Uncertainty quantification for kinetic models of ...wmatem.eis.uva.es/uqap/meeting_files/web_downloads/... · 6J.P. Bouchard, M. Mezard ’00; S. Cordier, L.Pareschi, G. Toscani ’05

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


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



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




Swarms, flocks, crowds, herds...




...collective behavior in socio-economy



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



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


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



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.



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


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


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


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.



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.



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



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



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.




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.




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

.




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

0.6

0.8

1

1.2

1.4

Cent. Diff.

Mid.

Milne

Exact

0.4 0.6 0.8

-5 0 5

w

-0.5

0

0.5

1

1.5

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




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]).



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.



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

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

w

f∞

0

f∞

0 ,0

f∞

0 ,2

−0.05 0 0.05

1.71

1.72

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.



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



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.




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).




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∞ ].



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.


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Uncertainty quantification for kinetic models of ...wmatem.eis.uva.es/uqap/meeting_files/web_downloads/... · 6J.P. Bouchard, M. Mezard ’00; S. Cordier, L.Pareschi, G. Toscani ’05