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Structure preserving stochastic Galerkin methods for

Fokker-Planck equations with background interactions

Mattia Zanella

Department of Mathematical Sciences “G. L. Lagrange”

Dipartimento di Eccellenza 2018-2022

Politecnico di Torino, Torino, Italy

mattia.zanella@polito.it

Abstract

This paper is devoted to the construction of structure preserving stochastic Galerkin

schemes for Fokker-Planck type equations with uncertainties and interacting with an ex-

ternal distribution called the background. The proposed methods are capable to preserve

physical properties in the approximation of statistical moments of the problem like nonnegat-

ivity, entropy dissipation and asymptotic behaviour of the expected solution. The introduced

methods are second order accurate in the transient regimes and high order for large times. We

present applications of the developed schemes to the case of ﬁxed and dynamic background

distribution for models of collective behaviour.

Keywords: uncertainty quantiﬁcation, stochastic Galerkin, Fokker-Planck equations, col-

lective behaviour.

MSC: 35Q70,35Q83,65M70.

1 Introduction

Uncertainty quantiﬁcation (UQ) for partial diﬀerential equations describing real world phenomena

gained an increased interest in recent years [7,10,12,15,16,26,27]. One of the main advantages

of UQ methods relies in its capability to provide a sound mathematical framework to replicate

realistic experiments. The introduction of stochastic parameters reﬂects our incomplete informa-

tion on the initial conﬁguration of a system, on its inner interactions forces and on the modelling

parameters as well.

In the context of kinetic equations, this issue can be translated on a general uncertainty

aﬀecting a distribution function of particles/agents, whose evolution is inﬂuenced by the presence

of a random variable θ, taking value in the set IΘ⊆R, and with known probability distribution

function Ψ(θ) : IΘ→R+. In particular, in the present manuscript we are interested in Fokker-

Planck type equations for the evolution of the distribution f=f(θ, v, t), v∈V⊆Rdv,θ∈IΘ

and t≥0 is the time. The introduced distribution represents the proportion of particles/agents

in [v, v +dv] at time t≥0 and for given value of uncertainty θ∈IΘ. In more details, we consider

the partial diﬀerential equation

∂tf(θ, v, t) = ∇v·[B[g](v , t)f(θ, v, t) + ∇v(D(v)f(θ, v, t))] ,(1)

where v∈V⊆Rdvand B[·] is the operator

B[g](v, t) = ZV

P(v, v∗)(v−v∗)g(v∗, t)dv∗,(2)

where g=g(v, t) is a background distribution, whose dynamics do not incorporate the presence of

the uncertain quantity θ∈IΘ. In applications to socio-economic and life sciences problems with

1

background interactions are very often considered to mimic the inﬂuence of environmental factors

on the agents’ dynamics. For example, the process of knowledge formation depends on social

factors that determine the progress in competence acquisition of individuals, see [20,21] and the

references therein. Similarly, in soft-matter physics biological particles like cells undergo various

heterogeneous stimuli forcing their observable motion [24]. Other examples have been studied in

opinion dynamics, economic processes for the formation of wealth distributions, and urban growth

theory, see [14] for a review.

We consider for (1) an uncertain initial distribution f(θ, v, 0), no-ﬂux boundary conditions

are considered on the boundaries of the domain to enforce conservation of the total mass of the

system. A clear understanding on the global behavior of the system governed by (1)-(2) is obtained

in terms of expected statistical quantities whose accurate and physically admissible description is

therefore of paramount importance.

Due to the increased dimensionality of the problem induced by the presence of uncertainties,

the issue of developing fast converging numerical methods for the approximation of statistical

quantities is of the highest importance. Among the most popular numerical methods for the UQ,

stochastic Galerkin (SG) methods gained in recent years increasing interest since they provide

spectral convergence in the random space under suitable regularity assumptions [1,18,27,28,29].

Similarly to classical spectral methods those methods generally require a strong modiﬁcation

of the original equation and can lead to the loss of structural properties like positivity of the

solution, entropy dissipation and hyperbolicity, when applied to hyperbolic and kinetic equations,

see [10,17]. The loss of structural properties of the solution induces an evident gap in its true

physical meaning. To overcome this problem, recently has been proposed a novel methods that

combines both Monte Carlo and SG methods [6,7] and which preserves spectral accuracy in the

random space.

