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Numerical studies of recovery chances for a simplified EIT problem TU Chemnitz - Faculty of Mathematics

Numerical studies of recovery chances for a simplified EIT problem

Christopher Hofmann joint work with Bernd Hofmann and Roman Unger

TU Chemnitz - Faculty of Mathematics

2nd November 2017

TUC · 2nd November 2017 · Christopher Hofmann 1 / 28 www.tu-chemnitz.de/mathematik/inverse probleme/

Content

• 1. Motivation
• 2. General EIT Model
• 3. Simplified Model
• 4. Numerical Examples
• 5. Conclusions
• 6. References

TUC · 2nd November 2017 · Christopher Hofmann 2 / 28 www.tu-chemnitz.de/mathematik/inverse probleme/

Motivation

Introduction to EIT

What is EIT?

◮ Apply current on the specimen’s boundary. ◮ Take voltage measurements on the boundary of the specimen. ◮ Reconstruct electric conductivity within the specimen from these measurements. ◮ Practical applications mostly in medical imaging. ◮ Focus often lies on the detection of inclusions.

Figure: Specimen to take EIT measurements.1

1Loyola, et al: Detection of spatially distributed damage in fiber-reinforced polymer composites

TUC · 2nd November 2017 · Christopher Hofmann 3 / 28 www.tu-chemnitz.de/mathematik/inverse probleme/

Motivation

Detection of mechanical strain in CNT/CNF

Case study motivated by an application on the detection of mechanical strain within a CNT/CNF specimen. Opposed to many previous approaches it is therefore assumed:

◮ ’Stripe’ structure with piecewise constant conductivities ◮ No a priori information on background conductivity TUC · 2nd November 2017 · Christopher Hofmann 4 / 28 www.tu-chemnitz.de/mathematik/inverse probleme/

Motivation

Overview of specimen

Figure: Specimen with electrodes and finite element grid.

TUC · 2nd November 2017 · Christopher Hofmann 5 / 28 www.tu-chemnitz.de/mathematik/inverse probleme/

General EIT Model

General EIT Model

◮ Ω ∈ R2, with smooth boundary ∂Ω ◮ Conductivity σ(x), x ∈ Ω, electric potential u(x), x ∈ Ω, ◮ σ∂νu |∂Ω can be interpreted as current ◮ Laplace equation holds in the interior of Ω:

∇ · (σ(x)∇u(x)) = 0 (+)

TUC · 2nd November 2017 · Christopher Hofmann 6 / 28 www.tu-chemnitz.de/mathematik/inverse probleme/

General EIT Model

Neumann-to-Dirichlet map

For practical applications it is desirable to apply current and to measure voltages/potential differences: Λσ : L2

⋄(∂Ω) → L2 ⋄(∂Ω),

g|∂Ω → ug|∂Ω with ug weak solution of the Laplace equation with Neumann boundary values σ∂νu |∂Ω = g |∂Ω with L∞

+ (Ω) := {σ ∈ L∞(Ω) : inf x∈Ω σ(x) > 0}

and L2

⋄(∂Ω) := {g ∈ L2(∂Ω) :

• ∂Ω

gds = 0} .

TUC · 2nd November 2017 · Christopher Hofmann 7 / 28 www.tu-chemnitz.de/mathematik/inverse probleme/

General EIT Model

Forward Operator

The forward operator in this model situation is then given by A : L∞

+ (Ω) → L(L2 ⋄(∂Ω)),

σ → Λσ. Inverse Problem: Retrieve σ(x), x ∈ Ω, from data of the current-to-voltage map Λσ.

TUC · 2nd November 2017 · Christopher Hofmann 8 / 28 www.tu-chemnitz.de/mathematik/inverse probleme/

Simplified Model

Simplified and Discretized Model

In practice it is obviously impossible to retrieve measurements on the whole boundary ∂Ω of Ω. Therefore we assume:

◮ K electrodes ǫk, K k=1 ǫk ⊂ ∂Ω ◮ Ik and Uk: current and voltage on the k-th electrode ◮ Steady state: K k=1 Ik = 0 (in- and outgoing currents add up to zero)

We further assume, that the conductivity σ is isotropic.

