Anderson Acceleration for Tiamat Anderson Acceleration Integration - - PowerPoint PPT Presentation

Anderson for Tiamat Alex Toth Tiamat Overview Anderson Acceleration Integration Numerical Tests

Single Fuel Rod Single Assembly

Conclusions

Anderson Acceleration for Tiamat

• A. Toth1,

C.T. Kelley1,

• R. Pawlowski2

1North Carolina State University 2Sandia National Laboratories

ICERM Workshop on Numerical Methods for Large-Scale Nonlinear Problems and Their Applications September 3, 2015

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Anderson for Tiamat Alex Toth Tiamat Overview Anderson Acceleration Integration Numerical Tests

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Conclusions

Outline of Topics

1

Tiamat Overview

2

Anderson Acceleration Integration

3

Numerical Tests Single Fuel Rod Single Assembly

4

Conclusions

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Anderson for Tiamat Alex Toth Tiamat Overview Anderson Acceleration Integration Numerical Tests

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Conclusions

Tiamat

Tiamat is a tool being developed in CASL for pellet-cladding interaction (PCI) analysis PCI is controlled by the complex interplay of the mechanical, thermal and chemical behavior of a fuel rod during operation Tiamat couples the single rod fuel performance code Bison-CASL with other tools in VERA which provide a whole core representation of fission density and coolant conditions in order to compute quantities of interest for identifying PCI failure.

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Anderson for Tiamat Alex Toth Tiamat Overview Anderson Acceleration Integration Numerical Tests

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Conclusions

VERA Code Suite

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Anderson for Tiamat Alex Toth Tiamat Overview Anderson Acceleration Integration Numerical Tests

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Conclusions

Components of Tiamat

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Anderson for Tiamat Alex Toth Tiamat Overview Anderson Acceleration Integration Numerical Tests

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Conclusions

Bison-CASL

Fuel performance code which models the thermo-mechanical behavior behavior of a single fuel rod Built on INL MOOSE framework, uses finite element geometric representation and JFNK to solve the governing systems of PDEs Used to compute key figures of merit (ex. max hoop stress) for identifying fuel rods requiring further detailed PCI calculations

Figure: Bison-CASL fuel rod hoop stress

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Conclusions

COBRA-TF (CTF)

Subchannel thermal hydraulics code maintained by Penn State University Utilizes a two-fluid, three-field representation of two-phase flow. Solves equations for: Continuous vapor (mass, momentum and energy) Continuous liquid (mass, momentum and energy) Entrained liquid drops (mass and momentum) Non-condensable gas mixture (mass) Only parallel to the assembly level

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Anderson for Tiamat Alex Toth Tiamat Overview Anderson Acceleration Integration Numerical Tests

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Conclusions

MPACT

Primary neutronics code in VERA, co-developed by ORNL and University of Michigan Includes several methods for solving the neutron transport equation, workhorse method is 2D/1D solver with coarse-mesh finite-difference acceleration Utilizes the subgroup method and embedded self-shielding method for cross section evaluation

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Conclusions

Data Transfer Kit (DTK)

Software designed to provide parallel services for scalable mesh/geometry searching and data transfer developed at ORNL Determines mapping for moving data between source and target arrays using the rendezvous algorithm

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Conclusions

PIKE

New Trilinos package for black box multiphysics coupling Provides interfaces for: single-physics model evaluators data transfers

• bservers

parallel distribution management local/global status tests Currently only includes Picard-based solvers

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Conclusions

Solution Process

1

Estimate hot full-power (HFP) state with CTF and MPACT

2

Model transition from cold zero-power (CZP) to hot zero-power (HZP) in Bison-CASL

3

Model transition from HZP to HFP in Bison-CASL

4

Model reactor state at HFP conditions for one or more time step

Figure: Bison-CASL ramp to HFP

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Anderson for Tiamat Alex Toth Tiamat Overview Anderson Acceleration Integration Numerical Tests

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Conclusions

Data Transfers

In the coupled HFP solve phase, Tiamat utilizes 5 data transfers Bison to MPACT: Fuel temperatures (Tf,B → Tf,M) MPACT to Bison-CASL: Fission heat generation (qM → qB) Bison-CASL to CTF: Heat flux (q′′

B → q′′ C)

CTF to Bison-CASL: Clad surface temperatures (Tc,C → Tc,B) CTF to MPACT: Coolant temperature and densities (Tw,C → Tw,M, ρw,C → ρw,M)

