Two fundamentally different computational approaches exist to simulate linear radiation transport equations, each with well-established schools of thought, advan-tages, and disadvantages. They are deterministic methods and Monte Carlo, or stochastic methods. More recently, hybrid deterministic-Monte Carlo methods have combined these approaches to increase the efficiency of Monte Carlo calculations. In this section we present a brief description and history of each of these methods and their applications to linear transport problems.
Deterministic methods entail the discretization of the entire problem domain on phase space grids. The solution is then represented on these subdomains, and a typically large, discrete, linear system of coupled equations is generated and solved.
Representing the solution on these subdomains is a topic unto itself. For example, in the spatial variable, one may choose from finite element or finite volume representa-tions; in the angular term, discrete ordinates or spherical harmonics expansions are frequently used. Once a decision has been made as to how to represent the discretized system, the resulting system of equations is “determined,” and a numerical solution of this system produces the global estimate of the radiation field. Here, “global” means
“over all of phase space.” The property that deterministic solutions are inherently global is one of the advantages of deterministic methods. However, truncation errors are introduced in the discretization process, or in any approximation made to reduce the number of equations to a level amenable for computation. Many practitioners of deterministic methods continue to look for ways to quantify these errors [3] [4]. Also, it has not always been known whether a particular deterministic method will main-tain its accuracy in all problem regimes. For instance, in thick, diffusive problems it may become impractical to impose spatial grids on the order of a mean free path of particle flight (this imposition is often necessary to maintain accuracy). Now, anal-ysis tools are available to deduce whether or not a particular deterministic method
will be accurate in the limit of thick, diffusive systems [5]. The hope is that if a particular deterministic method is accurate in the thick, diffusive limit and the thin, streaming limit, then it will be accurate elsewhere.
Another limitation of deterministic methods is that direct matrix inversions of the large, linear system are prohibitively inefficient; consequently, iterative procedures must be used. The most straightforward iterative procedures can converge very slowly, hence a host of acceleration schemes have been developed. An excellent overview of modern deterministic methods and some of their acceleration schemes is provided in references [6] and [7].
A further limitation of deterministic methods that has emerged more recently is their weak parallel scalability, which refers to the ability of a computational method to require less wall-clock time as the number of computer processors is increased.
Currently, any deterministic method that involves a discretization of the angular variable suffers from weak parallel scalability – instead of the wall-clock time de-creasing proportionately as the number of processors is increased, after a moderate number of processors are employed, the return from increasing the number of pro-cessors becomes marginal. This is significant in that many supercomputers today involve thousands, or even hundreds of thousands of processors. Enhancing the par-allel scalability of deterministic methods, even at the expense of serial performance, is another area of active research [8] [9] [10].
Linear radiation transport equations have a well-known Monte Carlo interpreta-tion and a rich history. Several references are available that provide the Monte Carlo method its mathematical foundation and apply it to certain linear, transport prob-lems [11] [12] [13]. Monte Carlo methods exploit pseudo-random number sequences to simulate the discrete interactions of individual radiation particles with the back-ground material. If the interactions of the Monte Carlo particles directly follow the mechanics of physical particles, then the simulation is designated an analog Monte Carlo method. Typically, many fewer Monte Carlo particles are simulated than the amount that occur in the physical problem; consequently, each Monte Carlo particle represents some multiple of physical particles. In an analog Monte Carlo method, this multiple is fixed. If a sufficiently large number of particles are simulated with their behaviors tabulated, then one can obtain estimates of their average behavior.
It is possible to show via the central limit theorem that the expected error in these estimates – the standard deviation of the sample mean – is reduced by a factor pro-portional to 1/√
N , where N is the total number of particles simulated. This means
of the linear transport problem, assuming that the underlying system is represented exactly. More practically, if one predetermines an acceptable level of error, it is then possible to probabilistically estimate a number of histories such that the solution is within this error tolerance (this number must be determined during the calcu-lation or after a trial calcucalcu-lation, in which an estimate of the largest coefficient of 1/√
N is generated throughout the solution region). However, the necessary number of histories may be too large for efficient computation. Monte Carlo methods also have the advantage that if the problem phase space can be represented on a single computer, than the parallelization is trivially and strongly scalable. Even if this is not the case, it is frequently possible to decompose the domain of the problem onto separate computers and maintain some degree of parallel scalability. Monte Carlo methods also suffer no theoretical restrictions in the expression of the underlying problem geometry, whereas deterministic methods may have difficulties adequately representing curved surfaces due to the regularity of the spatial grid. Analogously, energy and angular discretizations are not necessary if the continuum physics are suitably representable.
