uncertainty quantification for kinetic models of...
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Introduction Examples of mean-field models Uncertainty quantification Conclusions
Uncertainty quantification for kinetic modelsof collective behavior
Mattia Zanella
Department of Mathematics and Computer ScienceUniversity of Ferrara, Italy
http://www.mattiazanella.eu
Joint research with:L. Pareschi (University of Ferrara, Italy) J.A. Carrillo (Imperial College, UK)
Uncertainty Quantification for Applied Problems
BCAM, Bilbao 4-7 July 2016
Mattia Zanella (University of Ferrara) UQ for kinetic models July 6, 2016 1 / 28
Introduction Examples of mean-field models Uncertainty quantification Conclusions
Outline
1 IntroductionCollective behavior and self-organizationThe role of uncertainty
2 Examples of mean-field modelsOpinion dynamicsMarket economySwarming models
3 Uncertainty quantificationCollocation methodsNonlinear Chang-Cooper type schemesStochastic Galerkin methodsResidual distribution schemes
4 Conclusions
Mattia Zanella (University of Ferrara) UQ for kinetic models July 6, 2016 2 / 28
Introduction Examples of mean-field models Uncertainty quantification Conclusions
Collective behavior and self-organization
Collective behavior and self-organization
The description of emerging collective phenomena and self-organization insystems composed of large numbers of individuals has gained increasinginterest from various research communities in biology, robotics and controltheory, as well as sociology and economics 1.
To this set of problems belongs the description of the collective behaviors ofcomplex systems composed by a large enough number of individuals.Classical examples of such systems are groups of animals/humans with atendency to flock or herd, but also interacting agents in a financial market,potential voters during political elections, possible buyers of a given good orasset and connected members of a social network.
In this context, in order to study the formation of patterns and reducing thecomputational complexity of microscopic models ruling the dynamics of theindividual agents, it is of utmost importance to derive correspondingmesoscopic dynamics 2.
1F. Cucker, S. Smale,’07; M. R. D’Orsogna, A. L. Bertozzi et al.’06; E. Cristiani, B. Piccoli,A. Tosin,’10; M. Agueh, R. Illner, A. Richardson,’11; I.D.Couzin et al,’02
2J. A. Carrillo, M. Fornasier, G. Toscani, F. Vecil,’10; S.-Y. Ha, E. Tadmor,’08; P. Degond, S.Motsch,’07, L.Pareschi G. Albi ’12, L.Pareschi, G. Toscani ’13
Mattia Zanella (University of Ferrara) UQ for kinetic models July 6, 2016 3 / 28
Introduction Examples of mean-field models Uncertainty quantification Conclusions
Collective behavior and self-organization
Swarms, flocks, crowds, herds...
Mattia Zanella (University of Ferrara) UQ for kinetic models July 6, 2016 4 / 28
Introduction Examples of mean-field models Uncertainty quantification Conclusions
Collective behavior and self-organization
...collective behavior in socio-economy
Mattia Zanella (University of Ferrara) UQ for kinetic models July 6, 2016 5 / 28
Introduction Examples of mean-field models Uncertainty quantification Conclusions
The role of uncertainty
The role of uncertainty
How can we model the social dynamics, since social forces cannot beconsidered as universal and physical ones? How can we make use of the largeamount of data available (from the network for example)?
An essential step in the development of such models is represented by theintroduction of stochastic parameters reflecting the uncertainty in the termsdefining the interaction rules.
This is particularly relevant in many problems in the natural andsocio-economic sciences where the interaction rules are based on observationsand empirical evidence. In such cases we can have at most statisticalinformation on the modeling parameters.
In order to fully understand simulation results and to produce realisticpredictions, it is therefore essential to incorporate uncertainty from thebeginning of the modeling 3.
