Space Discretization

The realizations of the concentration distribution C are in the space L²(D). The computation requires space discretization. We need to choose a finite dimensional subspace ofL²(D) and assume that the realizations of the concentration distribution are in that subspace. This causes a discretization error. Usually the discretization error is ignored in numerical implementations. The discretized state estimation system is assumed to represent the reality. In this subsection we want to analyse the stochastic nature of the discretization error in the case of electrical impedance process tomography.

Let {Vm}^∞m=1 be a sequence of finite dimensional subspaces of L²(D) such that Vm ⊂ Vm+1 for all m ∈ N and ∪Vm = L²(D). Since L²(D) is a separable Hilbert space, there exists such a sequence, for example, Vm may be the subspace spanned by the m first functions in an orthonormal basis of L²(D). Let {ϕ^m_l }^N_l=1^m be an orthonormal basis of Vm for all m ∈ N. We denote by (·,·) the inner product in L²(D). We define the orthogonal projectionP_m :L²(D)→ Vm form∈Nby

P_mf =

Nm

X

l=1

(f, ϕ^m_l )ϕ^m_l

for allf ∈L²(D). The subspacesVmare appropriate discretization spaces ifP_mf → f inL²(D) asm→ ∞for allf ∈L²(D), i.e., the orthogonal projectionsPmconverge strongly to the identity operator.

LetX : (Ω,F,P)→L²(D) be a random variable. Then for all ω∈Ω P_mX(ω) =

Nm

X

l=1

(X(ω), ϕ^m_l )ϕ^m_l =

Nm

X

l=1

(X^m(ω))_lϕ^m_l

where X^m(ω) := ((X(ω), ϕ^m₁ ), . . . ,(X(ω), ϕ^m_Nm))^T is an R^Nm-valued random vari-able. We view X^m as a discretized version of the random variable X at the dis-cretization level m. If X is a Gaussian random variable with mean ¯x and covari-ance Γ, then X^m is also Gaussian [21, Theorem A.5]. Furthermore, the mean of X^m is EX^m = ((¯x, ϕ^m₁ ), . . . ,(¯x, ϕ^m_Nm))^T and the covariance matrix is defined by (CovX^m)_ij := (Γϕ^m_i , ϕ^m_j ) since

E(X^m)i(X^m)j =E(X, ϕ^m_i )(X, ϕ^m_j ) = (Γϕ^m_i , ϕ^m_j ) + (¯x, ϕ^m_i )(¯x, ϕ^m_j ) for all i, j= 1, . . . , N_m.

Evolution Equation

We want to discretize the discrete evolution equation (6.16). We use the discretiza-tion levelm. We form an evolution equation for the discreteR^Nm-valued stochastic process {C_k^m}ⁿ_k=0 where C_k^m := ((C_k, ϕ^m₁ ), . . . ,(C_k, ϕ^m_Nm))^T for allk = 0, . . . , n. By using the discrete evolution equation (6.16),

(C_k+1^m )i = (C_k+1, ϕ^m_i ) = (U(∆_k)C_k+W_k+1, ϕ^m_i )

= (U(∆_k)P_mC_k, ϕ^m_i ) + (U(∆_k)(I−P_m)C_k, ϕ^m_i ) + (W_k+1, ϕ^m_i )

=

Nm

X

l=1

(U(∆_k)ϕ^m_l , ϕ^m_i )(C_k^m)_l+ (E^m_k+1)_i+ (W_k+1^m )_i

for alli= 1, . . . , Nm andk= 0, . . . , n−1 almost surely where the discrete stochastic process

E_k+1^m = ((C_k,(I−Pm)U^∗(∆_k)ϕ^m₁ ), . . . ,(C_k,(I −Pm)U^∗(∆_k)ϕ^m_Nm))^T

represent the discretization error and W_k+1^m is the state noise vector. Thus the discretized evolution equation is

C_k+1^m =A^m_k+1C_k^m+E_k+1^m +W_k+1^m (6.18) for all k= 0, . . . , n−1 almost surely where the matrixA^m_k+1 is defined by

(A^m_k+1)ij := (U(∆_k)ϕ^m_j , ϕ^m_i ) (6.19) for all i, j = 1, . . . , N_m. The discretized evolution equation (6.18) is used in the evolution updating step of the Bayesian filtering. Thereby we need to define the statistical quantities of the discrete stochastic processes{E_k+1^m }ⁿ_k=0⁻¹and{W_k+1^m }ⁿ_k=0⁻¹. The state noise W_k+1 is a Gaussian random variable with mean 0 and covariance given by Formula (6.15) for allk= 0, . . . , n−1. Hence the state noise vectorW_k+1^m is Gaussian with mean 0 and covariance matrix

