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Identification of Current Sources in 3D Electrostatics

  • Aron SommerEmail author
  • Andreas Helfrich-Schkarbanenko
  • Vincent Heuveline
Living reference work entry

Abstract

Motivated by passive airborne geoexploration we consider a source identification problem. This problem setting arises in electrostatics and it turns out to be a linear, ill-posed inverse problem. After developing a theoretical framework for corresponding elliptic forward problem, an approach for reconstructing current sources from local electric potential data is illustrated. A pseudo-solution is achieved by means of Tikhonov regularization. The performance of the method is shown by three-dimensional synthetic and real-life numerical examples. For numerical modeling, we choose Method of Finite Elements provided by COMSOL Multiphysics and apply MATLAB for developing a reconstruction algorithm.

Keywords

Inverse Problem Direct Problem Tikhonov Regularization Unique Solvability Forward Problem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

1 Introduction

In general, inverse problems arise in many branches of mathematics and science including tomography, nondestructive testing, physics, geophysics, and many other fields. Their objective is a conversion of observed measurements into information about a (physical) system that we are interested in Kirsch (1996) and Rieder (2003). The inverse problem considers the “inverse” to the forward problem which relates the system parameters to the data that we observe. The set of inverse problems can be classified into linear/non-linear problems divided themselves into ill-posed/well-posed problems. Inverse problems are mostly ill-posed in sense of Hadamard (1915), who introduced this term w.r.t. ordinary differential equations. That means that at least one of the following three properties is violated – existence, uniqueness, stability of the solution – and it is this that makes inverse problems challenging and mathematically interesting. This class of problems had a profound influence on mathematics and led to the found of a new field of study – The Calculus of Variations. Furthermore, inverse problems have led to major physical advances, perhaps the most popular of which was the discovery of the planet Neptune after predictions made by Le Verrier and Adams on the basis of the inverse perturbation theory (Groetsch 1993).

We investigate a source identification problem (or inverse source problem) which arises in electrostatics motivated by passive airborne geoexploration. It turns out to be a linear and ill-posed inverse problem (Sommer 2012). By source identification we mean a reconstruction of the electric current density based on some local electric potential measurements for given electrical conductivity. The term passive exploration means that the investigated physical system contains its own excitation. In contrast, an active exploration implies a measurement methodology which contains an excitation device. The reconstruction of electrical conductivity based on some measurements is called parameter identification problem. It is a non-linear, ill-posed problem (Helfrich-Schkarbanenko 2011), and it is not an objective of this work.

In the following, we introduce briefly the physical system and the inverse problem we are interested in. A hydrocarbon reservoir exited by weak seismic activities generates characteristic electromagnetic waves which propagate through the medium. Hereby, the shape of the reservoir coincides with the current source density support. We aim to reconstruct this support based on local measurements of the electric field in the air.

Let \(\Omega \subset \mathbb{R}^{3}\) be a bounded domain representing the ground and the air layer, cf. Fig. 1. By σ we denote the electrical conductivity, by u the electric potential and \(-f := -\nabla \cdot J_{b}\) defines the density of the electric current stimulating u, where J b is the bound current density with \([J_{b}]_{SI} = Am^{-2}\) (Korovkin et al. 2007, Chap. 1.3). Starting with Maxwell equations and assuming time harmonic, low-frequency regime, cf. Sommer (2012), we obtain the Laplace equation
$$-\nabla \cdot (\sigma \nabla u)\ =\ f\quad \text{in}\;\Omega .$$
(1)
The Robin boundary condition on \(\partial \Omega \) completes (1) to an elliptic boundary value problem playing the starting point of our investigation. After turning it into a weak formulation, we apply the method of finite elements (FEM) for numerical modeling of the forward problem.
Fig. 1

Scenario for the problem

Facing the inverse problem, we have to reconstruct f in \(\Omega \) on the basis of some local electric potential data assuming σ is a given parameter. By local data we mean the restriction \(u\vert _{\Gamma }\) onto a curve \(\Gamma \subset \Omega \) which is a one-dimensional manifold, Fig. 1. The local character of the measurements leads to a non-injective forward operator, and thus the inverse problem is ill-posed, cf. analytical example in Sect. 3. That means the forward operator cannot be inverted in general. However, applying Tikhonov regularization we achieve a unique pseudo-solution.

In Hanke and Rundell (2011) the authors consider a relative problem for σ ≡ const in \(\overline{\Omega }\) with the Cauchy data on \(\partial \Omega \) in two- and three-dimensional case. They propose an algorithm for the determination of the support of f by solving a simpler “equivalent point source problem.” A corresponding full-space problem for homogeneous material parameters was solved for electromagnetic field via multipole expansions (Marengo and Devaney 1999). For the minimum L 2-norm current source distribution, the authors assumed measurements on a sphere containing the support of the stimulation. The reconstruction of current distribution σ ∇ u in a bounded three-dimensional domain from its magnetic field observed on the boundary was considered in Kress (2002).

Note that (1) models steady state, diffusive phenomena with source term f. Thus, the results achieved in this work are applicable for example onto steady state heat problems (then, u would be the temperature and σ the thermal diffusion coefficient) or underground steady state aquifers (then, u would represent the hydraulic head and σ the aquifer transmissivity coefficient), cf. Groetsch (1993, Sect. 3.5).

This chapter is organized as follows: In Sect. 2, we consider the strong formulation of the forward problem and derive its weak formulation fundamental for the FEM implementation. The ill-posedness of the inverse problem is demonstrated in Sect. 3. Furthermore, we set up a minimizing problem equivalent to the Tikhonov regularized inverse problem and prove its unique solvability applying Tikhonov result in combination with Sobolev Embedding Theorem and a diffeomorphism. Section 4 covers numerical implementation based on FEM followed by Sect. 5 containing some 3D synthetic and real-life examples and numerical investigations.

2 Direct Problem

The analysis of the forward problem, i.e., the determination of u for given f in a conducting medium is the first step of facing the corresponding inverse problem.

Let \(\Omega \subset \mathbb{R}^{3}\) be a bounded domain with C 2-boundary \(\partial \Omega \), which ensures a regularity property of u, see Theorem 1. By ν we denote the outer unit normal vector on \(\partial \Omega \). In general, we assume \(supp(f) \cap \Gamma = \varnothing \). Note that the model reduction from Maxwell equations to Laplace equation bounds the diameter \(L_{\Omega }\) of \(\Omega \) by satisfying
$$\omega \mu \sigma L_{\Omega }^{2}(1 + \tfrac{\omega \epsilon } {\sigma } )\ \ll \ 1,$$
where μ is the magnetic permeability and ω the maximum occurring frequency of the source term in Maxwell model. For further details see Sommer (2012). The strong formulation of the direct problem is given as follows:

Problem 1 (Strong Formulation). 