3D. Xiu ’10; M.P. Petterson, G. Iaccarino, J. Nordstrom ’15; S. Jin, D. Xiu and X. Zhu ’16;Shi Jin, Jingwei Hu ’16; L. Pareschi, M.Z. ’16
Mattia Zanella (University of Ferrara) UQ for kinetic models July 6, 2016 6 / 28
Introduction Examples of mean-field models Uncertainty quantification Conclusions
Opinion dynamics
Opinion dynamics
We consider the evolution of N agents where each agent has an opinionwi = wi(t) ∈ I, I = [−1, 1], i = 1, . . . , N and this opinion can change over timeaccording to 4
wi(θ, t) =1
N
N∑j=1
P (wi, wj , θ)(wj(θ, t)− wi(θ, t)),
wi(θ, 0) = w0i,
The function P (w, v, θ), such that the interaction potential −1 ≤ P (w, v, θ) ≤ 1,represents a measure of the agents’ propensity to change their opinion by theprocesses of agreement/disagreement and depends on a random inputθ ∈ (Ω,F , P ), θ ∼ p(θ).Remark. An additional opinion dependent noise term characterized by a functionD(wi), 0 ≤ D(wi) ≤ 1 can be added to the dynamics.
4F. Cucker, S. Smale ’07, G. Toscani ’06,Mattia Zanella (University of Ferrara) UQ for kinetic models July 6, 2016 7 / 28
Introduction Examples of mean-field models Uncertainty quantification Conclusions
Opinion dynamics
Mean-field limit
In the limiting case of infinitely many agents, models with mean-field coupling aredescribed by an evolution equation for the probability density f(θ, w, t) 5
∂tf(w, θ, t) + ∂w (K[f ](w, θ, t)f(w, θ, t)) =σ2
2∂2w(D2(w)f(w, θ, t)),
where
K[f ](w, θ, t) =
∫IP (θ, w,w∗)(w − w∗)f(θ, w∗, t) dw∗.
In some cases explicit steady states are known. For example if P = P (θ) andD = (1− w2) then u =
∫I fw dw is conserved in time and
f∞(w, θ) =C0,θ
(1− w2)2
(1 + w
1− w
)P (θ)m/(2σ2)
exp
−P (θ)(1− uw)
σ2 (1− w2)
where C0,θ is a normalization constant such that
∫I f∞ dw = 1.
5G. Toscani ’06; L. Boudin, F. Salvarani ’09; B. During, P.A. Markowich, J.F. Pietschmann,M.T. Wolfram ’09; G. Albi, L. Pareschi, M.Z. ’16
Mattia Zanella (University of Ferrara) UQ for kinetic models July 6, 2016 8 / 28
Introduction Examples of mean-field models Uncertainty quantification Conclusions
Market economy
Market economy
We consider an alignment dynamics analogous to the previous one where eachagent has a wealth wi = wi(t) ∈ R+, i = 1, . . . , N which can change over timeaccording to
dwi(θ, t) =1
N
N∑j=1
γ(wi, wj)(wj(θ, t)− wi(θ, t))dt+ σ(θ)η(wi)dWi(t),
wi(θ, 0) = w0i,
where Wi(t) are N independent components of a standard Wiener processes withvalues in R and σ(·) > 0 is the noise strength which depends on a random inputθ ∈ (Ω,F , P ), θ ∼ p(θ). The functions γ(·, ·), η(·) ∈ [0, 1] characterizes thesaving and the risk propensities respectively.
Mattia Zanella (University of Ferrara) UQ for kinetic models July 6, 2016 9 / 28
Introduction Examples of mean-field models Uncertainty quantification Conclusions
Market economy
Mean field limit
A mean-field model can be derived for systems composed by several agents andreads 6
∂tf(w, θ, t) + ∂w (G[f ](w, θ, t)f(w, θ, t)) =σ2(θ)
2∂2w(η2(w)f(w, θ, t)),
where
G[f ](w, θ, t) =
∫R+
γ(w,w∗)(w − w∗)f(θ, w∗, t) dw∗.
Steady states now present the formation of power-laws and for γ = 1 and η = wreads
f∞(w, θ) =(µ(θ)− 1)µ(θ)
Γ(µ(θ))w1+µ(θ)exp
(−µ(θ)− 1
w
)where µ(θ) = 1 + 2/σ2(θ) > 1 is the so-called Pareto exponent and we assumed∫R f∞(w, θ)w dw = 1.
