Inversion of the Laplace Transform from the Real Axis Using an Adaptive Iterative Method

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Volume 2009, Article ID 898195,38pages doi:10.1155/2009/898195

Research Article

Inversion of the Laplace Transform from

the Real Axis Using an Adaptive Iterative Method

Sapto W. Indratno and Alexander G. Ramm

Department of Mathematics, Kansas State University, Manhattan, KS 66506-2602, USA

Correspondence should be addressed to Alexander G. Ramm,[email protected] Received 24 July 2009; Accepted 19 October 2009

Recommended by Thomas Witelski

A new method for inverting the Laplace transform from the real axis is formulated. This method is based on a quadrature formula. We assume that the unknown functionftis continuous with

knowncompact support. An adaptive iterative method and an adaptive stopping rule, which yield the convergence of the approximate solution toft, are proposed in this paper.

Copyrightq2009 S. W. Indratno and A. G. Ramm. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

Consider the Laplace transform

Lfp:

_∞

0

e−ptftdtFp, Rep >0, 1.1

whereL:X0,b → L20,∞,

X0,b:

f∈L20,∞|suppf⊂0, b, b >0. 1.2

We assume in1.2thatfhas compact support. This is not a restriction practically. Indeed, if lim_{t→ ∞}ft 0, then|ft|< δfort > tδ, whereδ >0 is an arbitrary small number. Therefore, one may assume that suppf ⊂ 0, tδ, and treat the values off fort > tδas noise. One may also note that iff ∈L1₀_,_∞_{, then}

Fp:

_∞

0

fte−ptdt

_b

0

fte−pt_dt∞ b

fte−ptdt:F1

pF2

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and|F2p| ≤e−bpδ, where _∞

b |ft|dt≤δ. Therefore, the contribution of the “tail”fbtoff,

fbt:

⎧ ⎨ ⎩

0, t < b,

ft, t≥b,

1.4

can be considered as noise if b > 0 is large and δ > 0 is small. We assume in 1.2 that f ∈ L2₀_,_∞_{. One may also assume that} _f _∈ _L1₀_,_∞_{, or that}_|_ft_{| ≤} _c

1ec2t, where c1, c2

are positive constants. If the last assumption holds, then one may define the functiongt: fte−c21t_{. Then}gt∈_L1₀_,∞_{, and its Laplace transform}Gp Fp_c

21is known on

the intervalc21, c21bof real axis if the Laplace transformFpofftis known on the

interval0, b. Therefore, our inversion methods are applicable to these more general classes of functionsfas well.

The operator L : X0,b → L20,∞ is compact. Therefore, the inversion of the Laplace transform1.1is an ill-posed problemsee1,2. Since the problem is ill-posed, a regularization method is needed to obtain a stable inversion of the Laplace transform. There are many methods to solve 1.1 stably: variational regularization, quasisolutions, and iterative regularizationsee, e.g.,1–4. In this paper we propose an adaptive iterative method based on the Dynamical Systems MethodDSMdeveloped in2,4. Some methods have been developed earlier for the inversion of the Laplace transformsee5–8. In many papers the dataFpare assumed exact and given on the complex axis. In9it is shown that the results of the inversion of the Laplace transform from the complex axis are more accurate than these of the inversion of the Laplace transform from the real axis. The reason is the ill-posedness of the Laplace transform inversion from the real axis. A survey regarding the methods of the Laplace transform inversion has been given in6. There are several types of the Laplace inversion method compared in 6. The inversion formula for the Laplace transform that is well known

ft 1

2πi

σi∞

σ−i∞F

peptdp, σ >0, 1.5

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can be found in 20,21. Moreover, the Laplace transform inversion with perturbed data is not discussed in 15. In19 the authors develop an inversion formula, based on the eigenfunction expansion for the Laplace transform. The diﬃculties with this method arei the inversion formula is not applicable when the data are noisy,iieven for exact data the inversion formula is not suitable for numerical implementation.

The Laplace transform as an operator fromC0kintoL2, whereC0k{ft∈C0,∞| suppf ⊂ 0, k},k const > 0,L2 _: _L2₀_,_∞_,_{is considered in}₁₄_{. The finite di}_ﬀ_erence

method is used in14to discretize the problem, where the size of the linear algebraic system obtained by this method is fixed at each iteration, so the computation time increases if one uses large linear algebraic systems. The method of choosing the size of the linear algebraic system is not given in14. Moreover, the inversion of the Laplace transform when the data Fpis given only on a finite interval0, d,d >0, is not discussed in14.

The novel points in our paper are the following:

1the representation of the approximation solution2.69of the functionftwhich depends only on the kernel of the Laplace transform,

2the adaptive iterative scheme 2.72 and adaptive stopping rule 2.83, which generate the regularization parameter, the discrete dataFδp, and the number of terms in2.69, needed for obtaining an approximation of the unknown function ft.

We study the inversion problem using the pair of spacesX0,b, L20, d, whereX0,bis defined in1.2, develop an inversion method, which can be easily implemented numerically, and demonstrate in the numerical experiments that our method yields the results comparable in accuracy with the results, presented in the literature, for example, with the double precision results given in paper15.

The smoothness of the kernel allows one to use the compound Simpson’s rule in approximating the Laplace transform. Our approach yields a representation 2.69 of the approximate inversion of the Laplace transform. A number of terms in approximation2.69

and the regularization parameter are generated automatically by the proposed adaptive iterative method. Our iterative method is based on the iterative method proposed in22. The adaptive stopping rule we propose here is based on the discrepancy-type principle, established in23, 24. This stopping rule yields convergence of the approximation 2.69

toftwhen the noise levelδ → 0.

A detailed derivation of our inversion method is given inSection 2. InSection 3some results of the numerical experiments are reported. These results demonstrate the eﬃciency and stability of the proposed method.