(Cov(W_k+1^m ))ij =

µZ t_k+1 tk

U(t_k+1−s)QU^∗(t_k+1−s)ds ϕ^m_i , ϕ^m_j

¶

= Z t_k+1

tk

¡U(tk+1−s)QU^∗(tk+1−s)ϕ^m_i , ϕ^m_j ¢ ds for all i, j= 1, . . . , Nm and k= 0, . . . , n−1. We define the matrix Q^m_k,l(s) by

(Q^m_k,l(s))_ij := (U(t_k−s)QU^∗(t_l−s)ϕ^m_i , ϕ^m_j )

6.3. Electrical Impedance Process Tomography 113

for all i, j= 1, . . . , N_m,k, l= 0, . . . , n−1 and s∈[0, t_k∧t_l]. Then Cov(W_k+1^m ) =

Z t_k+1

tk

Q^m_k+1,k+1(s)ds (6.20) for all k = 0, . . . , n−1. Since the state noises W_k and W_l are uncorrelated for all k 6=l, by the Gaussianity the state noise vectors W_k^m and W_l^m are independent if k6=l.

We use our knowledge of the stochastic behaviour of the continuous evolution equa-tion (6.13) for the examinaequa-tion of the discretizaequa-tion errorE_k+1^m for allk= 0, . . . , n−1.

The concentration distribution C_k has a Gaussian modification with mean U(t_k)c₀ and covariance given by Formula (6.14) where t=t_k for all k= 0, . . . , n−1. Hence the discretization error E_k+1^m has a Gaussian version for all k = 0, . . . , n−1. The mean of the Gaussian version is given by

(EE_k+1^m )_i=E(C_k,(I−P_m)U^∗(∆_k)ϕ^m_i ) = (EC_k,(I −P_m)U^∗(∆_k)ϕ^m_i )

= (U(t_k)c0,(I−Pm)U^∗(∆_k)ϕ^m_i ) = (c0,U^∗(t_k)(I−Pm)U^∗(∆_k)ϕ^m_i ) for all i= 1, . . . , N_m and k= 0, . . . , n−1. Thus

(EE_k+1^m )_i = (c₀,U^∗(t_k+1)ϕ^m_i )−

Nm

X

l=1

(U(∆_k)ϕ^m_l , ϕ^m_i )(c₀,U^∗(t_k)ϕ^m_l ) for all i= 1, . . . , N_m because

U^∗(t)(I−Pm)U^∗(s)f =U^∗(t+s)f−

Nm

X

l=1

(U^∗(s)f, ϕ^m_l )U^∗(t)ϕ^m_l

=U^∗(t+s)f−

Nm

X

l=1

(U(s)ϕ^m_l , f)U^∗(t)ϕ^m_l

for all f ∈L²(D) and s, t∈[0, T]. Hence

EE_k+1^m =





(U(t_k+1)c0, ϕ^m₁ ) ...

(U(t_k+1)c0, ϕ^m_Nm)



+A^m_k+1





(U(t_k)c0, ϕ^m₁ ) ... (U(t_k)c0, ϕ^m_Nm)



 (6.21)

for all k= 0, . . . , n−1. The covariance matrix of the Gaussian version is given by (CovE_k+1^m )_ij

= (Cov(C_k)(I−Pm)U^∗(∆_k)ϕ^m_i ,(I−Pm)U^∗(∆_k)ϕ^m_j )

= (U(t_k)Γ₀U^∗(t_k)(I −P_m)U^∗(∆_k)ϕ^m_i ,(I−P_m)U^∗(∆_k)ϕ^m_j )+

+ µZ t_k

0 U(t_k−s)QU^∗(t_k−s)ds(I −P_m)U^∗(∆_k)ϕ^m_i ,(I−P_m)U^∗(∆_k)ϕ^m_j

¶

= (Γ₀U^∗(t_k)(I−P_m)U^∗(∆_k)ϕ^m_i ,U^∗(t_k)(I−P_m)U^∗(∆_k)ϕ^m_j )+

+ Z t_k

0

¡QU^∗(t_k−s)(I−P_m)U^∗(∆_k)ϕ^m_i ,U^∗(t_k−s)(I −P_m)U^∗(∆_k)ϕ^m_j ¢ ds

for all i, j= 1, . . . , N_m and k= 0, . . . , n−1. Since Γ₀ and Q are self-adjoint as the covariance operator of Gaussian random variables,