In the present manuscript we construct structure preserving methods for the SG formulation

of the problem in the case of background interactions. In order to do that we will take advantages

of structure preserving (SP) methods [22,23], that have been designed to preserve the mentioned

structural properties of the solution of nonlinear Fokker-Planck equations without restriction on

the mesh size. We consider applications of the developed schemes both in case of ﬁxed and dynamic

background.

The rest of the paper is organized as follows. In Section 2we brieﬂy introduce stochastic

Galerkin methods for the problem of interest where the interactions take place with respect to a

deterministic background, stability results are proved and discussed together with the analysis of

trends to asymptotic states. In Section 3we derive structure preserving methods in the Galerkin

setting, positivity conditions for explicit and semi-implicit schemes are discussed and we prove

entropy inequality for a class of one dimensional Fokker-Planck models. Several applications of

the schemes are ﬁnally considered in Section 4for several problems arising in the description of

collective phenomena in socio-economic and life-sciences. Some conclusions are reported at the

end of the manuscript.

2 Stochastic Galerkin methods for kinetic equations

For simplicity of presentation we consider the case dv= 1. We focus on real-valued distributions

depending on a one dimensional random input. Let (Ω, F, P ) be a probability space where as

usual Ω is the sample space, Fis a σ−algebra and Pa probability measure, and let us deﬁned a

random variable

θ: (Ω, F )→(IΘ,BR),

with IΘ⊂Rand BRis the Borel set. We focus on real-valued distributions of the form f(θ, v, t) :

Ω×V×[0, T ]→Rd. In the present section we derive a stochastic Galerkin approximation for

Fokker-Planck equation with uncertain initial distribution and background interactions (1).

Let us consider the linear space PMof polynomials of degree up to Mgenerated by a family

2

of orthogonal polynomials {Φh(θ)}M

h=0 such that

E[Φh(θ)Φk(θ)] = ZIΘ

Φh(θ)Φk(θ)Ψ(θ)dθ =kΦ2

h(θ)kL2(Ω)δhk ,

being δhk the Kronecker delta function. Assuming that Ψ(θ) has ﬁnite second order moment we

can approximate the distribution f∈L2(Ω,F, P ) in terms of the following chaos expansion

f(θ, v, t)≈fM(θ , v, t) =

M

X

k=0

ˆ

fk(v, t)Φk(θ),(3)

being ˆ

fk(v, t) the projection of finto the polynomial space of degree k, i.e.

ˆ

fk(v, t) = E[f(θ, v , t)Φk(θ)], k = 0, . . . , M.

Plugging fMinto (1) we obtain

∂tfM(θ, v, t) = ∂vB[g](v , t)fM(θ, v, t) + ∂v(D(v)fM(θ, v, t)).(4)

Hence, by multiplying (4) by Φh(θ) for all h= 0, . . . , M and after projection in each polynomial

space we obtain the following system of M+ 1 deterministic kinetic-type PDEs

∂tˆ

fh(v, t) = ∂vhB[g](v, t)ˆ

fh(v, t) + ∂v(D(v)ˆ

fh(v, t))i,(5)

with the initial conditions ˆ

fh(v, 0) = E[f(θ, v , 0)Φh(θ)].

The related deterministic subproblems can be tackled through suitable numerical methods and

the approximation of statistical quantities of interest are deﬁned in terms of the projections. In

particular we have

E[f(θ, v, t)] ≈ˆ

f0(v, t),(6)

whose evolution is given by (5) in the case h= 0. Thanks to the orthogonality in L2(Ω) of the

polynomials {Φh}M

h=0 we have

E[f(θ, v, t)2]−E[f(θ , v, t)]2≈E[(fM(θ, v, t))2]−E[fM(θ, v, t)]2

from (3) it corresponds to

E"M

X

k=0

ˆ

f2

k(v, t)Φ2

k(θ)+2

M

X

k=0

k−1

X

h=0

ˆ

fk(v, t)ˆ

fh(v, t)Φk(θ)Φh(θ)#−ˆ

f2

0(v, t).

