MAVRIC: Monaco with Automated Variance Reduction using Importance Calculations

D. E. Peplow and C. Celik

Introduction

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Monte Carlo particle transport calculations for deep penetration problems can require very long run times in order to achieve an acceptable level of statistical uncertainty in the final answers. Discrete-ordinates codes can be faster but have limitations relative to the discretization of space, energy, and direction. Monte Carlo calculations can be modified (biased) to produce results with the same variance in less time if an approximate answer or some other additional information is already known about the problem. If importances can be assigned to different particles based on how much they will contribute to the final answer, then more time can be spent on important particles, with less time devoted to unimportant particles. One of the best ways to bias a Monte Carlo code for a particular tally is to form an importance map from the adjoint flux based on that tally. Unfortunately, determining the exact adjoint flux could be just as difficult as computing the original problem itself. However, an approximate adjoint can still be very useful in biasing the Monte Carlo solution [Wag97]. Discrete ordinates can be used to quickly compute that approximate adjoint. Together, Monte Carlo and discrete ordinates can be used to find solutions to thick shielding problems in reasonable times.

The MAVRIC (Monaco with Automated Variance Reduction using Importance Calculations) sequence is based on the CADIS (Consistent Adjoint Driven Importance Sampling) and FW-CADIS (Forward-Weighted CADIS) methodologies [WH98] [Wag02] [HW03] [WBP07] MAVRIC automatically performs a three-dimensional, discrete-ordinates calculation using Denovo to compute the adjoint flux as a function of position and energy. This adjoint flux information is then used to construct an importance map (i.e., target weights for weight windows) and a biased source distribution that work together—particles are born with a weight matching the target weight of the cell into which they are born. The fixed-source Monte Carlo radiation transport Monaco [Pep11] then uses the importance map for biasing during particle transport, and it uses the biased source distribution as its source. During transport, the particle weight is compared with the importance map after each particle interaction and whenever a particle crosses into a new importance cell in the map.

For problems that do not require variance reduction to complete in a reasonable time, execution of MAVRIC without the importance map calculation provides an easy way to run Monaco. For problems that do require variance reduction to complete in a reasonable time, MAVRIC removes the burden of setting weight windows from the user and performs it automatically with a minimal amount of additional input. Note that the MAVRIC sequence can be used with the final Monaco calculation as either a multigroup (MG) or a continuous-energy (CE) calculation.

Monaco has a wide variety of tally options: it can calculate fluxes (by group) at a point in space, over any geometrical region, or for a user-defined, three-dimensional, rectangular grid. These tallies can also integrate the fluxes with either standard response functions from the cross section library or user-defined response functions. All of these tallies are available in the MAVRIC sequence.

Although it was originally designed for CADIS, the MAVRIC sequence is also capable of creating importance maps using both forward and adjoint deterministic estimates. The FW-CADIS method [WPM14] can be used for optimizing several tallies at once, a mesh tally over a large region, or a mesh tally over the entire problem. Several other methods for producing importance maps are also available in MAVRIC and are explored in MAVRIC Appendix C: Advanced Features.

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MAVRIC is an implementation of CADIS (Consistent Adjoint Driven Importance Sampling) using the Denovo SN and Monaco Monte Carlo functional modules. Source biasing and a mesh-based importance map, overlaying the physical geometry, are the basic methods of variance reduction. To make the best use of an importance map, the map must be made consistent with the source biasing. If the source biasing is inconsistent with the weight windows that will be used during the transport process, then source particles will undergo Russian roulette or splitting immediately, wasting computational time and negating the intent of the biasing.

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CADIS is well described in the literature, so only a  Batson Iii, John committed Jul 27, 2020 289 290 291 292 293 294 295 brief overview is given here. Consider a class source-detector problem described by a unit source with emission probability distribution function $$q\left(\overrightarrow{r},E \right)$$ and a detector response function $$\sigma_{d}\left(\overrightarrow{r},E \right)$$. To determine the total detector response, R, the forward scalar flux $$\phi\left(\overrightarrow{r},E \right)$$ must be known. The response is found by integrating the product of the detector response  Batson Iii, John committed Aug 21, 2020 296 function and the flux over the detector volume $$V_{d}$$:

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(1)$R = \int_{V_{d}}^{}{\int_{E}^{}{\sigma_{d}\left( \overrightarrow{r},E \right)}}\phi\left(\overrightarrow{r},E \right)\textit{dE dV.}$

