p-S) w(8(p))d8dv /f"t+p C1770))

(3.1)
for t E [0,1]. In this section, we will discretize this regularized equation through

standard collocation on the interval [0, 1]. For simplicity, we will assume that

R R
[0 9(p)dn(p)= [0 9(p)dp- (3-2)

Note that the simple Borel measure 77(p) used in (3.2) satisfies the general condition
as required in (2.8).
Let N = 1, 2, 3, - - -, We partition the interval [0, 1] into N equal-length subinter-

vals; i.e., let

At : l/jv,

i=iAt, i=0,1,---,N.

For i = 2,3, - -- ,N, let x,(t) be the indicator function on the interval (t,__1,t,-]. Let
X1(t) be the indicator function on the interval [t0,t1]. We further denote an ap-
proximation space of piecewise constant functions on [0, 1] as SN 2 span{ Xi, i =
1,2, - -- ,N}.

A standard discretization of equation (3.1) involves finding :17 E SN, i.e., :1: of the

form
N
t) = Zone). (3.3)
12:1
where the constants cp E R, p = 1, 2, - n ,N, are determined by requiring r to solve

60

 

the collocation equations

2/R [paw )(dsdpxt- )+/0R [1: LI:(.-):rt+p—s )r(s)dsdp=/0Rf6(t,-+p)dp (3.4)

for i = 1,2,--- ,N. We will further assume R = rAt, where r E {1,2,--- ,N} is

fixed, and we will use the apporximation

R 1‘
[0 90)) dp m At - 2901—1)-

i=1

We will now study equation (3.4) term by term for fixed i = 1, 2, - .. ,N. First, we

consider the right hand side of equation (3.4), and we have
R r-l
/ f6(tz‘ + p) (1.0 = At [f"(t.-) + f6(ti+l) + ' ° ' + f6(t,-+,._1)] 2 At ' Zf6(ti+ql°
o q:0

The first term on the left hand side of equation (3.4) becomes

2//p.1~(. s)dsdp-a:(t )
=2At(/OO J:s()ds+[)tl.r(s)ds+---+/0tr_lat(s)ds)-c,~

=2At[0+c1At+(c1+c2)At+~-+(c1+c2+---+c,_1)At]-c,~

r—1 1
w- (2; 2c...)

1:1 m:l

where we notice that the coefficient of c,- involves only c1,c2, - -- ,cr_1. The second
term of the left hand side is more complicated, and, depending on the value of i
in relation to r, we come to different forms for the term. In general, for any fixed

i=1,2,«-- ,N, we have

R ti r—l if
f / a:(t,- + p — s) 17(3) ds dp = AtZ/ :r(t,~+j — s).r(s) ds. (3.5)
0 P 3'20 ‘2'

61

 

Further investigation of equation (3.5) leads us to the following cases.

0 Ifi<r-—1,then

AIR/pti:r(ti+p—s)x(s)dsdp

= (At)2{[(c1c,- + - ° - + cic1)+(c2c.-+-~+ cicg) + - - - + cici]

— [6.4.1014] + (6.410142 + ci+2Ci+1)+"'+ (Ci+lcr—l + '°'+ Cr—lci+l)l}

 

i i r—l m
: (At) - Cl Ci+m—l "’ Cl Ci+1+m-—l a
171:1 lzm m=i+l l=i+l

which is nonlinear in c. and involves values beyond c,; more specifically, it

involves all values of c1, c2, - ~ ,c,._1.

oIfi=r—1,then

R z.
/ / :L‘(t,- + p — 8) 1(8) ds dp
0 p

Z (At)2](ClCi + ' ' ° + C1131) + ((3201 + ° ' ‘ + C162) + ° ’ ' + CiCl']

i i
: (At)? ° 2: E C! Ci+m—la

m=l (2771
which is also nonlinear in c,- and involves all values of c1, c2, - -- ,c,_1.

o Ifi 2 r, then

R t,-
/ / a:(t,~ + p - s) 27(3) ds dp
0 p

=(At)?-[0+(clc,+---+c,-c1)+(cgc,+---+c,c2)+---+(c,ci+°--+c,c,)]

2 (At)2 ' Z 2 Cl Ci+m—la (3.6)

m=l l=m

With this result, we notice that specifically for i :2 r, (3.6) is quadratic in c,

once c1, c2, - -- ,c,._1 are found. Even better, for i > r, (3.6) is linear in c,- once

62

 

c1,c2, - -- ,c, are found.