Therefore the variance of the solution is approximated in terms of the projections as follows

Var[f(θ, v, t)] ≈

M

X

k=0

ˆ

f2

h(v, t)E[Φ2

k]−ˆ

f2

0(v, t) (7)

We observe that the initial mass deﬁned by RVˆ

fh(v, 0)dv is conserved in time assuming no-ﬂux

boundary conditions, i.e.

B[g](v, t) + ∂vD(v)=0, v ∈∂V.

Let us introduce the vector ˆ

f(v, t) = ˆ

f0,..., ˆ

fM. If we deﬁne as kˆ

f(v, t)kL2the standard L2

norm of the vector ˆ

f(v, t)

kˆ

f(v, t)kL2="ZV M

X

h=0

ˆ

f2

h(v, t)!dv#1/2

,

3

then from the orthonormality of the introduced basis {Φh}M

h=0 in L2(Ω) we have that

kfM(θ, v, t)kL2(Ω) =kˆ

f(v, t)kL2,

where

kfM(θ, v, t)kL2(Ω) =ZIΘZV M

X

h=0

ˆ

fh(v, t)Φh(θ)!2

dvΨ(θ)dθ.

We can reformulate the problem (5) in a more compact form as follows

∂tˆ

f=∂vhBˆ

f+D∂vˆ

fi,

where B={Bij }M+1

i,j=1 and D={Di,j }N

i,j=1 are diagonal matrices with components

Bi,i =B[g](v, t) + ∂vD(v),Bi,j = 0

Di,i =D(v),Di,j = 0.

The following stability result can be established

Theorem 1. If k∂vB[g](v, t)kL∞≤CB, with CB>0, and if D≤CDwe have

kˆ

f(v, t)k2

L2≤et(CB+2CD)kˆ

f(v, 0)k2

L2

Proof. We multiply (5) by ˆ

fh(v, t) and integrate over V⊆R

ZV

∂t1

2ˆ

f2

h(v, t)dv =ZV

ˆ

fh(v, t)∂vhB[g](v, t)ˆ

fh(v, t) + ∂v(D(v)ˆ

fh(v, t))idv. (8)

From the integral on the right hand side of the above equation we have

ZV

ˆ

fh(v, t)∂vB[g](v, t)ˆ

fh(v, t)dv

=ZV

ˆ

f2

h(v, t)∂vB[g](v, t)dv −ZV

ˆ

fh(v, t)∂v(B[g](v, t)ˆ

fh(v, t))dv.

Hence, following estimate holds

M

X

h=0 ZV

ˆ

f2

h(v, t)∂vB[g](v, t)dv ≤CB

2kˆ

f(v, t)kL2.

Furthermore we have

ZV

ˆ

fh(v, t)∂2

vD(v)ˆ

fh(v, t)=ZV∂2

vˆ

fh(v, t)D(v)ˆ

fh(v, t)dv

≤ −CDZV∂vˆ

fh(v, t)2.

Finally, after summation h= 0, . . . , M of (8) we obtained

1

2kˆ

f(v, t)k2

L2≤CB

2kˆ

f(v, t)k2

L2− k∂vˆ

f(v, t)k2

L2

≤CB

2+CDkˆ

fkL2,

and thanks to the Gronwall theorem we conclude.

Remark 1. The background distribution g(v, t)is in general ruled by an additional PDE that

does not depend on the stochastic density function f(θ, v, t)and does not incorporate additional

uncertainties. In the case of evolving background we need to couple to (1)its dynamics.

4

2.1 Asymptotic behaviour

Assuming that the dynamics of the background g(v, t) admit a unique stationary state the asymp-

totic distribution of (1) is solution of the diﬀerential equation

B[g∞](v)f∞(θ, v) + ∂v(D(v)f∞(θ , v)) = 0,(9)

which gives

∂vf∞(θ, v)

f∞(θ, v)=−B[g∞](v) + D0(v)

D(v),

and therefore the analytical stationary distribution of the original problem reads

f∞(θ, v) = C(θ) exp −ZB[g∞](v) + D0(v)

D(v)dv,(10)

being C(θ)>0 a normalization constant depending only on the initial uncertainties of the problem.

