particle in fourier discretization of kinetic equations€¦ · particle in fourier discretization...

Vlasov Poisson with Particle in Fourier Fourier Filtering and Aliasing PIF/PIC Hybrid Outlook References

Particle in Fourier Discretizationof

Kinetic Equations

Jakob AmeresEric Sonnendrucker

Technische Universitat Munchen & Max–Planck–Institut fur Plasmaphysik

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Outline

1 Vlasov Poisson with Particle in FourierParticle DiscretizationControl Variate

2 Fourier Filtering and AliasingKEEN Waves

3 PIF/PIC HybridDrift Kinetic - ITG

4 Outlook

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Vlasov Poissonwith

Particle in Fourier

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The Vlasov Poisson (VP) System

∂f

∂t+ v · ∇x f − (E + v × B) · ∇v f = 0

−∆Φ = ρ− 1, E := −∇Φ

Phasespace(Plasma)-density f (x , v) : [0, L]d × Rn → [0,∞)

Charge density ρ(x , t) =∫f (x , v , t) dv (Marginal

distribution)

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Method of Characteristics

characteristics (V (t),X (t)) of the Vlasov equation

d

dtX (t) = V (t) (1)

d

dtV (t) = − (E (t,X (t)) + V (t)× B(t,X (t))) (2)

Initial value of density is transported for t ≥ 0.

f (t = 0,X (t = 0),V (t = 0)) = f (t,X (t),V (t)) ∀t ≥ 0

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Particle Discretization

Randomization

Probability density g > 0 with∫∫

g(t = 0, x , v) dxdv = 1.

Both distributions f and g follow the same Vlasov Equation.

∂f

∂t+ v · ∇x f + (E + v × B) · ∇v f = 0

∂g

∂t+ v · ∇xg + (E + v × B) · ∇vg = 0

Initial characteristic (X (t = 0),V (t = 0)) is randomlydistributed according to g(t = 0).

Weights as random variable

C (t) :=f (t,X (t),V (t))

g (t,X (t),V (t))=

f (0,X (0),V (0))

g (0,X (0),V (0))= const. ∀t

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Draw for k = 1, . . . ,Np markers

(xk (t = 0), vk (t = 0)) iid. ∼ (X (t = 0),V (t = 0))

“Draw” constant weights

ck :=f (t, xk (t), vk (t))

g(t, xk (t), vk (t))=

f (t = 0, xk (0), vk (0))

g(t = 0, xk (0), vk (0))

Particle discretization fp of f

f (t, x , v) ≈ fp(t, x , v) =1

Np

Np∑k=1

δ (x − xk (t)) δ (v − vk (t)) ck

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Moment estimation

Kinetic energy with the standard Monte Carlo estimator

T (t) =1

2

∫∫v2 f (t, x , v) dxdv

=1

2

∫∫v2 f (t, x , v)

g(t, x , v)g(t, x , v) dxdv

= E[

1

2V (t)2 f (t,X (t),V (t))

g(t,X (t),V (t))

]= E

[1

2V (t)2C

]

≈ 1

Np

Np∑k=1

vk (t)2ck

=1

Np

Np∑k=1

∫∫v2 δ (x − xk (t)) δ (v − vk (t)) ck dxdv

=1

2

∫∫v2 fp(t, x , v) dxdv

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Poisson equation and Fourier transform

−∆Φ = ρ− 1, E := −∇Φ

n − th Fourier mode of the charge density

Fρ(n) := ρn(t) :=

∫ L

0einx 2π

L ρ(x , t) dx =

∫R

∫ L

0einx 2π

L f (x , t) dxdv

En(t) = 2πinL ρn(t) n ∈ Z \ 0

Fourier series for the electric field (back transform)

E (t, x) =∑n 6=0

e−inx 2πL En(t)

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Fourier transform of the random particle density

f (t, x , v) ≈ fp(t, x , v) =1

Np

Np∑k=1

δ (x − xk (t)) δ (v − vk (t)) ck

ρn(t) = E[ein

2πL

X (t)C (t)]

