Multilevel acceleration of scattering-source iterations with application to electron transport

Đăng ngày 7/3/2019 10:04:48 PM | Thể loại: | Lần tải: 0 | Lần xem: 9 | Page: 11 | FileSize: 3.33 M | File type: PDF
Multilevel acceleration of scattering-source iterations with application to electron transport. 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.
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:
Acceleration/preconditioning strategies available in the SCEPTRE radiation transport code are described.
Received 1 June 2017
Received in revised form
20 July 2017
Accepted 13 August 2017
Available online 18 August 2017
A exible 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
Keywords:
Krylov subspace
Low-order discrete-ordinates solve
Spherical-harmonics solve
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 articial cross sections with large scattering ratios. These methods
were compared and evaluated by considering material discontinuities and scattering anisotropy.
Transport synthetic acceleration
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
1. Introduction
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-
angle solve to accelerate a ne-level (high-SN-order) SI solve.
sions before removal from the scattering source, which makes the
Preferably, a method is used for the coarse-level solve that con-
source iterations (SIs) in a sweep-based solution method slowly
verges rapidly for problems containing highly scattering media and
convergent. The SCEPTRE radiation transport code [1] has several
that effectively speeds up convergence of the SI of the ne-level
alternative solvers that do converge rapidly for electron transport
problem. If Trilinos tools are used for the coarse-level solves, it is
in many circumstances. In these alternative solvers, a discretized
possible to attain further acceleration by applying off-the-shelf
linear system is constructed involving both the spatial and angular
incomplete-factorization (IF) [3] or algebraic multigrid (AMG) [4]
variables, including the scattering source, which is then solved in
algorithms. This work is basically generalized transport synthetic
parallel with Krylov iterative methods. In SCEPTRE, the space/angle
acceleration (TSA) [5] with great exibility in performing the
linear system is solved by making use of capabilities contained in
coarse-level solves.
the Trilinos project [2], which includes iterative solvers and pre-
In addition, capability has been added to SCEPTRE by applying a
conditioners developed for the efcient parallel solution of dis-
generalized minimum residual (GMRES) algorithm for convergence
cretized linear systems for large-scale, scientic applications.
as an alternative to SI. Krylov iterative methods have been effec-
Convergence of the Krylov methods depends on the condition
tively applied to radiation transport problems for some time [6e9].
number of the system matrix and the clustering of the eigenvalues,
In this approach, sweep solves are used to build up a Krylov sub-
rather than on the scattering ratio, so that rapid convergence may
space that is used to minimize a residual and converge to a solution.
be obtained even when highly scattering media are present in the
This method is shown to be effective for electron transport and
problem. A main drawback of these methods is that they can be
otherapplications with largescattering ratios. TSA has been applied
memory intensive.
as a preconditioner to accelerate the convergence of the GMRES
iterations. The GMRES algorithm implemented in SCEPTRE will be
described in the following section.
* Corresponding authors.
The next section will provide more details about the TSA
E-mail
(W. Fan).
addresses:
(C.
Drumm),
methods, and then results will be provided for modeling electron
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/
=
=
=
0 0 0
1
1 1
C. Drumm, W. Fan / Nuclear Engineering and Technology 49 (2017) 1114e1124
1115
transport in a twisted-pair electrical cable, and a three-material-
operators and number of levels are handled by the Trilinos/MueLu
block cylinder with various combinations of uniform and nonuni-
package, and for IF the preconditioning is handled by the Trilinos/
form cross sections. Observed accelerations obtained are highly
Ifpack2 package. For the SAAF and LS coarse-level solves, which use
problem dependent, but speedup factors around 10 have been
CFE, the mapping back and forth between the DFE and CFE repre-
observed in typical applications.
sentations of the spatial dependence is handled by Trilinos/Tpetra
tools [2]. In SCEPTRE, the basic representation of the internal
2. Background
angular ux, xed source, and boundary angular ux is based on a
DFE representation in space. In constructing a linear system, a
Acceleration methods such as diffusion synthetic acceleration
mapping is set up using Trilinos/Tpetra mapping tools to transfer
(DSA) and TSA have been very effective at reducing the number of
the DFE information to a CFE representation in the FE assembly
iterations and/or solve time for a wide range of applications
phase. After obtaining the solution in the CFE space, the resulting
[10e13]. However, partially consistent DSA may perform poorly for
angular ux solution is mapped back to a DFE representation using
certain applications, and fully consistent DSA and TSA may effec-
the Import/Export tools in Tpetra.
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
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 exibility is desired in applying a
preconditioner.
The mono-energetic transport equation is
Z    
U,Vjþ stjðr;UÞ ¼ ss U /U j r;U dU þ Qðr;UÞ
(1)
The SCEPTRE radiation transport code has a unique ability of
allowing for any number of different types of solvers to be dened
with an imposed surface-source boundary condition
within a single transport calculation [1]. The primary application of
SCEPTRE is coupled photon/electron transport, so this exibility
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 ux, and Q is a known distributed xed source. G is the
external spatial boundaryand nis the unit outward normal on G. jb
is the imposed surface boundary condition for incoming directions.
Dening the transport operator T as
A wide variety of space/angle solvers are available in SCEPTRE,
such as those based on the rst-order (FO) form of the transport
T + ¼ U,V+ þ st+
(3)
equation using discontinuous nite 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 denite (non-SPD) linear system, so GMRES
and the scattering operator S as
S+ ¼ Z ssU0 /U+dU0
(4)
is used for these solvers. Also available are several solvers that
result in a SPD linear system, so that the highly efcient conjugate
the transport equation can be written compactly as
gradients (CGs) algorithm may be used. These methods include the
self-adjoint angular ux (SAAF) method and the least-squares (LSs)
T jðr;UÞ ¼ Sjðr;UÞ þ Qðr;UÞ
(5)
method, using continuous nite elements (CFEs), again for both PN
and SN [14,15]. Inparticular [14], describes the PN method applied to
The transport equation can be solved by SI
the FO transport equation with DFE spatial discretization, which in
this work is shown to be one of the most effective TSA methods
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 ne-level to coarse-level problem is basically
angular multilevel (AML) optionally combined with an algebraic
where jðkÞ is the known kth iterate of the angular ux, 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 ux is ob-
tained by an efcient sweeping algorithm.
In the TSA algorithm, the (kþ1)st iterate of the angular ux 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)
preconditioner, either IF [3] or AMG [4]. AMG is not expected to
perform well for hyperbolic problems, as encountered in this work.
the goal is to determine a correction to the (kþ½)th iterate of the
solution to accelerate convergence. The residual at the (kþ½)th
AMG results are included for completeness, and results conrm the
iteration is
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
   