Let \(f \in C(\Omega )\) with \(supp(f) \subset \Omega \), \(\sigma \in C^{1}(\Omega ) \cap C(\overline{\Omega })\) and \(g \in L_{>0}^{\infty }(\partial \Omega )\). Find \(u \in C^{2}(\Omega ) \cap C^{1}(\overline{\Omega })\) that fulfills the partial differential equations
$$\displaystyle\begin{array}{rcl} -\nabla \cdot (\sigma \nabla u)& =& f\quad \text{in }\Omega, \\ \frac{\partial u} {\partial \nu } + gu& =& 0\quad \text{on }\partial \Omega . \\ \end{array}$$
To tackle this problem numerically by FEM, we need its weak formulation. Hence, in weak sense the electric potential has to be set in a Sobolev space of an order reduced by one, i.e., \(u \in H^{1}(\Omega )\). The Sobolev space \(H^{m}(\Omega )\), see e.g., Brenner and Scott (1994), consists of all functions \(u \in L^{2}(\Omega )\) such that for every multi-index α with | α | ≤ m, the weak partial derivative D α u belongs to \(L^{2}(\Omega )\), i.e.,
$$H^{m}(\Omega )\ :=\ \left \{u \in L^{2}(\Omega ) : D^{\alpha }u \in L^{2}(\Omega )\ \forall \ \vert \alpha \vert \leq m\right \}.$$

Multiplying each side of Laplace’s equation by a test function \(v \in H^{1}(\Omega )\), integrating over \(\Omega \) and applying the integration by parts yields the weak formulation of Problem 1. That representation is more convenient for numerical implementation (Braess 2003).

Problem 2 (Weak Formulation). 

Let \(f \in L^{2}(\Omega )\) with \(supp(f) \subset \Omega \) and σ provided by Problem 1. Find \(u \in H^{1}(\Omega )\) that fulfills the integral equation
$$\displaystyle\int _{\Omega }\sigma \nabla u \cdot \nabla v\,dx +\displaystyle\int _{\partial \Omega }\sigma guv\,ds\ =\ \displaystyle\int _{\Omega }fv\,dx\quad \text{for all }v \in H^{1}(\Omega ).$$
(2)

Because of the measurement methodology, see the operator defined in (4), and the Sobolev Embedding Theorem 3 we need a stronger regularity for the inverse problem. Assuming some restrictions on σ, g, and the bound \(\partial \Omega \) following regularity theorem prescribes the space for the solution u.

Theorem 1.

Let \(\Omega \) be a bounded C 2 -domain. Assume σ is Lipschitz in \(\Omega \) with \(0 < \frac{1} {C} <\sigma (x) < C\) for all \(x \in \overline{\Omega }\), \(f \in L^{2}(\Omega )\) and 0 ≤ g(x) ≤ g0 a.e. on \(\partial \Omega \). Then \(u \in H^{2}(\Omega )\) and
$$\Vert u\Vert _{H^{2}(\Omega )}\ \leq \ C\left (\Vert u\Vert _{0} +\Vert f\Vert _{0}\right ).$$
A general form of this theorem can be found in Gilbarg and Trudinger (2001, Chap. 8.4) and Salsa (2008, Thm. 8.13, 8.14). For this more regular case we denote the forward operator mapping given source onto electric potential by
$$\Lambda :\, \left \{\begin{array}{ccc} L^{2}(\Omega )& \rightarrow &H^{2}(\Omega ), \\ f & \mapsto & u. \end{array} \right .$$
Because of the properties of an integral operator, \(\Lambda \) is a bounded linear operator (Sommer 2012). It represents the governing physics. So, the brief formulation of the Problem 2 reads: For given f solve the equation
$$\Lambda [f]\ =\ u.$$
(3)
The uniform ellipticity of the differential operator − ∇ ⋅(σ ∇ ⋅) is provided by the condition (Salsa 2008, Chap. 8.5 (8.46))
$$0 < \frac{1} {C} <\sigma (x) < C\text{ for all }x \in \overline{\Omega }.$$

The direct elliptic problem satisfies the Lax-Milgram lemma (Gilbarg and Trudinger 2001), therefore it is unique solvable. Details can be found in Sommer (2012). Analog results for general mixed boundary condition were achieved in Helfrich-Schkarbanenko (2011). Now we can turn to the inverse problem.

3 Inverse Problem

The usual way to solve an inverse problem is to minimize some functional of the discrepancy between the inverse problem solver and the data. In the following, we present an approach of finding f on the basis of incomplete information of u. The term incomplete means that the measurements are carried out on a subdomain of \(\Omega \). This restriction is common in the practice because one prefers a non-destructing imaging methodology. So we introduce an operator
$$T :\, \left \{\begin{array}{ccc} H^{2}(\Omega )& \rightarrow & L^{2}(\Gamma ), \\ u & \mapsto &u_{d} := u\vert _{\Gamma }, \end{array} \right .$$
(4)
which restricts \(u := \Lambda [f]\) to the local measurements \(u_{d} := u\vert _{\Gamma }\) on a curve \(\Gamma \subset \Omega \). We will show the well-definition of T in the proof of Theorem 6. The combination of \(\Lambda \) with T, cf. Fig. 2, leads to the following ill-posed inverse problem:
Fig. 2

Domains, co-domains and operators acting in Inverse Problem 3. Note that \(\Omega \subset \mathbb{R}^{3}\) and \(\Gamma \) is a curve in \(\Omega \)

Problem 3 (Inverse Problem). 

For given local measurements \(u_{d} \in L^{2}(\Gamma )\) find \(f \in L^{2}(\Omega )\) such that \(T\Lambda [f] = u_{d}\) holds.

This problem is ill-posed in sense of Hadamard (1915) because \(T\Lambda \) is non-injective. We demonstrate this in the following example considering an analytical radial symmetric case. Detailed computation can be found in Sommer (2012, Chap. 4.2.1).

Example. Our aim is to construct two radial symmetric sources f 1 ≢ f 2 such that their corresponding radial symmetric electric potentials u 1 and u 2 are identical in the air layer. This will clarify the non-injectivity of \(T\Lambda \).