Alternatively, if the adjoint scalar flux, $$\phi^{+}\left(\overrightarrow{r},E \right)$$, is known from the corresponding adjoint problem with adjoint source $$q^{+}\left(\overrightarrow{r},E \right) = \sigma_{d}\left(\overrightarrow{r},E \right)$$, then the total detector response could be found by integrating the product of the forward source and the adjoint flux over the source  Batson Iii, John committed Aug 21, 2020 305 volume, $$V_{s}$$:

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(2)$R = \int_{V_{s}}^{}{\int_{E}^{}{q\left(\overrightarrow{r},E \right)}}\phi^{+}\left( \overrightarrow{r},E \right)\textit{dE dV.}$

Unfortunately, the exact adjoint flux may be just as difficult to determine as the forward flux, but an approximation of the adjoint flux can still be used to form an importance map and a biased source distribution for use in the forward Monte Carlo calculation.

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Wagner [Wag97] showed that if an estimate of the adjoint scalar flux for the corresponding adjoint problem can be found, then an estimate of the response R can be made using (2). The adjoint source for the  Batson Iii, John committed Jul 27, 2020 315 adjoint problem is typically separable and corresponds to the detector  Batson Iii, John committed Aug 21, 2020 316 response and spatial area of the tally to be optimized:  Batson Iii, John committed Jul 27, 2020 317 318 319 $$q^{+}\left(\overrightarrow{r},E \right) = \sigma_{d}\left(E \right)g\left( \overrightarrow{r} \right)$$, where $$\sigma_{d}\left( E \right)$$ is a flux-to-dose conversion factor and $$g\left( \overrightarrow{r} \right)$$ is 1 in the tally  Batson Iii, John committed Aug 21, 2020 320 volume and is 0 otherwise. Then, from the adjoint flux  Batson Iii, John committed Jul 27, 2020 321 322 323 324 325 326 327 328 329 330 331 $$\phi^{+}\left( \overrightarrow{r},E \right)$$ and response estimate R, a biased source distribution, $$\widehat{q}\left( \overrightarrow{r},E \right)$$, for source sampling of the form

(3)$\widehat{q}\left(\overrightarrow{r},E \right) = \frac{1}{R}q\left(\overrightarrow{r},E\right)\phi^{+}\left( \overrightarrow{r},E \right)$

and weight window target values, $$\overline{w}\left( \overrightarrow{r},E \right)$$, for particle transport of the form

(4)$\overline{w}\left( \overrightarrow{r},E \right) = \frac{R}{\phi^{+}\left( \overrightarrow{r},E \right)}$
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can be constructed, which minimizes the variance in the forward Monte  Batson Iii, John committed Jul 27, 2020 333 334 335 336 337 338 339 340 Carlo calculation of R.

When a particle is sampled from the biased source distribution $$\widehat{q}\left( \overrightarrow{r},E \right)$$, to preserve a fair game, its initial weight is set to

(5)$w_{0}\left(\overrightarrow{r},E \right) = \frac{q\left(\overrightarrow{r},E \right)}{\widehat{q}\left( \overrightarrow{r},E \right)} = \frac{R}{\phi^{+}\left( \overrightarrow{r},E \right)}\,$

which exactly matches the target weight for that particle’s position and energy. This is the “consistent” part of CADIS—source particles are born  Batson Iii, John committed Aug 21, 2020 341 342 with a weight matching the weight window of the region/energy into which they are born. The source biasing and the weight windows work together.

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CADIS has been applied to many problems—including reactor ex-core detectors, well-logging instruments, cask shielding studies, and independent spent fuel storage facility models—and has demonstrated very significant speed-ups in calculation time compared to analog simulations.

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For a typical Monte Carlo calculation with multiple sources—each with a  Batson Iii, John committed Jul 27, 2020 352 353 354 probability distribution function $$q_{i}\left( \overrightarrow{r},E \right)$$ and a strength $$S_{i}$$, giving a total source strength of  Batson Iii, John committed Aug 21, 2020 355 $$S = \sum_{}^{}S_{i}$$—the source is sampled in two steps. First,  Batson Iii, John committed Jul 27, 2020 356 357 358 359 the specific source i is sampled with probability $$p\left( i \right) = \ S_{i}/S$$, and then the particle is sampled from the specific source distribution $$q_{i}\left( \overrightarrow{r},E \right)$$.

