reduced order modeling 687

A Reduced-Order Neutron Transport Model Separated in Space and Angle

Kurt A. Dominesey∗, Jaron P. Senecal, and Wei Ji†

Department of Mechanical, Aerospace, and Nuclear Engineering, Rensselaer Polytechnic Institute, 110 8th Street, Troy, NY∗[email protected], †[email protected]

INTRODUCTION

As radiation transport problems are posed in a high-dimensional phase space (defined in terms of space r=(x, y, z),angle Ω=(θ, µ), energy E, and time t) the corresponding com-putational difficulty scales rapidly, on the order O(NrNΩNE Nt)where N is the number of degrees of freedom in a given dimen-sion. This is known as the “curse of dimensionality” wherebymodels of dimension D have

∏Dd=1 Nd degrees of freedom

under traditional discretization techniques. This scaling lawcan render simulations of even moderate refinement costlyor intractable, often necessitating physical approximations orrestriction of the problem domain.

Reduced-Order Modeling (ROM) techniques present auseful alternative when such simplifications are inappropriateor undesirable. In particular, Proper Generalized Decomposi-tion (PGD), an a priori ROM strategy, is uniquely attractive,as the reduced-order basis is computed on-the-fly (rather thanfrom a series of pre-computed full-order solutions, as in aposteriori ROM methods). The application of PGD to neutralradiation transport therefore represents a promising area ofstudy which this work seeks to explore.

Preliminary work on PGD schemes for radiation transporthave been conducted, but efforts have so far been limited todiffusion theory. In particular, [1] presents a separated repre-sentation of transient 1D neutron diffusion in space and time,while [2] demonstrates a 2D, spatially-separated neutron dif-fusion method. More generally, in the realm of Boltzmanntransport (an advection-reaction problem not necessarily ap-plied to neutronics) [3] presents an upwinded PGD method forcollision-free advection decomposed in space, velocity, andtime (in 1D and 2D, as well as for steady-state and transientproblems).

In the present work, we establish a method for steady-state, monoenergetic neutron transport, moving beyond thework of [3] by the inclusion of collision and scattering op-erators, as well as an isotropic source term. Moreover, wepropose source iteration as an effective means of decouplingthe angular inscattering operator (avoiding the solution of adense matrix) and suggest Anderson acceleration as an avenueto expedite the convergence of a fixed-point solution strategy.

THEORY

The fundamental axiom of PGD is that a given solution uadmits an approximate separated representation as the finitesum of M products of d mode functions, Xd

m, where eachsuperscript d denotes a unique dimension of the solution

u(x1, x2, . . . xD) ≈M∑

m=1

D∏d=1

Xdm(xd). (1)

For the problem at hand, we seek the angular flux ψ(x, µ) interms of spatial modes Xm(x) and angular modes Um(µ),

ψ(x, µ) ≈M∑

m=1

Xm(x)Um(µ). (2)

Application to Neutron Transport

We begin with the steady-state, monoenergetic Boltzmanntransport equation in slab geometry with isotropic scatteringand an isotropic source1

µ∂ψ

∂x(x, µ) + Σt(x)ψ(x, µ) =

12

Σs(x)∫

Ωµ′

ψ(x, µ′)dµ′ +Q(x)

2.

(3)Multiplying by an arbitrary test function ϕ∗(x, µ) and inte-grating over our spatial and angular domains, Ωx and Ωµ, weachieve the following weak form. Note that we now movethe inscattering term to the left-hand-side of the equation,such that our right-hand-side consists exclusively of known orarbitrary quantities.∫

Ωµ

∫Ωx

µ∂ψ

∂x(x, µ)ϕ∗(x, µ)dxdµ

+

∫Ωµ

∫Ωx

Σt(x)ψ(x, µ)ϕ∗(x, µ)dxdµ

−

∫Ωµ

∫Ωx

12

Σs(x)∫

Ωµ′

ψ(x, µ′)dµ′ϕ∗(x, µ)dxdµ

=

∫Ωµ

∫Ωx

Q(x)2

ϕ∗(x, µ)dxdµ

(4)

In keeping with PGD methodology we assume our testfunction to be of the form

ϕ∗(x, µ) = X∗(x)U(µ) + X(x)U∗(µ) (5)

where X∗(x) and U∗(µ) are arbitrary functions. As Equation 4must be satisfied for any choice of X∗(x) and U∗(µ), our weakform can be distributed to form two independent equations.

It is at this point which we apply the PGD approximationof the flux, as in Equation 2, which we separate into knownand unknown terms. (For notational clarity, we will omit allfunctional dependencies of X, X∗, U, and U∗ moving forward.)