Monte Carlo methods are frequently applied to local problems, or problems in which the solution is desired in a small fraction of the total phase space. A classic example of this is a source-detector problem, in which a single detector sits far away from a radiation source, and the detector’s response to the radiation is desired. By contrast, in a global problem, the solution is desired throughout the entire problem phase space. For local problems especially, variance reduction techniques have been devised in which a more accurate (lower variance) estimate of the problem solution is obtained with fewer particle histories. These techniques introduce more computa-tional overhead per history, but, if employed properly, the overall variance reduction more than compensates for the additional overhead.
Variance reduction techniques for local problems generally exploit the locality of the solution phase space to increase the efficiency of each history. If we consider a source-detector problem, then the detector and the regions containing the flight paths of particles that are most likely to journey to the detector are important, whereas re-gions far removed from these flight paths are not. In this problem, the variance in the detector response may be reduced by ensuring that Monte Carlo particles follow these flight paths with a high probability, but without biasing the mean of the solution.
Methods such as these are termed nonanalog, since the Monte Carlo particles are not transported according to the same principles as do the physical particles. This is typically achieved by introducing the concept of particle weight, whereby Monte
Carlo particles are allowed to represent a variable fraction of physical particles. It is then possible to attempt to restrict the particle weights to some optimal value by effectively combining low weight particles to become average-weight (rouletting) and by splitting large weight particles into average-weight particles. Employing the mechanisms of splitting and rouletting in concert is termed weight windows. How-ever, determining the proper average weight throughout the problem phase-space requires much trial-and-error and an experienced, patient code user to adjust the parameters to an acceptable level. It is sometimes possible to automatically and dy-namically adjust these parameters during the Monte Carlo calculation [14]. We note that variance reduction is not confined to weight windows; many other techniques are available [2] [12] [13].
Variance reduction techniques for global problems that do not rely on a separate, deterministic calculation are meager, and Monte Carlo is not often used in global problems that have a highly varying problem solution; i.e., Monte Carlo is typically only applied to relatively thin global problems. Only the technique of “implicit capture” (also known as absorption weighting or survival biasing) is used. In this technique, particles are not allowed to undergo absorptions, rather, their weight is changed by the fraction that scatters during a collision event. This technique extends the lives and flight paths of the particle histories, but it is frequently not sufficient to generate accurate solutions in regions far away from particle sources.
Within the last 15 years, Monte Carlo methods for local problems have found success by employing a deterministic calculation to automatically set the Monte Carlo biasing parameters. In this thesis we refer to such schemes as hybrid deterministic-Monte Carlo methods. For instance, it has been shown that the “important” regions of a source-detector problem are proportional to the inverse of the adjoint solution, in which the detector region is employed as an adjoint source term [15]. Hybrid approaches exploit this fact by employing a deterministic calculation to obtain a global estimate of the adjoint solution, and then setting the inverse of the adjoint solution as the center of a weight window in a Monte Carlo calculation. It is not necessary to obtain a highly accurate adjoint solution, but generally, more accurate deterministic solutions yield more efficient Monte Carlo calculations.
The development of hybrid deterministic-Monte Carlo methods is recent, and with few exceptions [16], limited to local problems. Yet it is easy to conceive of global problems of interest to a Monte Carlo user: a source-multiple-detector problem, a shielding problem requiring internal heating rates, or a radiation dose estimate for a
regions of phase space by so much that traditional Monte Carlo methods falter: they could take lifetimes to reduce the problem-wide variance to an acceptable level. As of this writing, the only hybrid, global method available remedies this by setting weight windows according to a deterministic estimate of the forward solution [16] [17]. This method, developed by Cooper and Larsen, has been shown to potentially be orders of magnitude more efficient than Monte Carlo with implicit capture, which is the only variance reduction method currently available for global problems. In his thesis, Cooper considered steady-state, 2-D, energy-independent, linear neutron transport problems and 1-D, gray, nonlinear thermal radiative transfer (TRT) problems. One of the new contributions in this thesis will be an extension and modification of Cooper’s method to more complicated problems described by the 1-D, frequency-dependent, nonlinear TRT equations.