≈ ˆρn(t) :=

∫ ∫ein

2πL

x fp(x , v , t) dxdv

=1

Np

Np∑k=1

ck

∫ ∫ein

2πL

xδ(x − xk (t))δ(v − vk (t)) dxdv

=1

Np

Np∑k=1

ckein 2π

Lxk (t)

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Integrated Variance and Mean Squared Error

Variance of the Fourier coefficient estimator V[

ˆρn(t)]

The electric field estimator is random

E (t, x) =

n=Nf∑n 6=0,n=−Nf

e−inx 2πL

2π

inLˆρn(t)

Integrated mean squared error (IMSE/MISE)

IMSE(E (t, x)

)=

∫E[(

E (t, x)− E (t, x))2]

dx

=

∫V[E (t, x)

]dx︸︷︷︸

integrated variance

+

∫ (E[E (t, x)

]− E (t, x)

)2dx︸︷︷︸

bias2

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Variance and Bias

Integrated variance controlled by particle number

IVAR[E]∝ 1

Np

Bias reflects the spatial field discretization error

Bias2[E]

=

∫ (E[E (t, x)

]− E (t, x)

)2dx

=

∫ n=Nf∑n 6=0,n=−Nf

e−inx 2πL E[ˆEn(t)

]− E (t, x)

2

dx

=

∫ n=Nf∑n 6=0,n=−Nf

e−inx 2πL En(t)− E (t, x)

2

dx =∑|n|>Nf

En(t)

PIF Bias: Fourier Space (neglected modes)PIC Bias: Orthogonal V⊥h of the finite element space Vh 12 / 38



Landau damping

Domain [0, L]d , L = 2π, ε = 0.01

Initial condition

f (t = 0, x , v) =

1 + ε

d∑j=1

cos(xj

L

) (2π)−d2 e−

|v|22

Sampling distribution (prior)

g(t = 0, x , v) =1

Ld(2π)−

d2 e−

|v|22

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PIF: Linear Landau damping(1)

0 5 10 15 20 25 30 35 40 45 5010−12

10−7

10−2

103

time

Field Energy, Np=1E+06

1x1v=2d2x2v=4d3x3v=6d4x4v=8d

Symplectic RK3, 4t = 0.005 14 / 38




0 5 10 15 20 25 30 35 40 45 5010−22

10−19

10−16

10−13

time

Absolute `2 Momentum Error, Np=1E+06


Symplectic RK3, 4t = 0.005 15 / 38




0 5 10 15 20 25 30 35 40 45 50

10−10

10−8

10−6

10−4

time

Relative `2 Energy Error, Np=1E+06


Symplectic RK3, 4t = 0.005 16 / 38


Control Variate

Sampling the Difference in 1D

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Control Variate

Sampling the Difference in 2D

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Control Variate

Control Variate for Particle Discretization

Control Variate

h(x , v) := 1 · 1(√2π)d

e−|v|2

2

Redefine particle density,

fp(t, x , v) :=

1

Np

Np∑k=1

δ (x − xk (t)) δ (v − vk (t))

(ck −

h (xk (t), vk (t))

g(0, xk (0), vk (0))

)+

h(x , v)

g(t, x , v)

which does not change any expectation under the measure∫∫· g(t, x , v) dxdv but reduces the variance.

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Control Variate

Bump on Tail (1/2)

Initial condition Maxwellian with fast particles

f (t = 0, x , v) := (1− ε cos(k1 · x)− ε cos(k2 · x))

· (2π)−d2

(1 + a)

(e−|v|2

2 +a

σde−

(v−v0)2

2σ2

)Excite the 13− th and 14− th mode

k0 = 0.0157605, σ =1

2, L =

2π

k01d , v0 = (4.5 0 . . . )t ∈ Rd

k1 = 13k01d , k2 = 14k01d , ε = 10−4, a =2

9

Simulation for d = 1, . . . , 3, 4t = 0.5, Np = 5 · 106 with onlytwo modes k1, k2.