6J.P. Bouchard, M. Mezard ’00; S. Cordier, L.Pareschi, G. Toscani ’05Mattia Zanella (University of Ferrara) UQ for kinetic models July 6, 2016 10 / 28
Introduction Examples of mean-field models Uncertainty quantification Conclusions
Swarming models
Swarming models
We consider a large population of agents characterized by position xi ∈ R3 andvelocity vi ∈ R3 where each individual tries to mimic the velocity of the othersand have a preferred speed α > 0 7
dxi(θ, t)
dt= vi(θ, t),
dvi(θ, t) = αvi(θ, t)(|vi(θ, t)|2 − 1) +1
N
N∑j=1
a(xi, xj)(vj(θ, t)− vi(θ, t))dt
+√
2D(θ)dWi(t),
xi(θ, 0) = x0i, vi(θ, 0) = v0i,
where Wi(t) are N independent copies of standard Wiener processes with valuesin R and D(·) > 0 is the noise strength which depends on a random inputθ ∈ (Ω,F , P ), θ ∼ p(θ). The function a(·, ·) ∈ [0, 1] characterizes the alignmentdynamics.
7F. Cucker, S. Smale ’07; M. R. D’Orsogna, A. L. Bertozzi et al.’06, T.Vicsek et al. ’95.Mattia Zanella (University of Ferrara) UQ for kinetic models July 6, 2016 11 / 28
Introduction Examples of mean-field models Uncertainty quantification Conclusions
Swarming models
Mean-field limit
In the limit N →∞ the mean-field description for f = f(x, v, θ, t) gives
∂tf(x, v, θ, t) + v · ∇xf(x, v, θ, t)+∇v ·(αv(|v|2 − 1)f(x, v, θ, t)
+A[f ](θ, t)f(x, v, θ, t)) = D(θ)4vf(x, v, θ, t),
where
A[f ](θ, t) =
∫R3×R3
a(x, x∗)(v − v∗)f(x∗, v∗, θ, t) dv∗ dx∗.
In the simplified case where f = f(v, θ, t) independent of x, exact stationarysolutions can be computed. Taking a ≡ 1 we get
f∞(v, θ) = C exp− 1
D(θ)
[α|v|4
4+ (1− α)
|v|2
2− uf,∞(θ)v
],
where uf,∞(θ) =∫RN vf∞(v, θ)dv. At the deterministic level it has been shown 8
that a phase change phenomenon take place as diffusion decreases.8A. B. T. Barbaro, J. A. Canizo, J. A. Carrillo, P. Degond ’15, J. A. Carrillo, L. Pareschi,
M.Z. ’16Mattia Zanella (University of Ferrara) UQ for kinetic models July 6, 2016 12 / 28
Introduction Examples of mean-field models Uncertainty quantification Conclusions
Swarming models
A general setting
The examples of mean-field models just described follow in the general nonlinearFokker-Planck setting
∂tf(θ, w, t) = ∇w ·[(ξ(w) +B[f ](θ, w))f(θ, w, t) +∇w (C(θ, w)f(θ, w, t))
],
where w ∈ Λ ⊂ Rd, d ≥ 1 and C(θ, ·) ≥ 0,
B[f ](θ, w) =
∫Λ
b(θ, w,w∗)(w∗ − w)f(θ, w∗, t) dw∗,
depend on a random input θ ∈ (Ω,F , P ), θ ∼ p(θ).To simplify notations, in the sequel we will consider the one-dimensional cased = 1 with ξ = 0.
Mattia Zanella (University of Ferrara) UQ for kinetic models July 6, 2016 13 / 28
Introduction Examples of mean-field models Uncertainty quantification Conclusions
Collocation methods
Collocation methods
Collocation methods are those that require the residue of the governing equationsto be zero at discrete nodes (collocation points) in the computational domain.Let θj , j = 1, . . . ,M be a set of nodes in the random space. In a collocationmethod we enforce the Fokker-Planck equation in the node θj by solving
∂tf(θj , w, t) = ∂w
[(B[f ](θj , w))f(θj , w, t) + ∂w (C(θj , w)f(θj , w, t))
].