TUC · 2nd November 2017 · Christopher Hofmann 9 / 28 www.tu-chemnitz.de/mathematik/inverse probleme/

Simplified Model

Electrode Modelling

Shunt Electrode Model:

◮ No current flows outside electrodes: σ∂νu|∂Ω\K

k=1 ǫk = 0

◮ Current on electrode ǫk is equally distributed with overall current

Ik =

• ǫk σ∂νu|∂Ωds.

◮ σ∂νu|ǫk = Ik |ǫk|, with arclength |ǫk| of the electrode ǫk. ◮ It is further assumed that the potential on every electrode is constant,

i.e. u |ǫk = const.

TUC · 2nd November 2017 · Christopher Hofmann 10 / 28 www.tu-chemnitz.de/mathematik/inverse probleme/

Simplified Model

Discretized Forward Operator

The solution of Laplace problem (+) is not unique, it is assumed that the potentials add up to zero as well. With RK

⋄ = {x ∈ RK : K k=1 xk = 0} the mapping

Rσ : (Ik)K

k=1 ∈ RK ⋄

→ (Uk)K

k=1 ∈ RK ⋄

is then the basis for required sets of measurements.

TUC · 2nd November 2017 · Christopher Hofmann 11 / 28 www.tu-chemnitz.de/mathematik/inverse probleme/

Simplified Model

Discretized Forward Operator

◮ Discretize Ω ∈ R2 with triangular mesh with 32 boundary edges. ◮ K = 8 electrodes ǫi (i = 1, .., 8) ◮ Neumann boundary conditions on two neighbouring electrodes: σ∂νu|ǫi = 1 and

σ∂νu|ǫi+1 = −1

◮ Set Dirichlet boundary condition u(x) = 0 for one arbitrary chosen boundary

edge which is not an electrode to overcome non-uniqueness.

◮ Rotate electrodes where current flows and repeatedly solve the PDE until the

starting position is reached.

TUC · 2nd November 2017 · Christopher Hofmann 12 / 28 www.tu-chemnitz.de/mathematik/inverse probleme/

Simplified Model

Recalling the model geometry

Figure: Specimen with electrodes and finite element grid.

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Simplified Model

Discretized Forward Operator

Full set of measurements,

◮ Ω discretized into n ‘stripes’. ◮ Electrodes are rotated. ◮ The associated elliptic problem is solved in a repeated manner.

This results in the following nonlinear operator: σ = (σ1, ..., σn)T ∈ Rn → F(σ) ∈ R8×8 Note: Every column of F is one set of measurements.

TUC · 2nd November 2017 · Christopher Hofmann 14 / 28 www.tu-chemnitz.de/mathematik/inverse probleme/

Simplified Model

Least-Squares Minimization

Assume that σ∗ ∈ Rn

+ is the ‘true conductivity vector’ to be identified:

σδ

LS = arg min σ∈Q

F(σ) − F δ(σ∗)F (∗) where

◮ Q ⊂ Rn + is the set of admissible solutions. ◮ · F designates the Frobenius norm. ◮ F δ(σ∗) indicates noisy data associated with some noise level δ > 0. TUC · 2nd November 2017 · Christopher Hofmann 15 / 28 www.tu-chemnitz.de/mathematik/inverse probleme/

Simplified Model

Noise Model

Additive noise model: F δ(σ∗) = F(σ∗) + E with E = (εij) ∈ R8×8, ǫij i.i.d. and ǫij ∼ N(0, d2) Chose d so that: E F δ(σ∗) − F(σ∗)2

F

F(σ∗)2

F

• = δ2,

which leads to: d = δ

8F(σ∗)F . TUC · 2nd November 2017 · Christopher Hofmann 16 / 28 www.tu-chemnitz.de/mathematik/inverse probleme/

Numerical Examples

Two unknown conductivities

Ω discretized with a stripe structure, here n=2, with conductivities σ1 and σ2. Forward calculations for all admissible values in Q = {(σ1, σ2) ∈ [10, 75] × [5, 46]} .