(CovE^m_k+1)_ij

= (Γ₀U^∗(t_k+1)ϕ^m_i ,U^∗(t_k+1)ϕ^m_j )+

−

Nm

X

l=1

(U(∆_k)ϕ^m_l , ϕ^m_j )(Γ₀U^∗(t_k)ϕ^m_l ,U^∗(t_k+1)ϕ^m_i )+

−

Nm

X

l=1

(U(∆_k)ϕ^m_l , ϕ^m_i )(Γ0U^∗(t_k)ϕ^m_l ,U^∗(t_k+1)ϕ^m_j )+

+

Nm

X

l,p=1

(U(∆_k)ϕ^m_l , ϕ^m_i )(U(∆_k)ϕ^m_p , ϕ^m_j )(Γ0U^∗(t_k))ϕ^m_l ,U^∗(t_k)ϕ^m_p )+

+ Z t_k

0

¡QU^∗(tk+1−s)ϕ^m_i ,U^∗(tk+1−s)ϕ^m_j ¢ ds+

−

Nm

X

l=1

(U(∆_k)ϕ^m_l , ϕ^m_j ) Z t_k

0

(QU^∗(t_k−s)ϕ^m_l ,U^∗(t_k+1−s)ϕ^m_i )ds+

−

Nm

X

l=1

(U(∆_k)ϕ^m_l , ϕ^m_i ) Z t_k

0

¡QU^∗(t_k−s)ϕ^m_l ,U^∗(t_k+1−s)ϕ^m_j ¢ ds+

+

Nm

X

l,p=1

(U(∆k)ϕ^m_l , ϕ^m_i )(U(∆k)ϕ^m_p , ϕ^m_j ) Z t_k

0

¡QU^∗(tk−s)ϕ^m_l ,U^∗(tk−s)ϕ^m_p ¢ ds

for all i, j= 1, . . . , Nm and k= 0, . . . , n−1. We define the matrix Γ^m,k_0,l by

(Γ^m,k_0,l )_ij := (U(t_k)Γ₀U^∗(t_l)ϕ^m_i , ϕ^m_j )

for all i, j= 1, . . . , N_m and k, l= 0, . . . , n−1. Then

CovE_k+1^m = Γ^m,k+1_0,k+1 −(A^m_k+1Γ^m,k+1_0,k )^T −A^m_k+1Γ^m,k+1_0,k +A^m_k+1Γ^m,k_0,k (A^m_k+1)^T+ +

Z _tk

0

Q^m_k+1,k+1(s)ds− Z _tk

0

(A^m_k+1Q^m_k+1,k(s))^T ds+

− Z _tk

0

A^m_k+1Q^m_k+1,k(s)ds+ Z _tk

0

A^m_k+1Q^m_k,k(s)(A^m_k+1)^T ds

(6.22)

for all k= 0, . . . , n−1 where the integration is done componentwise.

Since C_k andW_k+1 are independent, also C_k^m and W_k+1^m as well asE_k+1^m and W_k+1^m are mutually independent for allk= 0, . . . , n−1. On the other hand,C_k^m andE_k+1^m are correlated for allk= 0, . . . , n−1. The correlation matrix ofC_k^m andE_k+1^m can be calculated by using the continuous evolution equation (6.13) for allk= 0, . . . , n−1.

6.3. Electrical Impedance Process Tomography 115

Then

(Cor(C_k^m, E_k+1^m ))_ij = (Cov(C_k)ϕ^m_i ,(I−P_m)U^∗(∆_k)ϕ^m_j )

= (U(t_k)Γ₀U^∗(t_k)ϕ^m_i ,(I−P_m)U^∗(∆_k)ϕ^m_j )+

+ µZ tk

0 U(t_k−s)QU^∗(t_k−s)ds ϕ^m_i ,(I−P_m)U^∗(∆_k)ϕ^m_j

¶

= (Γ₀U^∗(t_k)ϕ^m_i ,U^∗(t_k)(I−P_m)U^∗(∆_k)ϕ^m_j )+

+ Z tk

0

¡QU^∗(t_k−s)ϕ^m_i ,U^∗(t_k−s)(I −P_m)U^∗(∆_k)ϕ^m_j ¢ ds for all i, j= 1, . . . , Nm and k= 0, . . . , n−1. Thus

