Constant Conductivity Support Technical Notes

8/7/2019 Constant Conductivity Support Technical Notes

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APPLICATION OF CORE IDENTIFICATION TOSTARSHAPE CONSTANT CONDUCTIVITY

INCLUSION

AGAH D. GARNADIDEPARTMENT OF MATHEMATICS,

FACULTY OF MATHEMATICS AND NATURAL SCIENCES,BOGOR AGRICULTURAL UNIVERSITY

JL. MERANTI, KAMPUS IPB DARMAGA, BOGOR, 16680INDONESIA

Abstract. The problem of determining the interface separating astarshape support of constant conductivity inclusion from bound-ary measurement data of a solution of the corresponding PDEs isconsidered. An equivalent statement as a nonlinear integral equa-

tion is obtained. The problem is analyzed and implemented usingFourier method. Numerical experiments based on simplified itera-tively regularized Gauss-Newton method (sIRGNM) are presented.Keywords : impedance tomography, inverse source problem; non-linear integral equation; Fourier transform; simplified regularizedGauss-Newton,AMS subject classifications : 31B20, 35J05, 35R25, 35R30MSC 2000 : 31A25, 35R30, 65N21

Consider the following boundary value problem

u = S(x), in 1 u = 0 on 1, (0.1)

where 1 is a unit circular domain with boundary 1, and S 1 isthe support ofS, i.e. S = supp(S), and S denoted the characteristicfunction of S.

The inverse problem consists in identifying the shape of S given theNeumann data un of the solution on 1. Therefore, we define F as the

operator mapping q to un . Ring [9] studied S(x) = S in a unit circle,where S to be star-shaped with respect to the origin. He named theinverse problem as core identification. Hohage [4] addresses the sameproblem in general domain and for general S(x) without details.

In this work, we particularly interested in the case of S(x) of theform

S(x) = au0, 0 < a a constant, (0.2)where u0 is the solution of the following boundary value problem

u0 = 0, u0 = g on 1. (0.3)

This work extends the works of Ring [9], and studying in more detailsof a particular case of Hohage [4] work. Furthermore, this particular

1


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2 AGAH D. GARNADI

problem arises from identification of conductivity inclusion shape fromboundary measurement.

This report is organize into some sections and appendices:In the first section, we state the Inclusion Problem in EIT as Inverse

Source Problem/Core identification. The second section, we recalls theCore Identification results and recast it to the problem in our hand.We owe much of these results in this section to Ring [9]. Also we verifythe singular system of Frechet derivative at particular situation, whichdemonstrate the degree of ill-posedness of the problem.

At the end, we provides some numerical results, using a method thatutilize the previous result. Taking insights from the particular forms ofthe singular systems of discrete Frechet derivative which can be seenas truncated SVD of the Frechet derivative.

The appendices provided some issues to complement the main ar-ticle for the sake of conveniencies. In the first appendix we providesome technical tools and notations used. A short review related to thealgorithm used for the reconstruction is provided in the last appendix.

1. Object inclusion in Electrical Impedance Tomographyand inverse source problem of Poisson equation (0.1).

Consider the inverse boundary value problem of electric impedancetomography (EIT) in a bounded simply connected domain IR2 :Determine the conductivity : IR, with 0 < m = (x) = M 0 a constant, and for all j Z0

(|j| + |k|) ln R + cR + s ln(1 + j2) C(

1 + j2),

therefore

Cp(1 + j2)p/2

fp(

2j ).

So we conclude

fp(DF[q0]DF[q0])1Hs+p(T)Hs(T) Cp.Hence fp(DF[q0]

DF[q0]) is boundedly invertible.

Remark 2.7. The above result, highlights the degree of ill-posedness ofthe problem. The condition q0q = fp(DF[q0]DF[q0])w for some w Hs(T) is equivalent to the fact that q0q Hs+p(T). Moreover, thereare constants c, C such that cwHs(T) q0 qHs+p(T) CwHs(T)Remark 2.8. From the knowledge of the singular system ofDF[q0], we

could obtain the information of modified source condition needed forFrozen IRGN by following the argument in ([4]) in the discussion onsource condition for the inverse (constant) source problem case.

Remark 2.9. Likewise, it could highlights discussions on the numericalresults in the presence of stochastic error, as the analysis depends tosingular values expansion of DF[q0]


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14 AGAH D. GARNADI

Remark 2.10. In the case inclusion of the form a(r), that is a radialfunction, the forward map (1.10) will be of the form

F(q)(t) =

n=0,nZ

|n|2

20

ei n(ts)ei ks

q(s)0

r|n|+|k|2a(r)rdr

ds.