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Anderson for Tiamat Alex Toth Tiamat Overview Anderson Acceleration Integration Numerical Tests

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Conclusions

Picard Iteration - Block Gauss-Seidel Map

Algorithm 1: Gauss-Seidel Nonlinear Solve for Tiamat Given q0

M, T0 c,C, T0 w,C, ρ0 w,C, T0 f,B, q′′ B,0.

for k = 0, 1, . . . until converged do Transfer Bison-CASL to MPACT, Tk

f,B → Tk f,M

Transfer CTF to MPACT, Tk

w,C → Tk w,M and ρk w,C → ρk w,M

Using Tk

f,M, Tk w,M, and ρk w,M, solve MPACT and obtain qk+1 M

Transfer MPACT to Bison-CASL, qk+1

M

→ qk+1

B

Transfer CTF to Bison-CASL, Tk

c,C → Tk c,B

Using Tk

c,B and qk+1 B

, solve Bison-CASL and obtain Tk+1

f,B and

q′′

B,k+1

Transfer Bison-CASL to CTF, q′′

B,k+1 → q′′ C,k+1

Using q′′

C,k+1, solve CTF and obtain Tk+1 c,C ,Tk+1 w,C, and ρk+1 w,C

end

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Conclusions

Block Gauss-Seidel Map with Damping

Algorithm 2: Damped Gauss-Seidel Nonlinear Solve for Tiamat

Given q0

M, T0 c,C, T0 w,C, ρ0 w,C, T0 f,B, q′′ B,0.

for k = 0, 1, . . . until converged do Transfer Bison-CASL to MPACT, Tk

f,B → Tk f,M

Transfer CTF to MPACT, Tk

w,C → Tk w,M and ρk w,C → ρk w,M

Using Tk

f,M, Tk w,M, and ρk w,M, solve MPACT and obtain qk+1 M

Transfer MPACT to Bison-CASL, qk+1

M

→ qk+1

B

if k > 1 then Damp the transferred power, qk+1

B

= (1 − ω)qk

B + ωqk+1 B

end Transfer CTF to Bison-CASL, Tk

c,C → Tk c,B

Using Tk

c,B and qk+1 B

, solve Bison-CASL and obtain Tk+1

f,B and q′′ B,k+1

Transfer Bison-CASL to CTF , q′′

B,k+1 → q′′ C,k+1

Using q′′

C,k+1, solve CTF and obtain Tk+1 c,C ,Tk+1 w,C , and ρk+1 w,C

end

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Anderson for Tiamat Alex Toth Tiamat Overview Anderson Acceleration Integration Numerical Tests

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Conclusions

Tiamat Communication Layers

Issue: Applications live in independent processor space, so significant idle time for sequential solves

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Anderson for Tiamat Alex Toth Tiamat Overview Anderson Acceleration Integration Numerical Tests

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Picard Iteration - Block Jacobi Map

Algorithm 3: Jacobi Nonlinear Solve for Tiamat Given q0

M, T0 c,C, T0 w,C, ρ0 w,C, T0 f,B, q′′ B,0.

for k = 0, 1, . . . until converged do Transfer Bison-CASL to MPACT, Tk

f,B → Tk f,M

Transfer CTF to MPACT, Tk

w,C → Tk w,M and ρk w,C → ρk w,M

Transfer MPACT to Bison-CASL, qk

M → qk B

Transfer CTF to Bison-CASL, Tk

c,C → Tk c,B

Transfer Bison-CASL to CTF, q′′

B,k → q′′ C,k

Using Tk

f,M, Tk w,M, and ρk w,M, solve MPACT and obtain qk+1 M

Using Tk

c,B and qk B, solve Bison-CASL and obtain Tk+1 f,B and

q′′

B,k+1

Using q′′

C,k, solve CTF and obtain Tk+1 c,C ,Tk+1 w,C, and ρk+1 w,C

end

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Conclusions

Convergence criteria

Global convergence of the coupled system is determined by the following criteria Successful local convergence of each of the applications Bison-CASL: the change in the maximum fuel temperature across each of the fuel rods is less than some tolerance ǫT CTF: the change in the maximum clad temperature and maximum coolant temperature is less than ǫT MPACT: the relative change (in the l2 norm) in the power distribution is less than a tolerance ǫP, and the change in the dominant eigenvalue keff is less than a tolerance ǫk

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Conclusions

Advantages/Drawbacks of Picard Iteration

Advantages Simple to implement Few requirements for application codes Drawbacks Relatively slow convergence Poor robustness

0.2 0.4 0.6 0.8 1 2 4 6 8 10 12 14 16 18 20 Damping factor Iterations 80% Power 100% Power 120% Power

Figure: Picard iteration dependence

• n damping

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Anderson for Tiamat Alex Toth Tiamat Overview Anderson Acceleration Integration Numerical Tests

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Conclusions

Can We Do Better?