Assume \(\Omega := B_{R}(0) \subset \mathbb{R}^{3}\) is a ball with radius R > 0 and concentric layers, see Fig. 3. The ball B c (0) represents the ground and \(\Omega \setminus \overline{B_{c}(0)}\) the air layer. Inside each layer the functions σ and f are constant and especially σ ≡ 1 in \(\overline{B_{c}(0)}\).
Fig. 3

Two different current density sources, see black layers, generating the same potential field u in the air layer. (a) f 1 ≡ 1 in \(\overline{B_{b_{1}}(0)}\setminus B_{a_{1}}(0)\) and f 1 ≡ 0 else. (b) f 2 ≡ 2 in \(\overline{B_{b_{2}}(0)}\setminus B_{a_{2}}(0)\) and f 2 ≡ 0 else

Here, since σ is piecewise constant, the origin Strong Formulation 1 has to be reformulated as a Transmission Problem, see Sommer (2012) for details. Then, because of the fundamental solution of Laplace’s equation (Evans 2008, Thm. 2.2.1) and the simplicity of f and σ, the potentials u 1 and u 2 can be represented in an explicit way as follows:
$$u_{1}(x) = \left \{\begin{array}{ll} \frac{4} {3x}\big(b_{1}^{3} - a_{ 1}^{3}\big), &\text{for}\;c < \vert x\vert \leq R, \\ \frac{1} {3x}\big(b_{1}^{3} - a_{ 1}^{3}\big) + \frac{1} {c}\Big(b_{1}^{3} - a_{ 1}^{3}\Big), &\text{for}\;b_{ 1} \leq \vert x\vert \leq c, \\ \frac{1} {2}\big(b_{1}^{2} -\frac{2a_{1}^{3}} {3x} -\frac{x^{2}} {3} \big) + \frac{1} {c}\Big(b_{1}^{3} - a_{ 1}^{3}\Big),&\text{for}\;a_{ 1} < \vert x\vert < b_{1}, \\ \frac{1} {2}\big(b_{1}^{2} - a_{ 1}^{2}\big) + \frac{1} {c}\Big(b_{1}^{3} - a_{ 1}^{3}\Big), &\text{for}\;0 \leq \vert x\vert \leq a_{ 1}. \end{array} \right .$$
$$u_{2}(x) = \left \{\begin{array}{ll} \frac{8} {3x}\big(b_{2}^{3} - a_{ 2}^{3}\big), &\text{for}\;c < \vert x\vert \leq R, \\ \frac{2} {3x}\big(b_{2}^{3} - a_{ 2}^{3}\big) + \frac{2} {c}\Big(b_{2}^{3} - a_{ 2}^{3}\Big), &\text{for}\;b_{ 2} \leq \vert x\vert \leq c, \\ \big(b_{2}^{2} -\frac{2a_{2}^{3}} {3x} -\frac{x^{2}} {3} \big) + \frac{2} {c}\Big(b_{2}^{3} - a_{ 2}^{3}\Big),&\text{for}\;a_{ 2} < \vert x\vert < b_{2}, \\ \big(b_{2}^{2} - a_{2}^{2}\big) + \frac{2} {c}\Big(b_{2}^{3} - a_{ 2}^{3}\Big), &\text{for}\;0 \leq \vert x\vert \leq a_{ 2}. \end{array} \right .$$
Here the parameter a 1, b 1, a 2, and b 2 are still arbitrary with 0 < a 1 < b 1 < c < R <  and 0 < a 2 < b 2 < c. To achieve the identity u 1 ≡ u 2 in air layer we have to set
$$b_{1}^{3} - a_{ 1}^{3}\ =\ 2(b_{ 2}^{3} - a_{ 2}^{3}).$$
Obviously we obtain the non-injectivity of the operator \(T\Lambda \) for local measurements, since for c ≤ | x | ≤ R we have the identity
$$u_{1}(\vert x\vert ) = T\Lambda [f_{1}](x)\ =\ T\Lambda [f_{2}](x) = u_{2}(\vert x\vert ),$$
while f 1 ≢ f 2, see Fig. 4. This result leads to the following Lemma.

Lemma 1.

\(T\Lambda \) is non-injective.

We show that a solution of the Inverse Problem 3 exists, but it is not unique in general. In Devaney and Sherman (1982) the fields radiated by spherically symmetric time-harmonic sources satisfying the Helmholtz equation in homogeneous medium are used to illustrate how little can be learned about a source from knowledge of the radiated field outside of the source volume.
Fig. 4

Analytic solutions u 1 and u 2 of the radial symmetric problem generated by two different current sources. Chosen parameters: a 1 = 1, \(b_{1} = a_{2} = 4\), \(b_{2} = \root{3}\of{\tfrac{1} {2}(b_{1}^{3} - a_{ 1}^{3}) + a_{ 2}^{3}}\), c = 7 and R = 10

Note that the corresponding discrete version of Problem 3 is strongly under-determined (Sommer 2012). However, applying the Tikhonov regularization (Tikhonov 1963), which was independently developed by Phillips (1962) as well, we can enforce the uniqueness of a pseudo-solution. The idea of Tikhonov consists of perturbation of an operator via spectral shift to enforce the uniqueness of a pseudo-solution (Rieder 2003). Doing so, we obtain the following Regularized Inverse Problem 4 for local measurements, which plays the central role in this chapter.

Problem 4 (Regularized Inverse Problem). 

Let \(u_{d} \in L^{2}(\Gamma )\) be the local measurements on a curve \(\Gamma \subset \Omega \). Solve the minimization problem
$$\arg \min \limits _{{-18.0pt}f\in L^{2}(\Omega )}\left \{\tfrac{1} {2}\left \Vert T\Lambda [f] - u_{d}\right \Vert _{L^{2}(\Gamma )}^{2} + \tfrac{\alpha } {2}\|L[f]\|_{L^{2}(\Omega )}^{2}\right \}$$
(5)
for a bounded linear operator \(L : L^{2}(\Omega ) \rightarrow L^{2}(\Omega )\), which is continuously invertible on its image, and α > 0. We denote the solution by f L, α .

That means we are seeking for a f that minimizes the error in the data \(\|T\Lambda [f] - u_{d}\|_{L^{2}(\Gamma )}^{2}\) simultaneously considering an a priori information about the true solution involved by L.

3.1 Preliminary

Before we consider the unique solvability of Problem 4 we have to remember some definitions in differential geometry, as well as the cone property of a domain.

Definition 1 (Toponogov 2006, def. 1.2.1). 

A regular k-fold continuously differentiable curve (or path) \(\Gamma \) in the space \(\mathbb{R}^{3}\) is described by a homeomorphism (parametrization) \(\varphi : I \rightarrow \mathbb{R}^{3}\), where I : = [a, b] is the parameter interval, satisfying the following conditions:
  1. 1.

    \(\varphi \in C^{k}\), k ≥ 1;

     
  2. 2.

