A Dimension-adaptive Sparse Pseudo-SpectralProjection Method in Linear GyrokineticsIonut,-Gabriel Farcas, §, Tobias Goerler∗, Hans-Joachim Bungartz§, Tobias Neckel§§ Technical University of Munich, Chair of Scientific Computing, Boltzmannstr. 3, 85748 Garching, Germany, farcasi@in.tum.de∗ Max Planck Institute for Plasma Physics, Boltzmannstr. 2, 85748 Garching, Germany

A. MotivationA.1 Plasma micro-turbulence• goal: characterization of the turbulent transport in magnetically con-

fined fusion devices by gyrokinetic fully nonlinear simulations•modelling: 5D integro-differential Vlasov-Maxwell system of eqs.• here: restriction to linear physics

– less complicated testbed– insights into sensitivies of underlying micro-instabilities

Linear/local simulations

∂g

∂t= L[g]

• here: two particle species:deuterium ions and electrons• typical output of interest:

dominant eigenmode withmagnitude γ and frequency ω

3

Global gyrokinetic simulation of turbulence5

A.2 UQ in plasma micro-turbulence• strong temperature and density gradients in fusion plasmas → free

energy for micro-instabilities→ saturate via nonlinear coupling and de-velop a quasi-stationary state far from thermodynamic equilibrium• turbulent fluxes are highly sensitive to changes in temperature/density

gradients→ uncertainty quantification and sensitivity analysis neededfor both validation and prediction

Challenges• large number of stochastic inputs• significant computational demand

Our approach

• employ HPC-ready micro-turbulence simulation codes (here, GENE5)• construct non-intrusive, dimension-adaptive sparse grid surrogates• exploit sensitivity information to further tune the adaption

B. Methodology

Standard vs. adaptive sparse pseudo-spectral pro-jection. The multiindex sets are depicted in thetop part. In the bottom plot, the associated two-dimensional Leja grids are visualized.

Dimension adaptivity based on error work ratio (left)vs. error work ratio + directional variance (right). Inthe right plot, the algorithm does not refine further inthe directions where the available variance is belowa given tolerance tol2.

B.1 Sparse pseudo-spectral projection1

• sparse approximation based on pseudo-spectral projection operators• constructed on internal aliasing error-free spaces

• starting point: 1D projection operators P (1)1 , P

(1)2 , . . .

P(1)l : f (θ)→ P

(1)l f (θ) =

pl/2∑p=0

fpΨp(θ)

• {fp}pl/2p=0 evaluated via quadrature

fp ≈ Q(1)l (fΨp) =

Nl−1∑n=0

f (θln)Ψp(θln)ωln, pl/2 = (Nl − 1)/2

• internal aliasing error-free construction:⟨Ψi,Ψj

⟩= Q

(1)l (ΨiΨj), ∀Ψi,Ψj

• in d-dimensions: l = (l1, . . . , ld) ∈ Nd,∆(1)li

= P(1)li− P (1)

li−1, ∆

(l1)1 = P

(1)1

P(d)l (f )(θ) =

∑l∈LSG

(∆(1)l1⊗ . . .⊗∆

(1)ld

)(f )(θ) =∑l∈LSG

(∆(d)l )(f )(θ)

• LSG admissible: ∀l ∈ LSG : l− ej ∈ LSG, (ej)i=1,...,d = δij

• for a given level L, the standard LSG = {l ∈ Nd : |l|1 ≤ L + d− 1}• alternative: construct LSG adaptively

B.2 Leja sequences4

• sparse pseudo-spectral projection constructed on Leja points

u0 = 0.5

un = argmaxu∈[0,1]

|n−1∏i=0

(u− ui)|

• Leja sequences are nested• The number of points at level l, Nl, depends linearly on l: Nl = 2l−1

B.3 Dimension-adaptivity• construct LSG based on a tolerance tol and a maximum level Lmax• begin with LSG = {1d = (1, 1, . . . , 1)}• LSG = O ∪A, O - old index set, A - active set• A - set of multiindices that drive the adaption process

• add multiindices using g(||∆(d)l (f )(θ)||L2,Ml), Ml = cost

• ||∆(d)l (f )(θ)||2L2 =

∑k∈Kl f

2k, Kl =

⋃{k ∈ Nd : ki ≤ pli/2}

Error/work criterion

• g(||∆(d)l (f )(θ)||L2,Ml) = ||∆(d)

l (f )(θ)||L2/Ml

Error/work + directional variance criterion

• g(||∆(d)l (f )(θ)||L2,Ml) = ||∆(d)

l (f )(θ)||L2/Ml

• additionally, perform a Sobol’ decomposition of∑l∈A(∆

(d)l )(f )(θ)

and find the “available” variance Vi in all directions 1, . . . d

• stop adding multiindices in directions where Vi ≤ tol2i

C. Results: benchmark test case2• simplified setup: simple geometry • the dominant mode clearly changes with kyρref

C.1 Scenario I: three stochastic parametersstochastic parameter left bound right bound

density gradient 1.665 2.775ions temperature gradient 5.220 8.700

electrons temperature gradient 5.220 8.700

output of interest: the dominant modeγ (real part)

ω (imaginary part)

• dimension-adaptivity (Lmax = 100) based on:– error/cost criterion using tol = 10−5

– error/cost + directional derivative with tol = 10−5 and tol2i = 10−9, i = 1, 2

• error plot (left) vs. multiindex sets obtain after refinement for ω and kyρref = 0.4 (right)

• error/work + directional variance criterion→ significant computational savings for kyρref ≤ 0.5

• the quantification of uncertainty is challenging for kyρref ≥ 0.6

• total Sobol’ indices when refining for γ (left) vs. ω (right)

C.2 Scenario II: eight stochastic parametersstochastic parameter left bound right bound

β 0.59e− 03 0.73e− 03collision frequency νc 0.23e− 02 0.32e− 02

magnetic shear s 0.71 0.87safety factor q 1.33 1.47

density gradient 1.665 2.775ions temperature gradient 5.220 8.700

ions temperature Tions 0.95 1.05electrons temperature gradient 5.220 8.700

output of interest: the dominant modeγ (real part)

ω (imaginary part)

• dimension-adaptivity (Lmax = 12) based on:– error/cost + directional derivative with tol = 10−5 and tol2i = 10−9, i = 1, 2, . . . , 8

• error plot 8D setup (left), error plot 3D vs. 8D setup (right)

• total Sobol’ indices when refining for γ (left) vs. ω (right)

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[5] THE GENE CODE, 2017.