158
7 Integration of Functionals, PCM and Stochastic IntegralEquations
where the brackets around the integral sign denote a continuous infinity of integrals,
one for each r, and p[σ ] is the probability density of σ (r) at each r. Equation (7.1)
is what mathematicians call an ‘integral of a functional’ or ‘functional integration.’
Clearly, we cannot compute infinitely many integrals, so we discretize the
problem by discretizing the anomalous region into N cells (voxels), and then
replacing σ (r) by σ n , n = 1, . . . , N, where σ n is a random variable associated
with the nth cell. If there are N voxels, then (7.1) becomes
∞
−∞
· · ·
∞
−∞
Z(l, σ 1 , · · · , σ N )p(σ 1 , · · · , σ N )dσ 1 · · · dσ N .
(7.2)
N will generally be large, so we must consider high-dimensional numerical
quadrature routines to compute (7.2). This is where ‘Gaussian quadratures’ and
PCM come in [37, 42, 65, 69, 142].
The ‘quadrature rule’ determines a few points that are optimum for evaluating
(7.2), and these are the PCM points at which VIC-3D® must compute Z. That
is, the PCM points are points in (σ 1 , σ 2 , · · · , σ N )-space that go into the VIC3D®-model. We hope that the number of points is not huge, and will yield a good
approximation with fewer runs than a Monte Carlo study. The PCM points will
depend upon the orthogonal polynomials that are used in the approximation of the
integrand, and these polynomials depend upon the nature of the density function
p(σ 1 , · · · , σ N ).
7.2 Probability Densities and Numerical Procedures
We will work with certain probability densities of interest to us in the context
of VIC-3D®. Anomalous regions in VIC-3D®-models are defined by volumefractions, VF, from which conductivities of the voxels are computed (see (7.62) of
Appendix 1). The standard model that we will use is one in which each of the VFs
is independent of the other, but each has the same density function. Thus, we have a
problem with independent, identically distributed (iid) random variables. Only the
density function will change from problem to problem.
Uniform Density The first statistical model is one in which each cell VF has a
uniform density. This is clearly the simplest one to analyze from a statistical point
of view, and it easily falls within the numerical-quadrature model described above,
because orthogonal polynomials and their zeros (‘Gauss points’) that are compatible
with the uniform density are known and tabulated. The ‘quadrature weights’ for the
uniform density are tabulated, as well.
Quadratic Density In Appendix 1 we describe a model for random anisotropic
grain noise that arises from the random orientation of crystals with, say, 6 mm
symmetry, as in Ti64. The important result is (7.63), and the conclusion that
V F = cos 2 φ. In the model, however, it is cos φ that is uniformly distributed, so
7 Integration of Functionals, PCM and Stochastic IntegralEquations
where the brackets around the integral sign denote a continuous infinity of integrals,
one for each r, and p[σ ] is the probability density of σ (r) at each r. Equation (7.1)
is what mathematicians call an ‘integral of a functional’ or ‘functional integration.’
Clearly, we cannot compute infinitely many integrals, so we discretize the
problem by discretizing the anomalous region into N cells (voxels), and then
replacing σ (r) by σ n , n = 1, . . . , N, where σ n is a random variable associated
with the nth cell. If there are N voxels, then (7.1) becomes
∞
−∞
· · ·
∞
−∞
Z(l, σ 1 , · · · , σ N )p(σ 1 , · · · , σ N )dσ 1 · · · dσ N .
(7.2)
N will generally be large, so we must consider high-dimensional numerical
quadrature routines to compute (7.2). This is where ‘Gaussian quadratures’ and
PCM come in [37, 42, 65, 69, 142].
The ‘quadrature rule’ determines a few points that are optimum for evaluating
(7.2), and these are the PCM points at which VIC-3D® must compute Z. That
is, the PCM points are points in (σ 1 , σ 2 , · · · , σ N )-space that go into the VIC3D®-model. We hope that the number of points is not huge, and will yield a good
approximation with fewer runs than a Monte Carlo study. The PCM points will
depend upon the orthogonal polynomials that are used in the approximation of the
integrand, and these polynomials depend upon the nature of the density function
p(σ 1 , · · · , σ N ).
7.2 Probability Densities and Numerical Procedures
We will work with certain probability densities of interest to us in the context
of VIC-3D®. Anomalous regions in VIC-3D®-models are defined by volumefractions, VF, from which conductivities of the voxels are computed (see (7.62) of
Appendix 1). The standard model that we will use is one in which each of the VFs
is independent of the other, but each has the same density function. Thus, we have a
problem with independent, identically distributed (iid) random variables. Only the
density function will change from problem to problem.
Uniform Density The first statistical model is one in which each cell VF has a
uniform density. This is clearly the simplest one to analyze from a statistical point
of view, and it easily falls within the numerical-quadrature model described above,
because orthogonal polynomials and their zeros (‘Gauss points’) that are compatible
with the uniform density are known and tabulated. The ‘quadrature weights’ for the
uniform density are tabulated, as well.
Quadratic Density In Appendix 1 we describe a model for random anisotropic
grain noise that arises from the random orientation of crystals with, say, 6 mm
symmetry, as in Ti64. The important result is (7.63), and the conclusion that
V F = cos 2 φ. In the model, however, it is cos φ that is uniformly distributed, so