Alternative Adjoint Boltzmann Transport Operators in Alternative Hilbert Spaces in Spherical Geometry: Theory and Paradigm Applications

The solution of the customary adjoint Boltzmann equation for linear transport of particles and radiation, referred to as the “adjoint flux,” plays a prominent role in reactor physics, shielding, control, and optimization as a weighting function for cross-section processing, optimization, Monte Carlo acceleration procedures, and sensitivity and uncertainty analyses. All of the textbooks and scientific works published thus far use the same procedure to derive “the” adjoint Boltzmann operator, thereby conveying inadvertently the misleading impression that this traditional procedure is the only way to obtain “the” adjoint Boltzmann operator, and that the form of “the” adjoint operator thus derived is universally unique. None of the works published in the literature thus far touches on the fact that the customary textbook-form of the adjoint Boltzmann operator is actually derived in a particular Hilbert space, which is endowed with a specific inner product that is based on integrating spatially over the domain’s spatial volume such that Gauss’ divergence theorem holds. As this work will show, however, the Hilbert space that has been implicitly used in all of the published works thus far is not the only possible Hilbert space for deriving operators that are adjoint, in the respective Hilbert space, to the forward Boltzmann operator. Alternative Hilbert spaces may be used just as legitimately, and may actually be more suitable than the customary Hilbert space for computing adjoint functions to be used in inner products involving various forward and/or adjoint fluxes and forward and/or adjoint source terms.

By presenting paradigm illustrative examples in three-dimensional spherical coordinates, this work shows that although a unique form of the adjoint Boltzmann operator is obtained for each Hilbert space in which the respective adjoint operator is constructed, distinct Hilbert spaces will produce distinct adjoint Boltzmann operators accompanied by distinct forms of the corresponding bilinear concomitants on the respective spatial domain’s boundary. The fundamental practical reason for using alternative Hilbert spaces is to obtain alternative adjoint functions and/or Green’s functions that may be less singular than the customary adjoint function and/or Green’s functions (in the customary Hilbert space) and would consequently be computable numerically. As this work shows, such situations arise when attempting to use the adjoint sensitivity analysis methodology in the conventional Hilbert space for computing sensitivities to cross sections, isotopic number densities, etc., of responses of flux and/or power detectors placed near or at the center of the spherical coordinates. In such sensitivity analysis problems, the singularities of the conventional adjoint Boltzmann equation in the conventional Hilbert space may preclude its use, but the requisite sensitivities can nevertheless be computed efficiently using an alternative adjoint Boltzmann equation in an alternative Hilbert space. The consequences of this powerful breakthrough new concept of using alternative adjoint operators in alternative Hilbert spaces are highlighted by presenting a paradigm benchmark problem that admits a closed-form exact solution. This benchmark problem shows that the customary adjoint equation becomes singular at the sphere’s center, so the conventional adjoint flux is therefore noncomputable there, but the alternative adjoint transport equation in a judiciously chosen alternative Hilbert space is everywhere nonsingular and can therefore be used to compute the requisite sensitivities. By indicating the path for using alternative Hilbert spaces, this work opens new conceptual procedures for solving problems that have hitherto been unsolvable in the framework of the conventional adjoint particle transport equation.