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 → free energy for micro-

instabilities → saturate via nonlinear coupling and develop a quasi-stationary state far from thermodynamic equilibrium• turbulent fluxes → highly sensitive to changes in temperature/density

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

Challenges• large number of stochastic inputs (ranging from a few to hundreds)• significant computational demand (from a few minutes to a few hours/run)

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• perform the UQ simulations using multiple layers of parallelism

B. Methodology

Standard vs. adaptive sparse grid operations. Themultiindex sets are depicted in the top part. In thebottom plot, the associated two-dimensional Lejagrids are visualized.

Multiindices obtained after dimension-adaptivity us-ing the error work ratio criterion (left) vs. the errorwork ratio + directional variance strategy (right). Inthe right plot, the algorithm does not refine furtherin the directions in which the available variance isbelow a 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)

• construct LSG adaptively using a-posteriori heuristics

Interpolation-to-spectral-projection mapping• alternative approach: interpolation instead of spectral projection• for each subspace, find the equivalent spectral coefficients

B.2 Leja sequences4

• sparse grid operations constructed on Leja sequences

u0 = 0.5

un = argmaxu∈[0,1]

|n−1∏i=0

(u− ui)|

• here, we consider Leja sequences with growth rate–Nl = 2l − 1 – for pseudo-spectral projection (symmetrized points)–Nl = l – for interpolation

B.3 Dimension-adaptivity1• 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

• adaptively refine using the surplus (||∆(d)l (f )(θ)||L2 and the cost Ml

• ||∆(d)l (f )(θ)||2L2 =

∑k∈Kl f

2k, Kl =

⋃{k ∈ Nd : ki ≤ pli/2}

Error/work criterion• refine using ||∆(d)

l (f )(θ)||L2/Ml as local error indicator

Error/work + directional variance criterion• refine using ||∆(d)

l (f )(θ)||L2/Ml as local error indicator

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

(d)l )(f )(θ)

and find the “available” variances Vi in all directions i = 1, . . . d

•when Vi ≤ tol2i , do not add multiindices whose ith component ex-ceeds the maximum reached levelC. Results

C.1 Benchmark test case2 : 8 stochastic parameters• simple geometry • the dominant mode clearly changes with the wave number kyρref

stochastic parameter symbol left bound right boundplasma beta β 0.59e−03 0.73e−03

collision frequency νc 0.23e−02 0.32e−02magnetic shear s 0.710 0.870

safety factor q 1.330 1.470density gradient −(Lref/n)/(dn/dx) 1.665 2.775

ions temperature gradient −(Lref/T )/(dT/dx) 7.500 12.500ions temperature T 0.950 1.050

electrons temperature gradient −(Lref/T )/(dT/dx) 7.500 12.500

output of interestγ (growth rate)

• dimension-adaptivity (Lmax = 12) based on:– error/cost with tol = 5 · 10−3

– error/cost + directional derivative with tol = 5 · 10−3 and tol2i = 5 · 10−5, i = 1, 2, . . . , 8

Error plot (left), total number of grid points (right)

Total Sobol’ indices when using error/cost (left) vs. error/cost + directional derivative (right)

C.2 Real-world problem: 3 stochastic parameters• complex geometry • based on experimental data

stochastic parameter symbol left bound right bounddensity gradient −(Lref/n)/(dn/dx) 1.156 1.927

ions temperature gradient −(Lref/T )/(dT/dx) 2.096 3.494electrons temperature gradient −(Lref/T )/(dT/dx) 4.040 6.733

output of interestγ (growth rate)

• construct the sparse grid surrogate using two approaches:– pseudo-spectral projection– interpolation + interpolation-to-spectral-projection mapping• dimension-adaptivity (Lmax = 30) based on:

– error/cost criterion with tol = 5 · 10−3

– error/cost + directional derivative with tol = 5 · 10−3 and tol2i = 5 · 10−5, i = 1, 2, 3

• conclusion: interpolation + adaptivity based on directional variance clearly superior

Error plot (left), total number of grid points (right)

Total Sobol’ indices when using pseudo-spectral projection (left) vs. interpolation (right)

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