Reviewer's comment

Abstract

Author's reply ... Your reviewer has suggested that it might be advisable to obtain sensitivities by subtracting successive solutions obtained for small changes in the system parameters. He asserts that the solution would be obtained more cheaply and more quickly in this way. These judgments are incorrect.A major advantage of the methods indicated in the paper is that they can provide the sensitivity information for less cost and effort than by the direct comparison method. This is primarily due to the fact that quantities which appear in the calculation of a particular measure of the dynamic performance also appear in the calculation of the sensitivities. These quantities (operators may be a more appropriate term), which were needed to determine the original solution, may then be used directly in the calculation of the sensitivities and need not be recalculated. I think that this point is particularly clear in the case of the frequency-response calculation. The operator which is calculated to give the frequency response G(s) is -[A -sI] -1 (see equation 13). The only additional effort required to obtain the frequency-response sensitivities is to multiply three matrices which are either fixed or previously calculated in computing the frequency response. This is indicated in equation (16) of the paper:The matrix -[A -sI] -1 was determined in the frequency-response calculation. The matrix Aij is exactly known. The matrix G(s) is simply the frequency-response vector, which we have available from the frequency-response calculation. It is true that in some applications all components of the vector G(s) may not be required, but we have found that we can calculate the whole frequency response vector about as fast using these methods as we can calculate a single component of the vector using other methods which we have tried. One should note that the operator -[A -sI] -1 would be different if the sensitivities were calculated by the comparison method and would necessarily be reevaluated for each change.A similar situation exists in the case of the transient response and its sensitivity. A number of methods may be used to solve a differential equation such as equation (1) in the paper. We use a method based on the matrix exponential, which we have found to be quite efficient. The solution of is s and the solution of is s if f is constant. If we have an arbitrary, nonconstant f, we can build the solution as follows: Note that the same operators [e,4A' and (e-4A' -I)A-1] are used at each time-step. This procedure can be used to solve the sensitivity equation to give Note that the operators used to obtain the solution to the sensitivity equation are exactly the same as the ones used in the solution of the system equation. This means that we do not have to recalculate these operators, but can use directly those calculated in obtaining the transient response.We can also obtain a semiquantitative estimate of the advantage of the method proposed in the paper for calculating pole sensitivities. In equation (30), we found that it ...

The Discontinuous Asymptotic Telegrapher’s Equation (P1) Approximation

“…solved and tabulated the two-region Milne problem exactly [31][32][33], defining the exact jump conditions of both the asymptotic scalar flux (ρ2 /1 = φ A as /φ B as ) and the current density (j2 /1 = J A as /J B as ), the first two moments, as a function of the ω eff ( r, t) of the two media, ω A eff and ω B eff . We note that the two-region Milne problem was solved in many other studies, for example [30,34]. McCormick et.…”

“…Similar problem occurred in neutronics, with a sharp boundary of two different media, such as reactor-reflector problems [29]. This problem can be corrected by using the exact solution to obtain the exact scalar flux and the neutron current, on the boundary between the two media, yielding a discontinuous asymptotic diffusion theory [30][31][32][33][34]. This correction is the two-region extension to the classic radiative transfer Milne problem [1,6], that has its origins in the attempt to calculate the distribution of light emitted from the photosphere of a star.…”

“…However, in highly anisotropic media, such as sharp boundaries or strong sources, the asymptotic solutions fail to mirror exactly how the radiation behaves. For example, solving the problem of two adjacent semi-infinite half-spaces (the two-region Milne problem) [30][31][32][33][34], the exact solution is decomposed from an asymptotic part, which tends to the exact solution far from the boundary, and a transient part, which decays relatively fast from the boundary. Actually, this is a generalization of the classic Milne problem [1,6].…”

Two-region inverse transport analysis with solutions of the two-region milne problem

“…Also, the asymptotic scalar fluxes and currents corresponding to Eq. (6) are Equations (12) and (13) also require normally-directed outward radiance measurements on the surface; after use of the assumed boundary conditions, the equations become so we assume that these two quantities also are known.…”