Spherical harmonic solutions of the Boltzmann transport equation via discrete ordinates

The inverse source problem based on the radiative transfer equation in optical molecular imaging

“…First, the direction X is replaced with a set of discrete ordinates X k = (n k , g k , l k ) with full level symmetry [65,66,70,72]. The total number of ordinates X k with k 2 {1..K} is given by K ¼ NðN þ 2Þ and N the number of direction cosines of the S N method.…”

On SN-PN Equivalence

“…It is in this section that we show that other sources proposed in the literature are particular cases of our general expression for the second artificial source. We also include here an analysis of the so-called harmonic projection method (Brown et al, 2001), where a singular value decomposition technique is used to complete the PN basis and show that the source expression obtained with this method is identical to that earlier derived by Lathrop. We have relegated some of the material to the Appendices. The definition of an isotropic medium is given in Section 2 and reformulated in terms of group theory in Appendix A, where we discuss the meaning of "invariance under rotations" using elementary results from group theory, and also prove some invariance results for the SN equations.…”

“…A significant number of techniques have been proposed to diminish (Lathrop, 1971;Mordant, 1981;Briggs et al, 1975;Coppa et al, 1990;Morel et al, 2003;Widmer et al, 2008;Cao et al, 2008) or completely eliminate ray effects (Lathrop, 1971;Carlson, 1971;Reed, 1972;Miller and Reed, 1977;Brown et al, 2001) and the modern consensus is to modify the SN equations by adding an artificial source, so as to force the solution of the new equation to satisfy a related set of PN equations. The artificial source is shown to depend linearly on the solution of the modified equation and, therefore, the modification can also be viewed as adding an operator to the original SN equation (Brown et al, 2001). In any case, linearity implies that one can see this artificial source as composed of two components: the first one forces scattering to be rotationally invariant while the second makes the SN streaming operator to behave PN-like.…”

“…Reed's work was generalized in Miller and Reed (1977) where a general analysis and comparison of ray-effect mitigation techniques is also given. Recently, Brown et al (2001) reinterpreted Lathrop's technique by observing that if one restricts the order of the PN equations to a value K so that the SN quadrature formula exactly integrates all products of two spherical harmonics with orders less than or equal to K , then the technique used by Lathrop gives an exact SN-P K equivalence, although for an order K < K. 1 Moreover, on the premise that Reed's approach did not readily generalize to three-dimensional geometries, Brown et al, (2001) adopted a singular value decomposition (SVD) technique to complete the P K basis and establish a oneto-one mapping with the interpolation basis associated with the SN angular directions. Finally, they added an artificial source to the SN equation so that the P K solution could be extracted from the solution of the modified SN equation.…”

“…This comment applies also to boundary conditions which have not been included in our discussion. An example of a complete treatment comprising spatial discretization and boundary conditions is given in Brown et al (2001).…”