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Nuclear Engineering and Technology 49 (2017) 1114e1124
Contents lists available at ScienceDirect
Nuclear Engineering and Technology
journal homepage: www.elsevier.com/locate/net
Original Article
Multilevel acceleration of scattering-source iterations with application to electron transport
Clif Drumm*, Wesley Fan*
Radiation Effects Theory Department, Sandia National Laboratories, PO Box 5800, Albuquerque, NM 87185-1179, USA
a r t i c l e i n f o a b s t r a c t
Article history: Received 1 June 2017
Received in revised form 20 July 2017
Accepted 13 August 2017 Available online 18 August 2017
Keywords: Krylov subspace
Low-order discrete-ordinates solve Spherical-harmonics solve Transport synthetic acceleration
1. Introduction
Acceleration/preconditioning strategies available in the SCEPTRE radiation transport code are described. A ﬂexible transport synthetic acceleration (TSA) algorithm that uses a low-order discrete-ordinates (SN) or spherical-harmonics (PN) solve to accelerate convergence of a high-order SN source-iteration (SI) solve is described. Convergence of the low-order solves can be further accelerated by applying off-the-shelf incomplete-factorization or algebraic-multigrid methods. Also available is an algorithm that uses a generalized minimum residual (GMRES) iterative method rather than SI for convergence, using a parallel sweep-based solver to build up a Krylov subspace. TSA has been applied as a preconditioner to accelerate the convergence of the GMRES iterations. The methods are applied to several problems involving elec-tron transport and problems with artiﬁcial cross sections with large scattering ratios. These methods were compared and evaluated by considering material discontinuities and scattering anisotropy. Observed accelerations obtained are highly problem dependent, but speedup factors around 10 have been observed in typical applications.
© 2017 Korean Nuclear Society, Published by Elsevier Korea LLC. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
In this work, a method is described that combines the two so-
lution approaches, using a coarse-level (low-order-in-angle) space/
Electron transport is characterized by a large number of colli-sions before removal from the scattering source, which makes the source iterations (SIs) in a sweep-based solution method slowly convergent. The SCEPTRE radiation transport code [1] has several alternative solvers that do converge rapidly for electron transport in many circumstances. In these alternative solvers, a discretized linear system is constructed involving both the spatial and angular variables, including the scattering source, which is then solved in parallel with Krylov iterative methods. In SCEPTRE, the space/angle linear system is solved by making use of capabilities contained in the Trilinos project [2], which includes iterative solvers and pre-conditioners developed for the efﬁcient parallel solution of dis-cretized linear systems for large-scale, scientiﬁc applications. Convergence of the Krylov methods depends on the condition number of the system matrix and the clustering of the eigenvalues, rather than on the scattering ratio, so that rapid convergence may be obtained even when highly scattering media are present in the problem. A main drawback of these methods is that they can be memory intensive.
* Corresponding authors.
E-mail addresses: crdrumm@sandia.gov (C. Drumm), wcfan@sandia.gov
(W. Fan).
angle solve to accelerate a ﬁne-level (high-SN-order) SI solve. Preferably, a method is used for the coarse-level solve that con-verges rapidly for problems containing highly scattering media and that effectively speeds up convergence of the SI of the ﬁne-level problem. If Trilinos tools are used for the coarse-level solves, it is possible to attain further acceleration by applying off-the-shelf incomplete-factorization (IF) [3] or algebraic multigrid (AMG) [4] algorithms. This work is basically generalized transport synthetic acceleration (TSA) [5] with great ﬂexibility in performing the coarse-level solves.
In addition, capability has been added to SCEPTRE by applying a generalized minimum residual (GMRES) algorithm for convergence as an alternative to SI. Krylov iterative methods have been effec-tively applied to radiation transport problems for some time [6e9]. In this approach, sweep solves are used to build up a Krylov sub-space that is used to minimize a residual and converge to a solution. This method is shown to be effective for electron transport and otherapplications with largescattering ratios. TSA has been applied as a preconditioner to accelerate the convergence of the GMRES iterations. The GMRES algorithm implemented in SCEPTRE will be described in the following section.
The next section will provide more details about the TSA
methods, and then results will be provided for modeling electron
http://dx.doi.org/10.1016/j.net.2017.08.009
1738-5733/© 2017 Korean Nuclear Society, Published by Elsevier Korea LLC. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/ licenses/by-nc-nd/4.0/).
C. Drumm, W. Fan / Nuclear Engineering and Technology 49 (2017) 1114e1124 1115
transport in a twisted-pair electrical cable, and a three-material-block cylinder with various combinations of uniform and nonuni-form cross sections. Observed accelerations obtained are highly problem dependent, but speedup factors around 10 have been observed in typical applications.