    The rank of \(\varphi\) is maximal (equal to 1).

     

This definition except any intersections of \(\Gamma \) and \(\Gamma \setminus \{\varphi (a),\varphi (b)\}\) is a one-dimensional submanifold. Note that a regular curve \(\Gamma \) of class C k , k ≥ 1, is diffeomorphic to a line segment (Toponogov 2006). In rectangular Cartesian coordinate system \(\Gamma \) is determined by its so-called parametric functions \(\varphi (t) := (x(t)\), y(t), z(t)) ⊤ , where t ∈ [a, b]. The first condition in Definition 1 means that x, y and z belong to class C k , and the second condition means that the derivatives x , y , z cannot simultaneously vanish for any t.

Definition 2 (Adams and Fournier 2003, Def. 4.6). 

The domain \(\Omega \) satisfies the cone condition if there exists a finite cone \(\mathcal{C}\) such that each \(x \in \Omega \) is the vertex of a finite cone \(\mathcal{C}_{x}\) contained in \(\Omega \) and congruent to \(\mathcal{C}\).

For the unique solvability of the Regularized Inverse Problem 4 we apply the following Theorem, Remark and Sobolev embedding theorem. Firstly we introduce the Tikhonov functional
$$J_{L,\alpha }(x)\ :=\ \tfrac{1} {2}\|Kx - y\|_{Y }^{2} + \tfrac{\alpha } {2}\|Lx\|_{Z}^{2}\quad \text{for}\;x \in X$$
for a given bounded linear operator K : X → Y and y ∈ Y . So, the term in (5) is exactly of that form. The following Theorem establishes the relation between that functional and the corresponding normal equation.

Theorem 2.

Let both K : X → Y and L : X → Z be bounded linear operators between Hilbert spaces and K the adjoint operator to K. If α > 0 and L continuously invertible on its image, then the Tikhonov functional J L,α has a unique minimum x L, α ∈ X. This minimum x L,α is the unique solution of the normal equation
$$\big(K^{{\ast}}K +\alpha L^{{\ast}}L\big)x\ =\ K^{{\ast}}y.$$

So, solving the minimizing problem (5) is equivalent to dealing with the corresponding normal equation. Note that αL  ∗  L in (5) shifts the nonnegative eigenvalues of K  ∗  K away from zero and makes K  ∗  K +  αL  ∗  L invertible.

Remark 1.

For the case that L is the identity operator, the proof of Theorem 2 can be found in Kirsch (1996, Thm. 2.11). If L is continuously invertible on its image, then there exists β > 0 with
$$\beta \|x\|_{X}\ \leq \ \| Lx\|_{Z}\quad \text{for all}\;x \in X.$$

According to Rieder (2003, Thm. 8.1.15) this estimate is sufficient to extend the proof in Kirsch (1996, Thm. 2.11) for the general case given in Theorem 2. Note that the solution x L, α  ∈ X is unique and depends continuously on y ∈ Y .

Because of the measurement methodology we apply the Sobolev Embedding Theorem and generalize the results via a diffeomorphism.

Theorem 3 (Sobolev Embedding Theorem Adams and Fournier 2003, Thm. 4.12). 

Let \(\Omega \subset \mathbb{R}^{n}\) satisfy the cone condition and \(\Omega ^{k}\) is a k-dimensional domain generated by intersection of \(\Omega \) and k-dimensional plane in \(\mathbb{R}^{n}\) , where 1 ≤ k ≤ n. If one of both requirements
  1. 1.

    2m < n and n − 2m < k, or

     
  2. 2.

    2m = n

     
holds for \(j,m \in \mathbb{N}\) with j ≥ 0 and m ≥ 0, then the following embedding exists
$$H^{j+m}(\Omega )\ \hookrightarrow \ H^{j}(\Omega ^{k}).$$

Theorem 4 (Dobrowolski 2006, Thm. 6.8). 

Let \(x \in \Omega \), \(y \in \Omega ^{{\prime}}\) and \(d : \Omega ^{{\prime}}\rightarrow \Omega \) be a C m -diffeomorphism, so that \(\mathcal{D}u(y) = u(d(y))\) is the transformation of the function \(u : \Omega \rightarrow \mathbb{R}\) . Then the map \(\mathcal{D} : H^{m}(\Omega ) \rightarrow H^{m}(\Omega ^{{\prime}})\) is bijective and bounded with bounded inverse, which means that
$$c_{1}\|u\|_{m,\Omega }\ \leq \ \|\mathcal{D}u\|_{m,\Omega ^{{\prime}}}\ \leq \ c_{2}\|u\|_{m,\Omega }.$$

Theorem 5 (Transformation Formula Königsberger 2004, Thm. 9.1). 

Let \(\Omega \), \(\Omega ^{{\prime}}\) be open subsets of \(\mathbb{R}^{n}\) and \(d : \Omega ^{{\prime}}\rightarrow \Omega \) an invertible diffeomorphism. Then the function \(u : \Omega \rightarrow \mathbb{R}\) is integrable if and only if \(y\mapsto u(d(y))\vert \mathrm{det}(\mathrm{grad}(d)\vert \) is integrable. In this case the following equation holds:
$$\displaystyle\int _{d(\Omega ^{{\prime}})}u(x)\,dx\ =\ \displaystyle\int _{\Omega ^{{\prime}}}u(d(y))\,\vert \mathrm{det}(\mathrm{grad}(d(y)))\vert \,dy.$$
The term det (grad (d(y))) =: J(y) is called Jacobian or functional determinant of d.

Now we can present the main result of this chapter.

3.2 Main Result

Theorem 6.

Let \(\Gamma \subset \Omega \) be a regular curve of class C 2 . Then the regularized inverse problem  4 has a unique pseudo-solution \(f_{L,\alpha } \in L^{2}(\Omega )\) .

Remark 2.

Deriving the weak formulation 2 requires less regularity on \(\partial \Omega \) than C 2, namely \(\Omega \) has to be a Lipschitz domain, see Steinbach (2008, Def. 2.1). Furthermore, Theorem 3 assumes a domain that fulfills the cone condition, see Definition 2. We emphasize that a C 2 boundary of a bounded domain satisfies the Lipschitz condition, cf. Steinbach (2008, Def. 2.1), and every bounded Lipschitz domain satisfies the cone condition (Adams and Fournier 2003, Chap. 4).

If \(\Gamma \) does not start and end on the boundary \(\partial \Omega \), \(\Gamma \) could be extended to another curve, which fulfills this property. Then, the Theorem 6 is applicable again.

Proof.

