A general filter regularization method to solve the three dimensional Cauchy problem for...

A general filter regularization method to solvethe three dimensional Cauchy problem forinhomogeneous Helmholtz-type equations:

theory and numerical simulation

Quoc Viet Tran1, Huy Tuan Nguyen2, Van Thinh Nguyen1, and Duc Trong Dang3

1Department of Civil and Environmental Engineering, Seoul NationalUniversity, Republic of Korea.2Institute for Computational Science and Technology at Ho Chi MinhCity, Vietnam.3Department of Mathematics, University of Science, Vietnam NationalUniversity at HoChiMinh City, Vietnam.

Abstract

In this paper, we solve the Cauchy problem for an inhomogeneous Helmholtz-typeequation with homogeneous Dirichlet and Neumann boundary condition. The proposedproblem is ill-posed. Up to now, most investigations on this topic focus on veryspecific cases, and with Dirichlet boundary condition. Recently, we solve this problem in2D for an inhomogeneous modified Helmholtz equation (2012). This work is a continu-ous expansion of our previous results. Herein we introduce a general filter regularization(GFR) method, and then from the GFR we deduce two concrete filters, which area foundation to implement a numerical procedure. In addition, we develop a numericalmodel for solving this problem in three dimensional region. The proposed filter methodhas been verified by numerical experiments.

Keywords and phrases: Cauchy problem, Helmholtz, ill-posed, inhomogeneous source,regularization method.

1 Introduction

It is well known that the Helmholtz-type equations arise in many engineering applicationsrelated to propagating waves in different environments, such as acoustic, hydrodynamic andelectromagnetic waves [2, 3]. Such applications are interdisciplinary regarding material sci-ence, aerospace, chemical, nuclear and environmental engineering, medical instrumentation,etc. In most of real-world applications, the Helmholtz-type equations are usually solved in3D domains with inhomogeneous sources, and particularly the boundary conditions are oftenincomplete. A classical example of such problems is the Cauchy problem, in which boundaryconditions for the solution and its normal derivative are given only on a part of the solu-tion domain. Therefore a need of investigation on the Cauchy problem for inhomogeneous

Corresponding author. E-mail: [email protected]. Address: 599 Gwanak-ro, Gwanak-gu, Seoul 151-744Korea. Tel.: +82-2-880-7355, Fax: +82-2-873-2684

1

Helmholtz-type equations with Dirchlet, Neumann or mixed boundary conditions in 3D regionshas given a motivation for this study.

Let a, b, c ¡ 0 and Ω p0, aq p0, bq. We consider the Cauchy problem for 3D Helmholtz-type equation to determine u satisfied following system

∆u k2u fpx, y, zq, px, yq P Ω, z P r0, cs (1)

uzpx, y, 0q gpx, yq, px, yq P Ω (2)

upx, y, 0q ϕpx, yq, px, yq P Ω (3)

Bupx, y, zq 0, px, yq P BΩ, z P r0, cs (4)

where the boundary condition B given by (4) is corresponding to the homogeneous Dirichletboundary condition

upx, y, zq 0 , px, yq P BΩ, z P r0, cs (5)

or the homogeneous Neumann boundary condition

BuBnpx, y, zq 0 , px, yq P BΩ, z P r0, cs (6)

where n is the unit vector normal to BΩ, and k P C.In the past, there were many studies on the Cauchy problem for specific forms of the

Helmholtz-type equation (1). In case of the source function f 0 and the wave numberk k0 P R or k k0

?1, Eq. (1) is known as homogeneous Helmholtz or modified Helmholtzequation, which can be refered to [7, 8, 9, 10, 11]. In case of the wave number k 0 andthe source function f 0, Eq. (1) is known as Laplace equation, which can be refered to[12, 13, 14, 15, 16, 17, 18]. Recently, H.T.Nguyen et al. [19] considered Eq. (1) in 2D formodified Helmholtz equation with inhomogeneous source; i.e with k k0

?1 and f 0.The study on the Cauchy problem for inhomogeneous Helmholtz-type problem in 3D is

still limited. In this research, we continue to consider Eq. (1) in 3D with both homogeneousDirichlet and Neumann boundary conditions. In addition, we also solve the problem withk k0 P R and k k0

?1 covering both Helmholtz and modified Helmholtz equations. Beforeleading to the mathematical formulation of an ill-posed problem consisting of Eqs. (1)-(4), werecall following fundamentals of mathematics.

First, the inner product and the norm of L2pΩq are defined as follows

〈u, v〉 » a

0

» b

0

uv dx dy, u d» a

0

» b

0

|u|2 dx dy, (7)

And two complete orthonormal basis sets of L2pΩq are given by

φsspqpx, yq κpq sinpπxa

sin

qπyb

, p ¡ 0, q ¡ 0; (8)

φccpqpx, yq κpq cospπxa

cos

qπyb

, p ¥ 0, q ¥ 0; (9)

here

κpq c

1

abp2 δp0q p2 δ0qq, (10)

where δpq is Kronecker delta.

2

Second, the following eigenvalue problem#∆xyψ k2ψ λψ, px, yq P Ω,

Bψpx, yq 0, px, yq P BΩ

has infinite countable solutions pλpq, ψpqq according to the boundary condition B given by (5)as follows #

λpq pπa

2 qπb

2 k2

ψpq φsspq, for p ¡ 0, q ¡ 0 (11)

and given by (6) as follows#λpq

pπa

2 qπb

2 k2

ψpq φccpq, for p ¥ 0, q ¥ 0 (12)

For simplification, we set

A1 tpp, qq : λpq 0u , A2 tpp, qq : λpq 0u ,A3 tpp, qq : λpq ¡ 0u , A A1 Y A2 Y A3.

Henceforward, we denote Fourier coefficients of v vpx, yq by

vpq 〈v, ψpq〉 » a

0

» b

0

vpx, yqψpqpx, yq dx dy. (13)

Applying variable separable method, the solutions of problem (1)-(4) can be presented inthe following form

upx, y, zq ¸

pp,qqPA

upqpϕ, g, zqψpqpx, yq, (14)

where

upqpϕ, g, zq

$'''''''''''''''''''&'''''''''''''''''''%

coszaλpq

ϕpq

sinzaλpq

aλpq

gpq

1aλpq

» z

0

fpqpsq sinpz sq

aλpq

ds, if pp, qq P A1,

ϕpq z gpq » z

0

pz sqfpqpsq ds, if pp, qq P A2,

ez?λpq

ϕpq2

gpq

2aλpq

ez

?λpq

ϕpq2

gpq

2aλpq

1

2aλpq

» z

0

epzsq

?λpq epszq

?λpq

fpq psq ds, if pp, qq P A3.

(15)

We can see that the instability of (15) is caused by the fast increasing of ez?λpq as λpq tends

to infinity. Even though the exact Fourier coefficients pϕpq, gpq, fpqpsqq may tend to zero rapidly,a calculation of (14) is impossible because a given data set is usually diffused by varied factors,such as round-off, measurement errors, etc. A small perturbation in the data can arbitrarily

3

deduce a large error in the solution. Therefore, the problem is ill-posed and a regularization isin order.

