sensitivity analysis and error propagation for plasma edge ... fusion … · fusion 54 (2014)...
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Sensitivity Analysis and Error
Propagation for Plasma Edge CodesStatus and Challenges
W. Dekeyser, M. Blommaert, S. Carli, M. Baelmans
KU Leuven, Department of Mechanical Engineering
Role of plasma edge modeling
2
• Design of divertors and power exhaust scenarios
for next generation machines still an open question
o Limit power load to PFCs to acceptable levels
o Manage particle exhaust
o Ensure compatibility with burning plasma conditions in the
core
• Numerical codes (e.g. SOLPS-ITER) essential to
consistently model the complex plasma edge
o (Multi-)fluid plasma – kinetic neutral models
o Highly nonlinear, anisotropic, strongly coupled PDEs
o Coupling with PWI models, MHD equilibrium,…
o Coupled Finite Volume / Monte Carlo codes
ITER
[A.S. Kukushkin et al., Fusion Eng. Des. 96 (2011) 2865.]
Plasma edge codes as analysis tools
3
Profiles of
plasma
parameters
Magnetic
equilibrium
Simulation
(forward)
Vessel,
divertor
Parameters,
BCs,...
Fluxes to
PFCs
Relatively small
number of outputs,
well defined objectives
Required for model
validation,
simulation-based
design,...:
...
Very large number of inputs,
constraints, uncertainties,...
Plasma edge codes as optimization tools
4
Design/model
inputSimulation
(forward)
Analysis
Change in
input
Desired
changeAdjoint
simulation
Magnetic
equilibrium
sensitivity
information?
Inner target
Outer target
5
Divertor Shape optimization
Magnetic divertoroptimization
[W. Dekeyser et al., Nucl. Fusion 54 (2014) 073022, and M. Blommaert et al., Nucl. Fusion 55 (2015) 013001.]
Applications of optimization tools
Using adjoint techniques, entire
optimization problem solved at a cost of
only a few forward simulations!
Model calibration through optimization
• Cost function: match to “experimental data”
transport coefficients and plasma edge model constants
(the control variables)
state variables: plasma density, temperature,…
plasma edge model: set of PDEs and boundary conditions
• Efficient solution through adjoint sensitivity analysis
• Challenges: accounting for multiple diagnostics? Prior information
about uncertainties (Bayesian setting)? …
6
[M. Baelmans et al., PPCF 56 (2014) 114009.]
Exp. data
Simulation
Potential of sensitivity analysis for edge plasma model
validation
7
• Efficient solution of UQ problems through optimization
• Identification of dominant uncertainties, guide for parameter space reduction
• Efficient parametrization of input and output PDFs, efficient propagation of
uncertainty though the codes
• Construction of surrogate models
…but: several challenges to be addressed to enable application to realistic
problems!
Outline
8
• Motivation
• Sensitivities in the presence of MC noise
• Partially adjoint techniques for simulation chains
• Practical implementation in big codes
• Summary and outlook
Optimization for fluid-kinetic models
9
• Sources from kinetic neutrals, but ‘only flying left and right’ Can be solved with Monte Carlo or finite volume method
• Some essential features of SOL models are present
o Fluid-kinetic coupling, Monte Carlo noise, nonlinear source terms, …
Γ𝑛𝑖,0, 𝑛𝑖,0
(1 − 𝑅)Γ𝑛𝑖,𝑡
−𝐶𝑛𝑖
Upstream Target𝑅Γ𝑛𝑖,𝑡
ions
neutrals
plasma continuity and parallel momentum,
state variables
Target
Upstream
• Optimization of “divertor fluxes”:
Finite Difference (FD) sensitivities
10
• Reduced cost function/state solver as ‘black box’
• Sensitivity
o Cost scales with number of design variables
o Correlated random numbers to reduce variance
o Some decorrelation hard/impossible to avoid in practice
• Constrained optimization problem
• Reduced cost functional
• Chain rule for sensitivity computation
The adjoint approach to sensitivity calculation
11
Optimality conditions
12
• Lagrangian
• First order optimality conditions:
State equations
Adjoint equations
Design equations
Again a coupled FV-MC system!