On the other hand, the asymptotic solutions f∞

h(v) of (5) in each polynomial space of degree

h= 0, . . . , M are deﬁned by solving the following set of diﬀerential equations

B[g∞](v)ˆ

f∞

h+∂v(D(v)ˆ

f∞

h)=0, h = 0, . . . , M, (11)

whose stationary states are

ˆ

f∞

h(v) = Chexp −ZB[g∞](v) + D0(v)

D(v)dv(12)

being Chsuch that

ZIΘ

f∞(θ, v)Φh(θ)Ψ(θ)dθ =ˆ

f∞

h(v, t).

We can observe how if the initial state has deterministic mass RVf(θ, v, 0)dv = ¯ρ > 0 the asymp-

totic state of the problem given by (10) does not incorporate any uncertainty since the normaliza-

tion constant does not depend anymore on the uncertainty of the problem, meaning that C(θ) = ¯

C

for all θ∈IΘ. This fact reﬂects on the asymptotic state of each projection ˆ

f∞

h(v, t), h= 0, . . . , M ,

since E[¯

CΦh(θ)] = 0 for h > 0. Therefore in the case of deterministic initial mass we obtain

ˆ

f∞

h(v, t) =

¯

Cexp −ZB[g∞](v) + D0(v)

D(v)dvif h= 0

0 if h > 0,

and the variance of f(θ, v, t) vanishes asymptotically. In the general case of uncertain initial mass

the asymptotic state still depends on θ∈IΘ.

In the following we explicit the trend to equilibrium deﬁned by stochastic background interac-

tion models following the ideas in [13].

2.1.1 Constant background

Let us assume that the background is ﬁxed so that B[g](v, t) = B[g](v). In particular from (9) it

follows that the Fokker-Planck equation (1) with constant background can be rewritten as

∂tf(θ, v, t) = ∂vD(v)f(θ , v, t)∂vlog f(θ, v, t)

f∞(θ, v),

from which we obtain the evolution for F(θ, v, t) = f(θ,v,t)

f∞(θ,v)that reads

∂tF(θ, v, t) = D(v)∂2

vF(θ, v, t)− B[g](v)∂vF(θ, v, t),(13)

with no-ﬂux boundary conditions

D(v)f∞(θ, v)∂vF(θ , v, t)v∈∂V = 0.

The following result holds

5

Theorem 2. Let the smooth function Φ(x),x∈R+be convex. Then, if F(θ, v, t)is the solution

to (13)in V⊆Rand F(θ, v, t)is bounded for all θ∈IΘthe functional

H(f, f ∞)(θ, v, t) = ZV

f∞(θ, v)Φ(F(θ , v, t))dv

is monotonically decreasing in time and its evolution is given by

d

dtH(f, f ∞)(θ, v, t) = −I(f, f ∞)(θ, v, t),

where with Iwe denote the nonnegative quantity

I(f, f ∞) = ZV

D(v)f∞(θ, v)Φ00 (F(θ, v , t)) |∂vF(θ, v, t)|2dv.

Proof. The proof of this result follows the strategy adopted in [13] for all θ∈IΘ.

Now in the case Φ(x) = xlog(x) we obtain the relative Shannon entropy H(f, f ∞) which is a

functional depending on the uncertainties of the model. From the above result it follows that this

quantity is dissipated with the rate given for all θ∈IΘby

IH(f, f ∞) = ZV

D(v)f∞(θ, v)1

F(θ, v, t)|∂vF(θ , v, t)|2dv

and we have

d

dt ZV

f(θ, v, t) log f(θ, v, t)

f∞(θ, v)=−ZV

D(v)f(θ, v, t)∂vf(θ , v, t)

f(θ, v, t)−∂vf∞(θ , v)

f∞(θ, v)2

dv

In the stochastic Galerkin approximation the relative Shannon entropy for fM(θ, v, t, ) in (3)

reads

d

dt ZV

M

X

k=0

ˆ

fk(v, t)Φk(θ) log PM

k=0 ˆ

fk(v, t)Φk(θ)