Therefore, to determine the values for c,- where i 2: 1, 2, ~ - - ,N, we solve the following

N equations.

0 For i = 1, 2, - -- ,r -— 2, we have the first r — 2 equations.

r l l i i r—l m
(2(A.)2.::..)..( «A»? [)3 2....-.— z z:
(:1 "1:1 m=l l=m m=i+1 l=i+1

r+l

= At ' Z f6(ti+q)-
=0

 

Note that all values of c1, c2, - - - ,c,_1 are involved in any of these r—2 equations.

0 For i = r — 1, we have the (r — 1)-th equation.

r—1 1 r+1
(2(At)2-chm)( -c.+ )2: Zoom .zAt ZN (...)

mzl l: m q=0

Note that all values of c1, c2, - - ' ,c,._1 are also involved in this (r—1)-th equation.
Since we now have r — 1 equations with r —- 1 unknowns, we are ready to solve
c1, c2, - - - ,cr_1. We would expect this procedure numerically expensive since we

are basically solving a system of nonlinear equations for c1, c2, . - ~ ,c,_1.

0 Once c1, c2, - - - , c,._1 are found, we can solve for c,., - - - ,cN sequentially by

r—1 1 7+1
(2 (At)2' Zcm)( 'Ci+ )2': :Clci+m—1:At'q—Zof6(f(i+q)

(:1 m=1 m=l I: m

for i = r, r + 1, - - - ,N. As noted before, we still have to solve a nonlinear equa-
tion for or since the r-th equation is quadratic in 0,. However, once c1, c2, - - - ,c,
have already been determined, the remaining N — r equations can be sequen-

tially solved quickly since the ith equation is linear in c,- for i > r.

63

We have described so far a collocation scheme on the local regularization equa-
tion (2.6) for finding a step function that satisfies the equation (2.6) at N discrete
points. This method (hereafter referred to as Method 1) leads to a nonlinear system
of equations in recovering :1:(t) for t E [0, R], even though it is sequential and linear
in recovering :1:(t) for t E (R, 1]. Several cheaper and effective alternative methods to
recover :1:(t) on the interval [0,R] are presented below; their convergence was guaran-
teed by the two theorems in Section 2.3. While using equation (2.6) to recover a:(t)
on the interval (R, 1], these methods obtain :1:(t) on the interval [0, R] much more

efficiently.
0 Method 2: let :1:(t) = 53(0) for t E [0, R].
0 Method 3: let :z:(t) = :i:(0) + i'(0)t for t E [0, R].

0 Method 4: let 33(t) be the unique solution to the discretized autoconvolution
equation (2.60) on the interval [0, R], where no special regularization is used.

For details, we refer to Section 2.3.

3.2 Numerical Results

The examples in this section provide evidence of the effectiveness of the local regular-
ization methods on the autoconovolution equation. The local regularization methods
are also superior to the existing methods in that they don’t necessarily require the
value of 53(0) and they maintain the causal nature of the problem at least for the
majority of the domain. In order for easy comparison with the results of existing
methods [9] [12] , we try to recover a continuous 12 and a discontinuous one. We
select our true solution if: ahead of time, then generate the data function f by in-
tegration, f(t) = fotflt — s)'j‘(s) (is, for t E [0,1]. The perturbed data f6 was then

produced by adding uniformly distributed noise from the interval [-—6f(t), 6f(t)] to

64

the discrete values of f (t) for t 2 t,, where i = 1, 2, - - - ,N. In each of the examples,
we show the recovered solution against the true solution it, first without any regu-
larization, and then with regularization using Methods 1, 2, 3 and 4. In all figures
presented below, the true solution 1‘: is plotted as a dashed line, while the solid line

expresses the approximate solution computed according to the specified method.

3.2.1 Example with continuous 55 function

The true solution in this example is a continuous function

f(t) = 1—3(t—1/2)2, 05 t g 1,

with the true data then given by

3 3 3 1
——t5——t4+t3+—t2+—t, ogtgl.

