Uncertainty quantification for electrical impedance tomography

with resistor networks

L. Borcea1, F. Guevara Vasquez2 and A.V. Mamonov3

1Computational and Applied Mathematics, Rice University,

MS 134, 6100 Main St. Houston, TX 77005-1892, USA2Department of Mathematics, University of Utah,

155 S 1400 E RM 233, Salt Lake City, UT 84112-0090, USA3 Institute for Computational Engineering and Sciences, University of Texas at Austin,

1 University Station C0200, Austin, TX 78712, USA

E-mail: borcea@rice.edu, fguevara@math.utah.edu and mamonov@ices.utexas.edu

Abstract. We present a Bayesian statistical study of the numerical solution of the two dimensional

electrical impedance tomography problem, with noisy measurements of the Dirichlet to Neumann map.

The inversion uses parametrizations of the conductivity on optimal grids that are computed as part of

the problem. The grids are optimal in the sense that finite volume discretizations on them give spectrally

accurate approximations of the Dirichlet to Neumann map. The approximations are Dirichlet to Neumann

maps of special resistor networks, that are uniquely recoverable from the measurements. Inversion on optimal

grids has been proposed and analyzed in [9, 12]. However, the study of noise effects on the inversion has not

been carried out. In this paper we present a statistical study of both the linearized and the nonlinear inverse

problem. The linearization is about a constant conductivity. We take three different parametrizations of

the unknown conductivity perturbations, with the same number of degrees of freedom. We obtain that the

parametrization induced by the inversion on optimal grids is the most efficient of the three, because it gives

the smallest standard deviation of the estimates, uniformly in the domain. For the nonlinear problem we

compute the mean and variance of the maximum aposteriori estimates of the conductivity, on optimal grids.

For small noise, we obtain that the estimates are unbiased and their variance is very close to the optimal one,

given by the Cramer-Rao bound. For larger noise we use regularization and quantify the trade-off between

reducing the variance and introducing bias in the solution. The study considers the full measurement setup,

with access to the entire boundary, and the partial measurement setup, where only a subset of the boundary

is accessible. We also introduce an inversion algorithm on optimal grids for a two-sided partial measurements

setup that has not been considered before. The term two-sided refers to the accessible boundary consisting

of two disjoint parts.

1. Introduction

We study the inverse problem of electrical impedance tomography (EIT) in two dimensions, with noisy

measurements of the Dirichlet to Neumann (DtN) map. Explicitly, we seek the positive and bounded, scalar

valued coefficient σ(x) in the elliptic equation

∇ · [σ(x)∇u(x)] = 0, x ∈ Ω. (1.1)

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The domain Ω is bounded, simply connected, with smooth boundary B. By the Riemann mapping theorem

all such domains in R2 are conformally equivalent, so from now on we take for Ω the unit disk. We call σ(x)

the conductivity and u ∈ H1(Ω) the potential, satisfying the boundary conditions

u(x) = V (x), x ∈ B, (1.2)

for arbitrary V ∈ H1/2(B). The data are finitely many noisy measurements of the DtN map Λσ : H1/2(B)→H−1/2(B), which takes the boundary potential V to the normal boundary flux (current)

ΛσV (x) = σ(x)∂u(x)

∂n, x ∈ B. (1.3)

We consider both the full boundary setup, where ΛσV is measured all around the boundary B, and the partial

boundary setup, where the measurements are confined to an accessible subset BA ⊂ B, and the remainder

BI = B \ BA of the boundary is assumed grounded (V |BI = 0).

In theory, full knowledge of the DtN map Λσ determines uniquely σ, as proved in [35, 13] under some

smoothness assumptions on σ, and in [5] for bounded σ. The result extends to the partial boundary setup,

at least for σ ∈ C3+ε(Ω), ε > 0 as established in [24]. In practice, the difficulty lies in the exponential

instability of EIT. It is shown in [1, 7, 34] that given two sufficiently regular conductivities σ1 and σ2, the

best possible stability estimate is of logarithmic type

‖σ1 − σ2‖L∞(Ω) ≤ c∣∣log ‖Λσ1

− Λσ2‖H1/2(B),H−1/2(B)

∣∣−α , (1.4)

with some positive constants c and α. This means that even if the noisy data is consistent, i.e. it is in the

set of DtN maps, we need exponentially small noise to get a conductivity that is close to the true one.

In practice, we have finitely many measurements and we cannot expect to have exponentially small

noise. The inversion can be carried out only by imposing some regularization constraints on σ. Typical

regularization amounts to penalizing some norm of σ or its gradient [21] in a numerical optimization scheme

that fits the data in the least squares sense. The penalty is based on some prior assumption on the regularity

of σ, that may not be justified in general, and may lead to oversmoothed estimates with poor contrasts.

The computational cost is also high, specially for parametrizations with many degrees of freedom, such as

discretizations of σ on fine grids.

The stability of the numerical reconstructions of σ is strongly dependent on their parametrization. The

more parameters we seek, the more unstable the estimation, and the more need for regularization with

artificial penalties. We control partially the stability of the numerical estimates of σ by restricting their

number of degrees of freedom.

It is shown in [2] that if σ has finitely many degrees of freedom, more precisely if it is piecewise constant

with a bounded number of unknown values, then the stability estimates on σ are of Lipschitz type. However,

it is not clear how the Lipschitz constant grows depending on the distribution of the unknowns in Ω. For

example, it should be much easier to determine the value of σ near the boundary than in a small set in the

interior of Ω.

An important question in numerical inversion is how to find parametrizations of σ that capture the

trade-off between stability and resolution as we move away from the boundary, where the measurements are

made. On one hand, the parametrizations should be sparse, with a small number of degrees of freedom. On

the other hand, the parametrizations should be adaptively refined toward the boundary.

Adaptive parametrizations for EIT have been proposed in [28, 32] and in [3, 4]. The first approaches

use distinguishability grids that are defined with a linearization argument. The data is divided in certain

2

subsets that are then used to assemble the grids step by step by seeking the smallest single inclusion that is

distinguishable from a constant conductivity. For example, the grids in [28] are built in a disk shaped Ω one

layer at a time, by finding the smallest radius of a concentric inclusion that can be distinguished with the

n-th Fourier mode of the boundary excitation. The grids are refined near the boundary, as expected, but

the accuracy of the linearization is not understood and the regularization effect in numerical inversion has

not been studied in detail. The approach in [3, 4] is nonlinear. It considers piecewise constant conductivities

on subsets (zones) of Ω. The adaptivity consists in the iterative coarsening or refinement of the domain

segmentation (zonation), driven by the gradient of a least squares data misfit functional. The functional

is minimized for each update of the zonation, so the method can become computationally costly if many

updates of the conductivity are needed.

We follow the approach in [9, 37, 11, 12] and parametrize σ on optimal grids. The number of parameters

is limited by the noise level in the measurements and their geometrical distribution in Ω is determined as

part of the inversion. The grids are based on rational approximations of the DtN map. We call them optimal

because they give spectral accuracy of approximations of Λσ with finite volume schemes. The grids turn out

to be refined near the accessible boundary, where we make the measurements, and coarse away from it, thus

capturing the expected loss of resolution of the reconstructions of σ.

Optimal grids were introduced in [6, 19, 20, 26] for accurate approximations of the DtN map in forward

problems. The first inversion on optimal grids was proposed in [8], for Sturm-Liouville inverse spectral

problems in one dimension. The analysis of the inversion is in [10]. It shows that optimal grids provide a

necessary and sufficient condition for convergence of solutions of discrete inverse spectral problems to the

true solution of the continuum one. The numerical solution of EIT on optimal grids was introduced in

[9, 23] for the full boundary measurements case, and in [11, 12, 33] for partial boundary measurements. The

inversion in [9, 23, 11, 12, 33] is based on the rigorous theory of discrete inverse problems for circular planar

resistor networks [14, 15, 25, 17, 18]. The networks arise in five point stencil, finite volumes discretizations

of equation (1.1) on the optimal grids computed as part of the inversion. The networks are called critical

because they have no redundant connections, and they are uniquely determined by discrete measurements of

the continuum DtN map [27, 9]. Just as in the continuum EIT, the inverse problem for networks is ill-posed,

and there is a trade-off between the size of the network and the stability of the reconstruction.

In this paper we present a statistical study of the inversion algorithms on optimal grids, for noisy

measurements of the DtN map. We fix the number g of degrees of freedom, and analyze the effect of the

adaptive parametrization of σ on the uncertainty of the estimates. We use Bayesian analysis [22, 29] for the

linearized problem about a constant conductivity, and for the nonlinear problem. The noise is mean zero

Gaussian, and if its standard deviation is small, the only prior on σ is that it is positive and bounded. For

larger noise we use regularization (Gaussian priors), and study how the parametrization affects the trade-off

between the stability of the result and the bias.

The linearization study considers three different parametrizations of σ, with g degrees of freedom. The

first two are piecewise linear, on an equidistant grid and on the optimal grid. The only relation between

the second parametrization and resistor network inversion on optimal grids is the location of the grid nodes.

The third parametrization is also based on the optimal grid, but the basis functions are derived from the

linearization of the mapping introduced in [9, 23, 11, 12, 33] for estimating σ via the resistor networks. We

compute the standard deviation of the estimated fluctuations and show that the third parametrization is

clearly superior. It gives estimates with uniformly small standard deviation in Ω. The conclusion is that it is

3

not enough to distribute the parameters on the optimal grid to obtain good results. To control the stability

of the reconstructions, we also need to use proper basis functions.