PM

k=0 ˆ

f∞

k(v, t)Φk(θ)dv

=−ZV

D(v)

M

X

k=0

ˆ

fk(v, t)Φk(θ) ∂vlog PM

k=0 ˆ

fk(v, t)Φk

PM

k=0 ˆ

f∞

k(v, t)Φk(θ)!2

dv,

from which approximated statistical moments can be obtained by projection in the space deﬁned

by the polynomial basis

d

dt ZV

M

X

k=0

Hhk(v , t)ˆ

fk(v, t)dv =−ZV

D(v)

M

X

k=0

Ihk(v , t)ˆ

fk(v, t)dv,

being

Hhk =ZIΘlog fM(θ, v, t)−log fM ,∞(θ, v)Φh(θ)dθ,

Ihk =ZIΘ∂vlog fM(θ, v, t)−∂vlog fM ,∞(θ, v)2Φh(θ)dθ.

We observe that, due to the nonlinearities in the deﬁnition of the convex functional H(f, f ∞), a

coupled system of diﬀerential equations must be solved to estimate the expected trend to equi-

librium provided by the relative entropy functional. Nevertheless, at the Galerking level we have

no guarantee that the weighted Fisher information deﬁnes a positive quantity for the obtained

truncated distribution and, hence, that the entropy monotonically decreases.

6

On the other hand, the system of M+ 1 projections deﬁned in (5) can be rewritten for all

h= 0, . . . , M in the case of ﬁxed background as follows

∂tˆ

fh(v, t) = ∂v"D(v)ˆ

fh(v, t)∂vlog ˆ

fh(v, t)

ˆ

f∞

h(v)#,(14)

and therefore by introducing the ratio Fh=ˆ

fh(v,t)

ˆ

f∞

h(v)>0 we have

∂tFh=−B[g](v)∂vFh(v, t) + D(v)∂2

vFh(v, t).(15)

complemented with no-ﬂux boundary conditions. Then, in analogy with what we discussed above,

the following result holds.

Theorem 3. Let the smooth function Φ(x),x∈R+be convex. Then, if Fh(v, t)is the solution

to (15)in V⊆Rand Fh(v, t)is bounded the functional

H(Fh)(v, t) = ZV

ˆ

f∞

h(v)Φ(Fh(v, t))dv

is monotonically decreasing in time and its evolution is given by

d

dtH(Fh)(v, t) = −I (Fh)(v , t),

where with Iwe denote the nonnegative quantity

I(Fh(v, t)) = ZV

D(v)ˆ

f∞

h(v)Φ00(Fh(v, t)) |∂vFh(v, t)|2dv.

Now, in the case of relative Shannon entropy Φ(x) = xlog xwe obtain in each polynomial

space

d

dt ZV

ˆ

fh(v, t) log ˆ

fh(v, t)

ˆ

f∞

h(v)dv =−ZV

D(v)ˆ

fh(v, t) ∂vlog ˆ

fh(v, t)

ˆ

f∞

h(v)!2

dv. (16)

Therefore, each projection of f(θ, v , t) in the linear space of arbitrary degree h= 0, . . . , M con-

verges monotonically in time to its equilibrium ˆ

f∞

h(v). In particular this is true for the expected

quantities of the problem.

3 Structure preserving methods

In this section we introduce the class of so-called structure preserving (SP) numerical methods

for the solution of Fokker-Planck equations with nonlocal terms. These methods preserve the

fundamental structural properties of the problem like nonnegativity of the solution, entropy dis-

sipation and capture the steady state of each problem with arbitrarily accuracy, see [8,12,22,23].

The applications of the SP methods is here particularly appropriate since, thanks to background

interactions, the system of M+ 1 equations (5) is decoupled.

In the following we summarise the construction ideas at the basis of SP methods in dimension

d= 1, extension to general dimension can be found in [22].