Observe that in our case for starshape inclusion, it is necessary thatradially supp(a(r)) is an interval [0, b] [0, 1], 0 < b 1. This condi-tion rules out the following studies on non-uniqueness by :

(1) Counter example of constant radial object by Kang & Seo [7].(2) Identification of radial function by El Badia & Ha-Duong [1].

Assuming that a(r) is bounded over its support and with finite jumpdiscontinuity, the results for starshape constant conductivity supportstill valid without essential change in the proof.We dont pose the gen-eral radial case result in this work, as to maintain the simplicity ofexposition in mind.

Numerical Implementation

We may follow the implementation of [9] or [4] for star-shape supportof inverse source problem, for numerical implementation to reconstructsthe shape of conductivity inclusion. Rather than working in complexarithmetic as in [9], in this work we follow the numerical implemen-tation of [4], hence we need to recasts the forward operator and theFrechet derivative in terms of trigonometrical polynomials expression.To simplify the expression, we split into three cases.

Cosine boundary input. Case of uck(t) =rk

kcos(kt), or boundary

input of the form gck =cos(kt)

k . Forward map from q(t) to u

n:

Fck (q)(t) =1

nN0

n

(n + k)

(

20

q(s)n+k cos(k n)s ds) cos nt

(

20

q(s)n+k sin(k n)s ds) sin nt

. (2.14)

Derivative :

(DF

c

k [q]h)(t) =

1

nN0 n(2

0 q(s)

n+k1

h(s)cos(k n)s ds) cos nt

(

20

q(s)n+k1h(s)sin(k n)s ds) sin nt

. (2.15)

Sine boundary inputCase ofusk(t) =rk

k sin(kt), or boundary input of

the form gsk =sin(kt)

k .


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Forward map from q(t) to un

:

Fsk (q)(t) =1

nN0

n

(n + k)

(

20

q(s)n+k sin(k n)s ds) cos nt+

(2

0

q(s)n+k cos(k

n)sds) sin nt . (2.16) Derivative :

(DFsk [q]h)(t) =1

nN0

n

(

20

q(s)n+k1h(s)sin(k n)s ds) cos nt+

(

20

q(s)n+k1h(s) cos(k n)s ds) sin nt

. (2.17)

Finite linear combination of Cosine & Sine boundary input case.Case ofu0(t) =

kKN0

rk(gck cos(kt)+gsk sin(kt))/k,K = {k1, , kK}

or boundary input of the form g =

kKN0(gck cos(kt) + g

sk sin(kt))/k.

Forward map from q(t) toun :

F(q)(t) =

kKN0

1

nN0

n

(n + k)[

20

q(s)n+k(gck cos(k n)s + gsk sin(k n)s) ds ]cos nt+

[

20

q(s)n+k(gsk cos(k n)s gck sin(k n)s) ds ]sin nt

.

Derivative :(DF[q]h)(t) = kKN0(gck(DFck [q]h)(t) + gsk(DFsk [q]h)(t)).

=

kKN0

1

nN0

n

(n + k)[

20

q(s)n+k1h(s)(gck cos(k n)s + gsk sin(k n)s) ds ]cos nt+

[

20

q(s)n+k1h(s)(gsk cos(k n)s gck sin(k n)s) ds ]sin nt

.

Numerical Implementation (Discrete approximation). To solvethe inverse problem finding q from data u

n

= g, we use an itera-tive method called the simplified iteratively regularized Gauss-Newtonmethod (sIRGNM), defined by the iteration process

q(k+1) := qk+(F

[q0]F[q0]+kI)

1(F[q0](yF(qk))+k(qkq0)), k N0,

(2.18)where F[q0] is the Frechet derivative of F at initial guess q0, and kis regularization parameter. This method studied in details by Mahale


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16 AGAH D. GARNADI

& Nair [8], Jin [5], and George [3]. This method is a simplified versionof iteratively regularized Gauss-Newton method (IRGNM), which isfalls within the class of regularized Gauss-Newton method. The readermay consult the monograph by [6] or [2] on the subject of regularizedGauss-Newton method for solving nonlinear inverse problem.

To arrive at a finite dimensional system, we introduce the space TMof trigonometric polynomials of degree M, M is positive integers,and solve the following minimization problem

PN(F[q0]hk + F(qk) g)2L2 + khk + qk q02Hs,K = min! (2.19)over hk TM at each step. Here K and N are positive integers, PNis the orthogonal projection onto TN in L2([0, 2]) and f2Hs,K, s IN, is the approximation of f2Hs using the trapezoidal rule with Kequidistant grid points, i.e.

f2Hs,K :=1

2K

K1

k=0 |f(k

2K)|2 + |f(s)( k

2K)|2 .