2. Description of the Method

Letf∈X0,b. Then1.1can be written as

Lfp:

_b

0

e−ptftdtFp, 0≤p. 2.1

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Lmf

i:

_b

0

e−pitftdt_F_p

i

, i0,1,2, . . . , m, 2.2

pi :ih, i0,1,2, . . . , m, h: d

m, 2.3

andmis an even number which will be chosen later. Then the unknown functionftcan be obtained from a finite-dimensional operator equation2.2. Let

u, v _Wm :

m

j0

w_jmujvj, uWm :u, u _Wm 2.4

be the inner product and norm in Rm1_{, respectively, where} _wm

j are the weights of the compound Simpson’s rulesee10, page 58, that is,

w_jm:

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ h

3, j 0, m,

4h

3 , j 2l−1, l1,2, . . . , m

2,

2h

3 , j 2l, l1,2, . . . , m−2

2 ,

h d

m, 2.5

wheremis an even number. Then

Lmg, v_Wm

m

j0

w_jm

b

0

e−pjtgtdtvj

_b

0

gtm j0

w_jme−pjt_v

jdt

g,L∗_mv_X 0,b,

2.6 where L∗ mv m

j0

w_jme−pjt_v

j, v:

⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ v0 v1 .. . vm ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

∈Rm1_, _2.7

g, h_X 0,b:

_b

0

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It follows from2.2and2.7that

L∗ mLmg

t m

j0

w_jme−pjt

b

0

e−pjzgzdz_:_Tm_gt, _2.9

LmL∗mv

⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ _b 0

e−p0t m

j0

w_jme−pjt_v

jdt

b

0

e−p1t m

j0

w_jme−pjt_vjdt

.. .

_b

0

e−pmt

m

j0

w_jme−pjt_v

jdt ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

:Qm_v, _2.10

where

Qm ij:w

m

j

_b

0

e−pipjt_dt_wm

j

1−e−bpipj

pipj , i, j0,1,2, . . . , m. 2.11

From2.7we get RangeL∗_m span{w_jmkpj,·,0}m_j₀,where

Lemma 2.1. Letw_jmbe defined in2.5. Then

m

j0

w_jmd, 2.12

for any even numberm.

Proof. From definition2.5one gets

m

j0

w_jmw₀mwmm m/2

j1

w₂m_j−₁

m−2/2

j1

w₂m_j

2h 3

m/2

j1

4h 3

m−2/2

j1

2h 3 2h 3 2hm 3

hm−2

3 hm

d mmd.

2.13

Lemma 2.1is proved.

Lemma 2.2. The matrixQm_{, defined in}_2.11_{, is positive semidefinite and self-adjoint in}_Rm1_with

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Proof. Let

Hmij:

b

0

e−pipjt_dt 1−e

−bpipj

pipj ,

Dmij

⎧ ⎨ ⎩

w_im, ij,

0, otherwise,

2.14

w_jmare defined in2.5. ThenDmHmDmu, v Rm1 u, DmHmDmv Rm1,where

u, v _Rm1: m

j0

ujvj, u, v∈Rm1. 2.15

We have

Qmu, v Wm

m

j0

w_jmQmu jvj

m

j0

DmHmDmujvj

DmHmDmu, v Rm1 u, D_mH_mD_mv _Rm1

m

j0

ujDmHmDmvj m

j0

ujwjmHmDmvj

u, Qmv Wm.

2.16

Thus,Qm_{is self-adjoint with respect to inner product}_2.4_{. We have}

Hm_ij

_b

0

e−pipjt_dt

_b

0

e−pit_e−pjt_dt

φi, φj

X0,b, φit:e

−pit_,

2.17

where·,· _X₀_,bis defined in2.8. This shows thatHmis a Gram matrix. Therefore,

Hmu, u _Rm1 ≥0, ∀u∈Rm1. 2.18

This implies

Qmu, u Wm

Qmu, Dmu

Rm1HmDmu, Dmu Rm1 ≥0. 2.19

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Lemma 2.3. LetTm_{be defined in}_2.9_{. Then}_Tm_{is self-adjoint and positive semidefinite operator} inX0,bwith respect to inner product2.8.

Proof. From definition2.9and inner product2.8we get

Tmg, h X0,b

b

0

m

j0

w_jme−pjt

b

0

e−pjz_gzdz htdt

b

0

gzm j0

w_jme−pjz

b

0

e−pjthtdt dz

g, Tmh X0,b

.

2.20

Thus,Tm_{is a self-adjoint operator with respect to inner product}_2.8_{. Let us prove that}_Tm

is positive semidefinite. Using2.9,2.5,2.4, and2.8, one gets

Tmg, g X0,b

_b

0

m

j0

w_jme−pjt

_b

0

e−pjz_gzdz gtdt

m

j0

w_jm

_b

0

e−pjzgzdz

_b

0

e−pjtgtdt

m

j0

w_jm

b

0

e−pjz_gzdz

2

≥0.

2.21

Lemma 2.3is proved.

kp, t, z:e−ptz. 2.22

Let us approximate the unknownftas follows:

ft≈m j0

c_jmw_jme−pjt_T−1

a,mL∗mFm:fmt, 2.23

where pj are defined in2.3,Ta,m is defined in2.30, and c_jm are constants obtained by solving the linear algebraic system:

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whereQm_{is defined in}_2.10_,

cm_:

⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝

c₀m

c₁m .. .

cmm

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

, Fm_:

⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝

Fp0

Fp1

.. .

Fpm

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

. 2.25

To prove the convergence of the approximate solutionft, we use the following estimates, which are proved in4, so their proofs are omitted.

Estimates2.26and2.27are used in proving inequality2.88, while estimates2.28

and2.29are used in the proof of Lemmas2.9and2.10, respectively.

Lemma 2.4. Let Tm and Qm be defined in 2.9 and2.10, respectively. Then, fora > 0, the following estimates hold:

Q−_a,m1 Lm≤ 1

2√a, 2.26

aQ−_a,m1 ≤1, 2.27

T_a,m−1≤ 1

a, 2.28

T_a,m−1L∗_m≤ 1

2√a, 2.29

where

Qa,m:QmaI, Ta,m:TmaI, 2.30

Iis the identity operator andaconst>0.

