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!