Chapter 7
Integration of Functionals, PCM
and Stochastic Integral Equations
7.1 Theoretical Background
The mathematical theory of functional integration was developed in the fifties to be
applied to problems in probability and quantum field theory. We’re not interested
in the formal theory [38], but in its more practical form which can actually be used
to generate numbers. In particular, it leads to the probabilistic collocation method
(PCM) and other techniques for high-dimensional model representation (HDMR)
[37, 42, 65, 69, 142], which we intend to apply to eddy-current nondestructive
evaluation (NDE), using the volume-integral code, VIC-3D®, as our vehicle.
Let σ (r) be a random conductivity field, which means that at each point, r, there
exists a random variable, σ (ω, r), with a probability density. Here, ω, denotes the
‘random-set’ parameter that defines the ‘outcome’ of the experiment to determine
σ (r).
Given σ (r), we use VIC-3D® to compute an impedance, Z(l, σ (r)), where l is
the position of the probe coil. Clearly, before we can compute Z, we must know
σ (r) at all of its field points, which are infinite in number (a continuous infinity!).
Hence, Z is a ‘functional’ of σ (r), and because σ (r) = σ (ω, r) is a random field, Z
becomes a random variable at each of the probe coil points, l. We should write this
as Z(l, [σ ]) to denote that Z is a function of l, and a functional of σ (r). We could
include the parameter, ω, in Z to remind us that Z is a random functional of σ (r),
but that would be gilding the lilly with symbols.
If we want to find the average value of Z(l, [σ ]), we form
∞
−∞
Z(l, [σ ])p[σ ]d[σ ] ,
(7.1)
© 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_7
157
Integration of Functionals, PCM
and Stochastic Integral Equations
7.1 Theoretical Background
The mathematical theory of functional integration was developed in the fifties to be
applied to problems in probability and quantum field theory. We’re not interested
in the formal theory [38], but in its more practical form which can actually be used
to generate numbers. In particular, it leads to the probabilistic collocation method
(PCM) and other techniques for high-dimensional model representation (HDMR)
[37, 42, 65, 69, 142], which we intend to apply to eddy-current nondestructive
evaluation (NDE), using the volume-integral code, VIC-3D®, as our vehicle.
Let σ (r) be a random conductivity field, which means that at each point, r, there
exists a random variable, σ (ω, r), with a probability density. Here, ω, denotes the
‘random-set’ parameter that defines the ‘outcome’ of the experiment to determine
σ (r).
Given σ (r), we use VIC-3D® to compute an impedance, Z(l, σ (r)), where l is
the position of the probe coil. Clearly, before we can compute Z, we must know
σ (r) at all of its field points, which are infinite in number (a continuous infinity!).
Hence, Z is a ‘functional’ of σ (r), and because σ (r) = σ (ω, r) is a random field, Z
becomes a random variable at each of the probe coil points, l. We should write this
as Z(l, [σ ]) to denote that Z is a function of l, and a functional of σ (r). We could
include the parameter, ω, in Z to remind us that Z is a random functional of σ (r),
but that would be gilding the lilly with symbols.
If we want to find the average value of Z(l, [σ ]), we form
∞
−∞
Z(l, [σ ])p[σ ]d[σ ] ,
(7.1)
© 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_7
157