For each j, we have a deterministic problem since the value of the randomparameter θ is fixed. Therefore, solving the system poses no difficultyprovided one has a well-established deterministic algorithm.
The result of solving the above system is an ensemble of deterministicsolutions f (j)(w, t), j = 1, . . . ,M which can be post-processed to recover thevalues, for example, of the mean and the variance.
One possible choice is to select the nodes according to Gaussian quadraturerules. This is straightforward in the univariate case, whereas becomes moredifficult in the multivariate case.
Mattia Zanella (University of Ferrara) UQ for kinetic models July 6, 2016 14 / 28
Introduction Examples of mean-field models Uncertainty quantification Conclusions
Nonlinear Chang-Cooper type schemes
Nonlinear Chang-Cooper type schemes
In the deterministic setting a popular approach is based on Chang-Cooper typemethods9. Here we show how to extend the method to more general nonlinearFokker-Planck equations 10.We rewrite the collocation system as
∂tf(θj , w, t) = ∂wF [f ](θj , w)
where
F [f ](θj , w) = (B[f ](θj , w) + C ′(θj , w)) f(θj , w, t) + C(θj , w)∂wf(θj , w, t),
where we used the notation C ′(θj , w) = ∂wC(θj , w).The above equation is complemented with the initial data f(θj , w, 0) = f0(θj , w)and suitable boundary condition on w (typically zero flux). Note that thediscretization we will consider acts only on the variable w.
9J.S.Chang, G.Cooper ’70, E.W.Larsen, D.Levermore, G.C.Pomraning, J.G. Sanderson ’8510L.Pareschi, M.Z. ’16Mattia Zanella (University of Ferrara) UQ for kinetic models July 6, 2016 15 / 28
Introduction Examples of mean-field models Uncertainty quantification Conclusions
Nonlinear Chang-Cooper type schemes
The numerical scheme
Let us introduce a uniform grid wi, i = 0, . . . , N of mesh space ∆w. We denoteby wi±1/2 = wi ±∆w/2 and define
fi(θj , t) =1
∆w
∫ wi−1/2
wi+1/2
f(θj , w, t) dw.
Integrating the Fokker-Planck equation yields
∂
∂tfi(θj , t) =
Fi+1/2[f ](θj , t)−Fi−1/2[f ](θj , t)
∆w,
where Fi[f ](θj , t) is the flux function characterizing the numerical discretization.We assume a flux function as a combination of upwind and centered discretizationas in the classical Chang-Cooper flux
Fi+1/2[f ] =
((1− δi+1/2)(B[fi+1/2] + C ′i+1/2) +
1
∆wCi+1/2
)fi+1
+
(δi+1/2(B[fi+1/2] + C ′i+1/2)− 1
∆wCi+1/2
)fi,
where Ci+1/2 = C(θj , wi+1/2) and C ′i+1/2 = ∂wC(θj , wi+1/2).Mattia Zanella (University of Ferrara) UQ for kinetic models July 6, 2016 16 / 28
Introduction Examples of mean-field models Uncertainty quantification Conclusions
Nonlinear Chang-Cooper type schemes
Numerical flux at the steady state
To preserve the steady states we assume that the numerical flux vanishes when fis at the steady state. Imposing the numerical flux equal to zero we get
fi+1
fi=
−δi+1/2(B[fi+1/2] + C ′i+1/2) + 1∆wCi+1/2
(1− δi+1/2)(B[fi+1/2] + C ′i+1/2) + 1∆wCi+1/2
.
On the other hand the same computation directly on the real flux gives thedifferential equation
C(θj , w)∂wf(θj , w, t) = − (B[f ] + C ′(θj , w)) f(θj , w, t),
which in general cannot be solved, except is some special cases. Therefore, weintegrate the previous equation in the cell [wi, wi+1] to get∫ wi+1
wi
(1
f∂wf
)(θj , w, t) dw = −
∫ wi+1
wi
1
C(θj , w)(B[f ] + C ′(θj , w)) dw.
