Case and Inverse Problems Outline

1. Case and Inverse ProblemsOutline • Overview of inverse problems • Ken’s early contributions for a localized source in an infinite media • Ed Larsen and Chuck Siewert’s early breakthroughs for half-space and slab geometries • Azimuthal-angle-dependent slab geometry algorithms to determine anisotropic scattering coefficients

2. Outline, cont. · Multigroup algorithms · Non-plane geometry algorithms Objective of talk:To convince you that even minor contributions by Case have led to many INVERSE transport publications

3. Overview of inverse problems • Forward Problems: use properties of the medium, boundary conditions, and any sources to determine the angular flux • Inverse Problems: use the angular flux and or its angular moments to determine the properties of the medium, or boundary conditions, or any sources; often such problems are ill-conditioned

4. Overview of inverse problems, cont. • Case’s analytical approach for solving inverse problems can be called an “explicit” method, in contrast to a traditional “implicit” method involving some sort of iterative technique, e.g., the method of steepest descent or conjugate gradient methods.

5. Ken’s early contributions • Appendix H of Case and Zweifel’s 1967 monograph evaluates even-order spatially-integrated moments of the point- and plane-source densities due to a localized source in an infinite medium, as obtained by directly integrating the transport equation • In 1973 he introduced the phrase “inverse problem” into the transport theory literature while deriving analytical equations to determine the anisotropic scattering coefficients of the infinite medium from the spatially-integrated moments of the angular flux

6. Ken’s early contributions, cont. • Kuščer and I in 1974 used spatial moments of the flux to directly relate the mth azimuthal Fourier moment of the flux to the mth Legendre coefficient of the scattering function • In 1975 Case made some additions to his 1973 paper for a localized source in an infinite medium • BUT LOCALIZED SOURCES IN INFINITE MEDIA ARE TOTALLY IMPRACTICAL FOR APPLICATIONS!

7. Larsen and Siewert’s early half-space and slab geometry solutions • Larsen in 1975 derived a solution to determine the intensity and interior sources if the intensity at the surface of the half-space is known • Siewert in 1978 solved the multi-group problem for determining the properties of a slab from spatially-integrated moments inside the slab plus the angular moments of the emerging fluxes

8. Larsen and Siewert’s early contributions, cont. • Siewert in 1978 solved for the mean number of secondaries of an isotropically-scattering half-space using only angular moments of the emerging angular flux • In 1979 Siewert published 3 papers extending his ideas to determine 3 coefficients for quadratically-anisotropic scattering

9. Azimuthal-angle-dependent algorithms to determine anisotropic scattering coefficients • In 1979 I generalized Siewert’s slab-geometry equations to obtain two uncoupled sets of equations for determining—only in principle!—any number of anisotropic scattering coefficients • In 1981 Richard Sanchez and I obtained an even more general set of equations for determining the anisotropic scattering coefficients

10. Multigroup algorithms • Larsen in 1981 derived a transport theory solution for obtaining multi-group cross sections from surface outgoing angular fluxes and ingoing adjoint angular fluxes from multiple experiments • Sanchez and I followed in 1983 to obtain the diffusion theory solution for multi-group cross sections and diffusion coefficients • Siewert and also Sanchez and I in 1983 obtained the scattering coefficients for polarized radiative transfer (which is somewhat analogous to multi-group transport)

11. Non-plane geometry algorithms • In 1982 Siewert and Dunn solved the inverse problem for a variable illumination over the surface of a plane-parallel layer (e.g., for an incident searchlight illumination) • In 1984 Larsen formulated a solution to the time-dependent multi-group, homogeneous convex 3D medium for the scattering properties • In 2013 Machida at ICTT23 used rotated reference frames to solve 3D infinite medium problems

12. Summary • 50+ papers eventually were published on analytical methods for solving inverse transport problems • With a book appendix and a couple of papers Case initiated the exploration of inverse problems in transport theory