2. Background
Acceleration methods such as diffusion synthetic acceleration (DSA) and TSA have been very effective at reducing the number of iterations and/or solve time for a wide range of applications [10e13]. However, partially consistent DSA may perform poorly for certain applications, and fully consistent DSA and TSA may effec-tively reduce the iteration count, while shifting much of the work to the coarse-level solves. The effectiveness of acceleration methods
tends to be very problem dependent. In general, for problems with
operators and number of levels are handled by the Trilinos/MueLu package, and for IF the preconditioning is handled by the Trilinos/ Ifpack2 package. For the SAAF and LS coarse-level solves, which use CFE, the mapping back and forth between the DFE and CFE repre-sentations of the spatial dependence is handled by Trilinos/Tpetra tools [2]. In SCEPTRE, the basic representation of the internal angular ﬂux, ﬁxed source, and boundary angular ﬂux is based on a DFE representation in space. In constructing a linear system, a mapping is set up using Trilinos/Tpetra mapping tools to transfer the DFE information to a CFE representation in the FE assembly phase. After obtaining the solution in the CFE space, the resulting angular ﬂux solution is mapped back to a DFE representation using the Import/Export tools in Tpetra.
2.1. Description of the TSA algorithm
highly scattering regions, some type of acceleration is effective, but practical problems often have highly scattering regions and streaming regions, so some ﬂexibility is desired in applying a preconditioner.
The SCEPTRE radiation transport code has a unique ability of allowing for any number of different types of solvers to be deﬁned within a single transport calculation [1]. The primary application of
SCEPTRE is coupled photon/electron transport, so this ﬂexibility
The mono-energetic transport equation is
Z
U,Vjþ stjðr;UÞ ¼ ss U /U j r;U dU þ Qðr;UÞ (1)
with an imposed surface-source boundary condition
jðr;UÞ ¼ jbðr;UÞ; r2G; U,n<0 (2)
was needed because the convergence properties for problems involving neutral particles are so different from those involving charged particles, and even among different energy groups, convergence behavior may be quite variable. As noted previously, SCEPTRE includes a sweep-based SI solver as well as a number of solvers that use a Krylov iterative method to solve the space/angle
dependence simultaneously.
where r is the spatial position, Uis the particle direction of motion, st is the total cross section, ss is the scattering cross section, jis the angular ﬂux, and Q is a known distributed ﬁxed source. G is the external spatial boundaryand nis the unit outward normal on G. jb is the imposed surface boundary condition for incoming directions.
Deﬁning the transport operator T as
A wide variety of space/angle solvers are available in SCEPTRE, such as those based on the ﬁrst-order (FO) form of the transport equation using discontinuous ﬁnite elements (DFEs), for both the spherical-harmonics (PN) and discrete ordinates (SN) treatment of the angular variable. A DFE spatial discretization always results in a nonsymmetric positive deﬁnite (non-SPD) linear system, so GMRES is used for these solvers. Also available are several solvers that result in a SPD linear system, so that the highly efﬁcient conjugate gradients (CGs) algorithm may be used. These methods include the self-adjoint angular ﬂux (SAAF) method and the least-squares (LSs) method, using continuous ﬁnite elements (CFEs), again for both PN and SN [14,15]. Inparticular [14], describes the PN method applied to the FO transport equation with DFE spatial discretization, which in
this work is shown to be one of the most effective TSA methods
T + ¼ U,V+ þ st+ (3)
and the scattering operator S as
S+ ¼ Z ssU0 /U+dU0 (4)
the transport equation can be written compactly as
T jðr;UÞ ¼ Sjðr;UÞ þ Qðr;UÞ (5)
The transport equation can be solved by SI
T jðkþ1Þ ¼ SjðkÞ þ Q (6)
tested. The use of spatial DFE results in a nonsymmetric linear system, regardless of the form of the transport equation used, so only CFE spatial approximation is available for use with the SAAF and LS solvers.
As noted earlier, due to huge memory requirements, the space/ angle solvers are not generally practical except for fairly small problems. However, these methods are very effective as coarse-level accelerators for a SI solver. Because the space/angle linear systems are built using Trilinos [2] data structures, the rich assortment of preconditioners included in the Trilinos project, including AMG and IF, is accessible.
Mapping from the ﬁne-level to coarse-level problem is basically angular multilevel (AML) optionally combined with an algebraic preconditioner, either IF [3] or AMG [4]. AMG is not expected to perform well for hyperbolic problems, as encountered in this work. AMG results are included for completeness, and results conﬁrm the expected poor performance of AMG for this application. For AML, the restriction and prolongation operators involve the moment-to-discrete and discrete-to-moment operators, as will be described in
the next section. For AMG, the restriction and interpolation
where jðkÞ is the known kth iterate of the angular ﬂux, and jðkþ1Þ is the computed (kþ1)st iterate, with the iterations continuing until
convergence. With an SN discretization in angle and a DFE dis-cretization in space, the (kþ1)st iterate of the angular ﬂux is ob-
tained by an efﬁcient sweeping algorithm.