In the nonlinear statistical study we compute maximum a posteriori estimates of σ with the inversion

algorithms on optimal grids. These estimates are random, because of the noise. We assess their quality by

displaying pointwise in Ω their mean and standard deviation. We obtain that the resistor network based

inversion is efficient in the sense that it gives unbiased estimates of σ, with variance that is very close to

the optimal one, given by the Cramer-Rao bound [36]. This is for small noise. For larger noise we use

regularization priors that introduce some bias in the solution. We also compare the network based inversion

to the usual optimization approach that seeks the conductivity as the least squares minimizer of the data

misfit. For the optimization, the conductivity is piecewise linear with the same number g of degrees of

freedom, either on a uniform grid or on the optimal grid. Our numerical experiments indicate that for a

fixed uncertainty (standard deviation) in the reconstructions, the network based method gives reconstructions

that are closer in average to the true conductivity (i.e. with less bias). The conclusion for the non-linear

problem is similar to that for the linearized problem: the uncertainty in the reconstructions is reduced with

the network based inversion as compared to optimization on either equidistant or optimal grids.

Our statistical study considers both the full and partial measurement setups addressed in [9, 23, 11,

12, 33]. We also introduce a two-sided partial measurement setup that has not been studied before in the

context of inversion on optimal grids. It corresponds to an accessible boundary BA consisting of two disjoint

subsets of B.

The paper is organized as follows: We begin in section 2 with the description of the inversion approach

on optimal grids. The inversion uses critical resistor networks with topology designed specifically for each

measurement setup. The Bayesian estimation framework is described in section 3. The estimation results

are in section 4. We end with a summary in section 5.

2. EIT with resistor networks

The resistor networks used in our inversion algorithms are reduced models of (1.1) that are uniquely

recoverable from discrete measurements of the continuum DtN map, as described in section 2.1. They allow

us to estimate the conductivity σ parametrized on optimal grids, as explained in section 2.2. The network

topology (i.e. the underlying graph) and the optimal grids are strongly dependent on the measurement setup.

We review in sections 2.3 and 2.4 the full and partial boundary setups considered in [9, 23] and [12, 33],

respectively. Section 2.5 is concerned with the new results for the two-sided partial boundary measurements

case.

2.1. The discrete inverse problem for resistor networks

Resistor networks arise naturally in finite volume discretizations of equation (1.1), on staggered grids as in

Figure 1, with interlacing primary and dual lines that may be curvilinear. The potential is discretized at

the primary nodes Pi,j , the intersections of the primary grid lines, ui,j ≈ u(Pi,j). Each node Pi,j ∈ Ω is

surrounded by a dual cell Ci,j , with vertices (dual points) Pi± 12 ,j±

12, and boundary

∂Ci,j = Σi,j+ 12∪ Σi+ 1

2 ,j∪ Σi,j− 1

2∪ Σi− 1

2 ,j,

4

Figure 1. Illustration of finite volume discretization on a staggered grid. The primary gridlines are solid and the dual grid lines are dashed. The primary grid nodes are indicated by× and the dual grid nodes by . We show a resistor as a white rectangle with a midpoint at the intersection of a primary and dual line.

the union of the dual line segments Σi,j± 12

= (Pi− 12 ,j±

12, Pi+ 1

2 ,j±12) and Σi± 1

2 ,j= (Pi± 1

2 ,j−12, Pi± 1

2 ,j+12). The

finite volume discretization is a system of equations of form

γi+ 12 ,j

(ui+1,j − ui,j) + γi− 12 ,j

(ui−1,j − ui,j) + γi,j+ 12(ui,j+1 − ui,j) + γi,j− 1

2(ui,j−1 − ui,j) = 0. (2.1)

It is obtained by integrating (1.1) over the cells Ci,j , using the divergence theorem, and then approximating

the boundary fluxes with finite differences.

Equations (2.1) are Kirchhoff’s node law for the interior nodes of the resistor network with graph

Γ = (P, E). Here P = Pi,j is the set of primary nodes, given by the union of the disjoint sets PB and PIof boundary and interior nodes. Adjacent primary nodes are connected by edges, the elements of the set

E ⊂ P ×P. The network is the pair (Γ, γ), with positive valued conductance function γ : E → R+. It assigns

to an edge like Ei,j+ 12

= (Pi,j , Pi,j+1) the positive conductance γi,j+ 12, the inverse of the resistance.

Hereafter we denote the cardinality of PB by n. We also index the edges and the corresponding

conductances with (α, β) ∈(i, j ± 1

2

),(i± 1

2 , j)

. The relation between γα,β and σ depends on the

quadrature rule used in approximating the net fluxes through the boundary ∂Ci,j of the dual cell. For

example, the definition in [11, 12, 33] is∫Σi,j± 1

2

σ(x)∂u(x)

∂νdΣ(x) ≈ γi,j± 1

2(ui,j+1 − ui,j),

∫Σi± 1

2,j

σ(x)∂u(x)

∂νdΣ(x) ≈ γi± 1

2 ,j(ui+1,j − ui,j),

where

γα,β = σ(Pα,β)L(Σα,β)

L(Eα,β), (2.2)

and L denotes the arclengths of the primary and dual edges E and Σ. The definition in [9, Equations (11),

(12)] is specialized to tensor product grids in the disk shaped Ω. Either definition works well, as long as we

use them consistently in the inversion algorithm. See for example [11, Section 2.4] for a discussion of the

small differences induced by using one definition vs. the other.

5

The forward problem for a known network (Γ, γ) amounts to determining the potential function

U : P → R, with ui,j = U(Pi,j) satisfying equations (2.1) at the interior nodes, and Dirichlet boundary

conditions

u|PB = uB . (2.3)

The entries in the vector uB ∈ Rn may be related to the continuum boundary potential V as explained

below, in section 2.1.1.

The inverse problem for the network seeks the conductance function γ from the discrete DtN map Λγ .

The graph Γ is known, and the DtN map is a matrix in Rn×n that maps the vector uB of boundary potentials

to the vector JB of boundary current fluxes. Since we consider the two dimensional problem, all the graphs

Γ are circular planar graphs [14, 15], i.e. graphs that can be embedded in the plane with no crossing edges

and with all boundary nodes PB lying on a circle.

2.1.1. Discrete measurements of the continuum DtN map. To connect the discrete inverse problem

for the network (Γ, γ) to continuum EIT, we introduce a measurement operator Mn that defines an n × nmatrixMn (Λσ) from the continuum DtN map. The measurement operator is chosen so that it is consistent

with the DtN map of a circular planar resistor network, i.e. for any suitable conductivity, there is some

resistor network (Γ, γ) so that

Λγ =Mn (Λσ) . (2.4)

The properties characterizing the DtN map of a resistor network are explained in section 2.1.2. The particular

topologies of Γ considered in this paper are in sections 2.3–2.5.

One choice of the measurement operator consists of taking point values of the kernel of Λσ. Its

consistency with networks is shown in [27, 25]. Another choice is to lump fluxes over disjoint segments

of B that model electrode supports. Its consistency with networks is shown in [9, 23]. Other measurement

operators, based on more accurate electrode models, such as the complete electrode model [38], can be used

in our inversion approach, as long as they are consistent with DtN maps of planar resistor networks.

We use the measurement operator defined in [9, 23], and let χ1, . . ., χn be nonnegative “electrode”

functions in H1/2(B), with disjoint supports, numbered in circular order on B. We normalize them to

integrate to one on B. The operator Mn maps Λσ to the symmetric Λγ ∈ Rn×n, with off-diagonal entries

given by

(Λγ)i,j = (Mn(Λσ))i,j = 〈χi,Λσχj〉, i 6= j, (2.5)

where 〈·, ·〉 is the duality pairing between H1/2(B) and H−1/2(B). The diagonal entries are given by the law

of conservation of currents

(Λγ)i,i = (Mn(Λσ))i,i = −∑j 6=i

(Mn(Λσ))i,j , i = 1, . . . , n. (2.6)

2.1.2. Solvability of the inverse problem for resistor networks. The question of solvability of the

inverse problem for circular planar networks like (Γ, γ) has been resolved in [14, 15, 25, 17, 18], when Γ is a

circular planar graph. The answer is that Λγ determines uniquely the conductance function γ if the graph

Γ is critical and the data is consistent.

A graph is critical if it is well connected and if it does not contain any redundant edges. To define well

connectedness, let π(Γ) be the set of all circular pairs connected through Γ by disjoint paths. A circular

6

pair (Q1, Q2) consists of two sets Q1, Q2 ⊂ PB, of equal cardinality k, such that the nodes in Q1 and Q2 lie

on two disjoint arcs of B. The nodes are numbered in circular order, according to the orientation of B. A

circular pair is called connected if there exist k disjoint paths in Γ connecting the nodes in Q1 to those in

Q2. The graph is called well connected if all the circular pairs in π(Γ) are connected. Moreover, it is called

critical, if removing any edge in E breaks some connection in π(Γ).

The set of DtN maps of well connected resistor networks is defined in [15]. It consists of all symmetric

matrices Λγ ∈ Rn×n satisfying the law of conservation of currents Λγ1 = 0, and whose circular minors are

totally non-positive. A circular minor of Λγ is a submatrix (Λγ)Q1,Q2with row indices in Q1 and column

indices in Q2, where (Q1, Q2) is a circular pair. The total non-positivity means that the determinant of

−(Λγ)Q1,Q2is positive, for all the circular pairs (Q1, Q2).

It is shown in [9, Theorem 1] thatMn(Λσ) is the DtN map of a well connected network. That is to say,

the data are consistent. It remains to define the graph of the network, so that it is critical, and thus uniquely

determined by the measurements Mn(Λσ). The definition depends on the boundary measurement setup,

as explained in sections 2.3–2.5. In all cases, there is a precise relation between the number n of boundary

points and the number g of edges in the graph,

g =n(n− 1)

2. (2.7)

This relation says that there are as many unknown conductances in the network as there are degrees of

freedom in the DtN map Λγ ∈ Rn×n. Since Λγ is symmetric and its rows sum to zero, it is determined by

the n(n− 1)/2 entries strictly above the diagonal.