The remainder of this paper is organized as follows. In Section 2, the theorical foundationsof the general filter regularization (GFR) method and two specific regularization filters deducedfrom the GFR are presented. In Section 3, the numerical experiments including data input,numerical procedures, examples, results and discussions are described. Finally, the conclusionis shown in Section 4.

2 A general filter regularization method

In this section, we are constructing several regularization strategies for Eq.(14) from mea-sured data pϕε, gεq satisfy

ϕε ϕ ¤ ε, gε g ¤ ε, (16)

for a positive number ε. The primary idea of this method is to replace ez?λpq in Eq. (15) by

damping quantities Qpα, p, qqez?λpq to approximate u by uα which is defined by

uαpx, y, zq ¸

pp,qqPA

uαpqpϕε, gε, zqψpqpx, yq, (17)

where uαpq is given by

uαpqpw, v, zq

$'''''''''''''''''''&'''''''''''''''''''%

coszaλpq

wpq

sinzaλpq

aλpq

vpq

1aλpq

» z

0

fpqpsq sinpz sq

aλpq

ds, if pp, qq P A1,

wpq z vpq » z

0

pz sqfpqpsq ds, if pp, qq P A2,

Qpα, p, qqez?λpq

wpq2

vpq

2aλpq

ez

?λpq

wpq2

vpq

2aλpq

1

2aλpq

» z

0

Qpα, p, qqepzsq

?λpq epszq

?λpq

fpq psq ds, if pp, qq P A3.

(18)

for any z P p0, cs and w, v P L2pΩq. Herein the function Q is called a regularizing filter and αplays the role as the regularization parameter, which should be chosen with respect to ε. Undera priori assumption of u, the regularization strategy α αpεq is admitted if

αpεq Ñ 0 and uαp, , zq up, , zq Ñ 0 as εÑ 0

for any z P p0, cs. Now we establish a general regularizing filter Q in the following theorems.

2.1 Theorem 1 (General regularization filter)

Assuming that u is the exact solution of problem (1)-(4) and M is a real-valued functionsuch that ¸

pp,qqPA3

M2pλpqq|〈up, , zq, ψpq〉|2 1

λpq|〈uzp, , zq, ψpq〉|2

¤ E2

2, (19)

4

for any z P p0, cs, where E is a positive number. Let

Q : p0,8q A3 Ñ R

be a function in such a way that, for any α ¡ 0 there exist K1pαq and K2pαq satisfy

|Qpα, p, qq| ¤ K1pαqez?λpq , (20)

|Qpα, p, qq 1| ¤ K2pαqMpλpqq, (21)

for any pp, qq P A3 and z ¡ 0. Then uα defined by Eqs. (17) and (18), fulfills the followingestimate

uαp, , zq up, , zq ¤ C pK1pαq 1q ε EK2pαq, (22)

where C is a positive constant independent of ε.A choice α αpεq is admissable if the values of αpεq, εK1pαq and K2pαq tend to zero as ε

tends to zero.

Proof. The proof is split into two steps:

Step 1: Assigning S1 °

pp,qqPA

uαpqpϕε, gε, zq uαpqpϕ, g, zq212

, now we estimate S1.

From Eq. (18), by subtracting uαpqpϕ, g, zq from uαpqpϕε, gε, zq and using triangle inequality,we have

uαpqpϕε, gε, zq uαpqpϕ, g, zq ¤

$'''&'''%

ϕεpq ϕ z

gεpq g , if pp, qq P A1 Y A2,

|Qpα, p, qq| ez

?λpq 1

ϕεpq ϕ

2

gεpq g

2aλpq

, if pp, qq P A3,

where we applied following elementary estimates ez?λpq ¤ 1 for pp, qq P A3 and

coszaλpq

¤ 1,

sinzaλpq

aλpq

¤ z for pp, qq P A1.

Set λ minpp,qqPA3

λpq ¡ 0. Combine the latter estimate with (20) in regard to K1pαq ¥ 0, we get

S21

¸pp,qqPA1YA2

uαpqpϕε, gε, zq uαpqpϕ, g, zq2 ¸

pp,qqPA3

uαpqpϕε, gε, zq uαpqpϕ, g, zq2

¤¸

pp,qqPA1YA2

ϕεpq ϕpq z

gεpq gpq2 pK1pαq 1q2

¸pp,qqPA3

ϕεpq ϕpq

2

gεpq gpq

2aλpq

2

¤

1 c2 1

4

1 1

λ

pK1pαq 1q2

¸pp,qqPA

ϕεpq ϕpq2 gεpq gpq

2

¤ 1

2C2 pK1pαq 1q2 ϕε ϕ2 gε g2

¤ C2 pK1pαq 1q2 ε2,

where C max!

2?

1 c2,b

1 1λ

). Hence

S1 ¤ C pK1pαq 1q ε. (23)

5

Step 2: Assigning S2 °

pp,qqPA

uαpqpϕ, g, zq upqpϕ, g, zq212

, we now estimate S2.

Combining Eqs. (15) and (18), we have

uαpqpϕ, g, zq upqpϕ, g, zq ez?λpq pQpα, p, qq 1q

ϕpq2

gpq

2aλpq

1

2aλpq

» z

0

es?λpqfpqpsq ds

(24)

On other hand, for pp, qq P A3, we have

〈up, , zq, ψpq〉 ez?λpq

ϕpq2

gpq

2aλpq

ez

?λpq

ϕpq2

gpq

2aλpq

1

2aλpq

» z

0

epzsq

?λpq epszq

?λpq

fpq psq ds (25)

Differentiating (25) with respect to z gives

〈uzp, , zq, ψpq〉aλpq

ez?λpq

ϕpq2

gpq

2aλpq

ez

?λpq

ϕpq2

gpq

2aλpq

1

2aλpq

» z

0

epzsq

?λpq epszq

?λpq

fpq psq ds (26)

Adding (25) to (26) yields

ez?λpq

〈up, , zq, ψpq〉〈uzp, , zq, ψpq〉a

λpq

ϕpq

2 gpq

2aλpq

1

2aλpq

» z

0

es?λpqfpqpsq ds

(27)

Combining (24) with (27), we have

uαpqpϕ, g, zq upqpϕ, g, zq ¤ |Qpα, p, qq 1|

|〈up, , zq, ψpq〉| |〈uzp, , zq, ψpq〉|a

λpq

(28)

Applying (19) and (21) to (28), we have

S22

¸pp,qqPA

uαpqpϕ, g, zq upqpϕ, g, zq2 ¸

pp,qqPA3

uαpqpϕ, g, zq upqpϕ, g, zq2

¤ 2¸

pp,qqPA3

|Qpα, p, qq 1|2|〈up, , zq, ψpq〉|2 |〈uzp, , zq, ψpq〉|2

λpq

¤ E2K2

2pαq

Hence

S2 ¤ EK2pαq. (29)

From (14) and (17), applying (23) and (29) we deduce

uαp, , zq up, , zq ¸

pp,qqPA

uαpqpϕε, gε, zq upqpϕ, g, zq2

12

¤ S1 S2

¤ C pK1pαq 1q ε EK2pαqTheorem 1 has been completely proved.