Let us formulate an iterative method for obtaining the approximation solution offt with the exact dataFp. Consider the following iterative scheme:

unt qun−1t

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whereL∗is the adjoint of the operatorL, that is,

L∗_g_td

0

e−ptgpdp, 2.32

Tft:L∗Lft

b

0 d

0

kp, t, zdpfzdz

b

0

fz tz

1−e−dtzdz,

2.33

kp, t, zis defined in2.22,

Ta:aIT, a >0, 2.34

an:qan−1, a0>0, q∈0,1. 2.35

Lemma 2.5together withLemma 2.7withga aT−1

a fyields

Lemma 2.5. LetTabe defined in2.34,Lf F, andf ⊥ NL, whereNLis the null space of L. Then

aT_a−1f−→0 asa−→0. 2.36

Proof. Sincef ⊥ NL, it follows from the spectral theorem that

lim a→0a

2_T−1

a f

2

lim a→0

_∞

0

a2

as2d

Esf, f

P_N_Lf20, 2.37

where Es is the resolution of the identity corresponding to L∗L, and P is the orthogonal projector ontoNL.

Lemma 2.5is proved.

Theorem 2.6. LetLfF, andunbe defined in2.31. Then

lim

n→ ∞f−un0. 2.38

Proof. By induction we get

un n−1

j0

ω_jnT_a−_j1₁L∗F, 2.39

whereTais defined in2.34, and

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Using the identities

Lf F, T−1

a L∗LTa−1TaI−aI I−aTa−1, n−1

j0

ω_jn 1−qn,

2.41

we get

f−un f− n−1

j0

ω_jnf n−1

j0

ω_jnaj1Ta−j11f

qnf n−1

j0

ω_jnaj1Ta−j11f.

2.42

Therefore,

_f₋_un_≤_qn_fn−1 j0

ω_jnaj1Ta−j11f

. 2.43

To prove relation2.38the following lemma is needed.

Lemma 2.7. Letgxbe a continuous function on0,∞,c >0 andq∈0,1constants. If

lim

x→0gx g0:g0, 2.44

then

lim n→ ∞

n−1

j0

qn−j−1−qn−jgcqj1g0. 2.45

Proof. Let

Fln: l−1

j1

ω_jngcqj_, _2.46

whereω_jn:qn−j−qn1−j_{. Then}

_Fn₁_n₋_g₀_{≤ |}_Fln_|

n

jl

ω_jngcqj−g0

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Take >0 arbitrarily small. For suﬃciently large fixedlone can choosen> l, such that

Fln≤ ₂, ∀n > n, 2.48

because limn→ ∞qn 0.Fixl lsuch that|gcqj−g0| ≤/2 forj > l. This is possible

because of2.44. One has

Fln≤ ₂, n > n> l,

n

jl

ω_jngcqj−g0 ≤

n

jl

ω_jngcqj1−g0

n

jl

ω_jn−1

g0

≤ 2

n

jl

ω_jnqn−lg0

≤

2 g0q

n−l_≤_,

2.49

ifnis suﬃciently large. Here we have used the relation

n

jl

ω_jn1−qn1−l. _2.50

Since >0 is arbitrarily small, relation2.45follows.

Lemma 2.7is proved.

lim n→ ∞

n−1

j0

ω_jnaj1Ta−j11f

0. 2.51

This together with estimate2.43and conditionq∈0,1yields relation2.38.Theorem 2.6

is proved.

Lemma 2.9leads to an adaptive iterative scheme:

un,mnt qun−1,mn−1

1−qT_a−_n1_,m_nL∗_m_nFmn_, _u

0,m0t 0, 2.52

whereq∈0,1,anare defined in2.35,Ta,mis defined in2.30,AmLis defined in2.2, and Lemma 2.8. LetT andTm_{be defined in}_2.33_and_2.9_{, respectively. Then}

T−Tm_≤ 2bd5

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Proof. From definitions2.33and2.9we get

T−Tmft≤

_b 0 _d 0

kp, t, zdp− m

j0

w_jmkpj, t, z

fzdz

≤ b 0 d5

180m4_p∈max₀_,dtz

4_e−ptz

fzdz

_b

0

d5

180m4tz

4_f_z_dz_≤ d5

180m4

b

0

tz8_dz 1/2

f_X 0,b

d5

180m4

tb9₋_t9

9

1/2 _f

X0,b,

2.54

where the following upper bound for the error of the compound Simpson’s rule was used see10, page 58. Forf∈C4_x

0, x2l, x0< x2l,

_x₂_l

x0

fxdx−h 3

⎡ ⎣f04

l

j1

f2j−12

l−1

j1

f2jfx2l

⎤ ⎦

≤Rl, 2.55

where

fj:fxj, xjx0jh, j0,1,2, . . . ,2l, h x2l−x0

2l ,

Rl x2l−x0

5

1802l4

f4ξ, x0< ξ < x2l.

2.56

This implies

T−Tmf X0,b

≤ d5 540m4

2b10−2b10

10

1/2 f_X

0,b≤

2bd5

540√10m4fX0,b, 2.57

so estimate2.53is obtained.

Lemma 2.8is proved. Lemma 2.9. Let 0< a < a0,

mκa0 a

1/4

, κ >0. 2.58

Then

T−Tm≤ 2bd

5

540√10a0κ4

a, 2.59

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Proof. Inequality2.59follows from estimate2.53and formula2.58.

Fm:

⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝

Fp0

Fp1

.. .