Figure: Material ‘stripes’ with two unknown conductivities.

TUC · 2nd November 2017 · Christopher Hofmann 17 / 28 www.tu-chemnitz.de/mathematik/inverse probleme/

Numerical Examples

Two unknown conductivities

Discrepancy norm F(σ) − F(σ∗)F depending on σ = (σ1, σ2)T for σ∗ = (37.7 , 7.9)T .

Figure: Perspective drawing and level sets of discrepancy norm F(σ) − F(σ∗)F depending

• n σ = (σ1, σ2)T .

TUC · 2nd November 2017 · Christopher Hofmann 18 / 28 www.tu-chemnitz.de/mathematik/inverse probleme/

Numerical Examples

Two unknown conductivities

Discrepancy norm F(σ) − F(σ∗)F depending on σ = (σ1, σ2)T for σ∗ = (6 , 5)T .

Figure: Level sets of discrepancy norm F(σ) − F(σ∗)F depending on σ = (σ1, σ2)T .

TUC · 2nd November 2017 · Christopher Hofmann 19 / 28 www.tu-chemnitz.de/mathematik/inverse probleme/

Numerical Examples

Five unknown conductivities

Ω discretized in n=5 ’stripes’. Numerical evidence suggests that very different conductivies lead to very similar measurements:

Figure: left: σ(1) = (4.26, 17.33, 7.65, 0.99, 1.00)T , right: σ(2) = (4.27, 23.87, 4.34, 50.00, 28.99)T , F(σ(1)) − F(σ(2))2

F = 0.000099.

TUC · 2nd November 2017 · Christopher Hofmann 20 / 28 www.tu-chemnitz.de/mathematik/inverse probleme/

Numerical Examples

Five unknown conductivities

f(λ) := F(σ∗ + λ(σ(3) − σ∗)) − F δ(σ∗)F , λ ∈ [−5, 1], for σ∗ = (7.53, 22.23, 14.28, 4.26, 4.99)T and σ(3) = (7.53, 45.09, 12.63, 4.26, 4.99)T .

Figure: Graph of f(λ) for λ ∈ [−5, 1] without noise (δ = 0) and with 5% noise (δ = 0.05).

TUC · 2nd November 2017 · Christopher Hofmann 21 / 28 www.tu-chemnitz.de/mathematik/inverse probleme/

Numerical Examples

Five unknown conductivities

Level sets of F(σ) − F(σ∗)F , for σ∗ = (7.53, 22.23, 14.28, 4.26, 4.99)T with respect to the second and third coordinate.

Figure: Level sets Lc = {(σ2, σ3) : F(σ) − F δ(σ∗)F = c} with δ = 0 and δ = 0.1 for fixed σ∗

1, σ∗ 4 and σ∗ 5.

TUC · 2nd November 2017 · Christopher Hofmann 22 / 28 www.tu-chemnitz.de/mathematik/inverse probleme/

Numerical Examples

Error-to-noise ratios

Use Levenberg-Marquardt Algorithm to solve the least-squares problem: σδ

• pt = arg min

σ∈Q

F(σ) − F δ(σ∗)F . σdelta

• pt

denotes the solution found by the algorithm. The following examples are generated with:

◮ σ∗ = (37.7, 7.9, 10.7, 18.2, 5.6)T - exact conductivity solution ◮ σstart = (9, 32, 7, 1, 37)T - start value for the Levenberg-Marquardt iteration ◮ Q ≈ [1, 50]5

As the Jacobian has to be calculated in every step of the iteration process, which in turn requires multiple calculations of forward operator matrices, precalculated values for F in connection with multi-linear interpolation were used.