6

Remark 1. The priori assumption (19) is hold naturally in the case of Mpλpqq λpq. Indeed,let u be exact solution of the problem (1)-(4), and

φscpqpx, yq κpq sinpπxa

cos

qπyb

, p ¡ 0, q ¥ 0; (30)

φcspqpx, yq κpq cospπxa

sin

qπyb

, p ¥ 0, q ¡ 0; (31)

where κpq was given by (10). By applying the Green’s first identity, we have$'&'%

pπ

a

⟨uz, φ

sspq

⟩ ⟨uzx, φ

cspq

⟩qπ

b

⟨uz, φ

sspq

⟩ ⟨uzy, φ

scpq

⟩ , p ¡ 0, q ¡ 0

with the boundary condition B given by (5), and$'&'%

pπ

a

⟨uz, φ

ccpq

⟩ ⟨uzx, φ

scpq

⟩qπ

b

⟨uz, φ

ccpq

⟩ ⟨uzy, φ

cspq

⟩ , p ¥ 0, q ¥ 0

with the boundary condition B given by (6). Since φsspq

(, φccpq

(, φscpq

(, φcspq

(are complete

orthonormal basis sets of L2pΩq, then applying the Parseval’s identity yields¸pp,qqPA3

λpq |〈uz, ψpq〉|2 ¤ uzx2 uzy2 |k|2 uz2 . (32)

Furthermore, applying Green’s second identity, we have$''&''%

pπa

2 ⟨u, φsspq

⟩ ⟨uxx, φ

sspq

⟩qπb

2 ⟨u, φsspq

⟩ ⟨uyy, φ

sspq

⟩ , p ¡ 0, q ¡ 0

with the boundary condition B given by (5), and similarly we have$''&''%

pπa

2 ⟨u, φccpq

⟩ ⟨uxx, φ

ccpq

⟩qπb

2 ⟨u, φccpq

⟩ ⟨uyy, φ

ccpq

⟩ , p ¥ 0, q ¥ 0

with the boundary condition B given by (6). Thus we get

λpq 〈u, ψpq〉 ⟨uxx uyy k2u, ψpq

⟩, @pp, qq P A.

Hence, using Parseval’s identity and Schwarz’s inequality yielded¸pp,qqPA3

λ2pq |〈u, ψpq〉|2 ¤ 3uxx2 uyy2 |k|4 u2 . (33)

From (32) and (33), we deduce¸pp,qqPA3

λ2pq |〈up, , zq, ψpq〉|2 λpq |〈uzp, , zq, ψpq〉|2

¤ E2

2,

for any z P r0, cs, where

E Epzq c

6uxx2 uyy2 uzx2 uzy2 |k|2 uz2 |k|4 u2

(34)

In order to implement numerical procedure, we deduce from GFR two concrete filters.Applying Theorem 1 directly, we obtain Theorem 2 in the following section.

7

2.2 Theorem 2 (Two regularization filters)

Assuming the problem (1)-(4) has a unique solution u, there exist two filters and correspond-ing regularized solutions defined by Eqs. (17) and (18), named u1,α and u2,α, as follows

(1) Filter 1: (Quasi-boundary-type method [20, 21, 22, 23]) Let Q be

Qpα, p, qq 1

1 αλpq ez?λpq, α ¡ 0, pp, qq P A3. (35)

Then u1,α defined by (17) and (18) satisfies the following inequality

u1,αp, , zq up, , zq ¤ C

ε1m

λ ε

Epz 1q2

ln2

z2εm

, (36)

for any z P p0, cs, where E, C and λ are independent of ε. The admissible regularizationparameter is αpεq εm, m P p0, 1q.

(2) Filter 2: (Truncation method) Let Q be

Qpα, p, qq #

1, if λpq ¤ 1α2

0, if λpq ¡ 1α2

, α ¡ 0, pp, qq P A3. (37)

Then u2,α defined by (17) and (18) satisfies the following inequality

u2,αp, , zq up, , zq ¤ Cε1nc ε

E

n2 ln2 p1εq , (38)

for any z P p0, cs, where E and C are independent of ε. In this case, the admissibleregularization parameter is αpεq 1pn lnp1εqq, n P p0, c1q.

Proof. Let Mpλpqq λpq, E is given by (34).

Proof of Filter 1. SetK1pαq 1pλαq andK2pαq pz1q2 ln2z2α

, where λ minpPA3 λpq.

The constrain (20) is obtained as follows

Qpα, p, qq ez?λpq 1

ez?λpq αλpq

¤ 1

αλ K1pαq,

In order to verify constrain (21), we apply an elementary estimate

1

ez?λpq αλpq

¤ pz 1q2α ln2

z2α

, @pp, qq P A3. (39)

Indeed, let α ¡ 0, set

hpζq 1

ezζ αζ2, ζ ¡ 0.

We have

h1pζq 2αezζ

pezζ αζ2q2 z

2α ζezζ

.

8

It yields

z

2α ζezζ ¥ 0 z

2α eζ1ezζ1 , ζ ¤ ζ1,

z

2α ζezζ ¤ 0 z

2α ζ2ζ

z2 , ζ ¥ ζ2,

where

ζ1 1

z 1ln z

2α

, ζ2

z

2α

1z1

.

We can see that hpζq is increased as ζ ζ1 and decreased as ζ ¡ ζ2. Thus, Eq. (39) becomes

hpζq ¤ maxζPrζ1,ζ2s

hpζq maxζPrζ1,ζ2s

"1

ezζ αζ2

*¤ max

ζPrζ1,ζ2s

"1

αζ2

* 1

αζ21 pz 1q2α ln2

z2α

.From (39), we yield

|Qpα, p, qq 1| λpq α

ez?λpq αλpq

¤MpλpqqK2pαq.

By substituting K1 and K2 into (22), we deduce (36) in the following manneru1,αp, , zq up, , zq ¤ C pK1pαq 1q ε EK2pαq

¤ C

1

αλ 1

ε Epz 1q2

ln2z2α

¤ C

ε1m

λ ε

Epz 1q2

ln2

z2εm

.The regularization parameter is admissible as m P p0, 1q.

Proof of Filter 2. Set K1pαq ecα and K2pαq α2. We derive constrains (20) & (21) asfollows

Qpα, p, qq ez?λpq

#ez?λpq , if

aλpq ¤ 1

α

0, ifaλpq ¡ 1

α

¤ ezα ¤ ecα K1pαqand

|Qpα, p, qq 1| #

0, if αaλpq ¤ 1

1, if αaλpq ¡ 1

¤ α2λpq K2pαqMpλpqqHence (38) is deduced by substituting K1 & K2 into (22) as followsu2,αp, , zq up, , zq ¤ C pK1pαq 1q ε EK2pαq

¤ Cecα 1

ε Eα2

¤ Cε1nc ε

E

n2 ln2 p1εq .