Mattia Zanella (University of Ferrara) UQ for kinetic models July 6, 2016 17 / 28
Introduction Examples of mean-field models Uncertainty quantification Conclusions
Nonlinear Chang-Cooper type schemes
Exact flux at the steady state
We obtain
fi+1
fi= exp
(−∫ wi+1
wi
1
C(θj , w)(B[f ] + C ′(θj , w)) dw
).
Next we can approximate the integral on the right hand side with a suitablequadrature formula. To avoid singularities at the boundaries of the integrandfunction we can resort on open formula of Newton-Cotes type. For example, usingthe simple midpoint rule a second order approximation is obtained
fi+1
fi≈ exp
(− ∆w
Ci+1/2
(Bi+1/2[f ] + C ′i+1/2
)).
In this way we can construct a nonlinear scheme which preserves the steady statewith second order accuracy. Higher order accuracy of the steady state can berecovered using more general flux functions and more accurate quadratureformulas.
Mattia Zanella (University of Ferrara) UQ for kinetic models July 6, 2016 18 / 28
Introduction Examples of mean-field models Uncertainty quantification Conclusions
Nonlinear Chang-Cooper type schemes
Weight functions and properties
By equating the ratio fi+1/fi of the numerical and exact flux we recover thefollowing expression of the nonlinear weight functions
δi+1/2 =1
λi+1/2+
1
1− exp(λi+1/2), λi+1/2 =
∆w
Ci+1/2
(Bi+1/2[f ] + C ′i+1/2
).
We must evaluate
Bi+1/2[f ](θ) =
∫Λ
b(θ, wi+1/2, w∗)(w∗ − wi+1/2)f(θ, w∗, t) dw∗,
which, in general, lead to an O(N2) computational cost. This can be avoidedthanks to FFT algorithms when the dependence on wi+1/2 and w∗ factorizesin b(θ, wi+1/2, w∗).For explicit time discretizations, nonnegativity (and therefore stability) isobtained under the restriction ∆t ≤ ∆w/νn where
νn = maxi
(1− δi−1/2)Bni−1/2 − δi+1/2B
ni+1/2 +
1
∆wCi+1/2 +
1
∆wCi−1/2
.
Mattia Zanella (University of Ferrara) UQ for kinetic models July 6, 2016 19 / 28
Introduction Examples of mean-field models Uncertainty quantification Conclusions
Nonlinear Chang-Cooper type schemes
Numerical examples
-1 -0.5 0 0.5 1
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
Cent. Diff.
Mid.
Milne
Exact
-0.05 0 0.05
0 2 4 6 8 10
w
-0.2
0
0.2
0.4
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Cent. Diff.
Mid.
Milne
Exact
0.4 0.6 0.8
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w
-0.5
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f∞(w)
Cent. Diff.
Mid.
Milne
Exact
0.956 0.96 0.964
0 5 10 15 2010
-8
10-6
10-4
10-2
100
Cent. Diff.
Mid.
Milne
0 2 4 6 8 10 12
time
10-4
10-3
10-2
10-1
L1error
Cent. Diff.
Mid.
Milne
0 2 4 6 8 10 12
time
10-6
10-4
10-2
100
L1error
Cent. Diff.
Mid.
Milne
Figure: Left: Opinion model on [−1, 1] - P (·, ·) = 1, σ2 = 1, N = 50 and dt = dw2/4σ2.Center: Economy model o R+- γ(·, ·) = 1, σ2 = 1/10, L = 20, N = 200,dt = dw2/L2σ2. Right: Swarming model on R- L = 10, D = 0.3, α = 2
Mattia Zanella (University of Ferrara) UQ for kinetic models July 6, 2016 20 / 28
Introduction Examples of mean-field models Uncertainty quantification Conclusions
Nonlinear Chang-Cooper type schemes
Collocation
-0.5
1
0
0.8
0.5
1
w
0 0.6
P (θ)
1.5
0.4
-1 0.2
00.50
0.5
1
σ2(θ)w
110
1.5
1.520
0 2 4 6 8 10
M
10-6
10-5
10-4
10-3
10-2
10-1
L1errorforU
M 1
Central
Midpoint
Milne
1 1.5 2 2.5 3 3.5 4 4.5 5
M
10-4
10-3
10-2
L1errorforU
M 1
Central
Midpoint
Milne
Figure: Left: Opinion model - P (θ) = (1 + θ)/2, θ ∼ U([−1, 1]). Right: Market economymodel - σ2(θ) = 1 + θ/2, θ ∼ U([−1, 1]).