Fpm

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

∈Rm1_, _2.60

pj are defined in2.3. In the iterative scheme2.52 we have used the finite-dimensional operatorTm _{approximating the operator}_T_{. Convergence of the iterative scheme}_2.52_to

the solutionfof the equationLfFis established in the following lemma. Lemma 2.10. LetLf Fandun,mn be defined in2.52. Ifmnare chosen by the rule

mn

#

κ

$

a0

an

%1/4&

, anqan−1, q∈0,1, κ, a0>0, _2.61

wherexis the smallest even number not less thanx, then

lim

n→ ∞f−un,mn0. 2.62

Proof. Consider the estimate

f−un,mn≤f−unun−un,mn:I1n I2n, 2.63

whereI1n:f−unandI2n:un−un,mn. ByTheorem 2.6, we getI1n → 0 asn → ∞.

Let us prove that limn→ ∞I2n 0.LetUn : un−un,mn.Then, from definitions2.31and

2.52, we get

UnqUn−1

1−qT_a−_n1L∗F−T_a−_n1_,m_nL∗_m_nFmn

, U00. 2.64

By induction we obtain

Un n−1

j0

ω_jnT_a−_j1₁L∗F−T_a−_j1₁_,m_j₁Lmj1 _∗

Fmj1

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whereωjare defined in2.40. Using the identitiesLfF,LmfFm_,

T_a−1T T_a−1TaI−aI I−aT_a−1, T_a,m−1TmT_a,m−1 TmaI−aII−aT_a,m−1 ,

T−1

a,m−Ta−1Ta,m−1

T−Tm_T−1

a ,

2.66

one gets

Un

n−1

j0

ω_jnaj1

T_a−_j1₁_,m_j₁−T_a−_j1₁f

n−1 j0

ω_jnaj1Ta−j11,mj1

T−Tmj1

T_a−_j1₁f.

2.67

This together with the rule2.61, estimate2.28, andLemma 2.8yields

Un ≤ n−1

j0

ω_jnaj1Ta−j11,mj1

T−Tmj1_T−1 aj1f

≤ 2bd5 540√10a0κ4

n−1

j0

ω_jnaj1Ta−j11f .

2.68

Applying Lemmas2.5and2.7withga aT−1

a f, we obtain limn→ ∞Un0.

Lemma 2.10is proved.

2.1. Noisy Data

When the dataFpare noisy, the approximate solution2.23is written as

f_mδt m

j0

w_jmc_jm,δe−pjt_T−1

a,mL∗mFδm, 2.69

where the coeﬃcientsc_jm,δare obtained by solving the following linear algebraic system:

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Qa,mis defined in2.30,

cm,δ:

⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝

c₀m,δ

c₁m,δ .. .

cmm,δ

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

, F_δm:

⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ Fδ p0 Fδ p1 .. . Fδ pm ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

, 2.71

w_jmare defined in2.5, andpjare defined in2.3.

To get the approximation solution of the functionftwith the noisy dataFδp, we consider the following iterative scheme:

uδ_n,m_n quδ_n−₁_,m

n−1

1−qT_a−_n1_,m_nL∗_m_nFmn

δ , u δ

0,m0 0, 2.72

whereTa,mis defined in2.30,an are defined in2.35,q∈ 0,1,F_δm is defined in2.71, andmnare chosen by the rule2.61. Let us assume that

FδpjFpjδj, 0<δj≤δ, j 0,1,2, . . . , m, 2.73

whereδj are random quantities generated from some statistical distributions, for example, the uniform distribution on the interval−δ, δ, andδ is the noise level of the dataFp. It follows from assumption2.73, definition2.5,Lemma 2.1, and the inner product2.4that

F_δm−Fm2 Wm

m

j0

w_jmδ2_j ≤δ2 m

j0

w_jmδ2d. 2.74

Lemma 2.11. Letun,mn andu

δ

n,mnbe defined in2.52and2.72, respectively. Then

un,mn −u

δ n,mn ≤ √ dδ 2√an

1−qn_, _q_∈₀_,₁_, _2.75

whereanare defined in2.35.

Proof. LetU_nδ:un,mn−u

δ

n,mn. Then, from definitions2.52and2.72,

Uδ_nqU_n−δ ₁1−qT_a−_n1_,m_nL∗_m_nFmn−_Fmn

δ

, Uδ₀ 0. 2.76

By induction we obtain

Uδ_n n−1

j0

ω_jnT_a−_j1₁_,m_j₁Lmj1 _∗

Fmj1−_Fmj1 δ

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whereω_jnare defined in2.40. Using estimates2.74and inequality2.29, one gets

U_nδ≤'d n−1

j0

ω_jn δ 2√aj1 ≤

√ dδ 2√an

m

j0

ω_jn √

dδ 2√an

1−qn, 2.78

whereωjare defined in2.40.

Theorem 2.12. Suppose that conditions ofLemma 2.10hold, andnδsatisfies the following conditions:

lim

δ→0nδ∞, δlim→0

δ √_a nδ

0. 2.79

Then

lim δ→0

f−uδ_n_δ_,m

nδ

0. 2.80

Proof. Consider the estimate

f−uδ_n_δ_,m_nδ≤f−unδ,mnδ

unδ,mnδ−u

δ nδ,mnδ

. 2.81

This together withLemma 2.11yields

f−uδ_n_δ_,m_nδ≤f−unδ,mnδ

√ dδ 2√anδ

1−qn. 2.82

Applying relations2.79in estimate2.82, one gets relation2.80.

Theorem 2.12is proved.

In the following subsection we propose a stopping rule which implies relations2.79.

2.2. Stopping Rule

In this subsection a stopping rule which yields relations2.79inTheorem 2.12is given. We propose the stopping rule

Gnδ,mnδ ≤Cδ

ε_{< Gn,m}

n, 1≤n < nδ, C >

'

d, ε∈0,1, 2.83

where

Gn,mn qGn−1,mn−1

1−qLmnz

mn,δ−_Fmn

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· Wm is defined in2.4,

zm,δ_:m j0

c_jm,δw_jme−pjt_,

2.85

w_jm and pj are defined in2.5 and 2.3, respectively, and c_jm,δ are obtained by solving linear algebraic system2.70.