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Control Variate

Bump on Tail (2/2)

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Fourier Filtering and Aliasing

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Particle in Cell and Particle in Fourier

ψn(x) = S0(x)

=

1 if |x | ≤ 1

2

0 otherwiseψn(x) = einx 2π

L

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Aliasing with B-Splines

B-Spline Fourier

F(Sd )(ω) = sinc(ω

2

)d+1

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Fourier Filtering

Integrated variance (IVAR) for ρ(x) = 1, g(x) = 1L .

IVAR [ρ] IVAR[E]

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KEEN Waves

KEEN Waves

Initial Maxwellian electrostatic is driven by an external electricfield compactly supported in time

Characteristic phase locked modes

Only the first 10 modes are regarded as relevant (PIC andPIF)

Estimate modes with PIC and compare to PIF

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KEEN Waves

KEEN Waves - Aliasing

relative estimation error on ρ2 relative estimation error on ρ10

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KEEN Waves

KEEN Waves - Aliasing summary

Time average the relative estimated aliasing error for everyB-Spline degree

Fourier filtering of the high modes reduces aliasing

Estimate on low (converged) modes is unreliable (variancebias tradeoff)

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PIF/PIC Hybrid

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Discretization - Polar Plane and Cylinder

Ansatz function forl = 1, . . . ,Nr , m = −Nθ, . . . , 0, . . . ,Nθ, n = −Nϕ, . . . , 0, . . . ,Nϕis defined as

ψl ,m,n(r , θ, ϕ) = exp

(−i(mθ +

ϕ

Lϕ

))Λl (r).

0.2 0.4 0.6 0.8 1

0.2

0.4

0.6

0.8

1

radius r

radial B-Splines basis in polar plane

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Guiding Center and Drift Kinetic

Guiding Center, polar plane (r , θ), Diocotron Instability∂tρ+ (∇Φ)y∂xρ− (∇Φ)x∂yρ = 0 on Ω× [0,∞)

−4Φ = γρ

Φ(x , y) = 0 on ∂Ω

Drift Kinetic (gyroradius = 0)1, cylinder (r , θ, ϕ = z), ITG∂f∂t + ~vGC · ∇⊥f + v‖

∂f∂φ +

dv‖dt ·

∂f∂v‖

= 0

−∇⊥ · (n0(r)∇⊥Φ) + n0(r)Te (r)

(Φ− Φ

)=∫f dv‖ − n0(r)

Φ(r , θ, t) := 1Lϕ

∫ Lϕ0 Φ(r , θ, φ, t)dϕ

1Crouseilles et al. 2014, Grandgirard et al. 2006 , Hatzky et al. 200231 / 38


Diocotron instability

r− = 4, r+ = 5, rmax = 10, ε = 10−2.

ρ(t = 0, r , θ) =

1 + ε cos(lθ) for r− ≤ r ≤ r+

0 else.

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Diocotron - Polar/Cartesian

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Drift Kinetic - ITG

Drift Kinetic - ITG

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Outlook

Follow the characteristics with randomly distributed markers.

Assume randomly distributed density at every point in time.

Fourier filter on FEM basis suffers of aliasing (bias).

Particle in Fourier (PIF) with energy and additionallymomentum conservation

Discretize other geometries with B-Splines /Fourier-Hankel-Transform.

PIF is suitable for problems with small amount of physicallyrelevant modes e.g. (Kinetic) Alfven waves.

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Backup Slides

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ITG Linear Analysis - Eigenvalue problem (1)

The eigenvalue is not exactly excited.

f (t = 0, r , θ, ϕ) =n0(r) exp(−

v2‖

2Ti (r) )

(2πTi (r))12

·

(1 + εexp

(−−(r − rmax

2 )2

1.45

)cos

(2πn

Lϕ+ mθ

)(3)

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ITG Linear Analysis - Eigenvalue problem (2)

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References

Crouseilles, N. et al. (2014). “Semi-Lagrangian simulations onpolar grids: from diocotron instability to ITG turbulence”. url:https://hal.archives-ouvertes.fr/hal-00977342.