“0:10 2 4 16

For Figures 3.1—3.5, we use relative noise level 6 = 10—3, N = 100 and r = 4 for the
regularized problems. All sample solutions shown use exactly the same noisy data. In
Figure 3.1, we show the solution without any special regularization. As we can see,
the recovered solution contains strong oscillation, especially for t large in the domain
[0, 1]. Figures 3.2—3.5 show the regularized solution using Methods 1—4 as introduced
in the last section. All four methods are effective in producing stable solutions that
are very close to the true solution it. It is worth noting in Figure 3.3 that, even after
the poor start in approximating f: (as dictated by the method), the solution was able

to come back on track.

65

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 3.2. Solution obtained by Method 1, 6 2 10’3, N = 100, r = 4

66

 

 

 

 

 

 

 

Figure 3.3. Solution obtained by Method 2, 6 = 10—3, N = 100, r = 4

 

 

 

 

 

Figure 3.4. Solution obtained by Method 3, 6 = 10—3, N = 100, r = 4

67

 

 

 

 

 

 

Figure 3.5. Solution obtained by Method 4, 6 = 10—3, N = 100, r = 4

For Figures 3.6310, we use a bigger relative noise level 6 = 10‘2, N = 100. In
Figure 3.6, we show the solution without any special regularization. Figures 3.7—3.10
show the regularized solution using Methods 1—4. We use r = 4 for Method 1 and
r = 6 for Methods 2 4. We can observe the same effectiveness in the regularization

methods as in the case of noise level 6 = 10—3.

68

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 3.7. Solution obtained by Method 1, 6 = 10”, N = 100, r = 4

69

 

 

 

 

 

 

Figure 3.8. Solution obtained by Method 2, 6 = 10—2, N = 100, r = 6

 

 

 

 

 

Figure 3.9. Solution obtained by Method 3, 6 = 10-2, N = 100, r = 6

70

 

 

 

 

 

 

Figure 3.10. Solution obtained by Method 4, 6 = 10—2, N = 100, r = 6

3.2.2 Example with discontinuous a? function

In this example, we will use a step function as our true solution

0.5 iftE[0,0.5]
f(t) = 0.25 if tE (05.08]

0.75 if t e (0.8,1],

the true data f is then given by

0.25t if t 6 [0,05]
f(t) = 0.125 if t e (0.5, 0.8]

0.5t — 0275 if t e (0.8,1].

71

 

For all the figures shown in this example, we use relative noise level 6 = 10-3, N : 200
and r = 4 for the regularized problem. Again, all sample solutions shown use the
same noisy data. In Figure 3.11, we show the solution without any special regular-
ization. As we can see, the recovered solution contains increasingly strong oscillation
as t increases in the domain [0,1]. Figures 3.12-3.15 show the regularized solution
using Methods 1-4. All four methods effectively produce stable solutions, where the
location of jumps are determined quite precisely. We note in this example that all
four regularization methods produce nearly identical solutions. The reason is that our
four methods differ only in how they handle the interval [0, R], and our discontinuous
:E is constant on that interval, and all methods are reasonably good at reconstructing
the constant. These results are much better than that were presented in [9] [12].
It is worth noting that the true solution 5: here does not satisfy the assumptions of
Theorem 2.2. We point out that the assumptions in Theorem 2.2 are only sufficient
conditions for convergence, and convergence may also exist for the 27’s that don’t

satisfy those assumptions.

72

 

 

 

 

 

 

l—Jf

Figure 3.11. Solution without regularization, 6 = 10—3, N = 200

 

 

 

 

 

Figure 3.12. Solution obtained by Method 1, 6 2 10’3, N = 200, r = 4

73

 

 

 

 

 

Figure 3.13. Solution obtained by Method 2, (5 = 10—3, N = 200, r = 4

 

 

 

 

 

 

O)-
O
[\D
O
.b
O
O\
C)
CD
H

Figure 3.14. Solution obtained by Method 3, 6 = 10—3, N = 200, r = 4

74

 

 

 

 

 

Figure 3.15. Solution obtained by Method 4, 6 2 10'3, N = 200, r = 4

CHAPTER 4

Summary and Future Work

In this paper, we developed a local regularization theory for the autoconvolution
equation, a nonlinear Volterra problem. Several local regularization methods have
been established, all of which provide stable solutions for the autoconvolution prob-
lem. The methods also preserve the causal nature of the autoconvolution equation,
allowing for numerically fast sequential solution.