Mattia Zanella (University of Ferrara) UQ for kinetic models July 6, 2016 21 / 28
Introduction Examples of mean-field models Uncertainty quantification Conclusions
Stochastic Galerkin methods
Generalized Polynomial Chaos
We consider the linear space PM generated by ψh(θ)Mh=0 the orthogonalpolynomials of θ with degree up to M . They form an orthogonal basis ofL2(Ω,F , P ). The gPC approximation of the Fokker-Planck equation is given by
∂tfh(w, t) = ∂w
M∑m,k=0
Bmkh[fm](w, t)fk(w, t) + ∂w
M∑m=0
Cmh(w)∂wfm(w, t)
,
where fh(w, t) is the stochastic Galerkin projection of f into Ph and
Bmkh[fm](w, t) =
∫Λ
bmkh(w,w∗)(w∗ − w)fm(w∗, t) dw∗,
bmkh(w,w∗) =1
‖ψh‖2L2
∫Ω
b(θ, w,w∗)ψm(θ)ψk(θ)ψh(θ)dp(θ),
Cmh(w) =1
‖ψh‖2L2
∫Ω
C(θ, w)ψm(θ)ψh(θ)dp(θ).
At variance with the collocation approach the resulting equations for theexpansion coefficients are coupled and the Chang-Cooper approach cannot beapplied. Hence a new approach needs to be developed.
Mattia Zanella (University of Ferrara) UQ for kinetic models July 6, 2016 22 / 28
Introduction Examples of mean-field models Uncertainty quantification Conclusions
Stochastic Galerkin methods
gPC: Numerical results
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1−0.2
0
0.2
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1
1.2
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w
f∞
0
f∞
0 ,0
f∞
0 ,2
−0.05 0 0.05
1.71
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1.73
1.74
−5 −4 −3 −2 −1 0 1 2 3 4 50
0.2
0.4
0.6
0.8
1
1.2
1.4
w
D(θ) = θ/10 + 1/3, θ ∼ U ([−1, 1])
M = 0M = 1M = 2
0.85 0.9 0.95 1
−5 −4 −3 −2 −1 0 1 2 3 4 50
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
w
D(θ) =θ
2+ 1, θ ∼ U ([−1, 1])
M = 0M = 1M = 2
−0.4 −0.2 0 0.2 0.4
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 10
0.5
1
1.5
2
2.5
3
3.5
4
4.5x 10
−3
w
V ar (f∞
M)
M=1
M=2
−5 −4 −3 −2 −1 0 1 2 3 4 50
0.5
1
1.5
2
2.5
3
3.5
4x 10
−3
v
M=1M=2M=3
0.95 1 1.05
−5 −4 −3 −2 −1 0 1 2 3 4 50
0.2
0.4
0.6
0.8
1
1.2x 10
−4
w
M=1
M=2
M=3
−0.2 0 0.2
Figure: Left: gPC and exact steady states for the mean f0 for increasing M ≥ 0,P (θ) = 1 + θ/2 and the corresponding variance. Center/Right: swarming model withD(θ) = 1/3 + θ/10 and D(θ) = 1 + θ/2 respectively.
Mattia Zanella (University of Ferrara) UQ for kinetic models July 6, 2016 23 / 28
Introduction Examples of mean-field models Uncertainty quantification Conclusions
Residual distribution schemes
Residual distribution schemes
A general approach to construct steady state preserving methods is based on theuse of the knowledge of the analytic equilibrium state 11 .Suppose we have a differential problem of the form
∂tf = G(f),
where G(f) = 0 implies f = f∞, with f∞ a given equilibrium state.Let G∆w be an order q approximation of G(f), hereafter called the underlyingmethod, which originates the approximated problem
∂tf∆w = G∆w(f∆w).