Would like to utilize a method which converges more quickly than Picard, or is at least less sensitive to ad hoc damping factors Newton isn’t an option as the residual evaluation is too expensive for JFNK (Roger’s talk), and we can’t get derivatives from the applications Anderson acceleration is an ideal candidate, as it requires no more information to implement than Picard and only one function evaluation per iteration.

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Conclusions

Anderson Acceleration Algorithm

Algorithm 4: Anderson acceleration Given initial iterate u0, storage depth parameter m ∈ N, and mixing parameter β Set u1 = (1 − β)u0 + βG(u0) for k = 1, 2, . . . until converged do Set mk = min{m, k} Determine α(k) which solves:

min

α∈Rmk+1

• mk
• j=0

αjF(uk−mk+j)

• ,

such that mk

j=0 αj = 1, where F(u) = G(u) − u

Set uk+1 = (1 − β) mk

j=0 α(k) j uk−mk+j + β mk j=0 α(k) j G(uk−mk+j)

end

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Conclusions

Implementation

What do we choose for u? How do we define G? Let u be comprised of some subset of the transferred data Derive G from Picard iteration, i.e. take specified input u, perform one Picard iteration, and define the corresponding

• utput as G.

In order to leverage existing data transfer objects, we let apply Anderson acceleration by intercepting and overwriting transfer target arrays

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Conclusions

Block Gauss-Seidel Map

Given Tf,M, Tw,M, ρw,M, Tc,B

1

Using Tf,M, Tw,M, and ρw,M, solve MPACT and obtain ˆ qM

2

Transfer MPACT to Bison-CASL, ˆ qM → ˆ qB

3

Using Tc,B and ˆ qB, solve Bison-CASL and obtain ˆ Tf,B and ˆ q′′

B

4

Transfer Bison-CASL to MPACT, ˆ Tf,B → ˆ Tf,M

5

Transfer Bison-CASL to CTF, ˆ q′′

B → ˆ

q′′

C

6

Using ˆ q′′

C, solve CTF and obtain ˆ

Tc,C,ˆ Tw,C, and ˆ ρw,C

7

Transfer CTF to MPACT, ˆ Tw,C → ˆ Tw,M and ˆ ρw,C → ˆ ρw,M

8

Transfer CTF to Bison-CASL, ˆ Tc,C → ˆ Tc,B

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Anderson for Tiamat Alex Toth Tiamat Overview Anderson Acceleration Integration Numerical Tests

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Conclusions

Block Gauss-Seidel Map

Define: GGS     Tf,M Tw,M ρw,M Tc,B     =     ˆ Tf,M ˆ Tw,M ˆ ρw,M ˆ Tc,B    

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Conclusions

Block Jacobi Map

Given Tf,M, Tw,M, ρw,M, Tc,B, qB, q′′

C

1

Using Tf,M, Tw,M, and ρw,M, solve MPACT and obtain ˆ qM

2

Using Tc,B and qB, solve Bison-CASL and obtain ˆ Tf,B and ˆ q′′

B

3

Using q′′

C, solve CTF and obtain ˆ

Tc,C,ˆ Tw,C, and ˆ ρw,C

4

Transfer Bison-CASL to MPACT, ˆ Tf,B → ˆ Tf,M

5

Transfer CTF to MPACT, ˆ Tw,C → ˆ Tw,M and ˆ ρw,C → ˆ ρw,M

6

Transfer MPACT to Bison-CASL, ˆ qM → ˆ qB

7

Transfer CTF to Bison-CASL, ˆ Tc,C → ˆ Tc,B

8

Transfer Bison-CASL to CTF, ˆ q′′

B → ˆ

q′′

C

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Anderson for Tiamat Alex Toth Tiamat Overview Anderson Acceleration Integration Numerical Tests

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Conclusions

Block Jacobi Map

Define: GJAC         Tf,M Tw,M ρw,M Tc,B qB q′′

C

        =         ˆ Tf,M ˆ Tw,M ˆ ρw,M ˆ Tc,B (1 − ω)qB + ωˆ qB ˆ q′′

C

       

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Conclusions

Scaling of Variables

Issue: The different fields comprising u may exist on vastly different scales, and large magnitude fields may dominate the least-squares problem in Anderson acceleration We introduce scaled variables v = Mu, where M is a diagonal scaling matrix, and instead of u = G(u) apply Anderson to the scaled fixed-point problem v = MG(M−1v) ≡ H(v) For temperature and density unknowns we let Mi,i = (u0)i. For power unknowns, we scale by the the initial average power in the fuel rod. Heat flux unknowns are scaled by the global average initial heat flux.