(Cor(C_k^m, E_k+1^m ))_ij = (Γ₀U^∗(t_k)ϕ^m_i ,U^∗(t_k+1)ϕ^m_j )+

−

Nm

X

l=1

(U(∆_k)ϕ^m_l , ϕ^m_j )(Γ₀U^∗(t_k)ϕ^m_l ,U^∗(t_k)ϕ^m_i )+

+ Z _tk

0

¡QU^∗(t_k−s)ϕ^m_i ,U^∗(t_k+1−s)ϕ^m_j ¢ ds+

−

Nm

X

l=1

(U(∆_k)ϕ^m_l , ϕ^m_j ) Z _tk

0

(QU^∗(t_k−s)ϕ^m_l ,U^∗(t_k−s)ϕ^m_i )ds for all i, j= 1, . . . , N_m and hence

Cor(C_k^m, E^m_k+1)

= Γ^m,k+1_0,k −(A^m_k+1Γ^m,k_0,k )^T + Z _tk

0

¡Q^m_k+1,k(s)−(A^m_k+1Q^m_k,k(s))^T¢

ds (6.23) for all k= 0, . . . , n−1.

According to the discretized evolution equation (6.18) the random variableC_k+1^m has a Gaussian version. The mean of the Gaussian version is

EC_k+1^m =E¡

A^m_k+1C_k^m+E_k+1^m +W_k+1^m ¢

=A^m_k+1EC_k^m+EE_k+1^m (6.24) and the covariance matrix is

CovC_k+1^m = Cov¡

A^m_k+1C_k^m+E_k+1^m +W_k+1^m ¢

= Cov(A^m_k+1C_k^m) + Cor(A^m_k+1C_k^m, E_k+1^m )+

+ Cor(E_k+1^m , A^m_k+1C_k^m) + CovE_k+1^m + CovW_k+1^m

=A^m_k+1Cov(C_k^m)(A^m_k+1)^T +A^m_k+1Cor(C_k^m, E_k+1^m )+

+ Cor(C_k^m, E_k+1^m )^T(A^m_k+1)^T + CovE_k+1^m + CovW_k+1^m

(6.25)

for all k= 0, . . . , n−1.

Observation Equation

Since both the operator U and mapping g are non-linear, the space discretization of the observation equation (6.17) is far more difficult than the evolution equation (6.16), especially when we are interested in the discretization error. We assume

that the functiong is Fr´echet differentiable. Then we can linearize the observation equation (6.17). For allk= 1, . . . , n

V^k =U(g(C_k);I^k) +S^k

≈U(g(f);I^k) +U⁰(g(f);I^k)g⁰(f)(C_k−f) +S^k

=U(g(f);I^k)−U⁰(g(f);I^k)g⁰(f)f+U⁰(g(f);I^k)g⁰(f)C_k+S^k

where f ∈L²(D). The point f in which the linearization is done should be chosen wisely. It may be, for example, the mean of the initial value. The linearization induces error. However, in future we ignore the linearization error. We denoteb_k:=

U(g(f);I^k)−U⁰(g(f);I^k)g⁰(f)f and B_k := U⁰(g(f);I^k)g⁰(f) for all k = 1, . . . , n.

Thenb_k∈R^LandB_k:L²(D)→R^Lis a bounded linear operator for allk= 1, . . . , n.

The linearized observation equation is

V^k=B_kC_k+b_k+S^k (6.26)

for all k= 1, . . . , n. Then the discretized observation equation is

V^k =B_kPmC_k+B_k(I−Pm)C_k+b_k+S^k= [B_kϕ]C_k^m+Ek^m+b_k+S^k (6.27) for allk = 1, . . . , n where [B_kϕ] := [B_kϕ^m₁ . . . B_kϕ^m_Nm] is the L×N_m matrix whose l^th column is B_kϕ^m_l for all l = 1, . . . , N_m and E_k^m := B_k(I −P_m)C_k represents the discretization error. The discretized observation equation (6.27) is used in the observation updating step of the Bayesian filtering. We need to define the statistical quantities of the processes {E_k^m}ⁿ_k=1 and{V^k}ⁿ_k=1.

We use our knowledge of the stochastic behaviour of the continuous evolution equa-tion (6.13). We assume that the process S(t), t ≥ 0, is a Gaussian process inde-pendent of the process C(t), t ≥0. Then S^k is independent of C_k^m and E_k^m for all k = 1, . . . , n. On the other hand, C_k^m and E_k^m are correlated for all k = 1, . . . , n.