Given the discrete equilibrium state f∞∆w we define the residual equilibrium as
r∆w = G∆w(f∞∆w),
note that r∆w = O(∆wq) and define the new order q approximation
G∆w(f) := G∆w(f)− r∆w.
11L.Pareschi, T.Rey ’15Mattia Zanella (University of Ferrara) UQ for kinetic models July 6, 2016 24 / 28
Introduction Examples of mean-field models Uncertainty quantification Conclusions
Residual distribution schemes
Remarks and computational aspects
The method can be seen as an extension to general differential problem ofthe classical micro-macro decomposition used in kinetic theory.In order to be applied the residual distribution approach requires theknowledge of the asymptotic steady state for each mode
fh(w, t)→ f∞h (w),
of the gPC expansion which are not known in general. We can overcome thisdifficulty using as stationary modes the corresponding modes of the exactstationary solution
f∞,M (θ, w) =
M∑h=0
f∞h(w)Φh(θ).
By construction f∞h is a spectrally accurate approximation to f∞h .The simple residual equilibrium method just described can be improved inmany ways using flux limiter approaches. This permits to recovernonnegativity of the solution in a deterministic setting or in collocationmethods. For Galerkin methods nonnegativity usually is very challenging.
Mattia Zanella (University of Ferrara) UQ for kinetic models July 6, 2016 25 / 28
Introduction Examples of mean-field models Uncertainty quantification Conclusions
Residual distribution schemes
Errors: Opinion model
0 10 20 30 40 50
time
10-15
10-10
10-5
100
L1error
Cent. Diff.
Res. Dist.
0 2 4 6 8 10
M
10-15
10-10
10-5
100
L1error
Cent. Diff.
Cent. Diff. Collocation
Res. Dist.
Figure: Left: convergence in time of the L1 error. Right: convergence in M to thenumerical steady state. Central differences (red), Collocation (blue), Residualdistribution (black).
Mattia Zanella (University of Ferrara) UQ for kinetic models July 6, 2016 26 / 28
Introduction Examples of mean-field models Uncertainty quantification Conclusions
Residual distribution schemes
Swarming: effect of diffusion
We allow a variability of the 40% around the mean for the diffusion constant
D(θ) = D +2
5Dθ, θ ∼ U([−1, 1]).
−5 0 5−0.5
0
0.5
1
1.5
2
2.5
v
D = 0.1D = 0.2D = 0.3D = 0.4D = 0.5D = 0.6D = 0.7
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
0
0.2
0.4
0.6
0.8
1
D
uf
∞
On the left stationary mean state for several values of the diffusion D(θ) with mean D.
On the right statistical dispersion for E[uf∞ ].
Mattia Zanella (University of Ferrara) UQ for kinetic models July 6, 2016 27 / 28
Introduction Examples of mean-field models Uncertainty quantification Conclusions
Conclusions
In many models recently developed to describe collective dynamics includingthe effects of uncertainty is essential, since at most we have statisticalinformations on the parameters characterizing the interactions.
For mean-field models, the construction of numerical schemes which arecapable to preserve the steady states are essential in order to have a correctdescription of the dynamics.
Several open questions : nonnegativity for stochastic Galerkin methods,computational efficiency for multivariate random inputs, etc.
Future research directionsExtension of uncertainty to Boltzmann models (opinion, economy) andinhomogeneous equations.Limiting processes and asymptotic-preserving schemes with uncertainty:grazing type limits from Boltzmann to Fokker-Planck, macroscopic limits fromVlasov-Fokker-Planck to hyperbolic conservation laws.Control problems also play a relevant role in this field. Understanding theeffects of uncertainty over the control is essential to forceconsensus/alignment in the system.
Mattia Zanella (University of Ferrara) UQ for kinetic models July 6, 2016 28 / 28