The inverse problem for the resulting critical network (Γ, γ) can be solved with at least two approaches.

The first is direct layer peeling [14], which solves the nonlinear EIT problem in a finite number of algebraic

operations. It begins by determining the resistors in the outermost layer, then it peels it off and proceeds

inwards. The algorithm stops when the innermost layer of resistors is reached. The advantage of layer

peeling is that it is fast and explicit. The disadvantage is that it becomes quickly unstable, as the number

of layers grows. Moreover, the noise becomes a consistency issue, because noisy data may not lie in the set

of DtN maps of well connected networks. The second approach is based on optimization, that fits the noisy

measurements in the least squares sense. We use both approaches in our statistical analysis of the nonlinear

inverse problem. See section 4 for more details.

2.2. Inversion on optimal grids

We use henceforth the notation σ? for the true conductivity, to distinguish it from the estimates that we

denote generically by σ. The relations (2.2) between γ and σ? have been derived in the discretization of the

forward problem. We use them for the conductances of the network (Γ, γ) recovered from the measurements

Mn(Λσ?), in order to estimate σ?. This does not work unless we use a special grid in (2.2) [9, 10]. The idea

behind the inversion on optimal grids is that the geometrical factors L(Σα,β)/L(Eα,β) and the distribution

of the points Pα,β in (2.2) depend weakly on σ. Therefore, we can determine both the geometrical factors

and the grid nodes from the resistor network (Γ, γ(1)), with the same graph Γ as before, and DtN map

Λγ(1) =Mn(Λσ≡1). (2.8)

7

These are the measurements of the DtN map for constant conductivity σ ≡ 1, that we can compute, and

γ(1)α,β ≈ L(Σα,β)/L(Eα,β). We obtain the pointwise estimates

σ(Pα,β) ≈ γα,β

γ(1)α,β

, (2.9)

that we place in Ω at points Pα,β determined from a sensitivity analysis of the DtN map, as we explain next.

2.2.1. The sensitivity functions and the optimal grids. The distribution of points Pα,β in Ω is optimal

in the sense that

Λγ(σ) =Mn(Λσ), (2.10)

for conductances γ = γ(σ) related to the continuum σ as in (2.2), and for σ ≡ 1 (i.e., for γ(1) = γ(1)). We

define these points as the maxima of the sensitivity functions (Dσγα,β)(x) given below, evaluated at σ ≡ 1.

That is to say, the points at which the conductances are most sensitive to changes in the conductivity.

Let us enumerate the conductances in the network as γk, with k = 1, 2, . . . , g. Recall that g is the

total number of edges in the graph and it is related by (2.7) to the number n of boundary points. Taking

derivatives in (2.10) with respect to σ, we obtain after a calculation given in detail in [12, Section 4] that

(Dσγ) (x) =(DγΛγ |Λγ=Mn(Λσ)

)−1

vec (Mn(DKσ)(x)) , x ∈ Ω. (2.11)

The left hand side is a vector in Rg. Its k−th entry is the sensitivity of conductance γk with respect to

changes of σ. The Jacobian DγΛγ ∈ Rg×g is given by

(DγΛγ)jk =

(vec

(∂Λγ∂γk

))j

, (2.12)

and vec(A) denotes the operation of stacking in a vector in Rg the entries in the strict upper triangular part

of a matrix A ∈ Rn×n. Furthermore,

(Mn(DKσ))ij (x) =

∫B×B

χi(x)DKσ(x;x, y)χj(y)dxdy, i 6= j,

−∑k 6=i

∫B×B

χi(x)DKσ(x;x, y)χk(y)dxdy, i = j,

(2.13)

where DKσ(x;x, y) is the Jacobian of the kernel Kσ(x, y) of Λσ. It is given by

DKσ(x;x, y) = σ(x)σ(y)

(∇x

∂

∂νxG(x,x)

)·(∇x

∂

∂νyG(x, y)

), (2.14)

with G the Green’s function of the differential operator u→ ∇ · (σ∇u) with Dirichlet boundary conditions,

and νx the outer unit normal at x ∈ B.

Each conductance γk is associated with an optimal grid point Pα,β , as shown in (2.2). We write this

explicitly as k = k(α, β) and define the optimal grid points by

Pα,β = arg maxx∈Ω

(Dσγk(α,β))(x), evaluated at σ ≡ 1. (2.15)

8

2.2.2. The estimate of the conductivity on optimal grids. What we have computed so far allows

us to define an initial estimate σo(x) of the conductivity, as the linear interpolation of the values (2.9), on

the optimal grid defined by (2.15). Then, we improve the estimate using a Gauss-Newton iteration that

minimizes the objective function

O(σ) = ‖Qn [Mn (Λσ)]−Qn[dn]‖22 (2.16)

over search conductivity functions σ(x). Here we let dn =Mn (Λσ?) ∈ Rn×n be the data (the measurements

of the DtN map of the conductivity σ? that we wish to find) and the reconstruction mapping isQn : Dn → Rg+.

The mapping Qn is defined on the set Dn of DtN maps of resistor networks with n boundary nodes, and

its range is Rg+, the subset of vectors in Rg with positive entries. An evaluation of Qn implies solving the

discrete inverse problem for the resistor network (Γ, γ) with DtN map Mn (Λσ), and returning the discrete

values

σk = γk/γ(1)k , k = 1, . . . , g =

n(n− 1)

2. (2.17)

The objective function (2.16) is different than the usual output least squares data misfit between the

prediction

Fn(σ) =Mn (Λσ) (2.18)

and the measurements dn. We use Qn in (2.16) as a nonlinear preconditioner of the forward map Fn, as

explained in detail in [9, 23]. It is because of this preconditioning, and the good initial guess σo(x), that we

can obtain close estimates of σ by minimizing (2.16) in [9, 23] and in this paper. The estimates are computed

with a Gauss-Newton iteration that basically converges in one step [9, 23].

We enforce the positivity of σ by the change of variables κ = ln(σ), so that we work with the map

Gn(κ) = ln[Qn(Mn(Λexp(κ)))

]. (2.19)

The Gauss-Newton iteration is

κj = κj−1 + (DGn(κj−1))†

[ln[Qn(dn)]− Gn(κj−1)] , j = 1, 2, . . . , (2.20)

with initial guess κo = ln(σo). The index † denotes the Moore-Penrose pseudo-inverse, and the iteration

amounts to finding the update κj − κj−1 as the orthogonal projection of the residual onto the span of the

sensitivities, the column space of the transpose of DGn(κj−1). These sensitivities are easily related to those

computed in section 2.2.1, using the chain rule to deal with the change of variables κ = ln(σ).

2.3. Circular resistor networks for full boundary measurements

The full boundary measurements setup corresponds to having the disjoint n “electrode” functions χj in

(2.5)-(2.6) distributed all around B. For example, in [9, 23] and in the numerics section 4 we use

χj(x) = χ(x− xj), j = 1, 2, . . . , n, (2.21)

where in polar coordinates xj = (1, jhθ) are equidistant points on B, the unit circle, at spacing hθ = 2π/n.

The function χ is supported in (−π/n, π/n) and it is defined in detail in [9, Appendix B.2].

Resistor networks (Γ, γ) with circular graphs Γ = C(l, n) are natural reduced models of the problem

with full boundary measurements. The notation C(l, n) is consistent with that in [15, 16], where circular

networks have been analyzed in detail. It indicates that the graph has l layers, and n edges in each layer.

9

Figure 2. Circular resistor networks C(l, n) with critical graphs: l = (n−1)/2. The interiornodes are indicated with dots and the boundary nodes with crosses.

Figure 3. Pyramidal graphs Γn. Left: n = 6; right: n = 7. The interior nodes are indicatedwith and the boundary nodes vj , j = 1, . . . , n, with ×.

The edges may be in the radial direction, or transversal to it, as illustrated in Figure 2. Note that the

number of edges in the layers is the same as the number of boundary points.

For the network to be critical, and thus uniquely determined by Λγ = Mn(Λσ), there is a precise

relation between n and l. Explicitly, it is shown in [16, Proposition 2.3, Corollary 9.4] and [9, Theorem 2]

that the network is critical if and only if n is odd and l = (n − 1)/2. Note that the number of edges is

g = nl, consistent with (2.7). We take henceforth, for full boundary measurements, networks with graphs

C ((n− 1)/2, n), for odd n.

The computation of the optimal grids induced by the circular resistor networks is simplified by the

rotational symmetry of both the continuum and discrete problems for σ ≡ 1. We can use Fourier transforms

in angle and obtain the one dimensional reduced problem of finding the optimal radii, the placement of

the layers. The problem of finding the placement of the layers can in turn be reformulated as the rational

approximation of the transformed DtN map, as explained in [8, 10, 33, 23]. The sensitivity grids described

in section 2.2.1 are more general, in the sense that their construction does not rely on symmetries of the

problem. This is why we can use sensitivity grids for different measurement setups and various topologies,

as explained in [12]. The sensitivity grids are close to those obtained from rational approximations of the

DtN map in the full boundary measurement case, as shown in [12, Section 4.4].

2.4. Pyramidal resistor networks for one-sided partial boundary measurements

We call the partial boundary measurement setup one-sided when the accessible boundary BA consists of a

single segment of B. Circular networks with graphs C(l, n) can be used for this setup, after a coordinate

10

Figure 4. Two-sided resistor network Tn, n = 10. The boundary nodes vj , j = 1, . . . , n arenumbered in circular order and are indicated by ×. The interior nodes are indicated with .

transformation that is either conformal, or extremal quasiconformal, to minimize the anisotropy of the

transformed conductivity [11]. However, it is shown in [12] that there are more natural networks, with

pyramidal graphs Γ = Γn, that give much better reconstructions of σ from partial boundary measurements.