We observe that

Lmnz

mn,δ−_Fmn

δ Q

mn_cmn,δ−_Fmn

δ

Qmn

anIQmn

−1

Fmn

δ −F

mn

δ

Qmn_a

nI−anI

anIQmn

₋1

Fmn

δ −F

mn

δ

−ananIQmn−1_Fmn

δ −ancmn,δ.

2.86

Thus, the sequence2.84can be written in the following form:

Gn,mn qGn−1,mn−1

1−qancmn,δ

Wmn, G0,m0 0, 2.87

where · _Wm is defined in2.4, andcm,δsolves the linear algebraic system2.70.

It follows from estimates2.74,2.26, and2.27that

ancmn,δ

Wmn an

anIQmn

₋1

Fmn

δ

Wmn

≤ananIQmn−1_Fmn

δ −Fmn

Wmn

ananIQmn

−1

Fmn

Wmn

≤Fmn

δ −Fmn_Wmn

an

anIQmn

−1

Lmnf

Wmn

≤δ'd√anf_X 0,b.

2.88

This together with2.87yields

Gn,mn ≤qGn−1,mn−1

1−qδ'd√anf_X₀_,b

(18)

or

Gn,mn−δ

'

d≤qGn−1,mn−1−δ '

d1−q√anf_X

0,b. 2.90

Lemma 2.13. The sequence2.87satisfies the following estimate:

Gn,mn−δ

'

d≤

1−q√anf_X 0,b

1− √q , 2.91

whereanare defined in2.35.

Proof. Define

Ψn:Gn,mn−δ

√ d ,

ψn:

1−q√anf_X₀_,b.

2.92

Then estimate2.90can be rewritten as

Ψn≤qΨn−1 '

qψ_n−1, 2.93

where the relationan qan−1was used. Let us prove estimate2.91by induction. Forn0

we get

Ψ0−δ '

d≤

1−q√a0f_X₀_,b

1− √q . 2.94

Suppose that estimate2.91is true for 0≤n≤k. Then

Ψk1≤qΨk '

qψk≤ q 1− √qψk

'

qψk

√_q 1− √qψk

√_q 1− √q

ψk ψk1

ψk1

√_q 1− √q

√ ak √

ak1

ψk1

1

1− √qψk1,

2.95

where the relationak1qakwas used. Lemma 2.13is proved.

Lemma 2.14. Suppose

G1,m1> δ '

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whereGn,mn are defined in2.87. Then there exists a unique integernδ, satisfying the stopping rule

2.83withC >√d.

Proof. FromLemma 2.13we get the estimate

Gn,mn ≤δ

'

d

1−q√anf_X₀_,b

1− √q , 2.97

whereanare defined in2.35. Therefore,

lim sup

n→ ∞ Gn,mn ≤δ

'

d, _2.98

where the relation limn→ ∞an 0 was used. This together with condition 2.96yields the existence of the integernδ. The uniqueness of the integernδfollows from its definition.

Lemma 2.15. Suppose that conditions ofLemma 2.14hold andnδis chosen by the rule2.83, then

lim δ→0

δ √

anδ

0. 2.99

Proof. From the stopping rule2.83and estimate2.97we get

Cδε≤Gnδ−1,mnδ−1 ≤δ '

d

1−q√anδ−1fX0,b

1− √q , 2.100

whereC >√d, ε∈0,1. This implies

δCδε−1−√d √_an

δ−1

≤

1−qf_X 0,b

1− √q , 2.101

so, forε∈0,1, andanδ qanδ−1, one gets

lim δ→0

δ √_a nδ

lim δ→0

δ √_q√_a

nδ−1

≤lim δ→0

1−qδ1−ε_f X0,b

_√

q−qC−δ1−ε√_d 0. 2.102

Lemma 2.16. Consider the stopping rule2.83, where the parametersmnare chosen by rule2.61. Ifnδis chosen by the rule2.83then

lim

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Proof. From the stopping rule2.83with the sequenceGndefined in2.87one gets

qCδε₁₋_q_a nδ

cmnδ,δ

Wmnδ ≤qGnδ−1,mnδ−1

1−qanδ

cmnδ,δ

Wmnδ Gnδ,mnδ < Cδ

ε_,

2.104

wherecm,δ_{is obtained by solving linear algebraic system}_2.70_{. This implies}

0< anδ

cmnδ,δ

Wmnδ ≤Cδ

ε_. _2.105

Thus,

lim δ→0anδ

cmnδ,δ

Wmnδ 0. 2.106

IfFm_/_{0, then there exists a}_λm

0 >0 such that

Em λ₀mF

m_/₀_, _Em λ0 F

m_{, F}m

Wm :ξ

m_>₀_, _2.107

whereEsmis the resolution of the identity corresponding to the operatorQm:LmL∗m. Let

hmδ, α:α2Q_m,α−1 F_δm2

Wm, Qm,a:aIQ

m_. _2.108

For a fixed numbera >0 we obtain

hmδ, a a2Q_m,a−1 F_δm2 Wm

_∞

0

a2

as2d

EsmFδm, F

m

δ

Wm

≥

_λm

0

a2

as2d

EsmF_δm, F_δm

Wm

≥ a2 aλ02

_λm

0

dEsmF_δm, F_δm

Wm

a2_Em

λ₀mF

m

δ

2

Wm

aλ₀m2 .

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SinceE_λm

0 is a continuous operator, andF

m₋_Fm

δ Wm <

√

dδ, it follows from2.107that

lim δ→0

E_λm

0 F

m

δ , F

m

δ

Wm

E_λm 0 F

m_{, F}m

Wm >0. 2.110

Therefore, for the fixed numbera >0 we get

hmδ, a≥c2>0 2.111

for all suﬃciently smallδ >0, wherec2is a constant which does not depend onδ. Suppose

limδ→0anδ/0.Then there exists a subsequenceδj → 0 asj → ∞, such that

an_δj ≥c1>0,

0< mn_δj

⎡ ⎢ ⎢ ⎢ ⎡ ⎣κ a0 an_δj

1/4⎤ ⎦ ⎤ ⎥ ⎥ ⎥≤ # κ $ a0 c1

%1/4&

:c3<∞, κ, a0>0,

2.112

where rule 2.61was used to obtain the parameters mn_δj. This together with2.107and 2.111yields

lim j→ ∞hmnδj

δj, an_δj

≥ lim j→ ∞

a2 n_δj E m_nδj λ _mnδ j 0

F_δmnδj

j

2

Wmnδj

$

an_δj λ₀mnδj

%2

≥lim inf j→ ∞

c2₁

E m_nδj λ mnδ_j 0

Fmnδj

2

Wmnδj

$

c1λ m_nδj 0

%2 >0.