The regularization parameter is admissible as n P p0, c1q. Theorem 2 has been proved.

9

3 Numerical experiments

3.1 Modelling Data

In this paper, we assume that the exact data pϕ, gq were belong to H2pΩq. We apply anuniform rectangular grid with a resolution of m n in the horizontal xy-plane, which includesnodal points pxi, yjq determined by

xi pi 1qδx, δx a

m 1, i 1,m , (40)

yj pj 1qδy, δy b

n 1, j 1, n. (41)

Denote ϕi,j ϕpxi, yjq and gi,j gpxi, yjq. Let pgεi,j, ϕεi,jq be measured data in the grid withsome error levels of pointwise

ϕεi,j ϕi,j ε1,i,j, gεi,j gi,j ε2,i,j, (42)

where ε1,i,j and ε2,i,j play the role as random noise of measurement.

x2ix2i−1 x2i+1

y2j+1

y2j

y2j−1

Global coordinate

δx−δx

δy

−δy

1 2

34

5

6

7

8 9r

s

Local coordinate

1−1

−1

1

1 2

34

5

6

7

8 9

Figure 1: Quadrilateral 9-node element

Now using pgεi,j, ϕεi,jq, we approximate the exact data pϕ, gq. In order do that we modelanalytical expressions of pgε, ϕεq in pgεi,j, ϕεi,jq by quadrature interpolation scheme as shown infollowing steps:

First, a merit partition of Ω contained 9-node rectangular elements is chosen as follows

Γ tEij rx2i1, x2i1s ry2j1, y2j1s : 1 ¤ i ¤ m0, 1 ¤ j ¤ n0uwhere m0 pm 1q2 and n0 pn 1q2 (see Fig. 1).

Second, let v be an interpolating function of a given data set tvi,ju in the mannerthat, for any px, yq P Ω there exists an element Eij and a local coordinate pr, sq Pr1, 1s2 such that

px , yq P Eij , x 9

k1

xkNkpr, sq, y 9

k1

ykNkpr, sq, (43)

and

vpx, yq 9

k1

vkNkpr, sq, (44)

10

k Nodal point Mk Shape function Nk NkL2pp1,1q2q

1 M1p1,1q N1 14p1 rqp1 sqrs 4

15

2 M2p1,1q N2 14p1 rqp1 sqrs 4

15

3 M3p1, 1q N3 14p1 rqp1 sqrs 4

15

4 M4p1, 1q N4 14p1 rqp1 sqrs 4

15

5 M5p0,1q N5 12p1 r2qp1 sqs 8

15

6 M6p1, 0q N6 12p1 rqp1 s2qr 8

15

7 M7p0, 1q N7 12p1 r2qp1 sqs 8

15

8 M8p1, 0q N8 12p1 rqp1 s2qr 8

15

9 M9p0, 0q N9 p1 r2qp1 s2q 1615

Table 1: Nodes and shape functions on local rectangular r1, 1s r1, 1s.

where the shape functions Nk were listed in Table 1, the group of tvk, xk, yku is numbered inan element (see Fig. 1) in a fashion that

v1 v2i1,2j1, v2 v2i1,2j1, v3 v2i1,2j1, v4 v2i1,2j1,

v5 v2i,2j1, v6 v2i1,2j, v7 v2i,2j1, v8 v2i1,2j, v9 v2i,2j.

To be more precise, we emphasize that Eq. (43) can be rewritten as follows

x x2i rδx, y y2j sδy (45)

due to the 9-node rectangular element (see Fig 1).Now we assign gε gε and ϕε ϕε. Next, we define ϕ and g are interpolating functions of

data sets tϕi,ju and tgi,ju, respectively. From (42), (45) and Table 1, we obtain

ϕε ϕ2 m01¸i1

n01

j1

»Eij

|ϕε ϕ|2 dx dy m01¸i1

n01

j1

9

k1

» 1

1

» 1

1

|ε1,i,j|2 |Nkpr, sq|2 δxδy dr ds

64

25

m01¸i1

n01

j1

|ε1,i,j|2 δxδy 64

25

m01¸i1

n01

j1

|ε1,i,j|2 a

m 1

b

n 1

¤ 16

25abε20,

where ε0 maxi,j t|ε1,i,j|, |ε2,i,j|u. Thus, applying the same estimate to gε, we get

ϕε ϕ ¤ 4?ab

5ε0, gε g ¤ 4

?ab

5ε0.

On the other hand, the Bramble-Hilbert theorem (see Chapter 5 of [1]) guarantees the followingestimate

v v ¤ Cpδ2x δ2yqbvxx2 vyy2, (46)

for any v P H2pΩq, where the constant C CpΩq. Hence, we deduce

ϕε ϕ ¤ ϕε ϕ ϕ ϕ ¤ ε,

gε g ¤ gε g g g ¤ ε,

where

ε 4?ab

5ε0 Cpδ2x δ2yqmax

"bϕxx2 ϕyy2,

bgxx2 gyy2

*.

11

3.2 Quadrature formula for Fourier coefficients

In this section, we introduce numerical methods to calculate Fourier coefficients in Eq.(18). Due to the fluctuation of ψpq as p and q become large, the integrals require a specialtechnique (see Chapter 13 of [6]). Herein we applied a key idea suggested by Filon [4], in whichwe first identify discrete data of the integrands by their own quadrature interpolating functionsdescribed in Section 3.1, and then obtain an exact formula of the integrals (Eq. (13)).

Let a, b ¡ 0, according to ψpq with regardless to κpq, the integrals (13) can be written asfollows

vsspq » a

0

» b

0

vpx, yq sinpπxa

sin

qπyb

dx dy , p, q P N, (47)

vccpq » a

0

» b

0

vpx, yq cospπxa

cos

qπyb

dx dy , p, q P NY t0u . (48)

where v is defined by (44). Then we have

vsspq m01¸i1

n01

j1

» x2i1

x2i1

» y2j1

y2j1

vpx, yq sinppaxq sinpqbyq dx dy, (49)

vccpq m01¸i1

n01

j1

» x2i1

x2i1

» y2j1

y2j1

vpx, yq cosppaxq cospqbyq dx dy, (50)

where pa pπa, qb qπb and m0 pm 1q2, n0 pn 1q2. By substituting (44) and(45) into Eqs (49) and (50), and applying some trigonometric calculations, we obtain

vsspq δxδy

m01¸i1

n01

j1

9

k1

vkspisqjJ

k1 spicqjJ

k2 cpisqjJ

k3 cpicqjJ

k4

, (51)

vccpq δxδy

m01¸i1

n01

j1

9

k1

vkcpicqjJ

k1 cpisqjJ

k2 spicqjJ

k3 spisqjJ

k4

, (52)

where we denote

Jk1 » 1

1

» 1

1

Nkpr, sq cosppδrq cospqδsq dr ds, Jk2 » 1

1

» 1

1

Nkpr, sq cosppδrq sinpqδsq dr ds,

Jk3 » 1

1

» 1

1

Nkpr, sq sinppδrq cospqδsq dr ds, Jk4 » 1

1

» 1

1

Nkpr, sq sinppδrq sinpqδsq dr ds,

(53)and

pδ paδx, spi sin ppax2iq , cpi cos ppax2iq ,qδ qbδy, sqj sin pqby2jq , cqj cos pqby2jq .