It is shown in [12, Lemma 1] and in [15, Proposition 7.3] that pyramidal networks are critical for both

odd and even number n of boundary nodes. We illustrate in Figure 3 how they look for n = 6 and 7. They

are more natural to use with partial measurements because the two sides of the pyramid, where the boundary

nodes lie, can be mapped to the accessible segment BA of the boundary. The base of the pyramid consists

of interior nodes. They model the lack of penetration of the currents in the part of the domain near BI ,the inaccessible boundary. We do not need to know the potential at these nodes in order to determine the

network (Γn, γ).

2.5. Two-sided partial boundary measurements

We call the measurements two-sided when the accessible boundary BA consists of two disjoint segments.

This is a novel setup in the context of inversion on optimal grids, and we describe it here in detail.

The inversion is based on resistor networks with graph Γ = Tn, and n = 2m boundary nodes. There are

m nodes on each segment of the accessible boundary separated by the leftmost and rightmost interior nodes,

as illustrated in Figure 4, for m = 5. These interior nodes model the lack of penetration of the currents

in the parts of the domain close to the inaccessible boundary. The edges of Tn are called either horizontal

or vertical, according to their orientation in Figure 4. In general, Tn has m layers of vertical edges with m

edges in each layer, and m layers of horizontal edges with m − 1 edges in each layer. This makes the total

number of edges

g = m(2m− 1) =n(n− 1)

2,

the same as the number of degrees of freedom of the DtN map Λγ . The following proposition states that

the network Tn can be uniquely recovered from its DtN map according to the results of [15]. Its proof is not

included here as it is lengthy and similar to the proofs given in the appendices of [12].

Proposition 1. The circular planar network with graph Tn, n = 2m, m ∈ N is critical and well-connected.

11

2.5.1. Solution of the inverse problem for two-sided networks. The following layer peeling algorithm

for determining the network (Tn, γ) is similar to the one introduced in [12, Section 3] for pyramidal networks.

The network can be recovered with optimization as well.

Algorithm 1. To determine the conductance γ of the two-sided network (Tn, γ), with the given DtN map

Λγ ∈ Dn, perform the following steps:

(1) Peel the lower layer of horizontal resistors:

For p = m + 2,m + 3, . . . , 2m define the sets Z = p + 1, p + 2, . . . , p + m − 1 and C = p − 2, p −3, . . . , p−m. The conductance between vp and vq, q = p− 1 is given by

γ(vp, vq) = −Λp,q + Λp,C(ΛZ,C)−1ΛZ,q. (2.22)

Assemble a symmetric tridiagonal matrix A with off-diagonal entries −γ(vp, vp−1) and rows summing

to zero. Update the lower right m-by-m block of the DtN map by substracting A from it.

(2) Let s = m− 1.

(3) Peel the top and bottom layers of vertical resistors:

For p = 1, 2, . . . , 2m define the sets L = p − 1, p − 2, . . . , p − s and R = p + 1, p + 2, . . . , p + s. If

p < m/2 for the top layer, or p > 3m/2 for the bottom layer, set Z = L, C = R. Otherwise let Z = R,

C = L. The conductance of the vertical edge emanating from vp is given by

γ(vp, wp) = Λp,p − Λp,C(ΛZ,C)−1ΛZ,p. (2.23)

Let D = diag(γ(vp, wp)) and update the DtN map

Λγ = −D −D (Λγ +D)−1D. (2.24)

(4) If s = 1 go to step (7). Otherwise decrease s by 2.

(5) Peel the top and bottom layers of horizontal resistors:

For p = 1, 2, . . . , 2m define the sets L = p− 1, p− 2, . . . , p− s and R = p+ 2, p+ 3, . . . , p+ s+ 1. If

p < m/2 for the top layer, or p < 3m/2 for the bottom layer, set Z = L, C = R, q = p+ 1. Otherwise

let Z = R, C = L, q = p − 1. The conductance of the edge connecting vp and vq is given by (2.22).

Update the upper left and lower right blocks of the DtN map as in step (1).

(6) If s = 0 go to step (7), otherwise go to (3).

(7) Determine the last layer of resistors. If m is odd the remaining vertical resistors are the diagonal

entries of the DtN map. If m is even, the remaining resistors are horizontal. The first of the

remaining horizontal resistors γ(v1, v2) is determined from (2.22) with p = 1, q = m + 1, C = 1, 2,Z = m+ 1,m+ 2 and a change of sign. The rest are determined by

γ(vp, vp+1) =(Λp,E − Λp,C(ΛZ,C)−1ΛZ,E

)1, (2.25)

where p = 2, 3, . . . ,m−1, C = p−1, p, p+1, Z = p+m−1, p+m, p+m+1, E = p+m−1, p+m,and 1 is a vector (1, 1)T .

The algorithm is based on the construction of special solutions that confine the flow of current to a

certain part of the network. Thus, it allows the calculation of the conductance one edge at a time. The

construction of such special solutions is examined in detail in [14, 16, 12, 33] for circular and pyramidal

networks. In particular, formulas (2.22) and (2.23) are special cases of the “boundary edge” and “boundary

spike” formulas from [16, Corollaries 3.15 and 3.16].

12

Figure 5. Optimal grids for the two-sided network Tn with n = 16. Left: single gridfor one two-sided network. Right: grid for a network and its upside-down reflection. Theaccessible boundary segments are in solid red. The “electrode” centers are indicated by ×.The optimal grid nodes corresponding to vertical and horizontal resistors are indicated with and F, respectively.

2.5.2. The optimal grids. The grids are computed as explained in section 2.2.1, using the sensitivity

functions around σ ≡ 1. Note from Figure 4 that the graph Tn lacks the upside-down symmetry. Thus, it is

possible to come up with two sets of optimal grid nodes, by fitting the measured DtN map Λγ = Mn(Λσ)

once with a two-sided network and the second time with the network turned upside-down. This way, the

number of points in the reconstruction is essentially doubled. We do not use this approach in this paper,

but we mention it because it can be useful in applications. Figure 5 illustrates both types of grids. There is

no refinement in depth in the grid with double number of nodes. The refinement is only in the transversal

direction.

2.6. Noiseless reconstructions

We show in Figures 7 and 8 the reconstructions of the smooth and piecewise constant conductivities in

Figure 6. The reconstructions are obtained in the full and partial boundary setups, and for noiseless data.

The numerical study of the effect of noise on the reconstructions is in section 4.

Remark 1. We consider two test conductivity functions, shown in Figure 6. To facilitate the comparison of

different reconstruction methods, we use the same color scale for the true conductivity and the reconstructions.

We display in the top row of Figures 7 and 8 the initial guess σo(x) of the Gauss-Newton iteration

(2.20). It is the piecewise linear interpolation of the values σo(Pα,β) = σk(α,β), where σk are obtained from

(2.17), and the optimal grid nodes Pα,β are defined by (2.15). The function σo(x) is linear on the triangles

obtained by a Delaunay triangulation of the points Pα,β . In the partial boundary measurements case we

display σo(x) in the subdomain delimited by the accessible boundary and the segmented arc connecting the

innermost grid points. We set σo(x) to the constant value one in the remainder of the domain. The plots in

the bottom row in Figures 7 and 8 display the result of one step of the Gauss-Newton iteration described in

section 2.2.2.

13

0.5 1 1.5 2 0.5 1 1.5 2 2.5

Figure 6. Conductivities used in the numerical experiments. Left: smooth conductivity.Right: piecewise constant chest phantom.

The reconstructions with the full boundary data are obtained with a circular resistor network C(n−1

2 , n),

for n = 29. They are shown in the left column in Figures 7 and 8. The reconstructions with the one-sided

partial boundary measurements are in the middle column, and they are obtained with a pyramidal resistor

network Γn, for n = 16. The reconstructions with the two-sided boundary measurements are in the right

column. They are obtained with a resistor network Tn, for n = 16.

3. Bayesian estimation

Recall from section 2.2 the notation convention that distinguishes the true conductivity σ? from its estimates

denoted generically by σ(x). We first outline the parametrization of the estimates σ(x) in section 3.1.

The parametrization that reduces the uncertainty in the reconstructions is based on the resistor network

inversion and is described in section 3.2. When the data are noisy, the reconstructions are random variables

with probability density function given in section 3.3. We study maximum a posteriori estimates of the

reconstructions under certain priors, as explained in section 3.4.

3.1. Parametrizations of the conductivity

We parametrize the estimates σ(x) of the conductivity by a vector s = (s1, . . . , sg)T ∈ Rg,

σ(x) = [S(s)] (x), x ∈ Ω, (3.1)

using the operator S : Rg → C(Ω) that takes s to a continuous function in Ω. We consider three different

parametrizations, summarized in Table 1.

The first two parametrizations are linear in s. They correspond to piecewise linear interpolation on

grids with n boundary nodes and g = n(n− 1)/2 edges,

[S(s)] (x) =

g∑k=1

skφk(x), (3.2)

where φk are the piecewise linear basis functions. Consistent with the notation in (2.2), we index with

k = k(α, β) = 1, . . . , g, the edges connecting the primary grid points. To each edge we associate the

14

0.5

1

1.5

2

0.5

1

1.5

2

Figure 7. Reconstructions of the smooth conductivity function with noiseless boundarymeasurements. Left column: full boundary measurements. Middle column: one-sidedpartial measurements. Right column: two-sided partial boundary measurements. Thetop row shows the initial guess σo(x). The bottom row shows the result of one step ofthe Gauss-Newton iteration. The color scale is the same used for the true conductivity infigure 6.

point Pα,β at which we evaluate the conductivity perturbation. The basis functions for piecewise linear

interpolation are defined by their values at these points

φk(Pα,β) =

1, if k = k(α, β),

0, otherwise.(3.3)

The grids are either equidistant, with PUα,β in the middle of the edges (parametrization SLU ), or the optimal

ones POα,β (parametrization SLO), computed as explained in section 2.2.1.

