Parametric Model-Order Reduction for Radiation Transport Simulations Based on an Affine Decomposition of the Operators

Figures & data

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Algorithm 1: Offline computation of parameter-independent contribution to ${\mathit{W}}^T\mathit{T}(\vec{\theta }){\mathit{U}}_r$.

Input: ${\mathrm{W}}^{\mathrm{T}}$, the transposed projection matrix (shape $r\times N_{\mathrm{\Psi }}$)

Input: ${\mathrm{U}}_{\mathrm{r}}$, the POD modes (shape $N_{\mathrm{\Psi }}\times r$)

Result: Parameter-independent part of ${\mathit{W}}^T\mathit{T}(\vec{\theta }){\mathit{U}}_r$ is written to file.

file = NewFile();

for $k=1$ to $N_{mat}$ do

for $d=1$ to D do

for g = 1 to G do

${\mathrm{W}}^{\mathrm{T}}\mathrm{T}\mathrm{U}$ = ZeroMatrix(r, r)

for c in CellsInMaterial(k) do

mass_c = GetCellMass(c)

w^T_c = ExtractCellMatrix(W^T, c, d, g)

u_c = ExtractCellMatrix(Ur, c, d, g)

/* total cross section factored out, to be applied in online

phase */

tu_c = MatVecMult(mass_c, u_c)

W^TTU += MatMatMult(w^T_c, tu_c)

end

file.Write(k, d, g, W^TTU)

end

end

end

file.Close();

TABLE I Parameter Distributions for the Slab Benchmark

	Material
Parameter	Source [0, 0.1] $\mathrm{c}\mathrm{m}$	Scatterer (0.1, 0.8) $\cup$ (0.9, 1] $\mathrm{c}\mathrm{m}$	Absorber [0.8, 0.9] $\mathrm{c}\mathrm{m}$
$Q$ ($par$. ${\text{cm}}^{-3}$ . ${\mathrm{s}}^{-1}$)	$\mathcal{U}$ (0, 10)	0	0
${\mathrm{\Sigma }}_t$ (${\text{cm}}^{-1}$)	$\mathcal{U}$ (0, 1)		$\mathcal{U}$ (5, 10)
$c$	$\mathcal{U}$ (0.7, 1)		$\mathcal{U}$ (0.05, 0.1)

Fig. 1. 100 snapshots for the slab benchmark.

Fig. 2. Normalized singular values using 100 snapshots.

Fig. 3. Relative error norm of the ROM as a function of the number of basis vectors used for the slab benchmark. The boxes show the quartiles of the 100 relative error norms, while the whiskers extend to show the rest of the distribution.

Fig. 4. Speedup of the ROM as a function of the number of basis vectors used for the slab benchmark.

Fig. 5. ROM pointwise relative error (40-mode angle-dependent ROMs); 100 testing parameter realizations shown. Maximum pointwise error is 1.7%.

Fig. 6. Checkerboard benchmark geometry. The source material has the same cross sections as the scattering material.

TABLE II Parameter Distributions for the Checkerboard Benchmark

	Material
Parameter	Absorber	Scatterer
${\mathrm{\Sigma }}_a$ (${\mathrm{c}\mathrm{m}}^{-1}$)	$\mathcal{U}$ (7.5, 12.5)	$\mathcal{U}$ (0.0, 0.5)
${\mathrm{\Sigma }}_s$ (${\mathrm{c}\mathrm{m}}^{-1}$)	$\mathcal{U}$ (0.0, 5.0)	$\mathcal{U}$ (0.5, 1.5)
$Q$ ($par$ . ${\mathrm{c}\mathrm{m}}^{-3}$ . ${\mathrm{s}}^{-1}$)	0	$\mathcal{U}$ (0.0, 2.0)