In order to accurately approximate RHS of (51) and (52), we need to calculate Jkl exactly.Referring to Table 1 with respect to odd-even property of integrands, we can see followinggroup relationships

J11 J3

1 , J12 J3

2 , J13 J3

3 , J14 J3

4 ,

J21 J4

1 , J22 J4

2 , J23 J4

3 , J24 J4

4 ,

J51 J7

1 , J52 J7

2 , J53 J7

3 , J54 J7

4 ,

J61 J8

1 , J62 J8

2 , J63 J8

3 , J64 J8

4 ,

(54)

12

are satisfied at any p, q P N Y t0u. Thus we need to estimate only five terms J1l , J

2l , J

5l , J

6l , J

9l

pl 1, 4q and the others can be deduced from the group (54). Let

ϑpτq pτ 2 2q sin τ 2τ cos τ

τ 3, (55)

βpτq sin τ τ cos τ

τ 2. (56)

Applying integration by part to the group (53), we have

J11 ϑppδqϑpqδq, J1

2 ϑppδqβpqδq, J13 βppδqϑpqδq, J1

4 βppδqβpqδq,J21 ϑppδqϑpqδq, J2

2 ϑppδqβpqδq, J23 βppδqϑpqδq, J2

4 βppδqβpqδq,J51 4

pδβppδqϑpqδq, J5

2 4pδβppδqβpqδq, J5

3 0, J54 0,

J61 4

qδϑppδqβpqδq, J6

2 0, J63 4

qδβppδqβpqδq, J6

4 0,

J91 16

pδqδβppδqβpqδq, J9

2 0, J93 0, J9

4 0.

(57)

Some of Jkl within the group (57) require to be defined at pδ 0 or qδ 0, however theformulas (55) and (56) are not defined at τ 0, fortunately the functions ϑpτq, βpτq andγpτq : βpτqτ can be extended to be continuous functions on r0,8q, particularly at τ 0 byfollowing definitions:

ϑp0q : limτÑ0

ϑpτq 1

3,

βp0q : limτÑ0

βpτq 0 ,

γp0q : limτÑ0

βpτqτ

1

3.

Thus the group (57) is well-defined at any p, q P N Y t0u. Therefore, Fourier coefficients vsspqand vccpq from (51) and (52) can be determined.

Remark 2. In order to calculate (18), we need not only to estimate integrations in form of(47) and (48), but also to calculate the following integrals

fpqpzq » z

0

» a

0

» b

0

fpx, y, sqψpqpx, yq dx dy ds, @z P r0, cs.

For a fixed value of z P r0, cs, based on Fubini theorem and applying above Filon technique,the fpqpzq can be integrated by replacing vpx, yq in Eqs. (49) and (50) by

F px, y, zq » z

0

fpx, y, sq ds.

In order to approximate RHS of the latter equation, we use Gauss-Legendre quadrature methodas follows

F pxi, yj, zq » z

0

fpxi, yj, sq ds z

2

k0

k1

wk fxi, yj,

z

2pξk 1q

,

where ξk abscissas in r1, 1s and wk are associated weights. The computation of abscissas andweights is carried out by gauleg subroutine [6].

13

3.3 Numerical test

In this section, the proposed regularization methods are implemented numerically. Theefficiency of the methods is observed by comparing the result between numerical and exactsolutions. To do that, we constructed Eqs. (1)-(4) in such a way that they have a given exactsolution. Our numerical implementation is focused on calculation of Fourier coefficients in Eq.(13). We consider Eqs. (1)-(4) under three examples; the first and second examples are dealingwith Dirichlet boundary condition (5) and Neumann boundary condition (6), respectively.The third example is dealing with unbounded horizontal boundary and not-given boundaryconditions.

(a) Example 1 (b) Example 2 (c) Example 3

Figure 2: The graph of exact solutions upx, y, cq corresponding to each example

Before implementing numerical procedure, we introduce a relative root mean square errorbetween the regularized solutions uκ,αp, , cq and exact solution up, , cq as follows

δκpρq

b°mi1

°nj1 |uκ,αpxi, yj, cq upxi, yj, cq|2b°m

i1

°nj1 |upxi, yj, cq|2

, (58)

with κ 1, 2. In fact, δκ is dependent on the regularization parameter α which were determinedby Theorem 2

αpρq #ρ0.9c, for u1,α of Filter 1,

c0.9 lnp1ρq

, for u2,α of Filter 2,(59)

where ρ is a positive number.In order to illustrate the sensitivity of the computational accuracy to noise of the data,

we repeated calculations with variety of perturbed data. The perturbation was defined asε randp, q where each random term randp, q was uniformly determined on r1, 1s, and ε playsas an amplitude, i.e.

gεi,j gpxi, yjq ε randpxi, yjq, (60)

ϕεi,j ϕpxi, yjq ε randpxi, yjq, (61)

for any i 1,m, j 1, n, where xi, yj and m,n were described by (40) and (41). The aimof the numerical experiments is to observe the relative errors δκpρq as ρ tends to zero in twofollowing cases of ε:

14

1. ε 0. The couple of pϕεi,j, gεi,jq play a role as exact data. We expect that regularizedsolutions uκ,α stably converge to the exact solution u under a proper discretization.

2. ε 10r as r 2, 4, 6. The couple of pϕεi,j, gεi,jq play as measured data with a randomnoise. The regularized solutions uκ,α were expected to be closed to the exact solution u asρ tends to zero. However, after a certain value of ρ ρconv the numerical solution startsto diverge because the regularization was broken. This tendency has been predicted asabove theoretical results by the restriction (23). Consequently, the computational processwill be stopped at ρconv.

3.3.1 Example 1

Solving Eqs. (1)-(3) in which f , g and ϕ have been chosen in order that

upx, y, zq cospxyzq sinπxa

sin

πyb

is exact solution (see Fig. 2(a)). In this case, the wave number k is a real number, thereforethe problem (1)-(3) becomes Helmholtz problem with Dirichlet boundary condition,

up0, y, zq upa, y, zq upx, 0, zq upx, b, zq 0

Let a b 3 and c 1. The horizontal grid resolution of this calculation is p2s 1q p2s 1q p2sq for s 6, 7 and 8. The numerical results with wave number k 3 have been shownon Tables 2 & 3, and with wave number k 7 shown on Tables 4 & 5. Fig. 3 shows the graphof numerical solutions with the wave number k 7.