Using the underlying idea of local regularization, we formulated the local regu-
larized equation for the autoconvolution problem. We proved the convergence of the
solution produced by this regularized equation to the true solution of the autoconvo-
lution equation as the noise level in the data shrinks to zero. The convergence rate
we obtained is the same as that of the classic Tikhonov regularization and Lavrent’ev
regularization.

Simple collocation of the regularized equation leads to a sequential method over
the majority of the domain, i.e., on the interval (R, 1]; and a nonlinear system of
equations for t E [0, R]. This motivates us to look for other alternatives in seeking
the solution on the interval [0, R], while still maintaining the same regularized equa-
tion on the interval (R, 1]. We proved two theorems, which provided the convergence
of two alternative methods. However, both of those two methods require some knowl-

edge of the true solution (f(t) at t = 0. In practice, we do not always have access to

76

 

 

the true solution. A fourth method was then presented, where we simply solve the
unregularized discrete autoconvolution equation on the interval [0, R]. We demon-
strated also the convergence of this method which result from using this piecewise
constant function on [0, R].

Finally, we have shown numerical results which provide evidence that the local
regularization methods developed in this work are superior to the other existing reg-
ularization methods, especially in capturing sharp features in the solution. In fact,
the numerical results confirm the effectiveness of these local regularization methods,
even in cases not completely falling under the assumptions of the general theory we
developed here. It is our hope that, through further study, we will be able to weaken
the conditions imposed on the true solution 5: in Theorem 2.2 and Corollary 2.3 and
2.4.

One of the most commonly asked question with regard to the local regularization
methods is how one picks the regularization parameter r. It is currently an open
question for linear problems, and we are also seeking answers in the case of the
nonlinear autoconvolution equation.

The Discrepancy Principle is one of the most successful criteria in determining
the regularization parameter a in the Tikhonov regularization. To summarize the
Discrepancy Principle, we assume that the perturbed data f‘5 has an absolute noise
level (i, i.e., I] f 5 — f I] g 5. The Tikhonov theory states that for every choice of a > 0,

the Tikhonov problem (6) has a unique solution vi, and that the discrepancy

5d E ”A“: — f6”

is monotone in a. The discrepancy principle picks a such that

On
Q.

ll

fi

0))

with r 2 1 some constant. In practice, 7' is often picked as \/2.

In the following, we use the discrepancy principle in picking our best r for a given
relative noise level 6 and fixed discretization parameter N. Method 4 is used for all
numerical experiments that follow. We first investigate the example of the continuous
:E as presented in Section 3.2.1. Note that exactly the same noisy data is used for
results within the same table. In Table 4.1— 4.3, we present a comparison of the values
of discrepancy 6d and the absolute data noise 5 for different values of r at various
relative noise levels. Unlike the Tikhonov regularization, the discrepancy (id is not
exactly a monotone increasing function of the regularization parameter r. Therefore,
we predict a good r is such that the discrepancy 6,, first exceeds the absolute data
noise 5. The Discrepancy Principle then suggests r = 13 for 6 2 10*, r = 9
for 6 = 5 x 10—3, and r = 5 for 6 = 10‘3, as highlighted in the tables. It is a
satisfying observation that the suggested r decreases as the relative noise level in the
data decreases, since, naturally, less regularization is needed for less noisy data. To
further demonstrate the Discrepancy Principle does work in this case, we included
in Figure 4.1 the numerical result using the predicted r = 13 at relative noise level
6 = .01. We can see a significant improvement in recovering the true 5E than that in
Figure 3.10, where r = 6 was used. In fact, the relative root-mean-square (rms) error
of the reconstructed :1: is 0.0258251 with r = 13, which is just about half of the rms

error (0.0502236) using r = 6, when the same set of noisy data is used.

78

7.