Fig. 7. Full-order solutions for three of the snapshots. Parameter vectors are reported as ${\mathrm{\Sigma }}_a^{\mathbf{a}\mathbf{b}\mathbf{s}\mathbf{o}\mathbf{r}\mathbf{b}\mathbf{e}\mathbf{r}},{\mathrm{\Sigma }}_a^{\mathbf{s}\mathbf{c}\mathbf{a}\mathbf{t}\mathbf{t}\mathbf{e}\mathbf{r}\mathbf{e}\mathbf{r}},{\mathrm{\Sigma }}_s^{\mathbf{a}\mathbf{b}\mathbf{s}\mathbf{o}\mathbf{r}\mathbf{b}\mathbf{e}\mathbf{r}},{\mathrm{\Sigma }}_s^{\mathbf{s}\mathbf{c}\mathbf{a}\mathbf{t}\mathbf{t}\mathbf{e}\mathbf{r}\mathbf{e}\mathbf{r}},\mathrm{a}\mathrm{n}\mathrm{d}{Q}^{\mathbf{s}\mathbf{c}\mathbf{a}\mathbf{t}\mathbf{t}\mathbf{e}\mathbf{r}\mathbf{e}\mathbf{r}}$.

Fig. 8. Normalized singular values of the angular snapshots for each octant of the 300 snapshots, along with the singular values of the flux moment snapshots.

Fig. 9. Relative error norm of the ROM as a function of the number of basis vectors used for the checkboard benchmark.

Fig. 10. Speedup of the ROM as a function of the number of basis vectors used for the checkerboard benchmark.

Fig. 11. ROM pointwise relative error for one of the 50 validation parameter realizations (affine-decomposed ROM).

Fig. 12. Graphite benchmark geometry.

Fig. 13. Full-order solutions for three of the snapshots. The top and bottom rows correspond to energy groups 9 and 12, respectively, while each column corresponds to a snapshot. Parameter vectors are reported as quadruplets of boron fraction 1, boron fraction 2, boron fraction 3, and variable graphite density.

Fig. 14. Normalized singular values of the energy- and direction-agglomerated angular snapshots.

Fig. 15. Relative error norm of the ROM as a function of the number of basis vectors used for the graphite benchmark.

Fig. 16. Speedup of the ROM as a function of the number of basis vectors used for the graphite benchmark.

Fig. 17. RoI relative error norm of the ROM as a function of the number of basis vectors used.

Fig. 18. Pointwise relative error for one of the 100 validation parameter realizations. The top and bottom rows correspond to energy groups 9 and 12, respectively, while each column corresponds to the rank of the ROM.

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Algorithm A.1: Offline computation of parameter-independent contribution to W^TGUr.

Input: W^T, the transposed projection matrix (shape $r\times N_{\mathrm{\Psi }}$)

Input: Ur, the POD modes (shape $N_{\mathrm{\Psi }}\times r$)

Result: Parameter-independent part of W^TGUr is written to file.

file = NewFile();

for d = 1 to D do

for g = 1 to G do

W^TGU = ZeroMatrix(r, r)

for c = 1 to Ncells do

/* Stiffness term */

stiff_c = GetCellStiffness(c)

stiff_c_d = DotProduct(${\vec{\mathrm{\Omega }}}_d$, stiff_c)

u_c = ExtractCellMatrix(Ur, c, d, g)

gu_c = MatVecMult(stiff_c, u_c)

/* Surface terms */

cell_faces = GetCellFaces(c)

for face in cell_faces do

d_dot_n = DotProduct(face.normal, ${\vec{\mathrm{\Omega }}}_d$)

if d_dot_n $\ge$ 0 then

continue / / Skip outgoing faces

end

mass_F = GetCellIncomingSurfaceMass(c, face)

mass_F_dot = d_dot_n * mass_F

u_c_upstream_f = ExtractCellUpstreamMatrix(Ur, c, face, d, g)

u_c_f = ExtractCellMatrix(Ur, c, face, d, g)

gu_c += MatVecMult(mass_F_dot, u_c_upstream_f - u_c_f)

end

w^T_c = ExtractCellMatrix(W^T, c, d, g)

W^TGU += MatMatMult(w^T_c, gu_c)

end

file.Write(d, g, W^TGU)

end

end

file.Close();

Table

Algorithm A.2: Offline computation of parameter-independent contribution to W^TGUr.