In the first part of the proof we show the well-definition of the linear restriction operator T, defined in (4). In case of \(\Gamma \) is a line segment the proof consists of applying the Sobolev Embedding Theorem 3 (valid only for hyperplanes) and the Remark in the preliminary. To achieve the well-definition for general \(\Gamma \), we apply a smooth diffeomorphism d with properties given by Theorems 4 and 5 to diffeomorph \(\Gamma \) to a line segment and then apply the Sobolev Embedding Theorem 3. The second part of the proof consists in showing the uniqueness of the pseudo-solution f L, α by using the well-known Theorem 2 for the Tikhonov-Phillips regularization and the attached Remark.

Part 1. Without loss of generality we choose d as an automorphism, such that \(\overline{\Omega } = \overline{\Omega ^{{\prime}}}\). For the sake of simplicity let \(\Gamma \) be a curve whose starting and end point lie on the boundary \(\partial \Omega \). The preimage of the curve \(\Gamma \) under a C 2-diffeomorphism \(d : \Omega ^{{\prime}}\rightarrow \Omega \) has to be a line segment \(\Gamma ^{{\prime}}\) which starts and ends on the boundary \(\partial \Omega ^{{\prime}}\), as well. For clarification see Fig. 5.

By assumption \(u \in H^{2}(\Omega )\) holds. Applying Theorem 4 a generally nonlinear operator \(\mathcal{D} : H^{2}(\Omega ) \rightarrow H^{2}(\Omega ^{{\prime}})\) exists, which depends on d and has a bounded inverse. The connection between \(\mathcal{D}\) and d is, cf. Dobrowolski (2006, Sect. 6.3):
$$\mathcal{D}u(y)\ =\ u\big(d(y)\big) = u(x),$$
where \(x \in \Omega \), \(y \in \Omega ^{{\prime}}\) and hence \(\mathcal{D}u \in H^{2}(\Omega ^{{\prime}})\) holds. Since \(\overline{\Omega } = \overline{\Omega ^{{\prime}}}\) the regularity property of \(\partial \Omega \) transfers onto \(\partial \Omega ^{{\prime}}\). Now, applying the Sobolev Embedding Theorem 3 yields
$$H^{2}(\Omega ^{{\prime}})\ \hookrightarrow \ H^{0}(\Gamma ^{{\prime}}) = L^{2}(\Gamma ^{{\prime}}).$$
(6)
This embedding can be described by a bounded linear (embedding) operator
$$\mathcal{T} : H^{2}(\Omega ^{{\prime}}) \rightarrow L^{2}(\Gamma ^{{\prime}})$$
such that \(\mathcal{T}\mathcal{D}u(y) = \mathcal{D}u(y)\) for all \(y \in \Gamma ^{{\prime}}\) holds. Consequently \(\mathcal{T}\) is well-defined and \((\mathcal{D}u)\vert _{\Gamma ^{{\prime}}} = \mathcal{T}\mathcal{D}u \in L^{2}(\Gamma ^{{\prime}})\).
Fig. 5

Diffeomorphism d applied on \(\Gamma ^{{\prime}}\) and on \(\Omega ^{{\prime}}\), respectively

Now we have to show that \(u\vert _{\Gamma } = Tu \in L^{2}(\Gamma )\) holds w.r.t. given diffeomorphism d. For a C 2-diffeomorphism d the estimation
$$0 < \vert J(y)\vert < c$$
(7)
holds for all \(y \in \Omega ^{{\prime}}\) by Dobrowolski (2006, Chap. 6, p. 100). Thus, the length of \(\Gamma ^{{\prime}}\) and \(\Gamma \) is finite. This implies the following estimation.
$$\displaystyle\begin{array}{rcl} \|u\|_{L^{2}(\Gamma )}^{2}& = & \displaystyle\int _{ \Gamma }\vert u(x)\vert ^{2}\,dx \\ & = & \displaystyle\int _{d(\Gamma ^{{\prime}})}\vert u(x)\vert ^{2}\,dx \\ & \stackrel{\text{Theorem\ 5}}{=}& \displaystyle\int _{\Gamma ^{{\prime}}}\vert u(d(y))\vert ^{2}\vert \mathrm{det(grad(}d(y)))\vert \,dy \\ & = & \displaystyle\int _{\Gamma ^{{\prime}}}\vert u(d(y))\vert ^{2}\,\vert J(y)\vert \,dy \\ & \stackrel{(7)}{\leq } & c\displaystyle\int _{\Gamma ^{{\prime}}}\vert \mathcal{D}u(y)\vert ^{2}\,dy < \infty . \\ \end{array}$$

The last estimation follows from \((\mathcal{D}u)\vert _{\Gamma ^{{\prime}}} \in L^{2}(\Gamma ^{{\prime}})\) due to Sobolev Embedding Theorem 3. We show that \(u\vert _{\Gamma } \in L^{2}(\Gamma )\). Thus, the operator \(T : H^{2}(\Omega ) \rightarrow L^{2}(\Gamma )\) is well defined for \(C^{2}\)-smooth curves \(\Gamma \).

Part 2. Because of the fact that T and \(\Lambda \) are bounded linear operators, their composition
$$T\Lambda : L^{2}(\Omega ) \rightarrow L^{2}(\Gamma )$$
is also a bounded linear operator with estimation \(\|T\Lambda \| \leq \| T\|\,\|\Lambda \|\), so that Theorem 2 can be applied. Because L is assumed to be continuously invertible, the operator L  ∗  L has non-negative eigenvalues and the operator \(\Lambda ^{{\ast}}T^{{\ast}}T\Lambda +\alpha \, L^{{\ast}}L\) is continuously invertible for α > 0, as well (Rieder 2003, Chap. 4). Thus, according to Theorem 2, the unique pseudo-solution \(f_{L,\alpha } \in L^{2}(\Omega )\) of the regularized inverse Problem 4 can be computed for fixed L and fixed α by
$$f_{L,\alpha }\ =\ (\Lambda ^{{\ast}}T^{{\ast}}T\Lambda +\alpha L^{{\ast}}L)^{-1}\Lambda ^{{\ast}}T^{{\ast}}[u_{ d}],$$
(8)
which completes the proof.

Remark 3.

The first parameter L of the pseudo-solution f L, α can be chose by means of problem setting. We will touch this topic in Sect. 4.2. The determination of an optimal α turns out to be a challenging task, in particular w.r.t. the real-life scenario/data. For this purpose we apply the L-curve criterion, see Sect. 4.2.