A

6

5.1

 

8
9
10
11
12
013
14

Table 4.1. Relative noise level 6 = 10-2, N = 100, continuous i:

 

.0249610
.0236252
.0231554
.0226741
.0226101
.0219453
.0219144

A

 

.0188627
.0182476
.0194804
.0197377
.0211548
.0228909
.0182378

 

T (5 6d

7 .0130590 .0112185

8 .0124805 .0117506
0 9 .0118126 .0123196

10 .0115777 .0140569

Table 4.2. Relative noise level 6 = 5 x 10—3, N = 100, continuous 2‘:

 

A

6

 

5.:

 

 

CONC'DCfii-fis‘i

Table 4.3. Relative noise level 6 = 10—3, N = 100, continuous 57:

.00270788
.00269173
.00262241
.00261181
.00249610

79

 

.00265035
.00371632
.00508314
.00663777
.00821011

 

 

 

 

 

Figure 4.1. Solution obtained by Method 4, 6 = 10—2, N = 100, r = 13

We conduct a similar analysis in the case of a discontinuous :2, using the is pre-
sented in Section 3.2.2, and the effectiveness of the Discrepancy Principle is further
confirmed. As highlighted in the tables, we can see the best r’s predicted are r = 9
for 6 = 10—2, r = 6 for 6 = 5x10‘3 and r = 3 for 6 = 10‘3. In Figure 4.2, we show
the reconstructed solution using the predicted r = 3 at 6 2 10’3. While Figure 4.2
looks quite similar to Figure 3.15 where r = 4 was used, we can quantitively conclude
r = 3 is a better choice since the relative rms error is 0.072, which is slightly better
than the rms error (0.086) in the case of r = 4 when the same set of noisy data is

used.

80

A

6

6.:

 

Scoooxicacnpws

0.013285
0.0131227
0.0127907
0.0126117

0.012555
0.0125535
0.0122756

 

0.0120137

 

0.00754975
0.00831738
0.00890927
0.00966738
0.010586
0.0117132
0.0128592
0.0142196

Table 4.4. Relative noise level 6 = 10-2, N = 200, discontinuous :i:

A

6

5d

 

 

\lOBCfivlkOOfi

0.00664252
0.00656133
0.00639536
0.00630583
0.00627752

 

0.0040549
0.00482411
0.00568239
0.00675673
0.00801038

Table 4.5. Relative noise level 6 = 5 x 10‘3, N = 200, discontinuous i

A

(S

6.:

 

 

0.00133155
0.0013285
0.00131227

 

0.00113062
0.00202835
0.00308485

Table 4.6. Relative noise level 6 = 10-3, N = 200, discontinuous i‘

81

 

 

 

 

 

 

Figure 4.2. Solution obtained by Method 4, 6 = 10—3, N = 200, r = 3

We also conducted some numerical experiments to shed light on the question of
picking r for various values of N. In Figure 4.3, we have used Method 4 to reconstruct
the continuous function used in Section 3.2.1 with the relative noise level 6 = 10'3.
The relative rms error of the reconstructed a: is plotted as a function of N, for several
values of r. As expected, as N increases, the ideal choice of r increases as well. For a
given choice of r, generally, as N increases, the rms error decreases until an optimum
N, and then increases again. Each value of r thus has a “sweet spot,” where the rms
error is lower than for any other value of r. In this figure, for this function, we see
that r = 2 is optimal for 25 S N S 40, while r = 3 is optimal for 45 ,S N S 60, and

so on. We plan to study this question further.

82

xerr%

f error = 0.001

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

25 50 75 100 125 U 150 7 175 200

Figure 4.3. Relative error of numerical solution, for various values of N and r

Even though this study is the first time in our knowledge that local regulariza-
tion is extended to real nonlinear Volterra problems, it is applied to a very specific
nonlinear Volterra problem, the autoconvolution equation. We would like to extend

the nonlinear theory to a more general class of nonlinear Volterra problems.

83

BIBLIOGRAPHY

84

BIBLIOGRAPHY

[1] J. Baumeister, Deconvolution of appearance potential spectra, In: R.Kleinmann,

et al. (eds) Direct and inverse boundary value problems. Proc. Conf. Oberwolfach.
Lang, Frankfurt am Main (1991) 1-13.