How can we achieve low variance on the sensitivities?
Adjoint simulation
Contribution of particle to
source in cell
Adjoint particle contributes to
source in all of the cells it
crossed
more complex simulation
but exact correlation!
The discrete adjoint approach
13
Forward simulation
Contribution of particle to
source in cell
Accumulated weight
factor will be a function of
plasma properties in all
the cells crossed by the
particle
(Relative) standard deviation on sensitivity
14
[W. Dekeyser et al., Contrib. Plasma Phys. 58 (2018) 718.]
Performance of the optimization algorithm
15
Continuous adjoint Discrete adjoint(and correlated FD)
Convergence ‘on average’; reliable? Low noise, smooth convergence!
P
P
PP
Outline
16
• Motivation
• Sensitivities in the presence of MC noise
• Partially adjoint techniques for simulation chains
• Practical implementation in big codes
• Summary and outlook
Propagating sensitivities through simulation chains
17
Magnetic
equilibrium
Grid Simulation
(forward)
Adjoint
equilibrium
solver
Adjoint grid
generator
Adjoint
simulation
Analysis
sensitivity
information?
Propagating sensitivities through simulation chains
18
Magnetic
equilibrium
Grid Simulation
(forward)
Adjoint
simulation
Analysis
sensitivity
information?
FD
In-parts adjoint• Fast as sensitivities of slowest part
are adjoint
• Practical finite difference sensitivity
calculation for remainder
Sensitivities w.r.t. edge plasma model parameters
19
[M. Blommaert et al., NME 12 (2017) 1049.]
Outline
20
• Motivation
• Sensitivities in the presence of MC noise
• Partially adjoint techniques for simulation chains
• Practical implementation in big codes
• Summary and outlook
Implementation in full edge codes
21
• Challenges
o Dealing with “legacy code”
o Developer and user friendliness
o Maintainability
• Current research tracks for SOLPS-ITER
o Use of AD tools (“Automatic/Algorithmic Differentiation”): TAPENADE (INRIA)
• Link to discrete adjoint approach, very robust w.r.t. statistical noise
o Practical combination of adjoint and finite differences
(in-parts adjoint technique)
Proof-of-principle: forward AD in B2.5
22
• Case setup
o D only, fluid neutrals
o Input power PSOL = 31 MW, split equally between ions and
electrons
o Low recycling conditions, ce, ci = 6.0 m2 s-1
• Quantities of interest (@ targets):
o Max. electron temperature Te,max
o Max. heat load 𝑞𝑚𝑎𝑥′′
• Varied model parameters:
o Input power (Pe, Pi)
o Radial heat diffusion coefficients (ce, ci)
Verification of AD sensitivities
23
Relative error AD-central FD
Te,max 𝑞𝑚𝑎𝑥′′
Pe ~10-8 ~10-7
Pi ~10-7 ~10-7
ce ~10-9 ~10-6
ci ~10-9 ~10-6
Sensitivity of target profiles
24
• Pe strongly linked to Te, ions need collisions
• q’’ mainly driven by e- contribution
• c spreading of plasma power
Normalized sensitivities
e.g. 𝑆 = ቚ𝜕𝑇𝑒,𝑚𝑎𝑥
𝜕𝑃𝑒 𝑂𝑃∙
𝑃𝑒
𝑇𝑒,𝑚𝑎𝑥
[S. Carli et al., NME 18 (2019) 6.]
Outline
25
• Motivation
• Sensitivities in the presence of MC noise
• Partially adjoint techniques for simulation chains
• Practical implementation in big codes
• Summary and outlook
Summary and outlook
26
• Several challenges to compute accurate sensitivities of plasma edge code have
been addressed:
o Handling of statistical noise
o Complex simulation chains
o Dealing with big codes
• Sensitivities may be essential to enable UQ studies for plasma edge models
o Solving UQ problems through optimization
o Identifying dominant uncertainties over a parameter range (parameter space reduction)
o Efficient parametrization of input and output PDFs
o Construction of surrogate models
o …
Thank you for your attention!