2.113

This contradicts relation2.106. Thus, limδ→0anδ limδ→0a0q

nδ ₀_,_{that is, lim}

δ→0nδ∞.

It follows from Lemmas2.15and2.16that the stopping rule2.83yields the relations 2.79. We have proved the following theorem.

Theorem 2.17. Suppose all the assumptions ofTheorem 2.12hold,mnare chosen by rule2.61,nδ is chosen by rule2.83, andG1,m1 > Cδ,whereGn,mnare defined in2.87, then

lim δ→0

f−uδ_n_δ_,m

nδ

0. 2.114

2.3. The Algorithm

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1The dataFδpon the interval0, d,d >0, the support of the functionft, and the noise levelδ.

2Initialization: choose the parametersκ >0,a0>0,q∈0,1,ε∈0,1,C >

√ d, and setuδ

0,m00,G00,n1.

3Iterate, starting withn1, and stop when condition2.120 see belowholds, aana0qn,

bchoosemnby the rule2.61, cconstruct the vectorFmn

δ :

Fmn

δ

lFδ

pl, pllh, h d

mn, l0,1, . . . , m, 2.115

dconstruct the matricesHmnandDmn:

Hmnij:

_b

0

e−pipjt_dt 1−e

−bpipj

pipj

, i, j1,2,3, . . . , mn,

Dmnij

⎧ ⎨ ⎩

wmn

i , ij,

0, otherwise,

2.116

wherew_jmare defined in2.5,

esolve the following linear algebraic system:

anIHmnDmnc

mn,δ_Fmn

δ , 2.117

wherecmn,δ

ic mn,δ

i , fupdate the coeﬃcientcmn,δ

j of the approximate solutionuδn,mntdefined in

2.69by the iterative formula:

uδ_n,m_nt quδ_n−₁_,m_n₋₁t 1−q mn

j1

cmn,δ_wmn

j e−pjt, 2.118

where

uδ

0,m0t 0. 2.119

Stop when for the first time the inequality

Gn,mn qGn−1,mn−1an cmn,δ

Wmn ≤Cδ

ε _2.120

holds, and get the approximationfδ_t_uδ

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3. Numerical Experiments

3.1. The Parameters

κ, a

0

, d

From definition2.35and the rule2.61we conclude thatmn → ∞asan → 0.Therefore, one needs to control the value of the parametermn so that it will not grow too fast asan decreases. The role of the parameterκin2.61is to control the value of the parametermn so that the value of the parametermnwill not be too large. Since for suﬃciently small noise levelδ, namely,δ ∈10−16_,₁₀−6_{, the regularization parameter}_an

δ, obtained by the stopping

rule2.83, is at mostO10−9, we suggest to chooseκin the interval0,1. For the noise level δ ∈ 10−6_,₁₀−2_{one can choose}_κ _∈₁_,₃_{. To reduce the number of iterations we suggest to}

choose the geometric sequencean a0δαn, wherea0 ∈0.1,0.2andα∈0.5,0.9.One may

assume without loss of generality thatb 1, because a scaling transformation reduces the integral over0, bto the integral over0,1. We have assumed that the dataFpare defined on the intervalJ : 0, d. In the case the intervalJ d1, d, 0 < d1 < d, the constantdin

estimates2.59,2.74,2.75,2.78,2.90,2.91, and2.97are replaced with the constant d−d1. Ifb1, that is,ft 0 fort >1, then one has to takednot too large. Indeed, ifft 0

fort >1, then an integration by parts yieldsFp f0−e−pf1/pO1/p2_{, p} _{→ ∞}_._If

the data are noisy, and the noise level isδ, then the data becomes indistinguishable from noise forpO1/δ. Therefore it is useless to keep the dataFδpford > O1/δ. In practice one may get a satisfactory accuracy of inversion by the method, proposed in this paper, when one uses the data withd∈1,20whenδ≤10−2_{. In all the numerical examples we have used}_d

5. Given the interval0, d, the proposed method generates automatically the discrete data Fδpj,j 0,1,2, . . . , m, over the interval0, dwhich are needed to get the approximation of the functionft.

3.2. Experiments

To test the proposed method we consider some examples proposed in 5–7, 9, 12,13, 15,

16,19,25. To illustrate the numerical stability of the proposed method with respect to the noise, we use the noisy data Fδpwith various noise levels δ 10−2_{, δ} ₁₀−4_{, and} _δ

10−6_{. The random quantities}_δ

j in2.73are obtained from the uniform probability density function over the interval−δ, δ. In Examples3.1–3.12we choose the value of the parameters as follows:an0.1qn_,_q_δ1/2_{, and}_d_{5. The parameter}_κ_{1 is used for the noise levels}_δ

10−2_and_δ₁₀−4_{. When}_δ₁₀−6_{we choose}_κ₀_._{3 so that the value of the parameters}_m

nare not very large, namely,mn≤300. Therefore, the computation time for solving linear algebraic system2.117can be reduced significantly. We assume that the support of the functionft is in the interval0, bwithb 10. In the stopping rule2.83the following parameters are used: C √d0.01,ε 0.99. InExample 3.13the functionft e−t is used to test the applicability of the proposed method to functions without compact support. The results are given inTable 13andFigure 13.

