Chapter 1
A Bilinear Conjugate-Gradient Inversion
Algorithm
1.1 Optimization via Nonlinear Least-Squares
Standard methods for minimizing a real-valued function of several variables can be
divided into two general classes: those that require second derivative information,
usually referred to as Newton-type methods, and those requiring only first derivative
information, referred to as gradient methods. There are several excellent texts which,
in addition to discussing many of these methods in detail, also give suggestions
on when to use certain techniques. See, for example, the texts by Fletcher[36],
Hestenes[52] or Luenberger[64].
In this chapter, we concentrate on gradient techniques for minimizing Φ, the
norm of the residuals, for basically two reasons. First, the Töplitz-Hankel structure
of the operators in the original volume-integral equation allow us to use fast Fourier
transform techniques when doing matrix multiplications in solving the forward
problem, and secondly, the bilinearity of the entire system allows us to find the
gradient of Φ in closed form, as well as performing exact line searches when
minimizing Φ in a particular direction. We first presented this method in [102–
104]; it is known in the recent literature as the ‘contrast source inversion method’
[1, 10, 131].
1.2 A Bilinear Conjugate-Gradient Inversion Algorithm
Using Volume-Integrals
Bilinear Inversion Algorithm Consider a T/R configuration, in which a fixed
transmitting coil excites the anomaly, which is assumed to have a conductivity
vector, σ , and a receive coil scans the anomaly at positions, i = 1, . . . , N v , where
N v is the number of ‘views’. Conversely, we could assume a single transmitting coil,
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
H. A. Sabbagh et al., Advanced Electromagnetic Models for Materials
Characterization and Nondestructive Evaluation, Scientific Computation,
https://doi.org/10.1007/978-3-030-67956-9_1
3
A Bilinear Conjugate-Gradient Inversion
Algorithm
1.1 Optimization via Nonlinear Least-Squares
Standard methods for minimizing a real-valued function of several variables can be
divided into two general classes: those that require second derivative information,
usually referred to as Newton-type methods, and those requiring only first derivative
information, referred to as gradient methods. There are several excellent texts which,
in addition to discussing many of these methods in detail, also give suggestions
on when to use certain techniques. See, for example, the texts by Fletcher[36],
Hestenes[52] or Luenberger[64].
In this chapter, we concentrate on gradient techniques for minimizing Φ, the
norm of the residuals, for basically two reasons. First, the Töplitz-Hankel structure
of the operators in the original volume-integral equation allow us to use fast Fourier
transform techniques when doing matrix multiplications in solving the forward
problem, and secondly, the bilinearity of the entire system allows us to find the
gradient of Φ in closed form, as well as performing exact line searches when
minimizing Φ in a particular direction. We first presented this method in [102–
104]; it is known in the recent literature as the ‘contrast source inversion method’
[1, 10, 131].
1.2 A Bilinear Conjugate-Gradient Inversion Algorithm
Using Volume-Integrals
Bilinear Inversion Algorithm Consider a T/R configuration, in which a fixed
transmitting coil excites the anomaly, which is assumed to have a conductivity
vector, σ , and a receive coil scans the anomaly at positions, i = 1, . . . , N v , where
N v is the number of ‘views’. Conversely, we could assume a single transmitting coil,
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
H. A. Sabbagh et al., Advanced Electromagnetic Models for Materials
Characterization and Nondestructive Evaluation, Scientific Computation,
https://doi.org/10.1007/978-3-030-67956-9_1
3