It turns out that (2.19) is equivalent to a linear least squares problemfor the Fourier coeffficients of hk.

PN(F[q0]hk + F(qk)g) is easily evaluated by truncating the series

in (2.14) and (2.15). We approximate the integrals in these formulasusing the trapezoidal rule with Nt grid points. Since the L2norm ofa function f(t) = a0 +

Nj=1 aj cos(jt) + bj sin(jt) provided by

f2L2 := a20 +1

2

Nj=1

a2j + b2j ,

we obtain (2N+ 1) linear equations for the 2M+ 1 Fourier coefficientsofhk from the term PN(F[q0]hk + F(qk) g)2L2 in (2.19). The termhk + qk q02Hs,K yields another 2K equations. Hence in total weobtain a linear least squares problem with (2(N + K) + 1) equationsfor the (2M + 1) Fourier coefficients of hk.

We generate the synthetic using (2.14) over finer grid points for thetrapezoidal rule than the number of grid used in sIRGNM, to avoid anobvious inverse crime.

Test Cases. We consider two cases to be tested. The first examplewe chose a rose petal shaped domain described by the function

q1(t) := (0.5 + 0.1 cos(5 t)).The second example we chose a dented circular shaped domain de-

scribed by the function

q2(t) :=

0.5, t [0, 1/2] [ + 1/2, 2],0.5 0.1 exp(( 1

14(t)2)), t [ 1/2, + 1/2].


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The reconstruction and the initial guess of both domain are shownin figure 2.1. On both numerical experiments, we use M = 5, N =32, Nt = 128. The data generated using forwar operator, using n = 256grid points to avoid inverse crime.

Acknowledgment

This work funded by The Directorate General of Higher Education,Ministry of Education, The Government of Indonesia, under staff de-velopment programme in higher education establishment, years 2008-2011. The support by Graduirtenkolleg 1023 Identification of Math-ematical Models at the University of Goettingen, is here gratefullyacknowledged.

References

[1] A. El Badia and T. Ha-Duong, Some remarks on the problem of source iden-tification from boundary measurements, Inverse Problems,14(4),883,1998.

[2] A.B.Bakushinsky, M.Y. Kokurin & A. Smirnova, Iterative Methods for Ill-

Posed Problems, de Gruyter, Berlin, (2011).[3] S. George, On convergence of regularized modified Newtons method for non-

linear ill-posed problems, J. Inv. and Ill-posed Prob.,18(2), 133 (2010)[4] T. Hohage, 2001, On the numerical solution of a three dimensional inverse

medium scattering problem,Inv.Problem, v17,1743-1763[5] Q.N. Jin, On a class of frozen regularized Gauss-Newton methods for nonlinear

inverse problems,Math. Comp., 79, 2191 (2010).[6] B. Kaltenbacher, A. Neubauer & O. Scherzer, 2008, Iterative Regularization

Methods for Nonlinear Ill-Posed Problems, de Gruyter, Berlin.[7] H. Kang and J.-K. Seo, A note on uniqueness and stability in the inverse

conductivity problem with one measurement, Jour. of Korean Math. Soc. 38,781-791 (2001).

[8] P. Mahale and M.T. Nair, A simplified generalized Gauss-Newton method fornonlinear ill-posed problems, Math. Comp., 78, 171-184 (2009).

[9] W. Ring, 1995, Identification of a core from boundary data, SIAM Journal onApplied Mathematics, v55(3), 677-706.


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Figure 1. On the left we show the result of the firsttest case of conductivity star shape support and its do-main reconstruction for exact data. The first case shapesupport is q(t) := (0.5+0.1 cos(5t)), with harmonic ref-erence potential induced by boundary source g = cos(t).From top to bottom we show the domain reconstruc-tion at initial step, fifth step, and the fiftieth step. Onthe right column, we show the similar result for the sec-ond test case of conductivity star shape support andits domain reconstruction for exact data. The secondcase shape support where q(t) is a circle with inwarddent, with harmonic reference potential due to boundarysource g = cos(3 t). Similar to the first case, we showthe reconstruction from top to bottom at initial step,fifth step, and the fiftieth step.


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Technical Tools

Let H1(1) is the sobolev space of all functions u L2(1) for whichuxi

L2(1) for i = 1, 2, endowed with the inner product

u, v

H1(1) =

u, v

L2(1) +

u

x1

,u

x1 L2(1) +

u

x2

,u

x2 L2(1).