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Conclusions

Single Fuel Rod

First consider simulation of a single fuel rod at HFP Tests run with 12 processors, 10 allocated to MPACT, 1 for Bison-CASL, 1 for CTF We use convergence tolerances ǫP = 1e − 4, ǫT = 0.1◦C, ǫk = 1e − 5 8-group test cross sections are used in MPACT

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Conclusions

Sensitivity to Damping/Storage Depth

0.2 0.4 0.6 0.8 1 300 350 400 450 500 550 600 Damping Factor Solution Time (s) Picard Anderson−1 Anderson−2 Anderson−3

Figure: Run times for Gauss-Seidel map, varying storage depth parameter and damping level

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Conclusions

Convergence of the Fixed-Point Residual

2 4 6 8 10 10

−5

10

−4

10

−3

10

−2

10

−1

10 Iteration Relative fixed−point redisual Gauss−Seidel map, damping = 0.5 Picard Anderson−1 Anderson−2 Anderson−3 5 10 15 20 25 30 10

−5

10

−4

10

−3

10

−2

10

−1

10 Iteration Relative fixed−point redisual Jacobi map, damping = 0.5 Picard Anderson−1 Anderson−2 Anderson−3

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Conclusions

Agreement With Picard Solution

100 200 300 400 −5 −4 −3 −2 −1 1 2 3 4 x 10

−4

Height (cm) Relative difference Fuel Temperature Gauss−Seidel Jacobi 100 200 300 400 −1.5 −1 −0.5 0.5 1 1.5 x 10

−4

Height (cm) Relative difference Clad Temperature Gauss−Seidel Jacobi

Figure: Relative difference of Anderson-2 solutions from Picard solutions

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Conclusions

Agreement With Picard Solution

100 200 300 400 −2.5 −2 −1.5 −1 −0.5 0.5 1 x 10

−3

Height (cm) Relative difference Fission Rate Gauss−Seidel Jacobi 100 200 300 400 −3 −2.5 −2 −1.5 −1 −0.5 0.5 1 x 10

−3

Height (cm) Relative difference Heat Flux Gauss−Seidel Jacobi

Figure: Relative difference of Anderson-2 solutions from Picard solutions

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Conclusions

Sensitivity to Power Variation - GS Map

0.2 0.4 0.6 0.8 1 300 350 400 450 500 550 600 Damping Factor Solution Time (s) 80% Power Picard Anderson−2 0.2 0.4 0.6 0.8 1 300 350 400 450 500 550 600 Damping Factor Solution Time (s) 100% Power Picard Anderson−2 0.2 0.4 0.6 0.8 1 300 350 400 450 500 550 600 Damping Factor Solution Time (s) 120% Power Picard Anderson−2

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Conclusions

Sensitivity to Power Variation - Jacobi Map

0.2 0.4 0.6 0.8 1 300 350 400 450 500 550 600 650 700 750 Damping Factor Solution Time (s) 80% Power 100% Power 120% Power

Figure: Anderson-2 run times At each power level, Picard only converges in fewer than 30 iterations at

• ne tested damping level (Anderson iterations take between 10-20)

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Conclusions

Scaled vs Unscaled Fixed-Point Problem

0.2 0.4 0.6 0.8 1 5 10 15 Mixing parameter Iterations Gauss−Seidel Map Scaled Unscaled

(a) Block Gauss-Seidel map

0.2 0.4 0.6 0.8 1 5 10 15 20 25 Mixing parameter Iterations Jacobi Map Scaled Unscaled

(b) Block Jacobi map Figure: Iteration counts from applying Anderson-2 to the unscaled problem u = G(u) and the scaled problem v = MG(M−1v)

