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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, [email protected]∗ 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)
References[1] P. R. CONRAD AND Y. M. MARZOUK, Adaptive smolyak pseudospectral approximations, SIAM J. Sci. Comput., 35 (2013), pp. A2643 – A2670.
[2] A. M. DIMITS, G. BATEMAN, M. A. BEER, B. I. COHEN, W. DORLAND, G. W. HAMMETT, C. KIM, J. E. KINSEY, M. KOTSCHENREUTHER, A. H. KRITZ,L. L. LAO, J. MANDREKAS, W. M. NEVINS, S. E. PARKER, A. J. REDD, D. E. SHUMAKER, R. SYDORA, AND J. WEILAND, Comparisons and physics
basis of tokamak transport models and turbulence simulations, Physics of Plasmas, 7 (2000), pp. 969–983.
[3] T. GOERLER, Multiscale effects in plasma microturbulence, dissertation, Universität Ulm, Mar. 2009.
[4] A. NARAYAN AND J. D. JAKEMAN, Adaptive leja sparse grid constructions for stochastic collocation and high-dimensional approximation, SIAM J. Sci.Comput., 36 (2014), pp. A2952–A2983.
[5] THE GENE CODE, 2017.