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The source sampling can be biased at both levels: from which source to sample and how to sample each source. For example, the specific source can  Batson Iii, John committed Jul 27, 2020 362 363 364 365 366 367 368 369 370 be sampled using some arbitrary distribution, $$\widehat{p}\left( i \right)$$, and then the individual sources can be sampled using distributions $${\widehat{q}}_{i}\left( \overrightarrow{r},E \right)$$. Particles would then have a birth weight of

(6)$w_{0} \equiv \ \left(\frac{p\left( i \right)}{\widehat{p}\left( i \right)} \right)\left(\frac{q_{i}\left( \overrightarrow{r},E \right)}{{\widehat{q}}_{i}\left( \overrightarrow{r},E \right)} \right)\text{.}$

For CADIS, a biased multiple source needs to be developed so that the birth weights of sampled particles still match the target weights of the  Batson Iii, John committed Aug 21, 2020 371 importance map. For a problem with multiple sources—each with a  Batson Iii, John committed Jul 27, 2020 372 distribution $$q_{i}\left( \overrightarrow{r},E \right)$$ and a  Batson Iii, John committed Aug 21, 2020 373 strength $$S_{i}$$—the goal of the Monte Carlo calculation is to  Batson Iii, John committed Jul 27, 2020 374 375 compute some response $$R$$ for a response function $$\sigma_{d}\left( \overrightarrow{r},E \right)$$ at a given  Batson Iii, John committed Aug 21, 2020 376 detector,

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 Batson Iii, John committed Aug 21, 2020 378 (7)$R = \ \int_{V}^{}{\int_{E}^{}{\sigma_{d}\left( \overrightarrow{r},E \right)\phi \left( \overrightarrow{r},E \right)\textit{dE dV.}}}$
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Note that the flux $$\phi\left( \overrightarrow{r},E \right)$$ has contributions from each source. The response, $$R_{i}$$, from each specific source ($$S_{i}$$ with $$q_{i}\left( \overrightarrow{r},E \right)$$) can be expressed using just the flux from that source, $$\phi_{i}\left( \overrightarrow{r},E \right)$$, as

(8)$R_{i} = \ \int_{V}^{}{\int_{E}^{}{\sigma_{d}\left(\overrightarrow{r},E \right)\ \phi_{i}\left(\overrightarrow{r},E \right)\textit{dE dV .}}}$

The total response is then found as $$R = \sum_{i}^{}R_{i}$$.

For the adjoint problem, using the adjoint source of $$q^{+}\left( \overrightarrow{r},E \right) = \sigma_{d}\left( \overrightarrow{r},E \right)$$, the response $$R$$ can also be calculated as

(9)$R = \ \int_{V}^{}{\int_{E}^{}{\left\lbrack \sum_{i}^{}{S_{i}q_{i}\left( \overrightarrow{r},E \right)} \right\rbrack\ \phi^{+}\left( \overrightarrow{r},E \right)\textit{dE dV}}},$
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with the response contribution from each specific source being

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 Batson Iii, John committed Aug 21, 2020 395 (10)$R_{i} = \ \int_{V}^{}{\int_{E}^{}{\ {S_{i}q_{i}\left( \overrightarrow{r},E \right)\phi^{+}}\left( \overrightarrow{r}, E \right)\textit{dE dV.}}}$
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The target weights $$\overline{w}\left( \overrightarrow{r},E \right)$$ of the importance map are found using

 Batson Iii, John committed Aug 21, 2020 400 (11)$\overline{w}\left( \overrightarrow{r},E \right) = \frac{R/S}{\phi^{+}\left( \overrightarrow{r},E \right)\ }.$
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Each biased source $${\widehat{q}}_{i}\left( \overrightarrow{r},E \right)$$ pdf is found using

 Batson Iii, John committed Aug 21, 2020 405 (12)${\widehat{q}}_{i}\left(\overrightarrow{r},E \right) = \frac{S_{i}}{R_{i}}{q_{i}\left( \overrightarrow{r},E \right)\phi}^{+}\left( \overrightarrow{r},E \right)\ ,$
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and the biased distribution used to select an individual source is $$\widehat{p}\left( i \right) = \ R_{i}/\sum_{}^{}{R_{i} = R_{i}/R}$$.