[2] J .V. Beck, B. Blackwell, and Jr. C.R.St. Clair, Inverse Heat Conduction, Wiley-
Interscience, (1985).

[3] A. C. Cinzori, Continuous future polynomial regularization of 1-smoothing
Volterra problems, Inverse Problems 20 (2004) 1791-1806.

[4] A. C. Cinzori and P. K. Lamm, Future polynomial regularization of ill-posed
Volterra equations, SIAM J.Numer.Anal 37 (2000) 949-979.

[5] C. Cui, Local regularization methods for n-dimensional first-kind integral equa-
tions, Ph.D. Thesis, Michigan State University, 2005

[6] H. W. Engl, K. Kunisch, A. Neubauer Convergence rates for Tikhonov regular-
ization of nonlinear ill-posed problems, Inverse Problems 5 (1989) 524-540.

[7] G. Fleischer, B. Hofmann, On inversion rates for the autoconvolution equation,
Inverse Problems 12 (1996) 419-435.

[8] G. Fleischer, B. Hofmann, The local degree of ill-posedness and the autocon-
volution equation, Nonlinear Analysis, Theory, Methods 63 Application. Vol.30,
No.6, (1997) 3323-3332.

[9] G. Fleischer, R. Gorenfio, B. Hofmann, On the autoconvolution equation and
total variation constraints, Z. Angew. Math. Mech 79 (1999) 149-159.

[10] R. Gorenflo, B. Hofmann, On autoconvolution and regularization, Inverse Prob-
lems 10 (1994) 353-373.

[11] J. Janno, On a regularization method for the autoconvolution equation, Z.
Angew. Math. Mech 77 (1997) 393-394.

85

[12] J. Janno, Lavrent’ev regularization of ill-posed problems containing nonlinear
near-to—monotone operators with application to autoconvolution equation, In-
verse Problems 16 (2000) 333-348.

[13] P. K. Lamm, Approximation of ill-posed Volterra problems via predictor-
corrector regularization methods, SIAM J. Appl. Math. 195 (1995) 469-494.

[14] P. K. Lamm, Future-sequential regularization methods for ill-posed Volterra

equations: applications to the inverse heat conduction problem, J. Math. Anal.
Appl. 56 (1996) 524-541.

[15] P. K. Lamm, Regularized inversion of finitely smoothing Volterra operators:
predictor-corrector regularization methods, Inverse Problems 13 (1997) 375-402.

[16] P. K. Lamm, Variable-smoothing regularization methods for inverse problems,
Theory and Practice of Control and Systems ed A.Conte and A.M.Perdon (Sin-
gapore: World Scientific) (1999).

[17] P. K. Lamm, Variable-smoothing local regularization methods for first-kind in-
tegral equations, Inverse Problems 19 (2003) 195-216.

[18] P. K. Lamm, Full convergence of sequential local regularization methods for
Volterra inverse problems, Inverse Problems 21 (2005) 785-803.

[19] P. K. Lamm and Z. Dai, On local regularization methods for linear Volterra
problems and nonlinear equations of Hammerstein type, Preprint.

[20] P. K. Lamm and L. Eldén, Numerical solution of first-kind Volterra equations by
sequential T ikhonov regularization, SIAM J. Numer. Anal. 34 (1997)1432—1450.

[21] P. K. Lamm and T. Scofield, Sequential predictor-corrector methods for the
variable regularization of Volterra inverse problems, Inverse Problems 16 (2000)
373-399.

[22] P. K. Lamm and T. Scofield, Local regularization methods for the stabilization
for ill-posed Volterra problems, Numer. Funct. Anal. Opt 23 (2001) 913-940.

[23] R. Miller, Nonlinear Volterra integral equations, W.A.Benjamin, Inc (1971).
[24] H. Tanabe, Equations of evolution, LondonzPitman, (1979).

[25] W. Ring and J. Prix, Sequential predictor-corrector regularization methods and
their limitations, Inverse Problem 16 (2000) 619-634.

86

IIIIIIIIIIIIIIIIIIIIIIIIIIIII

llll]]l]]j]]]]l]]]l]]ll]l[]l]]][l