4 Numerical Implementation

After the theoretical framework, we concentrate on the numerical implementation of the forward and inverse problem as well. Due to the Céa lemma (Brenner and Scott 1994) we can discretize the forward weak problem, for example via FEM by means of Galerkin (Braess 2003) without disturbing the unique solvability. The Sobolev space \(H^{1}(\Omega )\) is modeled by a finite dimensional subspace \(H_{h}^{1}(\Omega )\) consisting of n basis functions {ϕ i } i = 1 n which are linear finite elements in this work. That means the direct and inverse problem has n degrees of freedom.

In Sects. 2 and 3 we assumed \(\Omega \) to have a C 2 boundary. We justify the modeling of \(\Omega \) by means of tetraeders since the boundary of a Lipschitz domain \(\Omega _{L}\) can be approximated arbitrarily close by a C 2 boundary via the convolution
$$\Omega _{\epsilon }\ :=\ \Omega = \Omega _{L} {\ast}\varphi _{\epsilon },$$
where \(\varphi _{\epsilon }\) is a mollifier satisfying \(\mathop{lim}_{\epsilon \rightarrow 0}\varphi _{\epsilon }(x) =\delta (x)\) with δ the Dirac impulse in \(\mathbb{R}^{3}\) (Adams and Fournier 2003, Chap. 2). In particular, the approximation of identity
$$\mathop{lim}_{\epsilon \rightarrow 0}\Omega _{\epsilon } =\mathop{ lim}_{\epsilon \rightarrow 0}\Omega _{L} {\ast}\varphi _{\epsilon } = \Omega _{L}$$
and the inclusion
$$\mathrm{supp}\Omega _{\epsilon } =\mathrm{ supp}(\Omega _{L} {\ast}\varphi _{\epsilon })\ \subset \ \mathrm{ supp}\Omega _{L} \oplus \mathrm{ supp}\varphi _{\epsilon },$$
hold, where ⊕ indicates the Minkowski addition.

In the following we emphasize by bold letters that the numerical linear operators are matrices and the discrete functions are represented by vectors. A detailed procedure can be found in Sommer (2012).

4.1 Direct Problem

Let \(H_{h}^{1}(\Omega )\) be the finite dimensional subspace of \(H^{1}(\Omega )\). The weak formulation 2 of the Forward Problem 1 is given in matrix notation by
$$\boldsymbol{A}\boldsymbol{u} =\boldsymbol{ M}\boldsymbol{f}\quad \text{for all}\;\phi _{k} \in H_{h}^{1}(\Omega ),$$
(9)
where \(\boldsymbol{A} \in \mathbb{R}^{n\times n}\) is the stiffness matrix with components
$$\boldsymbol{A}_{i,j} =\displaystyle\int _{\Omega }\sigma \nabla \phi _{i} \cdot \nabla \phi _{j}\,dx +\displaystyle\int _{\partial \Omega }g\,\phi _{i}\,\phi _{j}\,ds,$$
\(\boldsymbol{M} \in \mathbb{R}^{n\times n}\) is the mass matrix with entries
$$\boldsymbol{M}_{i,j} =\displaystyle\int _{\Omega }\phi _{i}\,\phi _{j}\,dx,$$
(10)
\(\boldsymbol{u} \in \mathbb{R}^{n}\) is the nodal electric potential vector and \(\boldsymbol{f} \in \mathbb{R}^{n}\) is the nodal current density source vector. Taking \(\boldsymbol{u} = (u_{1},\ldots,u_{n})^{\top }\in \mathbb{R}^{n}\) we show that \(\boldsymbol{A}\) is a positive definite matrix:
$$\displaystyle\begin{array}{rcl} \boldsymbol{u}^{\top }\boldsymbol{Au}& =& \displaystyle\sum _{ i=1}^{n}\displaystyle\sum _{ j=1}^{n}u_{ i}\Big(\displaystyle\int _{\Omega }\sigma \nabla \phi _{i} \cdot \nabla \phi _{j}\,dx +\displaystyle\int _{\partial \Omega }g\,\phi _{i}\,\phi _{j}\,ds\Big)u_{j} \\ & =& \displaystyle\int _{\Omega }\sigma \Big(\displaystyle\sum _{i=1}^{n}\nabla (u_{ i}\phi _{i})\Big) \cdot \Big (\displaystyle\sum _{j=1}^{n}\nabla (u_{ j}\phi _{j})\Big)\,dx +\displaystyle\int _{\partial \Omega }g\Big(\displaystyle\sum _{i=1}^{n}u_{ i}\phi _{i}\Big)\Big(\displaystyle\sum _{j=1}^{n}u_{ j}\phi _{j}\Big)\,ds \\ & =& \displaystyle\int _{\Omega }\sigma \vert \nabla u_{h}\vert ^{2}\,dx +\displaystyle\int _{ \partial \Omega }g\vert u_{h}\vert ^{2}\,ds\ \geq \ 0, \\ \end{array}$$
where \(u_{h} :=\sum _{ j=1}^{n}u_{j}\phi _{j} \in H_{h}^{1}(\Omega )\). Since σ, g > 0 a.e., bounded and measurable, equality holds if and only if \(\nabla u_{h} \equiv 0\) in \(\Omega \) and \(u_{h} \equiv 0\) on \(\partial \Omega \). Hence, \(\boldsymbol{u}^{\top }\boldsymbol{Au}\) is zero if \(\boldsymbol{u} \equiv 0\). So, \(\boldsymbol{u}^{\top }\boldsymbol{Au} > 0\) whenever \(\boldsymbol{u}\not =0\). That means the stiffness matrix \(\boldsymbol{A}\) is positive definite and thus invertible. Based on Eq. (9) we can now setup the discrete forward operator by
$$\boldsymbol{\Lambda } :=\boldsymbol{ A}^{-1}\boldsymbol{M}.$$

The existence of \(\boldsymbol{A}^{-1}\) and its positive definition follows from the positive definition of \(\boldsymbol{A}\) as well as from Céa Lemma and Brenner and Scott (1994). Analog to the arguments shown above the mass matrix \(\boldsymbol{M}\) is symmetric, positive definite, too. Hence, the product \(\boldsymbol{\Lambda }\) is invertible, but in general non-positive definite and non-symmetric.