The second choice of parametrization is the nonlinear mapping SNL(s) defined below, in equation (3.6),

in terms of the resistor network reduced model of (1.1). The mapping SNL(s) cannot be written in the form

(3.2). One way of comparing it to the linear case (3.2) is to consider a small perturbation δs ≡ (δs1, . . . , δsg)T

of some reference parameters s and linearize[SNL(s + δs)

](x) =

[SNL(s)

](x) +

g∑k=1

δskφk(x) + o(δs). (3.4)

Here the φk are the “columns” of the Jacobian DsSNL(s) of SNL with respect to s and evaluated at s. We

call the φk sensitivity basis functions.

15

0.5

1

1.5

2

2.5

0.5

1

1.5

2

2.5

Figure 8. Reconstructions of the piecewise constant chest phantom with noiseless boundarymeasurements. Left column: full boundary measurements. Middle column: one-sidedpartial measurements. Right column: two-sided partial boundary measurements. Thetop row shows the initial guess σo(x). The bottom row shows the result of one step ofthe Gauss-Newton iteration. The color scale is the same used for the true conductivity infigure 6.

Meaning of parameter s Linear Conductivity [S(s)](x)

SLU sk(α,β) = [SLU (s)](PUα,β) yes Is of the form (3.2) and is piecewise linear on a Delaunaytriangulation of the uniform grid PUα,β .

SLO sk(α,β) = [SLO(s)](POα,β) yes Is of the form (3.2) and is piecewise linear on a Delaunaytriangulation of the optimal grid POα,β (see section 2.2).

SNL sk = γk/γ(1)k no Defined implicitly in (3.5) in terms of the resistor network

(Γ,γ) matching data (see section 3.2).

Table 1. The parametrizations that we use in the numerical experiments have all thesame number of parameters g = n(n − 1)/2 corresponding to the number of independentmeasurements of the DtN map.

3.2. The nonlinear reconstruction mapping and the sensitivity basis functions

In the nonlinear resistor based inversion we parametrize σ with the vector

s = γ/γ(1) = (γ1/γ(1)1 , . . . , γg/γ

(1)g )T ∈ Rg+,

16

of pointwise estimates defined in (2.10). Here the γi are the conductances of the network whose DtN map

Λγ fits the measurements and the γ(1)i are the conductances so that Λγ(1) = Mn(Λσ≡1). The nonlinear

mapping SNL is defined implicitly as the operator giving an exact match between the resistor network with

conductances sγ(1) (product understood componentwise) and the measurements of the continuum DtN map.

That is to say,

F(SNL(s)) ≡ vec(Fn(SNL(s))

)= vec

(Λsγ(1)

). (3.5)

Here the operator F : Rg+ → C(Ω) is the composition of the stacking operation vec(·) (defined in section 2.2.1)

with the forward map Fn(σ) =Mn(Λσ). From the forward problem point of view, equation (3.5) means that

SNL(s) is a conductivity for which the finite volumes discretization of the elliptic equation (1.1) described

in section 2.1 matches the measurements exactly.

There are many functions that satisfy (3.5). We recall our inversion algorithm from section 2.2.2, and

define SNL using the limit of the Gauss-Newton sequence κjj≥0given by (2.20),

[SNL(s)](x) = exp(κ(x)), where κ = limj→∞

κj . (3.6)

The starting point in the Gauss-Newton iteration is κo(x) = ln(σo(x)), where σo(x) is the piecewise linear

interpolation of the discrete values s on the optimal grid defined in section 2.2.1. In practice, the evaluation

of (3.6) is computationally efficient because iteration (2.20) converges quickly, basically after one step, as

shown in [9, 23].

We emphasize that the conductivity parametrization SNL(s) is nonlinear, and thus cannot be written

as an expansion in basis of functions, as in (3.2). To compare it to parametrizations of the form (3.2), we

look at its linearization around reference parameters s to obtain (3.4). The basis functions φk are then the

“columns” of DsSNL(s), which can be determined by differentiating (3.5) with respect to s,

DsSNL(s) = DF†n(SNL(s))DγΛsγ(1)diag(γ(1))

= (diag(1/γ(1))Dσγ)†. (3.7)

The sensitivity functions Dσγ(x) are defined in section 2.2.1 and are evaluated at σ = SNL(s).

3.3. Bayesian statistics

Our goal is to estimate σ(x), parametrized as in (3.1), from the noisy measurements dn ∈ Rn×n of Λσ? . By

the symmetry of the DtN map and the condition of conservation of currents, we need only the g = n(n−1)/2

entries in dn, above the diagonal. We stack these entries in a vector in Rg, as we have done in section 2.2.1,

d = vec(dn) = vec (Mn(Λσ?)) + ε, ε ∈ N (0, C) . (3.8)

The notation N (0, C) states that the noise ε is Gaussian (multivariate normal), with mean zero and diagonal

covariance C. This is consistent with the definition (2.5)–(2.6) of the lumping measurement operator Mn,

because the “electrode” functions have disjoint supports.

We treat s as a continuum random variable, and denote by πpr(s) its prior probability density. The

likelihood function π(d|s) is the probability density of d conditioned to knowing s. Given our Gaussian

noise model, it takes the form

π(d|s) =1

(2π)g/2|C|1/2exp

[−1

2(F(S(s))− d)TC−1(F(S(s))− d)

], (3.9)

17

where |C| is the determinant of the covariance C. The estimation of s is based on the conditional (posterior)

density π(s|d). It is defined by Bayes’ rule [36, 22]

π(s|d) =π(s,d)

π(d)=

π(d|s)πpr(s)

π(d), (3.10)

where π(s,d) is the joint probability density of (s,d). The marginal

π(d) =

∫Rg

π(s,d)πpr(s)ds (3.11)

is just a normalization that plays no role in the estimation.

The prior density πpr(s) may introduce a regularization in the inverse problem [30, Chapter 3]. We

consider three different priors.

(1) Upper/lower bound prior: We use this prior alone to explore the effect of the parametrization on

the stability of the reconstructions, at small noise levels. It states that the conductivity is positive and

bounded. Let S be the set of parameters mapped by S to positive conductivity functions bounded by

σmax in Ω,

S = s ∈ Rg : [S(s)](x) ∈ (0, σmax), x ∈ Ω. (3.12)

The prior is

π(UL)pr (s) =

1S(s)

|S|, (3.13)

where 1S(s) is the indicator function that takes value one when s ∈ S, and zero otherwise, and |S| is the

volume of S,

|S| =∫Rg

1S(s) ds.

When we study maximum a posterior estimates of the conductivity in section 3.4, we set σmax to a large

enough value, and keep at the same time the number g of parameters low enough, for the constraint

[S(s)](x) ≤ σmax to be automatically satisfied. However, we do enforce the positivity of the estimates.

(2) Gaussian prior on the conductivity: This is a Tikhonov regularization prior that is useful at higher

noise levels [30, Chapter 3]. It says that in addition to the conductivity being positive and bounded, we

assume that σ has a normal distribution with mean σref . The fluctuations S(s)− σref are uncorrelated

from point to point, and the pointwise variance is α−1. The prior is defined by

π(GC)pr (s) ∼ π(UL)

pr (s) exp[−(α/2)‖S(s)− σref‖2L2(Ω)], (3.14)

where the symbol “∼” means equality up to a positive, multiplicative constant.

(3) Log-normal prior on the parameters: This is also a Tikhonov type regularization prior that is useful

at higher noise levels. It says that in addition to the conductivity being positive and bounded, the vector

of the logarithm of the parameters s is normally distributed, with mean log(sref) and covariance α−1I,

where I is the g × g identity matrix,

π(GP )pr (s) ∼ π(UL)

pr (s) exp[−(α/2)‖ log(s)− log(sref)‖22]. (3.15)

18

3.4. Maximum a posteriori estimation

We consider maximum a posteriori (MAP) estimates of the conductivity

σMAP(x) = [S(sMAP)](x), (3.16)

which maximize the conditional probability density π(s|d). The vector sMAP of parameters solves the

optimization problem

sMAP = arg mins∈Rg

[F(S(s))− d]TC−1[F(S(s))− d]− log(πpr(s)), such that [S(s)](x) ≥ 0. (3.17)

The MAP estimates sMAP are random, because they depend on the noise ε in the measurements. To

quantify their uncertainty, we approximate their variance using a large number M of independent samples

s(m)MAP, determined from (3.17) and data (3.8) with draws ε(m) ∈ N (0, C) of the noise,

Var[s] ≈ 1

M − 1

M∑m=1

[s

(m)MAP − 〈s〉

]2, 〈s〉 ≈ 1

M

M∑m=1

s(m)MAP. (3.18)

Then, we compare Var[s] with the optimal variance, which is the right hand side in the Cramer-Rao bound

[36, Corollary 5.23],

Es?[sMAP − Es?sMAP]2 ≥ bT (s?)I−1(s?)b(s?). (3.19)

The notation Es? indicates that the mean (expectation) depends on the true vector of parameters s?. The

bias factor b(s) is defined by

b(s) = ∇EssMAP, (3.20)

where ∇ denotes gradient with respect to s, and I(s) is the Fischer information matrix [36, Section 2.3]. It

measures how much information the data d carry about the parameter s?. The Fisher matrix is in Rg×g,with entries

Ii,j(s) = Es

∂si log π(d|s) ∂sj log π(d|s)

. (3.21)

Since the likelihood π(d|s) is Gaussian, we obtain under the natural assumption that the noise covariance Cis independent of s, that

I(s) = DsS(s)TDσF(S(s))TC−1DσF(S(s))DsS(s), (3.22)

where DsS(s) is the Jacobian of S(s) evaluated at s and DσF(σ) is the Jacobian of F(σ) evaluated at σ.