Grandgirard, V. et al. (2006). “A drift-kinetic Semi-Lagrangian 4Dcode for ion turbulence simulation”. In: Journal ofComputational Physics 217.2, pp. 395–423. issn: 0021-9991.doi: 10.1016/j.jcp.2006.01.023. url:http://www.sciencedirect.com/science/article/pii/

S0021999106000155.Hatzky, R. et al. (2002). “Energy conservation in a nonlinear

gyrokinetic particle-in-cell code forion-temperature-gradient-driven modes in I -pinch geometry”.In: Physics of Plasmas (1994-present) 9.3, pp. 898–912. doi:10.1063/1.1449889. url: http://scitation.aip.org/content/aip/journal/pop/9/3/10.1063/1.1449889.

Mathai, A. and S. B. Provost (1992). Quadratic Forms in RandomVariables. Vol. 126. Statistics Series. MARCEL DEKKER, INC.isbn: 0-8247-8691-2.

Rencher, A. (1998). Multivariate statistical inference andapplications. Wiley series in probability and statistics: Texts andreferences section. Wiley. isbn: 9780471571513. url:http://books.google.de/books?id=C7fvAAAAMAAJ.

Silverman, B. W. (1986). Density estimation for statistics and dataanalysis. Chapman and Hall.

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https://hal.archives-ouvertes.fr/hal-00977342

http://dx.doi.org/10.1016/j.jcp.2006.01.023

http://www.sciencedirect.com/science/article/pii/S0021999106000155

http://www.sciencedirect.com/science/article/pii/S0021999106000155

http://dx.doi.org/10.1063/1.1449889

http://scitation.aip.org/content/aip/journal/pop/9/3/10.1063/1.1449889

http://scitation.aip.org/content/aip/journal/pop/9/3/10.1063/1.1449889

http://books.google.de/books?id=C7fvAAAAMAAJ


Equilibrium example

f (x , v) = e−v2

21√2π

, b = 0, ,

(Σb)i ,j = E[W (X ,V )2ψi (X )ψj (X )]−E[W (X ,V )ψi (X )]E[W (X ,V )ψj (X )] = E[W (X ,V )ψi (X )ψj (X )] = L2(·M−Iψ)(4)

Drop Iψ because of ρ− 1 in the Poisson equation.

Σb = L2 ·M → Σa = L2K−1MK−†

⇒IVAR[Φh(x)] = L2Nh∑

i ,j=1

(K−1MK−†

)i ,jMi ,j

⇒IVAR[∇Φh(x)] = L2Nh∑

i ,j=1

(K−1MK−†

)i ,jKi ,j

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Quadratic form2

Corollary (Expected value of a quadratic form)

Let b ∼ N (b,Σb) and K−1 ∈ RNh×Nh symmetric.

E[b†K−1b] = E[b†]K−1E[b] + tr(K−1Cov(b))

= E[b†]K−1E[b] + tr(K−1Σb)

Corollary (Variance of a quadratic form)

Let b ∼ N (b,Σb) and K−1 ∈ RNh×Nh symmetric.

V[b†K−1b] = 2tr(K−1ΣbK

−1Σb

)+ 4E[b]K−1ΣbK

−1E[b]

2A. Mathai and S. B. Provost. Quadratic Forms in Random Variables. Vol.126. Statistics Series. MARCEL DEKKER, INC., 1992., pp. 51 & pp.75Mathai and Provost 1992

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Statistical Bias of Electrostatic Energy HE

Electrostatic Energy HE and its Galerkin discretization HE .

HE :=1

2

∫|E (x)|2 dx =

1

2

∫∇Φ · ∇Φ dx

HE := E[ψ(X )]†K−1E[ψ(X )] = b†K−1b = b†a

The standard estimator for the discretized Field Energy suffersfrom statistical bias.

HE := b†K−1b

E[Efield ] = E[b]†K−1E[b] + tr(K−1Σb)

= b†K−1b + tr(K−1Σb)

≈ b†K−1b + tr(K−1 1

NpΣb)︸︷︷︸

bias

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Moment matching - (initial) Maxwellian

Suppose v1, . . . , vNp ∼ v ∼ N (µ1, σ2) are iid.