TUC · 2nd November 2017 · Christopher Hofmann 23 / 28 www.tu-chemnitz.de/mathematik/inverse probleme/

Numerical Examples

Error-to-noise ratios

mean value random reconstruction mean random noise level noise level error error-to-noise ratio error-to-noise ratio δ

F (σ∗)−F δ(σ∗)F F (σ∗)F σopt−σ∗2 σ∗2 σopt−σ∗2 σ∗2

/δ

σopt−σ∗2 σ∗2

/F (σ∗)−F δ(σ∗)F

F (σ∗)F

0.0100 0.0181 0.0029 0.2925 0.1876 0.0250 0.0361 0.0045 0.1789 0.1403 0.0500 0.0662 0.0076 0.1516 0.1236 0.1000 0.1199 0.0142 0.1421 0.1181 0.1500 0.1791 0.0213 0.1421 0.1181 0.2000 0.2372 0.0285 0.1425 0.1184 0.2500 0.2935 0.0360 0.1441 0.1198

Table: List of stable error-to-noise ratios.

TUC · 2nd November 2017 · Christopher Hofmann 24 / 28 www.tu-chemnitz.de/mathematik/inverse probleme/

Numerical Examples

Condition Numbers

If Jh(σ∗) ∈ R64×5 is the approximated Jacobian to F(σ∗) and s1(σ∗) ≥ s2(σ∗) ≥ s3(σ∗) ≥ s4(σ∗) ≥ s5(σ∗) > 0 its singular values, then the condition number is given by κ := s1(σ∗) s5(σ∗) .

TUC · 2nd November 2017 · Christopher Hofmann 25 / 28 www.tu-chemnitz.de/mathematik/inverse probleme/

Numerical Examples

Condition Numbers for varying σ∗

condition number σ∗

1

σ∗

2

σ∗

3

σ∗

4

σ∗

5

6.74 5 5 5 5 5 6.76 50 50 50 50 50 14.73 3 4 5 6 7 14.87 30 40 50 60 70 12.55 300 400 500 600 700 4.07 1 30 1 30 1 4.42 10 300 10 300 10 3.99 100 3000 100 3000 100

Table: List of condition numbers of Jacobian Jh(σ∗) (h = 0.01) for varying σ∗.

TUC · 2nd November 2017 · Christopher Hofmann 26 / 28 www.tu-chemnitz.de/mathematik/inverse probleme/

Conclusions

Conclusions

The numerical case studies suggest the following assertions:

◮ The functional F(σ) − F δ(σ∗)F has no local minima. ◮ Least-squares approach (∗) has indeed a global minimum. ◮ With small discretizations, regularization by discretization is sufficient and no

further stabilization is required.

TUC · 2nd November 2017 · Christopher Hofmann 27 / 28 www.tu-chemnitz.de/mathematik/inverse probleme/

References

• C. Hofmann, B. Hofmann, R. Unger.

Numerical Studies of Recovery Chances for a Simplified EIT Problem. To appear in New Trends in Parameter Identification for Mathematical Models (Eds.: B. Hofmann, A. Leitao, J. P. Zubelli). Birkh¨ auser, Basel 2018.

• O. Kanoun et al.

Flexible carbon nanotube films for high performance strain sensors Sensors, 14: 10042–10071, 2014.

• B. Harrach, J.K. Seo.

Detecting inclusions in electrical impedance tomography without measurements. SIAM Journal on Applied Mathematics, 69(6):1662.1681, 2009.

• K. Paulson, W.Breckon, M. Pidcock.

Electrode modelling in electrical impedance tomography SIAM Journal on Applied Mathematics, 52(4):1012.1022, 1992.

TUC · 2nd November 2017 · Christopher Hofmann 28 / 28 www.tu-chemnitz.de/mathematik/inverse probleme/