When using the biased distribution used to select an individual source, $$\widehat{p}\left( i \right)$$, and the biased source distribution, $${\widehat{q}}_{i}\left( \overrightarrow{r},E \right)$$, the birth weight of the sampled particle will be

(13)$\begin{split} \begin{matrix}  Batson Iii, John committed Aug 21, 2020 414 415  w_{0} & \equiv & \left( \frac{p\left( i \right)}{\widehat{p}\left( i \right)} \right)\left( \frac{q_{i}\left( \overrightarrow{r}, E \right)}{{\widehat{q}}_{i}\left(\overrightarrow{r},E \right)} \right) \\ & = & \ \left( \frac{\frac{S_{i}}{S}}{\frac{R_{i}}{R}} \right) \left( \frac{q_{i}\left( \overrightarrow{r},E \right)}{\frac{S_{i}}{R_{i}}{q_{i}\left( \overrightarrow{r},E \right)\phi^{+}}\left( \overrightarrow{r},E \right)} \right) \\ & = & \frac{R/S}{{\phi}^{+}\left( \overrightarrow{r},E \right)\ }, \\  Batson Iii, John committed Jul 27, 2020 416 417 418 419 420 421 \end{matrix}\end{split}$

which matches the target weight, $$\overline{w}\left( \overrightarrow{r},E \right)$$.

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The CADIS methodology works quite well for classic source/detector problems. The statistical uncertainty of the tally that serves as the adjoint source is greatly reduced since the Monte Carlo transport is optimized to spend more simulation time on those particles that contribute to the tally, at the expense of tracking particles in other parts of phase space. However, more recently, Monte Carlo has been applied to problems in which multiple tallies need to all be found with low statistical uncertainties. The extension of this idea is the mesh tally—where each voxel is a tally for which the user desires low statistical uncertainties. For these problems, the user must accept a total simulation time that is controlled by the tally with the slowest convergence and simulation results where the tallies have a wide range of relative uncertainties.

The obvious way around this problem is to create a separate problem for each tally and use CADIS to optimize each. Each simulation can then be run until the tally reaches the level of acceptable uncertainty. For more than a few tallies, this approach becomes complicated and time-consuming for the user. For large mesh tallies, this approach is not reasonable.

Another approach to treat several tallies, if they are in close proximity to each other, or a mesh tally covering a small portion of the physical problem, is to use the CADIS methodology with the adjoint source near the middle of the tallies to be optimized. Since particles in the forward Monte Carlo simulation are optimized to reach the location of the adjoint source, all the tallies surrounding that adjoint source should converge quickly. This approach requires the difficult question of “how close.” If the tallies are too far apart, then certain energies or regions that are needed for one tally may be of low importance for getting particles to the central adjoint source. This may under-predict the flux or dose at the tally sites far from the adjoint source.

MAVRIC has the capability to have multiple adjoint sources with this problem in mind. For several tallies that are far from each other, multiple adjoint sources could be used. In the forward Monte Carlo, particles would be drawn to one of those adjoint sources. The difficulty with this approach is that typically the tally that is closest to the true physical source converges faster than the other tallies—–showing that the closest adjoint source seems to attract more particles than the others. Assigning more strength to the adjoint source further from the true physical source helps to address this issue, but finding the correct strengths so that all of the tallies converge to the same relative uncertainty in one simulation is an iterative process for the user.

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To converge several tallies to the same relative uncertainty in  Batson Iii, John committed Jul 27, 2020 455 one simulation, the adjoint source corresponding to each of those  Batson Iii, John committed Aug 21, 2020 456 457 458 459 460 tallies must be weighted inversely by the expected tally value. To calculate the dose rate at two points—–say one near a reactor and one far from a reactor—–in one simulation, then the total adjoint source used to develop the weight windows and biased source must have two parts. The adjoint source far from the reactor must have  Batson Iii, John committed Jul 27, 2020 461 462 463 more strength than the adjoint source near the reactor by a factor equal to the ratio of the expected near dose rate to the expected far dose rate.

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This concept can be extended to mesh tallies, as well. Instead of using a  Batson Iii, John committed Jul 27, 2020 465 466 467 468 469 470 471 uniform adjoint source strength over the entire mesh tally volume, each voxel of the adjoint source should be weighted inversely by the expected forward tally value for that voxel. Areas of low flux or low dose rate would have more adjoint source strength than areas of high flux or high dose rate.

An estimate of the expected tally results can be found by using a quick discrete-ordinates calculation. This leads to an extension of the CADIS  Batson Iii, John committed Aug 21, 2020 472 method: forward-weighted CADIS (FW-CADIS). First, a forward SN calculation is performed to  Batson Iii, John committed Jul 27, 2020 473 estimate the expected tally results. A total adjoint source is  Batson Iii, John committed Aug 21, 2020 474 constructed so that the adjoint source corresponding to each tally is  Batson Iii, John committed Jul 27, 2020 475 476 477 478 479 480 481 482 483 484 485 486 weighted inversely by those forward tally estimates. Then the standard CADIS approach is used—an importance map (target weight windows) and a biased source are made using the adjoint flux computed from the adjoint SN calculation.