ψ(x, µ) ≈M−1∑m=1

XmUm + XMUM (6)

1These assumptions of isotropy are merely for convenience. Anisotropicdistributions pose no special difficulty so long as they can be expressed asa finite series of separable functions in space and angle, such as a sphericalharmonics expansion.

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Transactions of the American Nuclear Society, Vol. 119, Orlando, Florida, November 11–15, 2018

Reduced Order Modeling

Omitting the subscript M on the functions XM and UM(as they are the same modes X and U in our test function,Equation 5), we substitute this approximation of ψ into ourweak form to obtain (after simplification)

X solver:∫Ωx

∂X∂x

X∗dx∫

Ωµ

µU2dµ

+

∫Ωx

Σt(x)XX∗dx∫

Ωµ

U2dµ

−

∫Ωx

12

Σs(x)XX∗dx∫

Ωµ

Udµ∫

Ωµ′

Udµ′

=

∫Ωx

Q(x)2

X∗dx∫

Ωµ

Udµ

−

M−1∑m=1

∫Ωx

∂Xm

∂xX∗dx

∫Ωµ

µUmUdµ

−

M−1∑m=1

∫Ωx

Σt(x)XmX∗dx∫

Ωµ

UmUdµ

+

M−1∑m=1

∫Ωx

12

Σs(x)XmX∗dx∫

Ωµ

Udµ∫

Ωµ′

Umdµ′

(7)

where the underlined terms are functions of the unknownangular mode U. Applying the same process to the U∗ term,we obtain

U solver:∫Ωx

∂X∂x

Xdx∫

Ωµ

µUU∗dµ

+

∫Ωx

Σt(x)X2dx∫

Ωµ

UU∗dµ

−

∫Ωx

12

Σs(x)X2dx∫

Ωµ

U∗dµ∫

Ωµ′

Udµ′

=

∫Ωx

Q(x)2

Xdx∫

Ωµ

U∗dµ

−

M−1∑m=1

∫Ωx

∂Xm

∂xXdx∫

Ωµ

µUmU∗dµ

−

M−1∑m=1

∫Ωx

Σt(x)XmXdx∫

Ωµ

UmU∗dµ

+

M−1∑m=1

∫Ωx

12

Σs(x)XmXdx∫

Ωµ

U∗dµ∫

Ωµ′

Umdµ′

(8)

where the underlined terms are now dependent on the unknownspatial mode X. Together, these two equations represent anonlinear problem which can be solved to find the Mth modes,given an appropriate linearizion strategy. In the present work,we choose Picard, or fixed-point, iteration due to its simplicityand widespread application in established PGD literature.

Picard Iteration

Picard iteration presents a natural and effective strategy todeal with Equations 7 and 8. In brief, we make an initial guessfor one mode (denoted X(0) or U(0)), and solve the oppositeequation (for U(1) or X(1), respectively) treating X(0) or U(0)

as a known. This process of iterating between dimensionscontinues until the error indicator, or residual, falls belowa prescribed tolerance. In the present work, we choose anindicator of

Res(k) =√||X(k) − X(k−1)||2 + ||U(k) − U(k−1)||2 (9)

for all Picard iterations k > 1 and guess the angular modeU(0) as a uniform (isotropic) distribution. In practice, theconvergence of our Picard scheme is found to be relativelyinsensitive to this initial guess.

Furthermore, as our PGD method seeks to enrich thesolution ψ by a product of modes X(k)U(k), it is prudent tonormalize one mode within each Picard iteration k. Otherwise,the magnitude of our modes may “drift” between iterations,with one mode growing as the other shrinks (while maintainingthe magnitude of the product X(k)U(k)). To mitigate this, wenormalize each U(k) mode as

U(k) =U(k)

||U(k)||2(10)

where U(k) denotes the unnormalized mode within Picard it-eration k (obtained by the solution of Equation 8 or by initialguess U(0)). This choice of normalization is relatively arbitrary,as any function which approximately preserves the magnitudeof U(k) between iterations would be appropriate.

Source Iteration

Owing to the inscattering factor∫

ΩµU∗dµ

∫Ωµ′

Udµ′ con-tained in the third term of Equation 8, solution of the angularmodes U(k) actually requires coupling over the entire angulardomain. Physically, this is due to the fact that neutrons mayscatter from one direction to any other (unlike in space, whereneutrons may only be transmitted to adjacent elements). Nu-merically, this requires the solution of a dense matrix ratherthan a sparse one, impairing the performance of the angu-lar solver by some factor scaling with the number of angulardegrees of freedom.