In the TSA algorithm, the (kþ1)st iterate of the angular ﬂux is
obtained by a two-step process: (1) a sweep solve, and (2)
computation and application of a correction term. Reindexing the result of a sweep solve as the (kþ½)th iteration
T j kþ 2 ¼ SjðkÞ þ Q (7)
the goal is to determine a correction to the (kþ½)th iterate of the solution to accelerate convergence. The residual at the (kþ½)th
iteration is
r kþ 2 ¼ Q ðT SÞj kþ 2 (8)
1116 C. Drumm, W. Fan / Nuclear Engineering and Technology 49 (2017) 1114e1124
or
r kþ 2 ¼ S j kþ 2 jðkÞ (9)
The error term, ε kþ12 , is the difference between the actual solution, j, and the (kþ½)th iterate
1 1
ε ¼ j j (10)
which can be computed from
ðT SÞε kþ 2 ¼ r kþ 2 (11)
Rather than solving Eq. (11) directly, which may be as difﬁcult as solving the original transport equation, the procedure is to map the ﬁne-level linear system for computing the correction term, Eq. (11), to a coarse-level linear system, which is solved for a coarse-level
correction term that is then mapped back to ﬁne-level and applied to the (kþ½)th iterate of the angular ﬂux. A coarse-level
linear system is deﬁned as
kþ12 1 1
rc ¼ ℛr ¼ Dfcr (16)
where ℛ is the restriction operator, which is Dfc, the ﬁne-to-coarse discrete-to-moment operator. The coarse-level correction term is mapped back to ﬁne level by
ε kþ12 ¼ Pεckþ 2 ¼ Mcf εckþ 2 (17)
where P is the prolongation operator, which is Mcf , the coarse-to-ﬁne moment-to-discrete operator.
2.1.2. Using SN as a coarse-level solver
For a coarse-level SN problem, the coarse-level residual is computed from
kþ12 1 1
rc ¼ ℛr ¼ MccDfcr (18)
where ℛ is the restriction operator, which is the product of Mcc, the coarse-to-coarse moment-to-discrete operator, and Dfc, the ﬁne-to-coarse discrete-to-moment operator. The coarse-level error term is
kþ12 kþ12 ðT c ScÞεc ¼ rc
where a coarse-level residual is computed from
then mapped back to the ﬁne level by
(12) 1 kþ12 kþ12
ε ¼ Pεc ¼ Mcf Dccεc (19)
rc kþ12 ¼ ℛrkþ12 (13)
where P is the prolongation operator, which is the product of Mcf , the coarse-to-ﬁne moment-to-discrete operator, and Dcc, the coarse-to-coarse discrete-to-moment operator.
where ℛ is a restriction operator for mapping from the ﬁne-level linear problem to a coarse-level linear problem. Eq. (12) can be
solved by any desired method for the coarse-level correction term.
2.2. Description of the preconditioned GMRES algorithm
The GMRES iterative method [16] solves a general linear system
Through numerical experimentation, an optimal convergence cri-terion (for minimum solver time) may be determined for solving the coarse-level system, often much more relaxed than that for the ﬁne-level solves.
The coarse-level correction term is mapped back to the ﬁne-
level problem by applying a prolongation operator
Ax ¼ b (20)
by constructing a Krylov subspace
h i
Km ¼ span r0;Ar0;A2r0;…;Am1r0 (21)
εkþ12 ¼ Pεckþ12 (14)
where P is a prolongation operator, for mapping the coarse-level problem back to the ﬁne-level problem. The (kþ1)st iterate of the
angular ﬂux is obtained by applying the correction term
and then by determining the optimal solution by minimizing a residual norm over the Krylov subspace. To apply the GMRES al-gorithm to solve the transport equation, following [6e9], we start with the monoenergetic transport equation
T j Sj ¼ Q (22)
and operate from the left with the inverse transport operator,
jðkþ1Þ ¼ j kþ 2 þ ε kþ 2
which results in
(15) I T 1S j ¼ T 1Q ¼ junc (23)
Speciﬁc restriction and prolongation operators for using either spherical harmonics or discrete ordinates as a coarse-level solver are described in the next sections.
2.1.1. Using PN as a coarse-level solver
For a coarse-level PN problem, the coarse-level residual mo-ments are computed from
where junc is the uncollided ﬂux. The GMRES algorithm is applied to solve Eq. (23) by identifying the components of the linear system
as