The concentration distribution C_k has a Gaussian modification with mean U(t_k)c₀ and covariance given by Formula (6.14) wheret=t_k for allk= 1, . . . , n. Hence the discretization errorE_k^m has a Gaussian version for allk= 1, . . . , n. The mean of the Gaussian version is

EEk^m =B_k(I −P_m)U(t_k)c₀ and the covariance matrix is

CovEk^m =B_k(I−P_m)U(t_k)Γ₀U^∗(t_k)(I−P_m)B^∗_k+ +

Z _tk

0

B_k(I−P_m)U(t_k−s)QU^∗(t_k−s)(I−P_m)B_k^∗ds

for all k = 1, . . . , n. The correlation matrix of C_k^m and E_k^m can be calculated by using the continuous evolution equation (6.13) for allk= 1, . . . , n. First of all,

Cor([B_kϕ]C_k^m,Ek^m) = Cor(B_kP_mC_k, B_k(I−P_m)C_k)

=B_kPmCov(C_k)(I−Pm)B_k^∗

=B_kP_mU(t_k)Γ₀U^∗(t_k)(I−P_m)B_k^∗+ +

Z t_k

0

B_kP_mU(t_k−s)QU^∗(t_k−s)(I−P_m)B_k^∗ ds

6.3. Electrical Impedance Process Tomography 117

for all k= 1, . . . , n. If the matrix [B_kϕ] is invertable,

Cor(C_k^m,Ek^m) = [B_kϕ]⁻¹Cor([B_kϕ]C_k^m,Ek^m) for all k= 1, . . . , n.

In the observation updating step of the Bayesian filtering we need the joint prob-ability distribution of C_k^m and V^k for all k= 1, . . . , n. By the continuous evolution equation (6.13) the random variable C_k^m has a Gaussian version. According to the discretized observation equation (6.27) the random variableV^k has a Gaussian ver-sion and the joint probability distribution of C_k^m and V^k is Gaussian. In addition, the mean of the Gaussian version of V^k is

EV^k=E³

[B_kϕ]C_k^m+Ek^m+b_k+S^k´

= [B_kϕ]EC_k^m+EEk^m+b_k+ES^k (6.28) and the covariance matrix is

CovV^k= Cov³

[B_kϕ]C_k^m+Ek^m+b_k+S^k´

= Cov([B_kϕ]C_k^m) + Cor([B_kϕ]C_k^m,Ek^m)+

+ Cor(Ek^m,[B_kϕ]C_k^m) + Cov(Ek^m) + Cov(S^k)

= [B_kϕ] Cov(C_k^m)[B_kϕ]^T + Cor([B_kϕ]C_k^m,Ek^m)+

+ Cor([B_kϕ]C_k^m,Ek^m)^T + Cov(Ek^m) + Cov(S^k)

(6.29)

for all k= 1, . . . , n. The correlation matrix of C_k^m and V^k is Cor(C_k^m, V^k) = Cor³

C_k^m,[B_kϕ]C_k^m+Ek^m+b_k+S^k´

= Cor(C_k^m,[B_kϕ]C_k^m) + Cor(C_k^m,Ek^m)

= Cov(C_k^m)[B_kϕ]^T + [B_kϕ]⁻¹Cor([B_kϕ]C_k^m,Ek^m)

(6.30)

for all k= 1, . . . , n.

Bayesian Filtering

The discretized state estimation system concerning the electrical impedance process tomography problem is

C_k+1^m =A^m_k+1C_k^m+E_k+1^m +W_k+1^m , k= 0, . . . , n−1, (6.31) V^k = [B_kϕ]C_k^m+Ek^m+b_k+S^k, k = 1, . . . , n. (6.32) The state noise vectorsW_k^mandW_l^mare mutually independent and also independent of C₀^m for all k 6= l. We assume that the observation noise vectors S^k are chosen such a way thatS^kandS^lare mutually independent and also independent ofC₀^m for allk6=l andS^k and W_l^m are mutually independent for all k, l= 1, . . . , n. Then the stochastic processes {C_k^m}ⁿ_k=0 and {V^k}ⁿ_k=1 form an evolution–observation model.

Therefore we may use the Bayesian filtering method.

