Accelerating the Calderon-Symmetric Formulation with the Adaptive Cross Approximation method 

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The EEG Problem

Let = ÐN i=1 i be a nested domain with Lipchitz boundaries ∂ i = 􀀀 ¯ i−1 Ñ ¯ i as in Fig. 2.1. We denote with ni the outward going normal to the surface 􀀀i , where 􀀀i = ¯ i Ñ ¯ i+1. Solving the EEG forward problem amounts to compute the potential V at given electrodes’ positions when the active brain current sources are known. Under quasi-static as- sumptions and isotropic conductivity, the EEG forward problem reads [64]: σV = ∇ · j (2.1)
where σ is the conductivity and j the current sources. The conductivity is assumed to be piecewise isotropic and homogeneous: in i , σ = σi . In the exterior domain, the conductivity is assumed to be 0. The current sources, as it is customary in literature [54], are assumed to be dipolar in nature. Hence, denoting with fi = ∇·j the electric source in i , we have fi = qi · ∇δri with qi the electric dipole moment and ri its position. Furthermore, the symbol [g]i = g− − g +, will refer to the jump of the function g at the interface 􀀀i , with g∓ the inner and outer trace of g at 􀀀i respectively. Then, the solvability of (2.1) is assured under the following boundary conditions [64]: [V]i = 0 8i ≤ N (2.2a) [σn · ∇V]i = 0 8i ≤ N.

The Symmetric Formulation for the EEG Forward Problem

Several BEM formulations have been proposed to solve the EEG forward problem [34, 45]. Among them, the symmetric formulation [45] is quite popular and known for providing high levels of accuracy [31]. In solving the EEG forward problem, an efficient strategy is to build the unknown potential V starting from two functions, a function u harmonic in R3 and a function v that takes into account the source term. The starting point of the symmetric formulation is to build ui in each domain such that ui = V − vi/σi in i and u = −vi/σi in R3 \ i , with vi the solution of (2.1) in an unbounded medium: vi(r) = ∫ i f (r′)G(r, r′)dr′. In this fashion, ui is harmonic in R3 \ ∂ ¯ i = 􀀀i−1 ∪ 􀀀i . Using the boundary conditions (2.2a) and (2.2b) as well as the representation theorem [69], two integral equations for the potential and its derivative can be obtained [45]. They read: σ−1 i+1 (vi+1)􀀀i − σ−1 i (vi)􀀀i = Di,i−1Vi−1 − 2DiiVi + Di,i+1Vi+1.

Discretization of the Operators

The numerical solution of an integral equation is often obtained by using a Boundary Elemen tMethod (BEM). Following a well-established strategy, the domain is tessellated into Nt triangular cells tk of area Ak and average length h. The set of vertices of the tessellation will be denoted by {vk }Nv k=1. Cells and vertices will form a mesh denoted by M . The number of triangles (respectively, vertices) of the surface 􀀀i will be denoted Nti (respectively, Nvi ). To discretize the unknown and to test the equations the following standard basis functions will be used. The piecewise constant functions in P0 are defined such that, P0k = 1/Ak in tk and 0 elsewhere. The piecewise linear functions are the set P1 = span{P1k }Nv k=1. The support of P1k , denoted by μP1k , is the set of triangles around vk such that P1k = 1 in vk and 0 on all other vertices. P0 and P1 functions are shown Fig. 2.3a and 2.3b respectively.

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A Calderon Preconditioner for the Symmetric Formulation

The high accuracy of the symmetric BEM formulation [2] has made of it a very popular tool for solving the EEG forward problem. However its system matrix suffers from illconditioning that can lead to the non-convergence of the employed iterative solver used to compute the solution [9]. Indeed, the operator S is compact [69]. This means that its spectrum will accumulate at zero when the mesh is refined and it will therefore have a condition number increasing inversely proportional to the average mesh length h. Moreover, the hypersingular operator N is an unbounded operator [69]. This implies that its condition number will also grow with 1/h. Since these operators are the diagonal blocks of the matrix Z in (2.9) and the off-diagonal blocks of the matrix are smoothers, it follows that the overall conditioning of Z will increase when the mesh discretization will increase (h → 0).
By leveraging on the Calderon identities, it is possible to build a preconditioner for the system matrix Z. The rationale behind our strategy can be understood by considering the continuous operators first. The two Calderon identities that are relevant for our approach read [65].

Table of contents :

1 Introduction 
1.1 Boundary element method
1.2 Maxwell equations
1.2.1 Static Case
1.2.2 Frequency domain
1.3 Static formulations
1.3.1 Electroencephalography forward problem
1.3.2 General formulation
1.3.3 Single layer approach
1.3.4 Double layer approach
1.3.5 Symmetric formulation
1.4 High frequency formulation
1.4.1 The PMCHWT formulation
2 A Calderon preconditioner for the symmetric formulation for the EEG forward problem 
2.1 Introduction
2.2 Background on the EEG Forward Problem
2.2.1 The EEG Problem
2.2.2 The Symmetric Formulation for the EEG Forward Problem
2.2.3 Discretization of the Operators
2.3 A Calderon Preconditioner for the Symmetric Formulation
2.4 Discretization of the Calderon Preconditioner and Solution of the Preconditioned Symmetric Formulation
2.5 Numerical Results
2.5.1 Accuracy Assessments
2.5.2 Condition Number Assessments
2.5.3 Assessments on a MRI-obtained head model
2.5.4 Discussion
2.6 Conclusion
3 Accelerating the Calderon-Symmetric Formulation with the Adaptive Cross Approximation method 
3.1 Introduction
3.2 Hierarchical partitioning
3.3 Compression of low-rank matrices
3.4 Computation of the compressed matrix
3.5 Acceleration of the Symmetric and Calderon-Symmetric formulations
3.5.1 Symmetric formulation
3.5.2 Calderon-Symmetric formulation
3.5.3 Results
3.6 Conclusion
4 A hierarchical preconditioner for the PMCHWT integral equation 
4.1 Introduction
4.2 Notation and Background
4.3 Hierarchical Preconditioners for the PMCHWT
4.4 Numerical Results
4.5 Conclusion
5 A new computational framework for 2D electromagnetic formulations 
5.1 Introduction
5.1.1 TM polarization
5.1.2 TE polarization
5.2 Discretization
5.3 Overview of the 2D-Library
5.4 Calderon preconditioners
5.5 Numerical results
5.6 Conclusions
6 Conclusion and FutureWork
List of Publications
Bibliography

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