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Conclusions

CASL Progression Problem P6a

Progression problem P6a simulates a single 17x17 PWR assembly at HFP Tests run with 64 processors, 32 allocated to MPACT, 31 to Bison-CASL, 1 for CTF We use convergence tolerances ǫP = 1e − 4, ǫT = 1◦C, ǫk = 1e − 5 Except when noted otherwise, results use 8-group test cross sections in MPACT

Figure: 17x17 lattice, fuel in blue, guide tubes in white, instrument tube in yellow

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Conclusions

Single Assembly Results

Iteration Count Time(s) Picard (Gauss-Siedel) 8 6111 Anderson-2 (Gauss-Seidel) 6 5237 Picard (Jacobi) 17 5675 Anderson-2 (Jacobi) 12 4919

Table: Single assembly test results, damping factor 0.5

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Conclusions

Application timing breakdown

Bison-CASL CTF MPACT Picard (Gauss-Siedel) 145 131 180 Anderson-2 (Gauss-Seidel) 155 126 193 Picard (Jacobi) 148 127 172 Anderson-2 (Jacobi) 149 137 190

Table: Average application solve times

As a result of good balance in the solve times, each Jacobi iteration takes on average approximately 40% of the solve time of a Gauss-Seidel iteration

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Conclusions

Agreement With Picard Solution

50 100 150 200 250 300 350 −1.5 −1 −0.5 0.5 1 1.5 x 10

−4

Height (cm) Relative difference Assembly Averaged Fuel Temperature Gauss−Seidel Jacobi 50 100 150 200 250 300 350 −2 −1.5 −1 −0.5 0.5 1 1.5 2 x 10

−5

Height (cm) Relative difference Assembly Averaged Clad Temperature Gauss−Seidel Jacobi

Figure: Relative difference of assembly averaged Anderson-2 solutions from Picard solutions

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Conclusions

Agreement With Picard Solution

50 100 150 200 250 300 350 −5 −4 −3 −2 −1 1 2 3 4 5 x 10

−4

Height (cm) Relative difference Assembly Averaged Fission Rate Gauss−Seidel Jacobi 50 100 150 200 250 300 350 −5 −4 −3 −2 −1 1 2 3 4 5 x 10

−4

Height (cm) Relative difference Assembly Averaged Heat Flux Gauss−Seidel Jacobi

Figure: Relative difference of assembly averaged Anderson-2 solutions from Picard solutions

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Conclusions

Anderson-2 Mixing Parameter Variation

Mixing Parameter Iteration Count Time(s) 0.25 11 7172 0.5 6 5237 0.75 8 6175 1.0 7 6029

Table: Results for Anderson-2 with Gauss-Seidel map, varying mixing parameter

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Conclusions

Varying Cross Section Libraries

Iteration Count Time(s) keff Picard (8-group) 8 6111 1.165224 Anderson-2 (8-group) 6 5237 1.165224 Picard (47-group) 8 13930 1.164680 Anderson-2 (47-group) 7 13190 1.164681

Table: Comparison of Picard and Anderson-2 for Gauss-Seidel map, varying cross section libraries

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Conclusions

Varying Cross Section Libraries

Bison-CASL CTF MPACT Picard (8-group) 145 131 180 Anderson-2 (8-group) 155 126 193 Picard (47-group) 147 139 961 Anderson-2 (47-group) 166 118 1042

Table: Average application solve times

With higher fidelity cross sections, MPACT takes roughly 75% of the Gauss-Seidel iteration time, so little gain from solving applications concurrently

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Conclusions

Conclusions

Anderson acceleration displays improved robustness over Picard iteration in that its performance is generally less sensitive to variation in parameters that need to be tuned for Picard In general, Anderson at worst converges as quickly as Picard with optimally chosen damping, generally provides marginal improvement Anderson with block Jacobi map can outperform Gauss-Seidel map, but good balance in application solve times is critical Scaling of the unknowns has been seen to be important in related calculations, and this merits further investigation in this context

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Conclusions

Acknowledgements

This research was supported by the Consortium for Advanced Simulation of Light Water Reactors (http://www.casl.gov), an Energy Innovation Hub (http://www.energy.gov/hubs) for Modeling and Simulation of Nuclear Reactors under U.S. Department of Energy Contract No. DE-AC05-00OR22725 This research used resources of the Oak Ridge Leadership Computing Facility at the Oak Ridge National Laboratory, which is supported by the Office of Science of the U.S. Department of Energy under Contract No. DE-AC05-00OR22725.

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Conclusions

Questions? Comments? Suggestions?

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