3.3.2 Example 2

Similar to Example 1, the exact solution of (1)-(3) was chosen as

upx, y, zq cos pzypb yqq cosπxa

(see Fig. 2(b)) which satisfies Neumann boundary condition,

uxp0, y, zq uxpa, y, zq uypx, 0, zq uypx, b, zq 0

Let a b 5 and c 1. The grid resolutions of this calculation are the same as in Example 1.The numerical results are shown in Tables 6 and 7 with the wave number k 1, and Tables 8and 9 with k 7. Fig. 4 shows the graph of numerical solutions with the wave number k 7.

3.3.3 Example 3

Considering Eqs. (1)-(3) in a specific case as the modified Helmholtz problem as follows

∆u k2u fpx, y, zq, px, yq P R2, z P r0, cs,upx, y, 0q ϕpx, yq, px, yq P R2,

uzpx, y, 0q gpx, yq, px, yq P R2,

where k 1, c ¡ 0. The same idea as in the example 1, the functions f , g and ϕ were chosen

in such a way that upx, y, zq °i0i1 e

γppxxiq2pyyiq2pzziq2q is an exact solution (see Fig.2(c)), where γ 20, i0 3, px1, y1, z1q p1.125, 1.125, 1.0q, px2, y2, z2q p1.875, 1.125, 1.0q and

15

px3, y3, z3q p1.5, 1.875, 1.0q. In this test case, starting from an unbounded domain in R2 wetry to reconstruct upx, y, cq in the bounded region Ω p0, aq p0, aq (a b 3) in such a waythat the values of g, ϕ and f are infinitesimal outside of Ω and Ωp2c, 2cq with c 1. Hereineither homogeneous Dirichlet or Neumann boundary condition can be set up for this example.Now we are seeking a solution of the following problem

∆u k2u fpx, y, zq, px, yq P Ω, z P r0, cs,upx, y, 0q ϕεpx, yq, px, yq P Ω,

uzpx, y, 0q gεpx, yq, px, yq P Ω,

satisfied either Dirichlet (5) or Neumann (6) boundary conditions, where the disturbed datapϕε, gεq described in Section 3.1 satisfy Eqs. (60) and (61). Each numerical result uκ,α associatedwith a specific boundary condition is expected to be a reasonable regularized solution. Figs. 5and 6 show the graph of numerical solutions with Dirichlet and Neumann boundary conditions,respectively. The grid resolutions of this calculation are the same as in two above Examples.

Tables 2, 4, 6, 8, 10 and 12 show relative error estimate δκpρq pκ 1, 2q in case of ε 0, i.e.for exact data pϕ, gq. As ρ tends to zero shown in these tables, the numerical calculations of allgrid resolutions are stable converged until ρ 106, thereafter the approximated solution uκ,α

is still converged with finer grid resolutions (m n 129 and 257), however it almost startedto diverge in most of cases with coarser grid resolutions (m n 65) (except in the example3, see Tables 10 and 12).

Tables 3, 5, 7, 9, 11 and 13 show relative error estimate δκpρq pκ 1, 2q in case of ε 10r

pr 2, 4, 6q, i.e. for disturbed data pϕε, gεq. In this case, the finest grid was used (m n 257).As shown in these tables, once ρ tends to zero and its value is still greater than or equal toε, the numerical solutions are still converged, however when ρ ε the numerical solutions arestarting to diverge.

Tables 4, 5 (Example 1) and 8, 9 (Example 2) showed relative error estimate δκpρq pκ 1, 2qin case of wave number k 7. The main difficulty in solving Eqs. (1)-(3) is dealing with a highwave number k, consequently a large number of grid points must be used to resolve the shortwave length. Therefore, in order to test the sensibility of the numerical computation to a highwave number, in the exact data we increase the wave number k from 3 to 7 in Example 1 andfrom 1 to 7 in Example 2. The results are showed the numerical solution is dependent on thegrid resolutions, and its convergence is dependent on the value of ρ.

In Figs. 3, 4, 5 and 6, the first and second columns show the graph and its contours of theregularized solutions of filter 1 (u1,α) and filter 2 (u2,α), respectively. As shown, once ρ tends tozero, we can see that if ρ ε, then uκ,α (κ 1, 2) start fluctuating which is shown by oscillatingcontours, afterward the solution begins to diverge, because the regularization strategies werebroken down.

4 Conclusion

The study on inverse Helmholtz-type problem with inhomogeneous source in 3D is stilllimited. Recently, H.T.Nguyen et al [19](2012) suggested two regularization methods for themodified Helmhotlz equation with inhomogeneous source and Dirichlet boundary condition in2D. In this paper, we solved inverse Helmholtz-type problem with inhomogeneous source in3D including both Dirichlet and Neumann boundary conditions. This work is a continuousdevelopment of our previous study (H.T.Nguyen et al [19]).

In theoretical results, we suggested a general method of regularization filter (Theorem 1),subsequently, we deduced to two concrete regularization filters (Theorem 2), which sets a

16

fundamental to solve the problem (1)-(4) numerically as shown in Examples 1, 2 and 3, andobtained the error estimation of logarithm type.

The numerical experiments cover both either Dirichlet (Example 1) or Neumann (Example2) boundary conditions. The numerical results prove the efficiency of the theoretical suggestion,i.e. regularized solutions stably converge to the exact solution. As shown in the numericalexperiments, for the exact data, the convergence of the numerical solution is dependent on thegrid resolution. However, for the disturbed data, generally in all cases once ρ tends to zero andits value is still greater than or equal to the value ε, the numerical solution is still converged,but afterward when ρ ε the numerical solution is starting to diverge. Therefore, in order tomake sure the numerical solution to be stable and converged, the value of ρ should be chosenas the same order of ε. This result has been confirmed by Theorem 2. As a result, we can usethe value of ρ (as ρconv) to estimate the accuracy of an observed data set (ε).

The numerical calculation of problems (1)-(4) in 3D is time consuming, particularly to dealwith higher value of the wave number k, we need to refine a computational grid enough to resolvea short wave length, consequently a large number of grid points must be used. Therefore, thecode of numerical calculation has been parallelized by OpenMP [5] in Fortran90.

In fact, it should be stated that our regularized solutions methods are based on the Fourierseries expression of solution, which may lead a limitation of the methods for applications in acomplicated domain where a solution cannot be expressed by a certain series. This issue willbe surveyed in a further research.

Acknowledgements

The authors gratefully acknowledge the financial support from the AdvancedResearch Center for River Operation and Management (ARCROM), Korea andthe Institute for Computational Science and Technology at Ho Chi Minh City,Vietnam. The authors also would like to thank the anonymous reviewers for theirvery valuable constructive comments to improve our manuscript.

References

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[3] Hogan, G., 1988, Migration of Ground Penetrating Radar Data: a Technique for LocatingSubsurface Targets. Proceedings of the Symposium on the Application of Geophysics toEngineering and Environmental Problems, U.S. Goeligical Survey, Golden, CO., U.S.A.,March 28-31. Society of Engineering and Mineral Geophysics, pp. 809-822.