To recover the sparsity of our angular operator, we pro-pose a kind of Jacobi iteration on the scattering kernel,∫

Ωµ′

U(k)dµ′ ≈∫

Ωµ′

U(k−1)dµ′ (11)

such that it becomes a known scalar rather than the integral ofan unknown vector. This is conceptually the same principleused to decouple the angular domain in SN methods, referredto as source iteration, with the caveat that we here updateour scattering kernel concurrently within the Picard couplingbetween space and angle. (There is no such Picard iteration inSN methods, as space and angle are discretized together.) Wemodify Equation 8 to reflect this source iteration, yielding

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Source-iterating U solver:∫Ωx

∂X∂x

Xdx∫

Ωµ

µU(k)U∗dµ

+

∫Ωx

Σt(x)X2dx∫

Ωµ

U(k)U∗dµ

=

∫Ωx

Q(x)2

Xdx +

∫Ωx

12

Σs(x)X2dx∫

Ωµ′

U(k−1)dµ′ ∫

Ωµ

U∗dµ

−

M−1∑m=1

∫Ωx

∂Xm

∂xXdx∫

Ωµ

µUmU∗dµ

−

M−1∑m=1

∫Ωx

Σt(x)XmXdx∫

Ωµ

UmU∗dµ

+

M−1∑m=1

∫Ωx

12

Σs(x)XmXdx∫

Ωµ

U∗dµ∫

Ωµ′

Umdµ′.

(12)

Anderson Acceleration

To mitigate the potentially slowed convergence of Pi-card iteration as a result of source iteration, we apply a fixed-point acceleration method known as (undamped) Andersonacceleration, or Anderson mixing. In short, this scheme is aquasi-Newton method which solves a least-squares problem tofind the optimal weights by which to combine (or “mix”) theprevious m iterations of a Picard calculation, where m is a user-supplied parameter. At present, we show results for m = 5 andformulate the least-squares problem as in [4]. Acceleration(“mixing”) is applied exclusively to the U(k) modes, effectivelyhastening convergence of the mapping U(k−1) → X(k) → U(k).As this procedure will adjust the magnitude of each U(k) mode,we now apply the normalization (Equation 10) both beforeand after each Anderson mixing.

Boundary Conditions

In PGD methods, Dirichlet boundary conditions are gen-erally provided by “seeding” a solution with known modefunctions which satisfy the desired conditions along the prob-lem boundaries. To preserve the solution along these edgesthroughout enrichment, all subsequent modes are required tovanish at these boundaries.

However, we now recognize a difficulty of our PGD imple-mentation: this “vanishing” cannot be enforced for a problemencompassing the entire angular domain, Ωµ ∈ [−1, 1]. Thisis due to the advective nature of (first-order) neutron transportwhich requires that, for a slab between x0 and x1, we imposethe boundary conditions

ψ(x, µ) =

ψx0 (µ), x = x0, µ > 0ψx1 (µ), x = x1, µ < 0

(13)

where ψx0 (µ) and ψx1 (µ) are known functions. In other words,we prescribe an incident flux on each side of the slab.

In keeping with the aforementioned PGD methodology,we would naively attempt to meet this condition by seedingour solution with some modes satisfying ψx0 and ψx1 andrequiring all following modes to vanish along x = x0, µ ∈(0, 1] and x = x1, µ ∈ [−1, 0). However, these boundariescannot be represented in our separated problem, as the X and Umodes are independent; we describe this scenario graphicallyin Figure 1.

µ

−1

0

1

xx0 x1

ψ+

ψ−

Known boundary Unknown boundary

Fig. 1. Schematic of the problem domain.

A natural strategy to reconcile this issue would be to in-stead consider two coupled PGD problems, for ψ+ and ψ−,representing the forward and backward streaming flux, respec-tively. This is similar to the approach taken in [3], whichconsiders only positive particle velocity and introduces anadditional binary dimension in space to account for the advec-tion direction (forward or backward). However, we here allowasymmetric angular distributions by considering directionalcomponents of both space and angle. Moreover, for particletransport in scattering media, coupling between the two direc-tional solutions is necessitated by the angular scattering kernel∫

Ωµ′U(µ′)dµ′, which encompasses the entire angular domain

Ωµ′ ∈ [−1, 1], such that∫Ωµ′

U(µ′)dµ′ =

∫Ωµ+

U+(µ+)dµ+ +

∫Ωµ−

U−(µ−)dµ− (14)

where U+ and U− are the angular modes of the forward andbackward fluxes, and µ+ ∈ (0, 1], µ− ∈ [−1, 0).

Finite Elements

In this work, we discretize Equations 7 and 8 (or 12) viacontinuous Galerkin finite elements with first-order Lagrangeshape functions. Future efforts may explore alternative dis-cretizations, as Equation 7 is an advection-reaction problem,and (depending on the physical characteristics) may exhibitspurious numerical oscillations when solved using continu-ous finite elements without stabilization. (Possible alternativemethods include Streamline Upwind Petrov Galerkin, SUPG,or Subgrid Scale Stabilized, SGS, formulations.) Moreover, asSN methods employ a discontinuous, 0th order discretization ofangle, similarly discretized PGD implementations may allowfor a more meaningful comparison with full-order solvers.