Moreover we define the Hilbert space

H(, 1) = {v H1(1) : v L2(1)}, (2.20)with inner product

u, vH(,1) = u, vH1(1) + u, vL2(1). (2.21)Points on the boundary of 1 are identified with their correspondingangle in polar coordinates, i.e. we set 1 = {t + 2Z : t R} =R/2Z =: T. For l 0, the Sobolev space Hl(T) is defined by

Hl(T) = {f L2(T) nZ

(1 + n2)l|f(n)|2 < }, (2.22)

where

f(n) =1

2

20

f(s)einsds, (2.23)

denotes the Fourier transform of f. Hl(T) is a Hilbert space withrespect to the inner product

f, gHl(T) =

nZ(1 + n2)lf(n)

g(n) (2.24)

By Hl(T)(l 0) we denote the dual of Hl(T). With the Fouriertransform defined on Hl(T) by f(n) := f, eintHl(T),Hl(T) we find

f, eintHl(T),Hl(T) =nZ

f(n)g(n) (2.25)for the duality pairing ., .Hl(T),Hl(T) and for f Hl(T) and g Hl(T). Moreover, Hl(T) is characterized by 2.22 and the inner producton Hl(T) is given by 2.25, in both cases with l replaced by l. Wehave

f = nZ f(n)eint (2.26)for f Hl(T), l R, where the series 2.26 converges in Hl(T). We

define the differential operator D : Hl(T) Hl1(T) by

Df =nZ

inf(n)eint. (2.27)


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It is a bounded linear operator with ker(D) given by the constantfunctions on T. The space C(T) of all infinitely differentiable functionson T is characterized by

C(T) = {f : T C |n|k|f(n) 0asn for all k N}. (2.28)Those facts follows as a special case of theorems on Sobolev spaces on

smooth compact manifolds as given for example in Wloka [1].

References

[1] Wloka,1987, Partial Differential Equations, Cambridge University Press.


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IRGN

We will consider here of Newton-type methods to solve the nonlinearproblem. These methods work by local linearization of the nonlinearforward operator F, and regularization on each Newton step.

We consider a general operator equation

F(q) = f, (2.29)

where X D(F) Y is an operator, which is Frechet differentiableon its domain D(F), and X, Y are Hilbert spaces. We assume thatF is one-to-one. Let q denote the exact solution of forward opera-tor (2.29). A Newton-type methods to solve (2.29) use the linearizedforward operator equation

DF[qk]hk = f F(qk), (2.30)to find update hk = qk+1

qk in each iteration step. As the linearized

forward operator equation still inherit ill-posedness of the nonlinear for-ward operator, at each iteration, instead of solving the linearized equa-tion (2.30), we solve Tikhonov regularization to the linearized equation(2.30) for each iteration:

hk = argminhkX(DF[qk]hk + F(qk) f2 + khk + qk q0, (2.31)with an initial guess q0. The sequence {k} in (2.31) satisfies

k > 0, 1 kn+1

r, limk

k = 0, (2.32)

for some r > n.A straight forward implementation of (2.31) involves to generating

the whole derivative matrix corresponding to DF[qk].However as mentioned previously, the equation (2.30) is ill-posed

which requires regularization to solve it. Solving the regularized (2.30)using IRGN is equivalent to finding the unique least-squares solutionsof

DF[qk]kI

hk =

f F(qk)

k(qk q0)

. (2.33)

With the operator

Gk :=

DF[qk]kI

: X Y X

and gk := (fF(qk), k(qkq0))T, the equation (2.33) can be writtenas Gkhk = gk.

The algorithm can be summarized as follows


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Compute F(q0);n = 0;while k Nmax

Solve Gkhk = gkqk+1 = qk + hk;k = k + 1;

Compute F(qk)Throughout all of the reconstruction, we choose initial regularization

parameter 0 > 0, sufficiently large, and we reduced the value at eachiteration, using a rule k = 0 q(k1), 0 < q < 1, k = 1, , Nmax ,where Nmax is the maximum iteration.

In practice, as the evaluation DF[qk] sometimes is quite expensiveat each step k, one approach is by approximating DF[qk] by Ak whichis easier to compute. In some case, even using fixed A := Ak for eachiteration. The formula of IRGNM than expressed as:

hk = argminhkX(DF[q0]hk + F(qk) f2 + khk + qk q0, (2.34)The algorithm can be summarized as follows

Compute F(q0);n = 0;while k Nmax

Solve G0khk = gkqk+1 = qk + hk;k = k + 1;

Compute F(qk)Where, the operator

G0k := DF[q0]

kI : X Y X.

This method called simplified IRGNM(sIRGNM), which only latelyaddressed theoretically by Mahale & Nair [8], Jin [5], and George [3].

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Constant Conductivity Support Technical Notes