We take as the true s? the solution of the optimization problem (3.17), with noiseless data and upper/lower

bound prior πpr ≡ π(UL)pr . The bias b(s?) is approximated via Monte Carlo simulations, from a large sample

of draws. The bound (3.19) is evaluated and displayed in section 4.4.

It is known [36, Section 5.1] that the Cramer-Rao bound is attained by the variance, for linear models

like (3.2). The variance depends of course on the operator (3.2) used in the estimation, as presented in section

4.3. The results say that when we take piecewise linear interpolations of the parameters s on the equidistant

or the optimal grids, we obtain much larger variances than if we use the sensitivity basis functions. That is

to say, the sensitivity basis functions lead to better estimates than the piecewise linear ones, even when we

interpolate on the optimal grids. We also show in section 4.4 that the nonlinear estimation based on resistor

networks is efficient, in the sense that the sample variance is very close to the Cramer-Rao bound.

19

4. Effect of parametrization on the uncertainty in the reconstructions: A numerical study

We study numerically the effect of noise (section 4.1) on the inversion with (a) resistor networks, and

(b) the conductivity parametrized with piecewise linear basis functions (see Table 1 for a summary of the

parametrizations). We consider in section 4.3 the linearized problem about the constant conductivity σ ≡ 1,

and in section 4.4 the nonlinear problem. The resistor networks are computed from the measurements of

the DtN map with either the direct, layer peeling method, or with the Gauss-Newton iteration described in

section 4.2.

4.1. Data and noise models

We simulate the DtN map Λσ? by solving equation (1.1), with a second order finite volume method on a

very fine, uniform, tensor product grid, with N n nodes on the boundary. The noise ε in (3.8) is given by

ε = vec (Mn(E)) , E ∈ RN×N . (4.23)

We use two noise models, defined in terms of the noise level ` and the symmetric N × N matrix η, with

Gaussian, identically distributed entries with mean zero and variance one. The entries in η on and above

the diagonal are uncorrelated. The first model scales the noise by the entries of the DtN map for constant

conductivity σ ≡ 1,

E = `Λ1 · η, (4.24)

where symbol · stands for componentwise multiplication. The scaling makes the noise easier to deal with,

and the model is somewhat similar to multiplicative noise. The second noise model is

E = `η

‖η‖, (4.25)

where ‖ ‖ is a matrix norm that approximates the continuum H1/2(B) → H−1/2(B) operator norm. It is

defined in Appendix A.

4.2. The Gauss-Newton iteration for determining the resistor networks

The direct, layer peeling algorithms described in [14, 12] and section 2.5.1 are fast, but highly unstable and

can be used only for very small noise. The instability of layer peeling comes from the ill-posedness of the

problem of finding the resistors in a network from their DtN map: it involves solving a discrete Cauchy

problem. Hence the larger the network, the more ill-posed the discrete inverse problem becomes. In [9], the

problem is regularized by using small networks to fit noisy data, and the size of the network is reduced as the

noise increases. However, regularization solely based on the network size is not sufficient for the larger noise

levels considered in the simulations. We use instead the more robust Gauss-Newton method described below,

which allows us to regularize with the log-normal prior described in section 3.3. The Gauss-Newton method

is more expensive than layer peeling (about 20 times more expensive for n = 7), but the computational cost

is reasonable because the dimension g of the vector of unknown conductances is small for noisy data, and

the Jacobian is relatively inexpensive to compute.

The Gauss-Newton method determines the log-conductances κ = log(γ) in the network (Γ,γ) (with

topology Γ fixed) by minimizing the objective functional

Θ(κ) = (vec(Λexp(κ))− d)TC−1(vec(Λexp(κ))− d) + α||κ− κref ||2, (4.26)

20

n=

11

10−4

10−3

10−2

10−1

100

(a) (b) (c)

Figure 9. Standard deviation of reconstructions for linearized problem for differentparametrizations of the unknown. (a): Piecewise linear basis functions on uniform grid(SLU ). (b): Piecewise linear basis functions on optimal grid (SLO). (c): Sensitivity basisfunctions (DsSNL(1)). The color scale is base-10 logarithmic and the noise level is ` = 0.01%(additive noise model (4.25)). The circular resistor network has n = 11 boundary pointsand the noise E is given at N = 100 equidistant points on the boundary. The fraction ofrealizations with active positivity constraints is: (a) 81.6%, (b): 61.1%, (c): 0.0%.

over κ ∈ Rg. Here log(γ) and exp(κ) are understood componentwise. Since we work with log-conductances

κ, the positivity of the conductances γ = exp(κ) is automatically satisfied. The first term in the functional

(4.26) measures the misfit between the measured data d modeled by (3.8), and the data produced by a

network with log-conductances κ. Here C is the covariance matrix of the measurements. The second term

in the functional (4.26) is the Tikhonov type regularization that comes from the log-normal prior described

in section 3.3, as explained in Remark 2. The larger the parameter α, the more regularization is introduced.

In our numerical experiments, the Gauss-Newton approximation of the Hessian of Θ(κ) is further

regularized by adding 10−4/C1,1 to its diagonal. The iterations are stopped either when the norm of the

gradient of Θ(κ) is 10−4 smaller than that at the initial iterate, or when the maximum number of iterations

(300) is reached. The initial iterate is κ = log(γ(1)), where γ(1) is given by (2.8) and we take as reference

log-conductances κref = log(γ(1)).

Remark 2. Recall the resistor network model matches the data exactly (3.5). Therefore:

(i) When α > 0, finding log-conductances that minimize the functional (4.26) is equivalent to finding the

maximum a posteriori estimate (3.17), with the resistor network based parametrization SNL (i.e. with

s = γ/γ(1), see section 3.2), and the log-normal prior πpr ≡ π(GP )pr defined in equation (3.15). The

regularization parameter α is the same in both (3.17) and (4.26). Moreover, when κref = log(γ(1)) in

(4.26), the reference parameters appearing in (3.15) are sref = 1 (vector of all ones with length g).

(ii) When α = 0, minimizing (4.26) is equivalent to finding the maximum a posteriori estimate (3.17) with

the resistor network based parametrization SNL and the upper/lower bound prior πpr ≡ π(UL)pr defined

in equation (3.13) (for a large enough upper bound σmax).

21

4.3. The linearized problem

The results in this section are for linearization at the constant conductivity σ(x) ≡ 1, with data

d = F(σ) + ε, F(σ) = vec (Mn(Λσ)) , (4.27)

and additive noise ε modeled as in (4.25). Notice that all three parametrizations that we consider can

represent exactly the constant conductivity σ ≡ 1 with a parameter vector s of all ones, i.e. S(1) = 1. Hence

the linearization of the forward map around s = 1 can be written as

F(S(1 + δs)) = F(1) +DσF(1)DsS(1)δs + o(δs), (4.28)

for a small perturbation δs of the parameters. If we discretize the conductivity on a fine grid with Nσ

points, the Jacobian DσF(1) is a g ×Nσ matrix. The Jacobian DsS(1) is an Nσ × g matrix with columns

given by the basis functions φk in (3.2) for the linear discretizations SLU , SLO (of course, when S is linear:

S = DsS(1)), and the sensitivity functions (3.4) for the non-linear SNL.

The estimate σMAP is calculated by solving the optimization problem (3.17), with the linearization (4.28)

of the forward map, and the upper/lower bound prior πpr ≡ π(UL)pr of equation (3.13). This optimization is

a quadratic programming problem that we solve using the software QPC [39]. The mean and variance of

σMAP are estimated with Monte Carlo simulations

Var[σ(x)] ≈ 1

M − 1

M∑m=1

[σ

(m)MAP(x)− 〈σ(x)〉

]2, 〈σ(x)〉 ≈ 1

M

M∑m=1

σ(m)MAP(x), (4.29)

using M = 1000 samples. We report in figure 9 the fraction of the realizations where the positivity constraints

are active.

We do not show the mean 〈σ(x)〉 because it is basically σ(x) for all the cases that we present below.

The standard deviation Var[σ(x)]1/2 is shown in Figure 9 for the case of full boundary measurements,

and the three choices of the basis functions φk described in section 3 and in table 1. The first choice is

piecewise linear on the grid with n boundary points, that is equidistant in both radius and angle (SLU ). The

second choice is piecewise linear on the optimal grid (SLO). The third choice is given by the sensitivity basis

functions (DsSNL(1)). The noise level is ` = 0.01%, with the additive noise model (4.25). Note that the

standard deviation is smaller and does not increase toward the center of the domain when we parametrize the

conductivity with the sensitivity basis functions. Note also that none of the realizations with the sensitivity

basis functions had active positivity constraints. The random fluctuations of σMAP in Figure 9 (b) lie mostly

within three standard deviations, and are much smaller than the background conductivity σ ≡ 1.

The piecewise linear parametrization on the equidistant grid gives a large, order one standard deviation

in the center of the domain. The positivity constraints are active in 81% of realizations. Surprisingly, the

piecewise linear parametrization on the optimal grid is not better. Its standard deviation is large, of order

one in most of the domain, and the positivity constraints are active in 61% realizations. This shows that it is

not enough to distribute the parameters on the optimal grid. We also need to parametrize the reconstructions

using the sensitivity basis functions.

The same conclusion can be reached from Figure 10, where we display the condition number of the

matrix DσF(1)DsS(1), as a function of the number n of boundary points. The condition number increases

exponentially with n, as expected from the exponential ill-posedness of the problem. However, the rate of

increase is smaller for the parametrization with the sensitivity basis functions.