4.2 Inverse Problem

The set \(\Gamma _{h}\) of measure grid points
$$\Gamma _{h}\ :=\ \{ p\;\vert \;p_{i}\;\text{is measuring grid point},i = 1,\ldots,k\}$$
is a discrete FE model for the path \(\Gamma \). We start with the implementation of the operator T given in (4). Let \(\boldsymbol{T} \in \mathbb{R}^{k\times n}\), where k ≪ n is the number of measuring grid points. Its components are:
$$\boldsymbol{T}_{i,j} := \left \{\begin{array}{ll} 1,&\text{if the measuring grid point}\;i\;\text{has} \\ &\text{the global grid point number}\;j, \\ 0,&\text{else}, \end{array} \right .$$
such that
$$\boldsymbol{u}_{d} :=\boldsymbol{ u}\vert _{\Gamma _{h}}\ =\ \boldsymbol{ T\Lambda }\,\boldsymbol{f}.$$
(11)
Applying (8) the pseudo-solution \(\boldsymbol{f}\!_{L,\alpha }\) can be computed by
$$\boldsymbol{f}\!_{L,\alpha }\ =\ \big (\boldsymbol{\Lambda }^{\top }\boldsymbol{T}^{\top }\boldsymbol{T\Lambda } +\alpha \boldsymbol{ L}^{\top }\boldsymbol{L}\big)^{-1}\boldsymbol{\Lambda }^{\top }\boldsymbol{T}^{\top }\boldsymbol{u}_{ d},\quad \alpha > 0,$$
(12)
for experimental data \(\boldsymbol{u}_{d}\) or synthetic measurements \(\boldsymbol{u}_{d}\) generated by (11). In general the pseudo-solution \(\boldsymbol{f}\!_{L,\alpha }\) depends on the penalty operator \(\boldsymbol{L}\) and on the regularization parameter α. To find the optimal α we apply the L-curve criterion (Hansen 1998, Chap. 4.6), which is a heuristic parameter choice rule. It investigates the graph \((\|T\Lambda [f_{L,\alpha }] - u_{d}\|_{L^{2}(\Gamma )}^{2},\|L[f_{L,\alpha }]\|_{L^{2}(\Omega )}^{2})\) as a function of \(\alpha \in [\alpha _{\min },\alpha _{\max }]\) for maximal curvature w.r.t. α identifying the optimal one. For the discretized problem the L-curve consists of a finite set of points
$$\left (\begin{array}{c} \|\boldsymbol{T\Lambda }[\boldsymbol{f}]_{L,\alpha _{i}} -\boldsymbol{ u}_{d}\|_{2}^{2} \\ \|\boldsymbol{L}[\boldsymbol{f}]_{L,\alpha _{i}}\|_{2}^{2} \end{array} \right ),\quad i = 1,\ldots,m.$$
Numerical experiments show that the penalty operator \(\boldsymbol{L}\) has a strong influence on the pseudo-solution \(\boldsymbol{f}\!_{L,\alpha }\). Some possible \(\boldsymbol{L}\)-matrices are listed in Table 1.
Table 1

A set of \(\boldsymbol{L}\)-matrices; \(\boldsymbol{I} \in \mathbb{R}^{n\times n}\) is the identity matrix. \(\boldsymbol{M} =\langle \phi _{i},\phi _{j}\rangle _{L^{2}(\Omega )}\) is given in (10), \(\boldsymbol{G} =\langle \nabla \phi _{i},\nabla \phi _{j}\rangle _{L^{2}(\Omega )}\) and \(\boldsymbol{D}\) from (13) are symmetric, positive definite matrices

Regularization

Impact on pseudo-solution

\(\boldsymbol{L}^{\top }\boldsymbol{L}\)

 

Zero-order

Minimal amplitude

\(\boldsymbol{I}\)

 

Zero-order with L 2-norm

Minimal amplitude involving grid element size

\(\boldsymbol{M}\)

 

Zero-order with damping

A priori information about depth

\(\boldsymbol{D}\)

 

First-order with L 2-norm

Minimal gradient involving grid element size

\(\boldsymbol{G}\)

 
If the vertical position of the current source support is available, we can formulate a very effective penalty operator \(\boldsymbol{D}\). It forces the pseudo-solution to damp above a chosen depth τ and to vanish above γ, so that its support has to exist below the depth value τ. Let p i z be the z-coordinate of the i-th grid point p, then we chose a diagonal matrix \(\boldsymbol{D}\) with corresponding matrix elements
$$\boldsymbol{D}_{i,i} := \left \{\begin{array}{ll} \beta, &\text{if}\;p_{i}^{z} \geq \gamma, \\ - \frac{\beta -1} {(\tau -\gamma )^{2}} \,(p_{i}^{z}-\gamma )^{2}+\beta,&\text{if}\;\gamma < p_{i}^{z} <\tau, \\ 1, &\text{if}\;\tau \leq p_{i}^{z}, \end{array} \right .$$
(13)
where γ is the stop band depth, \(\tau\) is the cutoff depth and β is the damping factor. We chose these matrix entities of \(\boldsymbol{D}\) to represent a continuous monotonic damping function depending on the depth of each grid point. Note that all eigenvalues of \(\boldsymbol{D}\) are greater than one.

For the numerical implementation we apply the physics modeling and simulation software COMSOL Multiphysics 4.2 based on FEM to generate the mesh and to solve the elliptic direct problem in weak form 2. The inverse problem solver is implemented in MATLAB R2012b, which is connected to COMSOL by COMSOL Matlab Live-Link to import the necessary components like the stiffness matrix \(\boldsymbol{A}\) into MATLAB environment.

In Colton and Kress (1992, pp. 133, 304) the authors coin the expression inverse crime to denote the act of employing the same model to generate, as well as to invert, synthetic data. Moreover, they warn against committing the inverse crime, “in order to avoid trivial inversion” and go on to state: “it is crucial that the synthetic data be obtained by a forward solver which has no connection to the inverse solver.” Thus, the inverse problem mesh is essentially coarser than the direct problem mesh and it does not contain any information about the structure of the source. The question whether there is a risk of committing the inverse crime in real-life problems is answered by the following. We see no such risk since the forward problem solver for real data is unknown (Wirgin 2008).

