Complementary Variational Principles and Their Application to Neutron Transport Problems

Several variational principles are developed which give upper and lower bounds for the linear functional (S, ψ), where ψ is the solution of the inhomogeneous equation Hψ = S with H a self-adjoint, positive-definite, linear operator. Some of the principles bound this functional only with respect to small or local variations, whereas others give bounds for arbitrary variations. Several of our results coincide with those of other authors widely scattered throughout the literature, and we show that these principles have a common origin. Other results given are new. Examples of the use of these principles are taken from the field of neutron transport theory, and we use both the linear Boltzmann or transport equation and the diffusion equation. One interesting result is that certain ``exact'' values of the extrapolated endpoint for the Milne problem which have been reported in the literature fall, due to numerical inaccuracies, outside the bounds computed here.