22

5 10 15 20 2510

0

105

1010

1015

Figure 10. Comparison of the condition number of the matrix DσF(1)DsS(1). The abscissais the number n of boundary nodes and the ordinate is in logarithmic scale. The green line isfor the fine grid forward map (S = identity). The red line is for the piecewise linear functionson the equidistant grid in radius and angle (SLU ). The blue line is for the piecewise linearfunctions on the optimal grid (SLO). The black line is for the sensitivity basis functions(SNL).

The standard deviation Var[σ(x)]1/2 is shown in Figure 11 for the case of one and two-sided boundary

measurements. We use a much smaller noise level (` = 10−8, additive model (4.25)) in the partial

measurements case than in the full data case because we compute the standard deviation for bigger networks

(n = 16). That is for the same size of the networks as in the reconstructions in Figures 7 and 8. Since the

condition number of the linearized problem grows exponentially as we increase n, only very small levels of

noise can be used for n = 16. We present the results for two choices of the basis functions. The piecewise

linear parametrization on the optimal grid is in the top row, and the parametrization with sensitivity basis

functions is in the bottom row. We reach the same conclusion as before. The sensitivity basis functions give

a smaller standard deviation, that does not increase toward the inaccessible region. The piecewise linear

parametrization on the optimal grid gives a large standard deviation near the inaccessible region in the

one-sided case, and in the whole domain in the two-sided case. The positivity constraints are active in most

realizations for the piecewise linear parametrization. They are not active for the parametrization with the

sensitivity basis function.

Remark 3. To facilitate comparison of different inversion methods we use the same color scale for the

standard deviations that are reported in Figures 9, 11, 12, 14 and 16. When we report reconstructions or

mean reconstructions (Figures 7, 8, 12, 14, 15, 16 and 17) we use the same color scales as for the true

conductivities appearing in Figure 6.

4.4. The nonlinear problem

We study the statistics (mean and standard deviation) of the maximum a priori estimates of the parameters

sMAP and the conductivities σMAP = S(sMAP), which come from minimizing the functional (3.17). The

23

10−4

10−3

10−2

10−1

100

10−4

10−3

10−2

10−1

100

Figure 11. The standard deviation (base-10 logarithmic scale) for the one-sided (left)and two-sided (right) boundary measurements setup. The resistor networks have n = 16boundary points and the noise E with level ` = 10−8% is given at N = 83 (left) and N = 104(right) equidistant points on the accessible boundary. Top: piecewise linear basis functionson the optimal grid. Bottom: sensitivity basis functions. The accessible boundary is shownin solid red.

study can be done for both the full and partial boundary measurement setup, but we present here only the

full measurements case. We consider first, in section 4.4.1, very small noise so that we can use the fast direct

resistor inversion algorithm to find the minimizer of (3.17), with S ≡ SNL and upper/lower bound prior

πpr ≡ π(UL)pr . The speed of the algorithm allows us to compute the Cramer-Rao lower bound in a reasonable

amount of time. We do not calculate the bound for larger noise, where we use regularized Gauss-Newton

to determine the resistors, because of the extensive computational cost. However, we do show in section

4.4.2 the bias and relative standard deviation of the reconstructions, and we also compare in section 4.4.3

the results to those of traditional output least squares with Tikhonov regularization of the conductivity

(Gaussian prior).

4.4.1. Statistics of resistor network inversion using layer peeling The results in this section are

for the MAP estimates sMAP, solving the optimization problem (3.17), with the resistor network based

parametrization SNL, and upper/lower bound prior πpr ≡ π(UL)pr . We present the mean 〈s〉, the bias

Bias[s] = s? − 〈s〉 and the standard deviation (Var[s])1/2 of the estimates. We consider a very small noise

level ` = 0.1% (noise model (4.24)), so that we are able to minimize (4.26) with α = 0 directly, using the

layer peeling algorithm. There is no regularization term in the prior.

24

0.5 1 1.5 2 0 1 2 3 4 5×10−4 10−410−310−210−1 100 -2% 0% 2% 4% 6%

0.5 1 1.5 2 2.5 0 1 2 3 4 5×10−4 10−410−310−210−1 100 -2% 0% 2% 4% 6%

(a) (b) (c) (d)

Figure 12. Results with the direct resistor finding algorithm: (a) mean, (b) absolutebias, (c) standard deviation relative to the mean (d) relative difference between varianceand Cramer-Rao bound relative to Cramer-Rao bound (in percentage). The noise model is(4.24) and the level is ` = 0.1%. The circular resistor network has n = 11 boundary pointsand the noise matrix E is given at N = 100 equidistant points on B. All parameters arelinearly interpolated on the optimal grid.

0

1

smoth piecewise constant

Figure 13. The estimated bias factor (3.20) for the conductivities in Figure 6 is close to theidentity.

Note that in the Cramer-Rao bound, the Fischer matrix (3.21) can be calculated analytically, but the

bias term (3.20) is estimated with Monte Carlo simulations. This is the expensive part of the computation,

because we need a large number of samples to estimate the mean. In addition, each component of the vector

s is perturbed to approximate the partial derivatives in (3.20) via finite differences. We use M = 1000

25

samples, and the bias is relative to s?, the solution of the optimization problem (3.17) with noiseless data,

S ≡ S(NL) and πpr ≡ π(UL)pr . The partial derivatives in (3.20) are approximated with finite differences with

a step size of 0.01.

The bias factor is shown in Figure 13, and it is close to the identity matrix. That is to say, the

estimates are unbiased. Figure 12 shows (a) the mean 〈s〉, (b) Bias[s] and (c) the relative standard deviation

(Var[s])1/2/ 〈s〉, where the division is understood componentwise. The last column (d) shows the difference

in percentage between Var[s] and the Cramer-Rao bound in (3.19), normalized pointwise by the Cramer-Rao

bound. We evaluate the Cramer-Rao bound by setting the bias factor (3.20) to the identity, which is a good

approximation (Figure 13). Note that the difference between the variance and the Cramer-Rao bound is

very small, indicating that the estimation is efficient. The result in column (d) should be non-negative. We

have some negative numbers, probably due to insufficient sampling in Monte Carlo, but they are so small in

absolute value that we can treat them as essentially zero.

4.4.2. Statistics of conductivity estimates using resistor network parametrization We consider

here the second noise model (4.25), with noise levels ` = 0.01% and ` = 1%. These are the levels used in [31],

and we use them to compare our results with those in [31]. Because solving (3.17) with only the upper/lower

bound prior does not give reliable estimates, we use the log-normal prior π(GP )pr , which is equivalent to finding

the resistors in a network by minimizing (4.26) (see Remark 2).

Figure 14 shows (a) the mean 〈σ(x)〉, (b) Bias[σ(x)] and (c) the relative standard deviation

(Var[σ(x)])1/2/ 〈σ(x)〉, for the noise level ` = 0.01%. The bias is

Bias[σ](x) = σ?(x)− 〈σ(x)〉 , (4.30)

where σ? is the solution of the optimization problem (3.17), with noiseless data, no regularization (α = 0) and

parametrization S(NL) corresponding to a resistor network with n = 11 boundary nodes. The regularization

parameter α = 10−6 is chosen so that both the bias and the relative standard deviation are small. The

choice of the regularization parameter is discussed in more detail in section 4.4.3. The realizations of the

maximum a posteriori estimates shown in Figure 15 are close to the mean because the standard deviation

of the reconstructions is below 10−3.

For the higher noise level ` = 1%, we present in Figure 16 the results for a smaller network, with n = 7

boundary nodes. Again we chose the regularization parameter α = 10−3, in such a way that both the bias

and standard deviation are small. The relative standard deviation is less than 10%, and the realizations

of σMAP(x) shown in Figure 15 resemble the mean in Figure 16. These realizations are comparable to the

reconstructions in [31]. The reconstructions with n = 11, noise level ` = 1% and an appropriate choice of

the regularization parameter are qualitatively similar to those with n = 7 shown in Figure 16, and thus are

not included here.

4.4.3. Resistor network parametrization compared to other parametrizations We now study the

interplay between regularization and parametrization. We solve the optimization problem (3.17) with the

three parametrizations in Table 1. At the noise levels considered here, the same as in the section 4.4.2,

we need regularization to get reliable estimates of the conductivity. As was the case in section 4.4.2, we

regularize the resistor network based parametrization S(NL) with the log-normal prior πpr ≡ π(GP )pr defined

in (3.15). For the output least squares approach, with piecewise linear parametrizations S(LU) and S(LO),

we regularize with the Gaussian prior πpr ≡ π(GC)pr defined in (3.14), with reference conductivity σref ≡ 1.

26

0.5 1 1.5 2 0 0.1 0.2 10−4 10−3 10−2 10−1 100

0.5 1 1.5 2 2.5 0 0.2 0.4 0.6 10−4 10−3 10−2 10−1 100

(a) (b) (c)

Figure 14. Results for ` = 0.01% additive noise (4.25) and a regularization parameterα = 10−6 in (4.26). The circular resistor network has n = 11 boundary points and E isgiven at N = 100 equidistant points on B. Top: the smooth conductivity shown on the leftin Figure 6. Bottom: the piecewise constant conductivity shown on the right in Figure 6.(a) mean, (b) bias, (c) relative standard deviation.

Moreover, we stopped the iterations when either the maximum number of iterations is reached (70) or when

the norm of the gradient of the objective function at the current iterate becomes 10−4 that at the initial

iterate. We also added 10−8/C1,1 to the diagonal of the Gauss-Newton approximation to the Hessian. Recall

that C is the covariance of the measurements.