5 Numerical Examples

Consider a ground cuboid \(\Omega := (0,20) \times (0,30) \times (-6,2)\) with length unit km. Its discretization contains n ≈ 1. 8 ⋅104 degrees of freedom. The conductivity σ is set to a constant value \(4 \cdot 10^{-2}\,Sm^{-1}\) in the ground and \(10^{-5}\,Sm^{-1}\) in the air. The curve \(\Gamma _{h}\), on which the measurement takes place, is represented by six lines, see Fig. 6. We model a dipole current source density by
$$\boldsymbol{f}(x) := \left \{\begin{array}{rl} 1,&\text{for}\;x \in (8,12) \times (11,15) \times (-3,-2), \\ - 1,&\text{for}\;x \in (8,12) \times (15,19) \times (-3,-2), \\ 0,&\text{else}, \end{array} \right .$$
where \([f]_{SI} = A \cdot m^{-3}\) since f =  ∇ ⋅J b . The first step is generating synthetic data \(\boldsymbol{u}_{d}\) by (11). So, we have to solve the system of linear equations (9) representing the Direct Problem 3, see Fig. 6.
Fig. 6

Illustration for the Direct Problem (9). The current source \(\boldsymbol{f}\) generates the measurements \(\boldsymbol{u}_{d}\), see the colors at six lines in the air layer (in parts faded out)

Subsequently, we generate another (coarser) mesh for computing the pseudo-solution \(\boldsymbol{f}\!_{L,\alpha }\) from these synthetic data \(\boldsymbol{u}_{d}\). Assuming the minimal depth of the source, we use the penalty operator \(\boldsymbol{L}^{\top }\boldsymbol{L} :=\boldsymbol{ D}\) and obtain the results presented in Fig. 7. In general the reconstruction of the depth position of the current source density is a challenging task. Thus, in our experiments we applied the operator \(\boldsymbol{D}\) for penalty purpose.
Fig. 7

Illustration for Inverse Problem (12). Data \(\boldsymbol{u}_{d}\), see six lines in the air layer of \(\Omega \) and the pseudo-solution \(\boldsymbol{f}\!_{L,\alpha }\), see the iso-surfaces in the middle. Note that f was designed as a dipol

5.1 Noise in the Data

In the previous computations we considered exact data. Here we aim to show the behavior of the reconstruction algorithm w.r.t. the noise in the data. Assuming additive white Gaussian noise
$$u^{\delta } := u+\delta$$
with noise level δ, we compute the pseudo-solution \(\boldsymbol{f}\!_{L,\alpha }^{\delta }\), see Fig. 8b, d. Only the components \(\boldsymbol{f}_{L,\alpha,i}^{\delta }\) which fulfill
$$0.3 \cdot \min \boldsymbol{ f}_{L,\alpha }^{\delta } >\boldsymbol{ f}_{ L,\alpha,i}^{\delta }\quad \text{or}\quad \boldsymbol{f}_{ L,\alpha,i}^{\delta } > 0.3 \cdot \max \boldsymbol{ f}_{ L,\alpha }^{\delta }$$
are visible. We notice that even a signal-to-noise ratio of 10 dB does not cause a big noise in the solution, cf. Fig. 8b, d. Note that the regularization parameter α has to be adopted for every noise level individually. Otherwise the computation of the pseudo-solutions would fail in general.
Fig. 8

Measurements at six lines with corresponding reconstruction depending on SNR. The regularization parameter α is adapted to the SNR. (a) Measurements \(\boldsymbol{u}_{d}\) with SNR = 30 dB. (b) Reconstructed \(\boldsymbol{f}\) for SNR = 30 dB. Points represent FE nodes. (c) Measurements \(\boldsymbol{u}_{d}\) with SNR = 10 dB. (d) Reconstructed \(\boldsymbol{f}\) for SNR = 10 dB. Points represent FE nodes

To model a channel with additive white Gaussian noise we applied the MATLAB function awgn(u, snr,  measured ). It measures the energy of the desired vector signal u and adds white Gaussian noise to it. The variable snr describes the signal-to-noise ratio (SNR) in dB.

5.2 Real-Life Data

In (airborne) electromagnetic exploration one measures the amplitude of the magnetic field or its component(s) instead of electric potential. In general, the conversion from magnetic field to electric potential is non-linear. This conversion can be represented by a bounded linear operator via the linear approximation, such that the presented approach based on the Tikhonov regularization can be applied again (Sommer 2012). Numerical investigation shows that the choice of the optimal regularization parameter α is more difficult than in the synthetic case (Sommer 2012). In detail, the L-curve criterion yields unfortunately non-unique pseudo-solutions. We find out that applying \(\boldsymbol{L}^{\top }\boldsymbol{L} =\boldsymbol{ G}\), see Table 1 for the penalty term in (12), leads to a smooth enough L-curve. Similar to synthetic case, the measurements in practice are stable against relatively high noise levels. One example of reconstruction from experimental data is shown in Fig. 9. The number of degrees of freedom for the corresponding FE model is 14 ⋅103. Applying \(\boldsymbol{L}^{\top }\boldsymbol{L} =\boldsymbol{ G}\) for the penalty term yields a compact support of \(\boldsymbol{f}\!_{L,\alpha }\). The parameter α is chosen by means of L-curve criterion, Fig. 9a.
Fig. 9

Reconstruction results (b) for real-life measurement data. Third red marked cross in (a) represents the optimal α. Measurements took place at six flight lines in North-South direction (faded out) in the middle of the air layer

6 Summary and Outlook

We discussed a linear inverse problem derived from an elliptic boundary value problem that arises in electrostatics. It turned out that the direct problem is well-posed and uniquely solvable by Lax-Milgram Lemma. The corresponding restricted inverse problem, i.e., identification of current source density f from local electric potential data u d , is not unique solvable and consequently it is ill-posed. The uniqueness of a pseudo-solution was achieved by means of the Tikhonov regularization. For the corresponding theoretical framework we applied the Sobolev Embedding Theorem and a smooth diffeomorphism. In particular, the penalty term of the Tikhonov functional incorporates a priori information about the solution f and so stabilizes the reconstruction. Remember that (1) models steady state, diffusive phenomena and not only the electrostatic phenomena.

The FEM is an attractive tool to model, discretize and solve the continuous problem numerically. For mesh generating and assembling the stiffness matrix we took COMSOL Multiphysics software. The inverse problem solver was developed in MATLAB environment. Because of the high mesh resolution it makes sense to use high performance computers dealing with this problem.

The numerical experiments confirm that different penalty operators deliver different pseudo-solutions. So, further analysis of this relation and providing the algorithm with suitable adapting penalty term is one focus of our research. Moreover, in context of inverse problem the electrical conductivity σ in the ground is actually unknown and has to be reconstructed, as well. This leads to a non-linear ill-posed inverse problem (of parameter identification class) which we can face by means of iterative Tikhonov regularized methods. Doing so we could increase the quality of the current source density reconstruction.

Notes

Acknowledgements

We thank Jörg Bäuerle for his fruitful comments.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Aron Sommer
    • 1
    Email author
  • Andreas Helfrich-Schkarbanenko
    • 2
  • Vincent Heuveline
    • 2
  1. 1.Institut für Informationsverarbeitung (TNT)Leibniz Universität HannoverHannoverGermany
  2. 2.Institute for Applied and Numerical Mathematics 4Karlsruhe Institute of Technology (KIT)KarlsruheGermany

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