In the evolution updating step of the Bayesian filtering it is assumed that we know the conditional probability density ofC_k^mwith respect to some measurementsD_k:=

{v¹, v², . . . , v^k}. We need to calculate the conditional probability density of C_k+1^m

with respect to the dataD_k. We suppose that the conditional expectationE(C_k^m|D_k) is a Gaussian random variable with mean ¯c_kand covariance matrix Γ_k. According to the discretized evolution equation (6.31) we are able to present the joint distribution ofC_k^mand C_k+1^m conditioned on the measurementsD_kand know that it is Gaussian.

By Theorems 6.3 and 6.4 and Formulas (6.24) and (6.25) the Gaussianity of the joint probability density implies that the conditional marginal probability density of C_k+1^m is Gaussian with mean

¯

c_k+1 :=A^m_k+1¯c_k+EE_k+1^m (6.33) and covariance matrix

Γ_k+1 :=A^m_k+1Γ_k(A^m_k+1)^T +A^m_k+1Cor(C_k^m, E_k+1^m )+

+ Cor(C_k^m, E_k+1^m )^T(A^m_k+1)^T + CovE_k+1^m + CovW_k+1^m . (6.34) Thus the evolution updating step is defined if we are able to calculate the vector EE_k+1^m and matrices A^m_k+1, CovE_k+1^m , CovW_k+1^m and Cor(C_k^m, E_k+1^m ) given by For-mulas (6.19)–(6.23) for all k= 0, . . . , n−1.

In the observation updating step of the Bayesian filtering it is assumed that we know the conditional probability density of C_k+1^m with respect to some measured data D_k := {v¹, v², . . . , v^k}. A new measurement v^k+1 is obtained. We need to calculate the conditional probability density of C_k+1^m with respect to measure-ments D_k+1 := {v¹, v², . . . , v^k+1}. We suppose that the conditional expectation E(C_k+1^m |D_k) is a Gaussian random variable with mean ¯c_k+1 and covariance matrix Γ_k+1. By Theorems 6.3 and 6.4 the conditional probability density of C_k+1^m with respect to the dataD_k+1 is Gaussian with mean

˜

c_k+1 := ¯c_k+1+ Cor(C_k+1^m , V^k+1) Cov(V^k+1)⁻¹(v^k+1−EV^k+1) (6.35) and covariance matrix

Γe_k+1 = Γ_k+1−Cor(C_k+1^m , V^k+1) Cov(V^k+1)⁻¹Cor(C_k+1^m , V^k+1)^T. (6.36) Thus the observation updating step is defined if we are able to calculate the vector EV^k+1 and matrices Cov(V^k+1) and Cor(C_k+1^m , V^k+1) given by Formulas (6.28)–

(6.30) for all k= 0, . . . , n−1.

The evaluation of the matrices needed in the Bayesian filtering method depends on the discretization space Vm, analytic semigroup {U(t)}t≥0, function c₀, operators Γ₀,Qand B_k, vectorb_k and statistics of the observation noiseS for allk= 1, . . . , n.

Usually the discretization space Vm is chosen such a way that the projectionPm is fairly easy to calculate. For example, the basis functionsϕ^m_l have compact supports and they are piecewise polynomial. The function c₀ and operator Γ₀ represent our prior knowledge of the concentration distribution. The mean c0 should illustrate the expected concentration distribution in the pipe and hence it depends heavily on the application. Since the diffusion is a smoothing operation, we may assume that the initial state is rather smooth. Thus the covariance operator Γ0 should have some smoothness properties. Our certainty of the time evolution model is coded into the Wiener process and hence into the operatorQ. The choice ofQdepends on the application. The crucial factor in the evaluation of the matrices is the analytic semigroup{U(t)}t≥0. Since it is defined by Formula (2.3), only in some special cases we can present the analytic semigroup in a closed form. In Subsection 6.3.5 we

6.3. Electrical Impedance Process Tomography 119

study the one dimensional version of the problem. Then the analytic semigroup is a convolution operator. The operatorB_k and vectorb_k for allk= 1, . . . , nare related to the measurement situation. We need to be able to solve the complete electrode model for a known concentration distribution and also to calculate the Fr´echet de-rivatives of mappings U and g for that concentration distribution. The function g depends on the application. At least for strong electrolytes and multiphase mixtures relations between the conductivity and concentration distribution are studied and discussed in the literature. The observation noise S represents the accuracy of the measurement equipment.

In document A MATHEMATICAL MODEL FOR ELECTRICAL IMPEDANCE PROCESS TOMOGRAPHY (sivua 123-131)