For the comparison we use two metrics. The first one is the L2 norm of the true bias

TrueBias[σ] =

(∫Ω

(σtrue(x)− 〈σ(x)〉)2dx

)1/2

,

as a percent of the true conductivity σtrue. This measures the fidelity (in average) of our reconstructions. The

second metric is the L2 norm of the standard deviation relative to L2 norm of the mean of the reconstructions

RelStd[σ] =

(∫Ω

Var[σ(x)]dx)1/2(∫

Ω〈σ(x)〉2 dx

)1/2,

27

0.5

1

1.5

2

0.5

1

1.5

2

2.5

Figure 15. Realizations of σMAP(x) for n = 11 boundary points, ` = 0.01% additive noise(4.25) and regularization parameter α = 10−6 in (4.26).

which measures the stability of our reconstructions.

We report in Figure 18, for different values of the regularization parameter α in (3.14) and (3.15), the

true bias versus the relative standard deviation. Note the typical L-curve shape, which reflects the trade-

off between accuracy (small bias) and stability (small standard deviation). When the regularization is not

sufficient, the bias is small but the standard deviation is large (vertical branch). When the problem is over

regularized, the bias is large and the standard deviation is small (horizontal branch). The “best” choice of

the regularization parameter would be near the “corner” of the L shape.

The first row in Figure 18 shows that for noise level ` = 1% and for n = 7 boundary nodes, the resistor

network approach outperforms the linear interpolation methods because for a fixed standard deviation, the

bias is smaller. This is specially noticeable in the case of the piecewise constant conductivity. Interestingly,

the uniform grid is slightly better than the optimal grid in this case. For n = 11 boundary nodes and

` = 0.01% (second row), all approaches appear to give comparable results, with the resistor network approach

giving bias smaller by a few percent, especially if we allow a standard deviation above 0.1%.

We emphasize that solving (3.17) with the resistor network approach (i.e. solving (4.26)) is negligible

compared to computing the sensitivity functions DsS(NL)(s). Thus, the resistor network approach takes

about the same time as one step of Gauss-Newton to solve (3.17) with the piecewise linear parametrizations.

In the computations for Figure 18, the mean number of iterations for these linear parametrizations was at

least 8, and varied depending on the regularization parameter, the grid and the conductivity. Therefore, the

resistor network method is at least 8 times faster than the one using conventional discretization.

5. Summary

We presented a numerical study of the effects of noise on resistor based inversion algorithms on optimal

grids. The algorithms were introduced in [9, 23, 11, 12, 33], and we reviewed them here briefly. We have

three measurements setups. The first assumes that the entire boundary B of the domain Ω is accessible.

The other two are for partial boundary measurements confined to the accessible boundary BA ⊂ B. One

setup assumes one sided measurements, with BA consisting of a segment of B. The inversion algorithm is

28

0.5 1 1.5 2 0 0.02 0.04 0.06 0.08 10−4 10−3 10−2 10−1 100

0.5 1 1.5 2 2.5 0 0.1 0.2 0.3 10−4 10−3 10−2 10−1 100

(a) (b) (c)

Figure 16. Results for ` = 1% additive noise (4.25) and a regularization parameter α = 10−3

in (4.26). The circular resistor network has n = 7 boundary points and E is given atN = 100 equidistant points on B. Top: the smooth conductivity shown on the left in Figure6. Bottom: the piecewise constant conductivity shown on the right in Figure 6.(a) mean,(b) bias, (c) relative standard deviation.

introduced in [12, 33]. The other setup is two sided, with BA consisting of two disjoint segments of B. It has

not been studied before, so we presented here in detail.

The inversion amounts to defining a reconstruction mapping S(NL)(s), that takes a vector s = γ/γ(1)

of ratios of positive conductances of a network (Γ,γ) and a reference network (Γ,γ(1)), to continuous

conductivity functions defined in Ω. The network has a special graph Γ (a circular planar graph), that is

adapted to the measurement setups. It is critical, in the sense that it has precisely the number of connections

needed, and no redundant ones, in order to be determined uniquely by the measurements of the Dirichlet to

Neumann (DtN) map. The definition of S(NL)(s) involves a Gauss-Newton iteration that is very different

from the usual one which minimizes in least squares sense the data misfit.

The Bayesian statistical study of the linearized inverse problem presented in this paper considers three

29

0.5

1

1.5

2

0.5

1

1.5

2

2.5

Figure 17. Realizations of σMAP(x) for n = 7 boundary points, ` = 1% additive noise (4.25)and regularization parameter α = 10−3 in (4.26).

different parametrizations of the unknown conductivity with g degrees of freedom. The first two are piecewise

linear interpolations on an equidistant grid and on the optimal grid, respectively. The third parametrization

is with the sensitivity functions of the nonlinear reconstruction mapping S(NL)(s). The piecewise linear

parametrization gives large variances of the maximum a posteriori estimates σMAP(x), even if we use the

optimal grids. The parametrization based on the sensitivity functions is superior because it gives lower

variances of σMAP(x), that do not increase toward the inaccessible part of the domain.

The Bayesian study for the non-linear problem shows that when no additional prior (regularization) is

introduced, and the noise is very small, the resistor network based inversion has small bias and the variance of

the maximum a priori estimates σMAP(x) is very close to the optimal one, given by the Cramer-Rao bound.

For larger noise, we regularize the resistor based inversion with a log-normal prior on the conductances,

and we compare the results with those of output least squares with piecewise linear parametrizations of

the conductivity, on uniform and optimal grids. The least squares approach is regularized with a Gaussian

prior on the conductivity. The study assumes noise levels that have been considered before, in [31]. Both

the resistor based inversion and the least squares approach give a trade-off between accuracy (small bias)

and statistical stability (small standard deviation of the estimates). However, the resistor based inversion

consistently outperforms output least squares, giving a smaller bias for a fixed standard deviation in the

reconstructions. Moreover, the quality of the reconstructions is comparable to that in [31].

Our regularization priors are very simple. If additional prior information is available, the results of the

resistor based inversion can be greatly improved, by enlarging in a controlled way the space of the Gauss-

Newton iterates described above, beyond the span of the sensitivity functions of the reconstruction mapping.

This was shown for example in [9, Section 7.1].

From the computational point of view, the inversion with resistor networks can be done at roughly the

cost of one Gauss-Newton iteration for a conventional output least squares method, with the same number

of degrees of freedom of the parametrization. In our numerical experiments, the resistor network inversion

was at least eight times faster.

30

smooth pcws const.

n=

7,`

=1%

5 10 15 200

5

10

15

20

20 30 40 500

5

10

15

20

25

n=

11,`

=0.

01%

5 10 15 200

0.05

0.1

0.15

0.2

20 30 40 500

0.1

0.2

0.3

0.4

Figure 18. The true bias (in abscissa and in percent) against the relative standard deviation(in ordinate and also in percent) for different values of the regularization parameter. Thecurves correspond to the resistor network approach (black), the linear interpolation on theoptimal grid (blue) and the uniform grid (red). The number of realizations used for thisstudy was N = 100. The regularization parameter was α = 10−j/2, with j = 1, 2, . . . , 6for the linear interpolation (red and blue) and with j = 1, 2, . . . , 12 for the resistor networkapproach (black). The regularization parameter used for the reconstructions in figures 14and 16 is indicated with a black star.

Acknowledgements

The work of L. Borcea was partially supported by the National Science Foundation grants DMS-0934594,

DMS-0907746 and by the Office of Naval Research grant N000140910290. The work of F. Guevara Vasquez

was partially supported by the National Science Foundation grant DMS-0934664. The work of A. Mamonov

was partially supported by the National Science Foundation grants DMS-0914465 and DMS-0914840. The

authors were also partially supported by the National Science Foundation and the National Security Agency,

during the Fall 2010 special semester on inverse problems at MSRI, Berkeley. We are grateful to Vladimir

Druskin for sharing his deep insight of optimal grids.

31

Appendix A. The norm used in the noise model

Since B is the unit circle, we associate it with the angle interval [0, 2π]. Consider the Fourier series operator

U : `2 → L2[0, 2π], defined by

(U f)(θ) =1√2π

∞∑k=−∞

f(k)eikθ, (A.1)

and its adjoint

(U?f) (k) =1√2π

∫ 2π

0

dθ f(θ)e−ikθ. (A.2)

The fractional Sobolev norm Hs of f can be written as a weighted `2 norm ‖ ‖2,

‖f‖Hs = ‖W sU?f‖2, (A.3)

where

(W sv)k =(1 + k2

)s/2v(k), k ∈ Z. (A.4)

Let A : H1/2(B)→ H−1/2(B) be a linear operator. Its operator norm is given by

‖A‖H1/2(B)→H−1/2(B) = supf 6=0,f∈H1/2

‖W−1/2U?f‖2‖W 1/2U?f‖2

= supg 6=0,g∈`2

‖W−1/2U?AUW−1/2g‖2‖g‖2

. (A.5)

In particular, when A = Λ1, we have Λ1 = UKU?, where Kv(k) = |k|v(k), and

‖λ1‖H1/2→H−1/2 = 1. (A.6)

We approximate the operator norm (A.5) by the norm ‖ ‖, as follows. In an abuse of notation, let Λσ

and Λ1 be the restrictions of the continuum DtN maps to the N uniformly distributed fine grid points on

the boundary. Consider the spectral decomposition

2π

NΛ1 = UΣU?, U?U = I. (A.7)

The approximate norm is given by

‖Λσ‖ = supg 6=0,g∈RN

2π

N

‖(I + Σ2)−1/4U?ΛσU(I + Σ2)−1/4g‖2‖g‖2

, (A.8)

which is equivalent to finding the largest eigenvalue in magnitude of the matrix appearing in the numerator

above. By construction, we have ‖Λ1‖ = 1.

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