r kþ 2 ¼ Q  ðT  SÞj kþ 2
(8)
the next section. For AMG, the restriction and interpolation
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
i
h
1 1


kþ
kþ
2
2
kþ
kþ
2
2


1 1
1 1
kþ
kþ
2
2
kþ
2


1 1
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)
kþ12  1   1 
rc ¼ r ¼ Dfcr (16)
where is the restriction operator, which is Dfc, the ne-to-coarse
discrete-to-moment operator. The coarse-level correction term is
mapped back to ne level by
   
ε kþ12 ¼ Pεckþ 2 ¼ Mcf εckþ 2 (17)
which can be computed from
where P is the prolongation operator, which is Mcf , the coarse-to-
ne moment-to-discrete operator.
   
ðT  SÞε kþ 2 ¼ r kþ 2
(11)
2.1.2. Using SN as a coarse-level solver
For a coarse-level SN problem, the coarse-level residual is
Rather than solving Eq. (11) directly, which may be as difcult as
solving the original transport equation, the procedure is to map the
ne-level linear system for computing the correction term, Eq. (11),
to a coarse-level linear system, which is solved for a coarse-level
computed from
kþ12  1   1 
rc ¼ r ¼ MccDfcr
(18)
correction term that is then mapped back to ne-level and
applied to the (kþ½)th iterate of the angular ux. A coarse-level
linear system is dened as
where is the restriction operator, which is the product of Mcc, the
coarse-to-coarse moment-to-discrete operator, and Dfc, the ne-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
(12)
then mapped back to the ne level by
 1  kþ12 kþ12
ε ¼ Pεc ¼ Mcf Dccεc
(19)
rc kþ12 ¼ rkþ12
(13)
where P is the prolongation operator, which is the product of Mcf ,
the coarse-to-ne moment-to-discrete operator, and Dcc, the
coarse-to-coarse discrete-to-moment operator.
where is a restriction operator for mapping from the ne-level
2.2. Description of the preconditioned GMRES algorithm
linear problem to a coarse-level linear problem. Eq. (12) can be
solved by any desired method for the coarse-level correction term.
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
Ax ¼ b
(20)
the coarse-level system, often much more relaxed than that for the
ne-level solves.
by constructing a Krylov subspace
The coarse-level correction term is mapped back to the ne-
level problem by applying a prolongation operator
h i
Km ¼ span r0;Ar0;A2r0;;Am1r0
(21)
εkþ12 ¼ Pεckþ12
(14)
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
where P is a prolongation operator, for mapping the coarse-level
problem back to the ne-level problem. The (kþ1)st iterate of the
T j Sj ¼ Q
(22)
angular ux is obtained by applying the correction term
and operate from the left with the inverse transport operator,
   
jðkþ1Þ ¼ j kþ 2 þ ε kþ 2
(15)
which results in
 
I  T 1S j ¼ T 1Q ¼ junc
(23)
Specic restriction and prolongation operators for using either
spherical harmonics or discrete ordinates as a coarse-level solver
are described in the next sections.
where junc is the uncollided ux. The GMRES algorithm is applied
to solve Eq. (23) by identifying the components of the linear system
as
2.1.1. Using PN as a coarse-level solver
For a coarse-level PN problem, the coarse-level residual mo-
<A0I  T 1S
x0j
b0junc
(24)
ments are computed from
We have implemented the GMRES(m) algorithm with restart,
HƯỚNG DẪN DOWNLOAD TÀI LIỆU

Bước 1:Tại trang tài liệu slideshare.vn bạn muốn tải, click vào nút Download màu xanh lá cây ở phía trên.
Bước 2: Tại liên kết tải về, bạn chọn liên kết để tải File về máy tính. Tại đây sẽ có lựa chọn tải File được lưu trên slideshare.vn
Bước 3: Một thông báo xuất hiện ở phía cuối trình duyệt, hỏi bạn muốn lưu . - Nếu click vào Save, file sẽ được lưu về máy (Quá trình tải file nhanh hay chậm phụ thuộc vào đường truyền internet, dung lượng file bạn muốn tải)
Có nhiều phần mềm hỗ trợ việc download file về máy tính với tốc độ tải file nhanh như: Internet Download Manager (IDM), Free Download Manager, ... Tùy vào sở thích của từng người mà người dùng chọn lựa phần mềm hỗ trợ download cho máy tính của mình  
9 lần xem

Multilevel acceleration of scattering-source iterations with application to electron transport. 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..

Nội dung

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

Tài liệu liên quan