For a comparison with the exact solutions we use the mean absolute error

MAE :

⎡ ⎢ ⎣

*100

j1

fti−fδ m_nδti

2

100

⎤ ⎥ ⎦

1/2

, tj0.010.1

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−0.4

−0.2 0.2 0.4 0.6 0.8 1 1.2

0

0 2 4 6 8 10

t

Exact solution

δ10−2 _δ₁₀−6

δ10−4

Figure 1:Example 3.1: the stability of the approximate solution.

Table 1:Example 3.1.

δ MAE mnδ Iter. CPU timesecond anδ

1.00×10−2 ₉_.₆₂_×₁₀−2 ₃₀ ₃ ₃_.₁₃_×₁₀−2 ₂_.₀₀_×₁₀−3

1.00×10−4 ₅_.₉₉_×₁₀−2 ₃₂ ₄ ₆_.₂₅_×₁₀−2 ₂_.₀₀_×₁₀−7

1.00×10−6 ₄_.₇₄_×₁₀−2 ₅₄ ₅ ₃_.₂₈_×₁₀−1 ₂_.₀₀_×₁₀−10

where ftis the exact solution and fδ

m_nδtis the approximate solution. The computation timeCPU timefor obtaining the approximation offt, the number of iterationsIter., and the parametersmnδ andanδ generated by the proposed method is given in each experiment

see Tables1–12. All the calculations are done in double precision generated by MATLAB.

Example 3.1see15. We have

f1t ⎧ ⎪ ⎨ ⎪ ⎩

1, 1 2 ≤t≤

3 2,

0, otherwise,

F1

p

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

1, p0,

e−p/2₋_e−3p/2

p , p >0.

3.2

The reconstruction of the exact solution for diﬀerent values of the noise levelδ is shown in

Figure 1. When the noise levelδ 10−6_,_{our result is comparable with the double precision}

results shown in15. The proposed method is stable with respect to the noiseδas shown in

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−0.2 0 0.2 0.4 0.6 0.8 1 1.2

0 2 4 6 8 10

t

Exact solution

δ10−2 _δ₁₀−6

δ10−4

δ MAE mnδ Iter. CPU timeseconds anδ

1.00×10−2 ₁_.₀₉_×₁₀−1 ₃₀ ₂ ₃_.₁₃_×₁₀−2 ₂_.₀₀_×₁₀−3

1.00×10−4 ₈_.₄₇_×₁₀−2 ₃₂ ₃ ₆_.₂₅_×₁₀−2 ₂_.₀₀_×₁₀−6

1.00×10−6 ₇_.₄₁_×₁₀−2 ₅₄ ₅ ₄_.₃₈_×₁₀−1 ₂_.₀₀_×₁₀−12

Example 3.2see13,15. We have

f2t ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

1

2, t1,

1, 1< t <10,

0, elsewhere,

F2

p

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

9, p0,

e−p−e−10p

p , p >0.

3.3

The reconstruction of the functionf2tis plotted inFigure 2. In15a high accuracy result is

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−0.05 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

0 2 4 6 8 10

t

Exact solution

δ10−2 _δ₁₀−6

δ10−4

1.00×10−2 ₂_.₄₂_×₁₀−2 ₃₀ ₂ ₃_.₁₃_×₁₀−2 ₂_.₀₀_×₁₀−3

1.00×10−4 ₁_.₀₈_×₁₀−3 ₃₀ ₃ ₃_.₁₃_×₁₀−2 ₂_.₀₀_×₁₀−6

1.00×10−6 ₄_.₀₂_×₁₀−4 ₃₀ ₄ ₄_.₆₉_×₁₀−2 ₂_.₀₀_×₁₀−9

Example 3.3see6,12,13,16,19. We have

f3t ⎧ ⎪ ⎨ ⎪ ⎩

te−t, 0≤t <10, 0, otherwise,

F3

p 1−e −p110

p12 −

10e−p110 p1 .

3.4

We get an excellent agreement between the approximate solution and the exact solution when the noise levelδ10−4and 10−6as shown inFigure 3. The results obtained by the proposed method are better than the results given in13. The mean absolute error MAE decreases as the noise level decreases which shows the stability of the proposed method. Our results are more stable with respect to the noiseδ than the results presented in19. The value of the parametermnδis bounded by the constant 30 when the noise levelδ10−

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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0 2 4 6 8 10

t

Exact solution

δ10−2 _δ₁₀−6

δ10−4

1.00×10−2 ₁_.₅₉_×₁₀−2 ₃₀ ₂ ₃_.₁₃_×₁₀−2 ₂_.₀₀_×₁₀−3

1.00×10−4 ₈_.₂₆_×₁₀−4 ₃₀ ₃ ₉_.₄₀₀_×₁₀−2 ₂_.₀₀_×₁₀−6

1.00×10−6 ₁_.₂₄_×₁₀−4 ₃₀ ₄ ₁_.₂₅₀_×₁₀−1 ₂_.₀₀_×₁₀−9

f4t ⎧ ⎪ ⎨ ⎪ ⎩

1−e−0.5t_, ₀_≤_{t <}₁₀_,

0, elsewhere,

F4

p

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

82e−5_, _p₀_,

1−e−10p

p −

1−e−p1/210

p0.5 , p >0.

3.5

As in our Example 3.3 when the noise δ 10−4 _{and 10}−6 _{are used, we get a satisfactory}

agreement between the approximate solution and the exact solution. Table 4 gives the results of the stability of the proposed method with respect to the noise levelδ. Moreover, the reconstruction of the function f4t obtained by the proposed method is better than

the reconstruction of f4t shown in 13, and is comparable with the double precision

reconstruction obtained in15.