Input: W^T, the transposed projection matrix (shape $r\times N_{\mathrm{\Psi }}$)

Input: Ur, the POD modes (shape $N_{\mathrm{\Psi }}\times r$)

Result: Parameter-independent part of W^TGUr is written to file.

file = NewFile();

for k = 1 to Nmat do

for d = 1 to D do

for g = 1 to G do

W^TTU = ZeroMatrix(r, r)

for c in CellsInMaterial(k) do

w^T_c = ExtractCellMatrix(W^T, c, d, g)

u_c = ExtractCellMatrix(Ur, c, d, g)

/* total cross section factored out, to be applied in online

phase */

mass_c = GetCellMass(c)

tu_c = MatVecMult(mass_c, u_c)

W^TTU += MatMatMult(w^T_c, tu_c)

end

file.Write(k, d, g, W^TTU)

end

end

end

file.Close();

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Algorithm A.3: Offline computation of parameter-independent contribution to ${\mathit{W}}^T\mathit{M}\sum (\vec{\theta })\mathit{D}{\mathit{U}}_r$

Input: W^T, the transposed projection matrix (shape $r\times N_{\mathrm{\Psi }}$)

Input: Ur, the POD modes (shape $N_{\mathrm{\Psi }}\times r$)

Result: Parameter-independent part of ${\mathit{W}}^T\mathit{M}\sum (\vec{\theta })\mathit{D}{\mathit{U}}_r$ is written to file.

file = NewFile();

for k = 1 to Nmat do

for d = 1 to D do

for $d\prime$ = 1 to D do

for g = 1 to G do

for $g\prime$ = 1 to G do

for m = 1 to Nmom do

${\mathrm{W}}^{\mathrm{T}}\mathrm{M}\text{diams}\mathrm{D}\mathrm{U}$ = ZeroMatrix(r, r)

for c in CellsInMaterial(k) do

u_c = ExtractCellMatrix(Ur, c, $d\prime$, $g\prime$)

/* D${ }_{d\prime }$ applied for moment m */

du_c = ComputeDirectionContributionToMoment(u_c, $d\prime$, m)

/* scattering moment factored out, applied in online

phase */

mass_c = GetCellMass(c)

$\sigma$du_c = MatVecMult(mass_c, du_c)

m$\sigma$du_c = ApplyMomentToDiscrete($\sigma$du_c, m, d)

w^T_c = ExtractCellMatrix(W^T, c, d, g)

${\mathrm{W}}^{\mathrm{T}}\mathrm{M}\text{diams}\mathrm{D}\mathrm{U}$ += MatMatMult(wT c, m$\sigma$du_c)

end

file.Write(k, d, $d\prime$, g, $g\prime$, m, W^TTU)

end

end

end

end

end

end

file.Close();

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Algorithm A.4: Offline computation of parameter-independent contribution to ${\mathit{W}}^T\vec{Q}(\vec{\theta })$.

Input: W^T, the transposed projection matrix (shape $r\times N_{\mathrm{\Psi }}$)

Result: Parameter-independent part of ${\mathit{W}}^T\vec{Q}(\vec{\theta })$ is written to file.

file = NewFile();

for k = 1 to Nmat do

for d = 1 to D do

for g = 1 to G do

W^TQ = ZeroVector(r)

for c in CellsInMaterial(k) do

/* source strength factored out, to be applied in online

phase */

load_c = GetCellLoad(c)

w^T_c = ExtractCellMatrix(W^T, c, d, g)

W^TQ += MatVecMult(w^T_c, load_c)

end

file.Write(k, d, g, W^TQ)

end

end

end

file.Close();