NUMERICAL RESULTS

Moving now to characterize the method performance, wemodel a problem (discretized using 500 mesh points in the xand µ dimensions and defined as in Table I) with no known

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solution2. While omitted for brevity, two analytical bench-marks (transport in void and pure scattering with an isotropicincident flux), were simulated with favorable results (errors onthe order of 10−15 cm−2s−1 after one mode), rendering our cur-rent numerical discussion appropriate. Note that this problem(the solution of which is plotted in Figure 2) only considersforward neutron transport, Ωµ ∈ (0, 1], such that backscatteredparticles are effectively lost; bidirectional problems (coupledvia Equation 14) are under development and will be discussedin forthcoming works.

To measure increasing convergence with enrichment, weemploy the indicator ||Xm||∞ · ||Um||∞, the maximum localchange in the solution from the latest modes m. We notefrom Figure 3 that convergence is unaffected by both sourceiteration and Anderson acceleration (as each Picard processconverges to the same criterion regardless), yet Anderson ac-celeration allows each pair of modes to be computed in feweriterations than the standard Picard strategy, shown in Table II.

TABLE I. Problem definition (scattering with loss of backscat-tered particles)

Spatial domain x ∈ [0,1] cmTotal cross-section Σt(x) = 1 cm−1

Scattering cross-section Σs(x) = 1 cm−1

Isotropic source Q(x) = 0 cm−3s−1

Picard tolerance εPicard = 10−4

Seeded spatial mode X1(x) =

1 cm−2s−1, x = 00 cm−2s−1, x > 0

Seeded angular mode U1(µ) = 1Computed modes M − 1 = 50

x [cm]

0.0 0.20.4

0.60.8

1.0

µ

0.00.2

0.40.6

0.81.0

ψ[c

m−2

s−1]

0.20.30.40.50.60.70.80.91.0

Fig. 2. Forward neutron transport in a slab, computed by PGD.

TABLE II. Number of Picard iterations per mode, by strategyMean, Relative Simple Picard Anderson AccelerationCoupled Angle 14.98, 1.000 7.12, 0.475Source Iteration 15.84, 1.057 7.96, 0.531

CONCLUSIONS

In summary, we have derived and demonstrated a reduced-order PGD method for monoenergetic neutron transport across

2Constant values of Σt(x), Σs(x), and Q(x) are assumed simply for clarityof demonstration; spatially-dependent material properties (as in transportthrough heterogeneous media) do not pose any special difficulty.

0 10 20 30 40 50

10−2

10−1

100

||Xm|| ∞·||Um|| ∞

Standard

Source Iterated

Anderson Accelerated

Anderson Accelerated Source Iterated

0 10 20 30 40 50Mode m

0

5

10

15

20

25

30

Pic

ard

Iter

atio

ns

Standard

Source Iterated

Anderson Accelerated

Anderson Accelerated Source Iterated

Fig. 3. Method convergence and performance, by strategy.

a slab with isotropic scattering. Numerical results suggestsource iteration is an effective scheme to decouple the angulardomain (in the present monodirectional method), while An-derson acceleration is found to reduce the number of requiredPicard iterations by approximately a factor of two. Futurework will focus on implementing bidirectional transport, ex-ploring alternative discretizations for each one-dimensionalsolver, and extending the method to 2D geometries.

ACKNOWLEDGMENTS

The first and second authors are supported by the NuclearRegulatory Commission Fellowship Program under the GrantsNRC-HQ-60-17-G-0006 and NRC-HQ-13-G-38-0035. Thiswork is partially supported by the Department of Energy Officeof Nuclear Energy under Award Number DE-NE0008707.

REFERENCES

1. A. L. ALBERTI and T. S. PALMER, “A-Priori ReducedOrder Modeling for Transient Neutron Diffusion,” in “25thICTT,” Monterey, CA (October 2017).

2. J. P. SENECAL and W. JI, “A Reduced-Order Model forthe Solution of Diffusion Equations,” Trans. Am. Nucl. Soc.,118, 463–466 (2018).

3. F. CHINESTA, E. ABISSET-CHAVANNE, A. AMMAR,and E. CUETO, “Efficient Stabilization of Advection TermsInvolved in Separated Representations of Boltzmann andFokker-Planck Equations,” Communications in Computa-tional Physics, 17, 975–1006 (2015).

4. H. FANG and Y. SAAD, “Two classes of multisecant meth-ods for nonlinear acceleration,” Numerical Linear Algebrawith Applications, 16, 197–221 (2009).

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