µ1 := E[v ], µ1 =1

Np

Np∑k=1

vk , µ2 := σ2−µ21 = E[v2], µ2 =

1

Np

Np∑k=1

v2k

Search for a preferably simple transformationT : R→ R, v∗ = T (v) such that

µ∗1 :=1

Np

Np∑k=1

T (vk ) = µ1 and µ∗2 :=1

Np

Np∑k=1

T (vk )2 = µ2.

Linear Ansatz for T yields for all k = 1, . . . ,Np

v∗k = T (vk ) = (v∗k − µ1)c + m1, c :=

√(m2 −m2

1)

(µ2 − µ21).

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Moment matching - δf

wk := ck −h(xk , vk )

g(xk , vk )

δµn =1

Np

Np∑k=1

wk (vk )n, vn =1

Np

Np∑k=1

(vk )n, δλn =1

Np

Np∑k=1

(wk )n

Linear Ansatz for matching δµ1, δµ2 by manipulation of theweights only.

w∗k = T (wk ) :=δµ1v2 − δµ2v1

δµ1v2 − δµ2v2

wk −δµ1δµ2 − δµ2δµ1

δµ1v2 − δµ2v1

Problem: The velocity moments can be set/kept, but massconservation δλ1 = δλ1 = 0 is lost.

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Fast FEM based covariance estimates

Idea: Estimated charge density ρh is probability density. With X ρ.

(Σb)i ,j = Covρ[ψi (X ), ψj (X )]

= Eρ[(ψi (X )− E[ψi (X )]) (ψj (X )− E[ψj (X )])

]= Eρ

[ψi (X )ψj (X )

]− Eρ [ψi (X )]Eρ [ψj (X )] (5)

Eρ[ψi (X )ψj (X )

]=

∫ψi (x)ψj (x)ρ(x)dx

≈∫ψi (x)ψj (x)ρh(x) dx =

∫ψi (x)ψj (x)(M−1b)ψ(x) dx (6)

Coefficient estimator for ρ bj = 1Np

∑Np

k=1 ψj (xk ), mass matrix M.

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Principal Components

Discriminate signal from noise in b by using the Covariancematrix Σb.

Large eigenvalue of Σb corresponds to large Variance →signal.

Propagate covariance of solution a and select significantcomponents.

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Statistics: Fourier filtered PIC v.s. PIF

A low-pass filter reduces variance in PIC to levels as in PIF. ButPIC suffers from high frequency (sub grid) noise aliasing.

Why use PIF? −→ Energy and momentum conservation.

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Variational Field solve - Galerkin Discretization (2)

n,m = 1, . . . ,Nh

Right hand sidebm :=

∫ρ(x)ψm(x) dx =

∫ ∫f (x , v)ψm(x) dx dv

Stiffness matrix Kn,m := 〈5ψn,5ψm〉L2([0,L])

Mass matrix Mn,m := 〈ψn, ψm〉L2([0,L])

Vectorize

x 7→ ψ(x) :=

ψ1(x)...

ψNh(x)

, b :=

∫ L

0ρ(x)ψ(x) dx

Discrete approximation Φh of the solution Φ.

Φ(x) ≈ Φh(x) =

(K−1b︸︷︷︸

:=a

)t

ψ(x) = atψ(x)

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Variational Field solve - Univariate normal bm

In the framework of probability theory bm random deviate.

bm =

∫ L

0ρ(x)ψm(x) dx =

∫ ∞−∞

∫ L

0f (x , v)ψm(x) dxdv

=

∫ ∞−∞

∫ L

0

f (x)

g(x)︸︷︷︸=W (x ,v)

ψm(x) g(x , v) dxdv

=Eg [W (X ,V )ψm(X )︸︷︷︸:=bm

]

Estimate bm = E[bm] for large Np with the Monte-Carlo estimatorand the Law of large numbers

bm ≈ bm =1

Np

Np∑k=1

wkψ(xk ), bm ∼ N(bm,

V[bm]

Np

).

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Variational Field solve - Multivariate normal b

Covariance Cov [b] = Σb of the random vector b.