For example, if the goal is to calculate a detector response function $$\sigma_{d}\left( E \right)$$ (such as dose rate using flux-to-dose-rate conversion factors) over a volume (defined by $$g\left( \overrightarrow{r} \right)$$) corresponding to mesh tally, then instead of simply using $$q^{+}\left( \overrightarrow{r},E \right) = \sigma_{d}\left( E \right)\ g(\overrightarrow{r})$$, the adjoint source would be

 Batson Iii, John committed Aug 21, 2020 487 (14)$q^{+}\left( \overrightarrow{r},E \right) = \frac{\sigma_{d}\left( E \right)\text{g}\left( \overrightarrow{r} \right)}{\int_{}^{}{\sigma_{d}\left( E \right)\phi\left( \overrightarrow{r},E \right)}\textit{dE}}\ ,$
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where $$\phi\left( \overrightarrow{r},E \right)$$ is an estimate of  Batson Iii, John committed Aug 21, 2020 489 the forward flux, and the energy integral is over the voxel at $$\overrightarrow{r}$$.  Batson Iii, John committed Jul 27, 2020 490 491 492 493 The adjoint source is nonzero only where the mesh tally is defined ($$g\left( \overrightarrow{r} \right)$$), and its strength is inversely proportional to the forward estimate of dose rate.

The relative uncertainty of a tally is controlled by two components:  Batson Iii, John committed Aug 21, 2020 494 (1) the number of tracks contributing to the tally and (2) the  Batson Iii, John committed Jul 27, 2020 495 496 497 498 499 500 501 502 503 504 505 506 507 508 shape of the distribution of scores contributing to that tally. In the Monte Carlo game, the number of simulated particles, $$m\left( \overrightarrow{r},E \right)$$, can be related to the true physical particle density, $$n\left( \overrightarrow{r},E \right),$$ by the average Monte Carlo weight of scoring particles, $$\overline{w}\left( \overrightarrow{r},E \right)$$, by

(15)$n\left( \overrightarrow{r},E \right) = \ \overline{w}\left( \overrightarrow{r},E \right)\text{m}\left( \overrightarrow{r},E \right).$

In a typical Monte Carlo calculation, tallies are made by adding some score, multiplied by the current particle weight, to an accumulator. To calculate a similar quantity related to the Monte Carlo particle density would be very close to calculating any other quantity but without including the particle weight. The goal of FW-CADIS is to make the Monte Carlo particle density, $$m\left( \overrightarrow{r},E \right)$$,  Batson Iii, John committed Aug 21, 2020 509 510 uniform over the tally areas, so an importance map must be developed that represents the importance of achieving uniform Monte Carlo particle  Batson Iii, John committed Jul 27, 2020 511 512 513 514 515 516 517 density. By attempting to keep the Monte Carlo particle density more uniform, more uniform relative errors for the tallies should be realized.

Two options for forward weighting are possible. For tallies over some area where the entire group-wise flux is needed with low relative uncertainties, the adjoint source should be weighted inversely by the forward flux, $$\phi\left( \overrightarrow{r},E \right)$$. The other  Batson Iii, John committed Aug 21, 2020 518 option, for a tally in which only an energy-integrated quantity is desired,  Batson Iii, John committed Jul 27, 2020 519 is to weight the adjoint inversely by that energy-integrated  Batson Iii, John committed Aug 21, 2020 520 521 quantity,$$\int_{}^{}{\sigma_{d}\left( E \right)\phi\left( \overrightarrow{r},E \right)}\text{\ dE}$$. For a tally in which the total flux is desired, then the response in the  Batson Iii, John committed Jul 27, 2020 522 523 524 adjoint source is simply $$\sigma_{d}\left( E \right) = 1$$.

To optimize the forward Monte Carlo simulation for the calculation of some quantity at multiple tally locations or across a mesh tally, the  Batson Iii, John committed Aug 21, 2020 525 adjoint source must be weighted by the estimate of that quantity.  Batson Iii, John committed Jul 27, 2020 526 527 528 529 530 531 532 For a tally defined by its spatial location $$g\left( \overrightarrow{r} \right)$$ and its optional response $$\sigma_{d}\left( E \right)$$, the standard adjoint source would be $$q^{+}\left( \overrightarrow{r},E \right) = \sigma_{d}\left( E \right)\text{g}\left( \overrightarrow{r} \right)$$. The forward-weighted adjoint source, $$q^{+}\left( \overrightarrow{r},E \right)$$, depending on what quantity is to be optimized, is listed below.

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