In this example whenδ10−6_and_κ₀_._{3 the value of the parameter}_mn

δ is bounded

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−0.2

−0.1 0 0.1 0.2 0.3 0.4 0.5 0.6

0 2 4 6 8 10

t

Exact solution

δ10−2 _δ₁₀−6

δ10−4

1.00×10−2 ₄_.₂₆_×₁₀−2 ₃₀ ₃ ₆_.₃₀₀_×₁₀−2 ₂_.₀₀_×₁₀−3

1.00×10−4 ₁_.₂₅_×₁₀−2 ₃₀ ₃ ₉_.₃₈_×₁₀−2 ₂_.₀₀_×₁₀−6

1.00×10−6 ₁_.₈₆_×₁₀−3 ₅₄ ₄ ₃_.₁₃_×₁₀−2 ₂_.₀₀_×₁₀−9

Example 3.5see5,7,13. We have

f5t √ 2

3e−t/2_sin_t√₃_/₂

F5

p

1−cos10√3/2e−10p0.5

+

p0.523/4, −

2p0.5e−10p0.5_sin₁₀√₃_/₂

√

3+p0.523/4, .

3.6

This is an example of the damped sine function. In 5,7 the knowledge of the exact data Fp in the complex plane is required to get the approximate solution. Here we only use the knowledge of the discrete perturbed dataFδpj,j 0,1,2, . . . , m,and get a satisfactory result which is comparable with the results given in 5,7when the level noise δ 10−6_.

The reconstruction of the exact solutionf5tobtained by our method is better than this of

the method given in 13. Moreover, our method yields stable solution with respect to the noise levelδas shown inFigure 5andTable 5. In this example whenκ0.3 the value of the parametermnδis bounded by 54 for the noise levelδ10−

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−0.2 0 0.2 0.4 0.6 0.8 1 1.2

0 2 4 6 8 10

t

Exact solution

δ10−2 _δ₁₀−6

δ10−4

1.00×10−2 ₄_.₁₉_×₁₀−2 ₃₀ ₂ ₄_.₇₀₀_×₁₀−2 ₂_.₀₀_×₁₀−3

1.00×10−4 ₁_.₆₄_×₁₀−2 ₃₂ ₃ ₉_.₃₈_×₁₀−2 ₂_.₀₀_×₁₀−6

1.00×10−6 ₁_.₂₂_×₁₀−2 ₅₄ ₄ ₃_.₁₃_×₁₀−2 ₂_.₀₀_×₁₀−9

f6t ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

t, 0≤t <1, 3

2− t

2, 1≤t <3, 0, elsewhere,

F6

p

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

3

2, p0,

1−e−p1p

p2

e−3pe2p2p−1

2p2 , p >0.

3.7

Example 3.6 represents a class of piecewise continuous functions. FromFigure 6the value of the exact solution at the points where the function is not diﬀerentiable cannot be well approximated for the given levels of noise by the proposed method. When the noise level δ 10−6_{, our result is comparable with the results given in}₁₅_._{Table 6}_{reports the stability}

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−0.05 0 0.05 0.1 0.15 0.2 0.25 0.3

0 2 4 6 8 10

t

Exact solution

δ10−2 _δ₁₀−6

δ10−4

1.00×10−2 ₁_.₅₂_×₁₀−2 ₃₀ ₂ ₄_.₆₀₀_×₁₀−2 ₂_.₀₀_×₁₀−3

1.00×10−4 ₂_.₆₀_×₁₀−3 ₃₀ ₃ ₉_.₃₈_×₁₀−2 ₂_.₀₀_×₁₀−6

1.00×10−6 ₂_.₀₂_×₁₀−3 ₃₀ ₄ ₃_.₁₃_×₁₀−2 ₂_.₀₀_×₁₀−9

f7t ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

−te−t−e−t1, 0≤t <1,

1−2e−1, 1≤t <10,

0, elsewhere,

F7

p

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

3 e −19

$

1−2 e

%

, p0,

e−1−pe1p−e1p2p32p/pp12 e−2e−1−p−10pe10p_−ep_/p

, p >0.

3.8

When the noise levelδ 10−4 _and_δ ₁₀−6_{, we get numerical results which are comparable}

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−0.2

−0.1 0 0.1 0.2 0.3 0.4 0.5 0.6

0 2 4 6 8 10

t

Exact solution

δ10−2 _δ₁₀−6

δ10−4

1.00×10−2 ₂_.₇₄_×₁₀−2 ₃₀ ₂ ₁_.₁₀₀_×₁₀−2 ₂_.₀₀_×₁₀−3

1.00×10−4 ₃_.₅₈_×₁₀−3 ₃₀ ₃ ₃_.₁₃_×₁₀−2 ₂_.₀₀_×₁₀−6

1.00×10−6 ₅_.₀₄_×₁₀−4 ₃₀ ₄ ₄_.₆₉_×₁₀−2 ₂_.₀₀_×₁₀−9

f8t ⎧ ⎪ ⎨ ⎪ ⎩

4t2_e−2t_, ₀_≤_{t <}₁₀_,

0, elsewhere,

F8

p

84e−102p+₋₂₋₂₀₂_p₋₁₀₀₂₋_p2,

2p3 .

3.9

The results of this example are similar to the results ofExample 3.3. The exact solution can be well reconstructed by the approximate solution obtained by our method at the levels noise δ 10−4 _and_δ ₁₀−6 _see_{Figure 8}_._{Table 8} _{shows that the MAE decreases as the noise}

(32)

−1 0 1 2 3 4 5

0 2 4 6 8 10

t

Exact solution

δ10−2 _δ₁₀−6

δ10−4

1.00×10−2 ₂_.₀₇_×₁₀−1 ₃₀ ₃ ₆_.₂₅_×₁₀−2 ₂_.₀₀_×₁₀−6

1.00×10−4 ₇_.₁₄_×₁₀−2 ₃₂ ₄ ₃_.₄₄_×₁₀−1 ₂_.₀₀_×₁₀−9

1.00×10−6 ₂_.₅₆_×₁₀−2 ₅₄ ₅ ₃_.₇₅_×₁₀−1 ₂_.₀₀_×₁₀−12

f9t ⎧ ⎪ ⎨ ⎪ ⎩

5−t, 0≤t <5,

0, elsewhere,

F9

p

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

25

2 , p0,

e−5p₅_p₋₁

p2 , p >0.

3.10