(Σb)n,m := Cov [bn, bm] = E [(bn − E[bn])(bm − E[bm])]

The MC estimator b is approximately Multivariate normal

b ≈ b ∼ N (b,Σb), Σb ≈Σb

Np

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Variational Field solve - Estimators

For given x ∈ [0, L], estimate for the Potential

Φ(x) ≈ Φh(x) ≈ Φh(x) =

K−1b︸︷︷︸:=a

t

ψ(x) = atψ(x)

Estimate for the Electric field E = −∇Φ,

Eh(x) ≈ Eh(x) = −at ∇ψ(x)

Kernel density estimate (KDE) of ρ =∫fdv by e.g. L2 projection

with mass matrix M

ρ(x) ≈ ρh(x) ≈ ρh(x) =(M−1

)tbψ(x).

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Overview - Notations

Random deviate b

Expected value b

Monte Carlo Estimator b

Discretization of function Φ as Φh and its Monte Carloestimator Φh

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Variance and Covariance estimation (standard)

A first but computationally expensive estimate of Variance andCovariances of b is.

V[bm] ≈ 1

Np − 1

Np∑k=1

(wkψm(Xk )− bm

)2

Cov [bn, bm] ≈ 1

Np − 1

Np∑k=1

(wkψn(Xk )− bn

)(wkψm(Xk )− bm

)

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(Linear) Covariance propagation

Theorem (Linear covariance propagationa)

aA. Rencher, Multivariate statistical inference and applications, Wiley seriesin probability and statistics, Wiley, 1998, p.16 Rencher 1998

Let A be a linear operator represented by a matrix A ∈ RNh×Nh

and b a random variable with Covariance matrix Σb. Then c = Abis a random variable with covariance matrix

Σc = AΣbA† ⇔ Σc = A

(AΣ†b

)†For coefficients a = K−1b of the Poisson equations solution

Σa = K−1ΣbK−† = K−1

(K−1Σ†b

)†.

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Variance and Mean squared Error 3

For fixed x ∈ [0, L].

Var [Φh(x)] = ψ(x)†Σaψ(x)

⇒ Var [Eh(x)] = Var [−∇Φh(x)] = ∇ψ(x)†Σa∇ψ(x)

Φh as an estimator for Φh is unbiased

E[Φh(x)] = E[(K−1b)tψ(x)] = (K−1E[b])tψ(x)

= (K−1b)tψ(x) = Φh(x).

But Φh as an estimator for Φ is biased.

MSE (Φh(x)) =E[(Φh(x)− Φ(x))2]

=V[Φh(x)] +(E[Φh(x)]− Φ(x)

)2

=V[Φh(x)] + (Φh(x)− Φ(x))2︸︷︷︸bias2

3Silverman 198638 / 38


Integrated Variance and Integrated Mean squared Error

IVAR[Φh(x)] :=

∫Var [Φh(x)] dx

IMSE[Φh(x)

]:=

∫MSE (Φh(x)) dx

= E[∫ (

Φh(x)− Φh(x))2

dx

]= IVAR[Φh(x)] +

∫(Φh(x)− Φ(x))2︸︷︷︸

(L2 error )2

dx (7)

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Integrated Variance of Potential Φ and Field E = −∇Φ

Mi ,j :=

∫ψiψj dx , Ki ,j =

∫∇ψi · ∇ψj dx

IVAR[Φh(x)] =

∫Var [Φh(x)] dx =

∫ψ(x)†Σaψ(x) dx

=

Nh∑i ,j=1

(Σa)i ,j

∫ψi (x)ψj (x) dx =

Nh∑i ,j=1

(Σa)i ,j Mi ,j

IVAR[∇Φh(x)] =

∫Var [∇Φh(x)] dx =

∫∇ψ(x)†Σa∇ψ(x) dx

=

Nh∑i ,j=1

(Σa)i ,j Ki ,j

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Diocotron Instability - Various Tests

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Fourier filtering as Variance Reduction technique

Motivation: Select only physically relevant modes.

Here modes are eigenfunctions of 4Select eigenvectors of K−1 with largest eigenvalue.

Disregard higher modes.

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particle in fourier discretization of kinetic equations€¦ · particle in fourier discretization...

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