[4] L. N. G. Filon, On a quadrature formula for trigonometric integrals, Proc. Roy. Soc. Ed-inburgh, v. 49, 1929, pp. 38-47.

[5] The OpenMP, API specification for parallel programming, http://openmp.org/

[6] W.H. Press et al., Numerical Recipes in Fortran 90, 2nd ed., Cambridge University Press,New York, 1996.

17

[7] H.H. Qin, T. Wei, Modified regularization method for the Cauchy problem of the Helmholtzequation, Applied Mathematical Modelling, Volume 33, Issue 5, May 2009, Pages 23342348

[8] H.H. Qin, T. Wei, Two regularization methods for the Cauchy problems of the Helmholtzequation, Applied Mathematical Modelling, Volume 34, Issue 4, April 2010, Pages 947967

[9] H.H. Qin and T. Wei, Quasi-reversibility and truncation methods to solve a Cauchy problemfor the modified Helmholtz equation, Math. Comput. Simulation 80 (2009), no. 2, 352–366

[10] G. Alessandrini, L. Rondi, E. Rosset, S. Vessella, The stability for the Cauchy problem forelliptic equations, Inverse Probl. 25 (2009) 123004. 47p

[11] T. Reginska and U. Tautenhahn (2009), Conditional stability estimates and regularizationwith applications to Cauchy problems for the Helmholtz equation, Num. Funct. Anal. andOptimiz. 30, 1065

[12] D. Lesnic, L. Elliott and B.D. Ingham, The boundary element solution of the Laplace andbiharmonic equations subjected to noisy boundary data Int. J. Numer. Methods Eng. 43479-492, 1998.

[13] M. Charton, H.-J. Reinhardt, Approximation of Cauchy problems for elliptic equationsusing the method of lines, WSEAS Trans. Math. 4/2 (2005) 64-69.

[14] Z. Qian, C.L.Fu and X.T. Xiong, Fourth-order modified method for the Cauchy problemfor the Laplace equation, Journal of Computational and Applied Mathematics, 192: 205-218(2006).

[15] Qian, Zhi; Fu, Chu-Li; Li, Zhen-Ping, Two regularization methods for a Cauchy problemfor the Laplace equation, J. Math. Anal. Appl. 338 (2008), no. 1, 479–489.

[16] J.-J. Reinhardt, H. Han, D.N. Hao, Stability and regularization of discrete approximationto the Cauchy problem for the Laplaces equation , SIAM Numer. Anal. 36 (1999) 890905.

[17] D.N. Hao, P.M. Hien, Stability results for the Cauchy problem for the Laplace equation ina strip, Inverse Problems 19 (2003) 833-844.

[18] N.H. Tuan, D.D. Trong and P.H. Quan, A note on a Cauchy problem for the Laplaceequation: regularization and error estimates. Appl. Math. Comput. 217 (2010), no. 7,2913–2922

[19] Huy Tuan Nguyen, Quoc Viet Tran, Van Thinh Nguyen, Some remarks on a modifiedHelmholtz equation with inhomogeneous source, Applied Mathematical Modelling, 2012.(http://dx.doi.org/j.apm.2012.03014)

[20] Trong, Dang Duc; Quan, Pham Hoang; Tuan, Nguyen Huy. A quasi-boundary value methodfor regularizing nonlinear ill-posed problems. Electron. J. Differential Equations 2009, No.109, 16 pp.

[21] Denche, M.; Bessila, K. A modified quasi-boundary value method for ill-posed problems. J.Math. Anal. Appl. 301 (2005), no. 2, 419–426.

[22] Feng, Xiao-Li; Eldn, Lars; Fu, Chu-Li. A quasi-boundary-value method for the Cauchyproblem for elliptic equations with nonhomogeneous Neumann data. J. Inverse Ill-PosedProbl. 18 (2010), no. 6, 617–645

18

[23] Zhang, Hongwu. Modified quasi-boundary value method for Cauchy problems of ellipticequations with variable coefficients. Electron. J. Differential Equations 2011, No. 106, 10pp.

19

k 3 m n 65 m n 129 m n 257

ρ δ1pρq δ2pρq δ1pρq δ2pρq δ1pρq δ2pρq

101 3.519E-01 2.990E-01 3.519E-01 2.990E-01 3.519E-01 2.990E-01

102 1.671E-01 1.304E-01 1.671E-01 1.304E-01 1.671E-01 1.304E-01

103 6.547E-02 1.642E-02 6.547E-02 1.642E-02 6.547E-02 1.642E-02

104 1.734E-02 6.414E-03 1.734E-02 6.417E-03 1.734E-02 6.417E-03

105 5.839E-03 2.993E-03 5.839E-03 2.994E-03 5.839E-03 2.994E-03

106 2.857E-03 1.791E-03 2.855E-03 1.745E-03 2.855E-03 1.745E-03

107 1.739E-03 2.514E-03 1.685E-03 1.137E-03 1.684E-03 1.126E-03

Table 2: Example 1, a b 3, c 1, k 3, noise ε 0.

k 3 ε 102 ε 104 ε 106


101 3.518E-01 2.990E-01 3.519E-01 2.990E-01 3.519E-01 2.990E-01

102 1.668E-01 1.304E-01 1.671E-01 1.304E-01 1.671E-01 1.304E-01

103 6.809E-02 3.576E-02 6.546E-02 1.642E-02 6.547E-02 1.642E-02

104 1.633E-01 2.616E-01 1.735E-02 6.940E-03 1.734E-02 6.417E-03

105 diverged diverged 1.264E-02 3.588E-02 5.845E-03 3.012E-03

106 diverged diverged 7.341E-02 diverged 3.020E-03 3.327E-03

107 diverged diverged diverged diverged 4.834E-03 diverged

Table 3: Example 1, a b 3, c 1, k 3, m n 257, noise amplitude ε 10r,r 2, 4, 6.

k 7 m n 65 m n 129 m n 257


101 1.828E-02 1.971E-02 1.828E-02 1.971E-02 1.828E-02 1.971E-02

102 1.214E-02 9.908E-03 1.215E-02 9.910E-03 1.215E-02 9.910E-03

103 7.296E-03 5.075E-03 7.298E-03 5.078E-03 7.298E-03 5.078E-03

104 4.353E-03 3.123E-03 4.356E-03 3.127E-03 4.356E-03 3.127E-03

105 2.694E-03 1.897E-03 2.696E-03 1.899E-03 2.696E-03 1.899E-03

106 1.761E-03 1.224E-03 1.759E-03 1.192E-03 1.759E-03 1.192E-03

107 1.288E-03 2.027E-03 1.207E-03 8.285E-04 1.206E-03 8.177E-04

Table 4: Example 1, a b 3, c 1, k 7, noise ε 0.

k 7 ε 102 ε 104 ε 106


101 1.845E-02 1.972E-02 1.828E-02 1.971E-02 1.828E-02 1.971E-02

102 1.341E-02 1.082E-02 1.215E-02 9.911E-03 1.215E-02 9.910E-03

103 2.332E-02 1.890E-02 7.302E-03 5.083E-03 7.298E-03 5.078E-03



106 diverged diverged diverged diverged 1.852E-03 3.426E-03

107 diverged diverged diverged diverged 4.268E-03 diverged


20

(a) u1,αpx, y, cq, ρ 102 (b) u2,αpx, y, cq, ρ 102

(c) u1,αpx, y, cq, ρ 103 (d) u2,αpx, y, cq, ρ 103

(e) u1,αpx, y, cq, ρ 104 (f) u2,αpx, y, cq, ρ 104

Figure 3: Example 1, k 7, noise amplitude ε 102, m n 257. Divergencephenomena occurs as ρ becomes less than ε.

21

k 1 m n 65 m n 129 m n 257


101 4.961E-01 4.842E-01 4.986E-01 4.847E-01 4.999E-01 4.849E-01

102 3.321E-01 2.103E-01 3.300E-01 2.084E-01 3.290E-01 2.074E-01

103 1.595E-01 7.199E-02 1.568E-01 7.126E-02 1.554E-01 7.090E-02

104 5.062E-02 9.555E-03 4.902E-02 9.298E-03 4.833E-02 9.214E-03

105 1.416E-02 5.451E-03 1.254E-02 2.242E-03 1.221E-02 2.179E-03

106 1.017E-02 1.357E-02 3.778E-03 1.543E-03 3.376E-03 1.268E-03

107 3.909E-02 diverged 3.134E-03 6.607E-03 1.335E-03 6.763E-04

Table 6: Example 2, a b 5, c 1, k 1, noise amplitude ε 0.

k 1 ε 102 ε 104 ε 106


101 4.999E-01 4.849E-01 4.999E-01 4.849E-01 4.999E-01 4.849E-01

102 3.286E-01 2.075E-01 3.290E-01 2.074E-01 3.290E-01 2.074E-01

103 1.548E-01 8.534E-02 1.555E-01 7.090E-02 1.554E-01 7.090E-02

104 1.140E-01 3.071E-01 4.835E-02 9.558E-03 4.833E-02 9.214E-03

105 4.995E-01 diverged 1.350E-02 2.207E-02 1.221E-02 2.196E-03




k 7 m n 65 m n 129 m n 257


101 3.296E-02 3.745E-02 3.258E-02 3.710E-02 3.239E-02 3.691E-02

102 1.950E-02 1.116E-02 1.908E-02 1.097E-02 1.891E-02 1.088E-02

103 9.375E-03 4.769E-03 8.959E-03 4.605E-03 8.826E-03 4.552E-03

104 5.016E-03 2.721E-03 4.372E-03 2.329E-03 4.265E-03 2.291E-03

105 4.089E-03 3.562E-03 2.220E-03 1.353E-03 2.073E-03 1.308E-03

106 9.685E-03 diverged 1.545E-03 1.905E-03 1.095E-03 5.562E-04


Table 8: Example 2, a b 5, c 1, k 7, noise amplitude ε 0.

k 7 ε 102 ε 104 ε 106


101 3.240E-02 3.692E-02 3.239E-02 3.691E-02 3.239E-02 3.691E-02

102 1.906E-02 1.166E-02 1.891E-02 1.088E-02 1.891E-02 1.088E-02

103 1.576E-02 3.189E-02 8.827E-03 4.560E-03 8.826E-03 4.552E-03

104 6.952E-02 diverged 4.327E-03 3.310E-03 4.265E-03 2.292E-03





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Figure 4: Example 2, k 7, noise amplitude ε 102, m n 257. Divergencephenomena occurs as ρ becomes less than ε.

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k 101 m n 65 m n 129 m n 257


101 4.581E-01 4.340E-01 4.580E-01 4.339E-01 4.580E-01 4.339E-01

102 3.916E-01 3.804E-01 3.915E-01 3.802E-01 3.915E-01 3.802E-01

103 3.479E-01 3.416E-01 3.477E-01 3.415E-01 3.477E-01 3.414E-01

104 2.889E-01 2.424E-01 2.888E-01 2.422E-01 2.887E-01 2.422E-01

105 1.923E-01 1.479E-01 1.922E-01 1.478E-01 1.921E-01 1.478E-01

106 1.109E-01 1.011E-01 1.107E-01 1.012E-01 1.107E-01 1.012E-01

107 8.344E-02 9.158E-02 8.341E-02 9.178E-02 8.341E-02 9.179E-02

Table 10: Example 3, Dirichlet condition, a b 3, c 1, k 1, noise amplitude ε 0.

k 101 ε 102 ε 104 ε 106


101 4.578E-01 4.340E-01 4.580E-01 4.339E-01 4.580E-01 4.339E-01

102 3.916E-01 3.803E-01 3.915E-01 3.802E-01 3.915E-01 3.802E-01

103 3.512E-01 3.478E-01 3.477E-01 3.414E-01 3.477E-01 3.414E-01

104 4.551E-01 8.181E-01 2.889E-01 2.423E-01 2.887E-01 2.422E-01




Table 11: Example 3, Dirichlet condition, a b 3, c 1, k 1, m n 257, noiseamplitude ε 10r, r 2, 4, 6.

k 1 m n 65 m n 129 m n 257


101 4.608E-01 5.027E-01 4.607E-01 5.020E-01 4.607E-01 5.016E-01

102 3.895E-01 3.819E-01 3.894E-01 3.815E-01 3.894E-01 3.814E-01

103 3.490E-01 3.579E-01 3.489E-01 3.575E-01 3.489E-01 3.573E-01

104 2.876E-01 2.623E-01 2.875E-01 2.613E-01 2.875E-01 2.609E-01

105 1.933E-01 1.479E-01 1.931E-01 1.472E-01 1.931E-01 1.468E-01

106 1.114E-01 1.019E-01 1.112E-01 1.017E-01 1.112E-01 1.016E-01

107 8.312E-02 9.209E-02 8.306E-02 9.206E-02 8.305E-02 9.197E-02

Table 12: Example 3, Neumann condition, a b 3, c 1, k 1, noise amplitudeε 0.

k 1 ε 102 ε 104 ε 106


101 4.607E-01 5.016E-01 4.607E-01 5.016E-01 4.607E-01 5.016E-01

102 3.896E-01 3.814E-01 3.894E-01 3.814E-01 3.894E-01 3.814E-01

103 3.519E-01 3.699E-01 3.489E-01 3.573E-01 3.489E-01 3.573E-01

104 4.493E-01 8.199E-01 2.877E-01 2.610E-01 2.875E-01 2.609E-01




Table 13: Example 3, Neumann condition, a b 3, c 1, k 1, m n 257, noiseamplitude ε 10r, r 2, 4, 6.

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Figure 5: Example 3, Dirichlet condition, m m 257, noise amplitude ε 104.Divergence phenomena occur as ρ becomes less than ε.

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Figure 6: Example 3, Neumann condition, m m 257, noise amplitude ε 104.Divergence phenomena occurs as ρ becomes less than ε.

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