3.1 Derivation of the SP method

For all h= 0, . . . , M we may rewrite (5) in ﬂux form as follows

∂tˆ

fh(v, t) = ∂vF[ˆ

fh](v, t),

7

where

F[ˆ

fh](v, t) = C[g](v, t)ˆ

fh(v, t) + D(v)∂vˆ

fh(v, t),

and C[g](v, t) = B[g](v, t) + ∂vD(v). Let us introduce a uniform grid vi∈V, such that vi+1 −vi=

∆v > 0 and let vi±/2=vi±∆v/2. We consider the conservative discretization

d

dt ˆ

fh,i(t) = Fh,i+1/2(t)− Fh,i−1/2(t)

∆v, t ≥0 (17)

being Fh

i±1/2a numerical ﬂux having the form

Fh,i+1/2=˜

C[g]i+1/2˜

fh,i+1/2+Di+1/2

ˆ

fh,i+1 −ˆ

fh,i

∆v,(18)

where ˜

fh,i+1/2= (1 −δi+1/2)ˆ

fh,i+1 +δi+1/2ˆ

fh,i.

Hence, we aim at ﬁnding the weight functions δi+1/2and ˜

C[g]i+1/2such that the scheme pro-

duces nonnegative solutions without restrictions on the mesh size ∆v, and is able to capture with

arbitrary accuracy the steady state of the (5) for all h= 0 ...,M.

We observe that for a vanishing numerical ﬂux we obtain

ˆ

fh,i+1

ˆ

fh,i

=

−δi+1/2˜

Ci+1/2+Di+1/2

∆v

(1 −δi+1/2)˜

Ci+1/2+Di+1/2

∆v

.

At the analytical level we obtained from (11) in Section 2.1 that

D(v)∂vˆ

fh(v, t) = −(B[g](v, t) + ∂vD(v)) ˆ

fh(v, t),

which admits the quasi state approximation for all h= 0, . . . , M

Zvi+1

vi

1

ˆ

fh(v, t)∂vˆ

fh(v, t)dv =−Zvi+1

vi

1

D(v)(B[g](v, t) + ∂vD(v))dv,

that is ˆ

fh(vi+1, t)

ˆ

fh(vi, t)= exp −Zvi+1

vi

1

D(v)(B[g](v, t) + ∂vD(v))dv.(19)

Equating ˆ

fh(vi+1, t)/ˆ

fh(vi, t) and ˆ

fh,i+1/ˆ

fh,i and setting

˜

C[g]i+1/2=Di+1/2

∆vZvi+1

vi

1

D(v)(B[g](v, t) + ∂vD(v))dv,

we can determine weight functions

δi+1/2=1

λi+1/2

+1

1−exp(λi+1/2)∈(0,1),(20)

where

λi+1/2=Zvi+1

vi

1

D(v)(B[g](v, t) + ∂vD(v))dv =∆v˜

Ci+1/2

Di+1/2

.

It is worth pointing out that by construction the numerical ﬂux of the SP scheme vanishes

when the analytical ﬂux is equal to zero. The long time behavior of (11) is described with the

accuracy with which we evaluate the weights (20). In the following we will show that suitable

restrictions on the time discretization can be deﬁned to guarantee positivity preservation of the

SP scheme. Moreover, we will show that the scheme dissipates the numerical entropy with a rate

which is coherent with what we observed in Section 2.1.

8

Remark 2. The obtained weights do not depend on the degree of the linear space since they are

equal for all h= 0, . . . , M . Furthermore, in the case of interaction with a constant background,

i.e. B[g](v, t) = B[g](v), we can compute explicit stationary state ˆ

f∞

h(v)for all h= 0 ...,M, see

equation (11)together with boundary conditions. Hence, thanks to the knowledge of the stationary

state in each polynomial space we have

ˆ

f∞

h,i+1

ˆ

f∞

h,i

= exp −Zvi+1

vi

1

D(v)(B[g](v) + ∂vD(v))dv= exp −λ∞

i+1/2.

Which leads to

λ∞

i+1/2= log ˆ

f∞

h,i

ˆ

f∞

h,i+1 !,

and

δ∞

i+1/2=1

log( ˆ

f∞

h,i)−log( ˆ

f∞

h,i+1)+

ˆ

f∞

h,i+1

ˆ

f∞

h,i+1 −ˆ

f∞

h,i

.

In this case the SP scheme do not introduce additional source of errors at the steady state. We

highlight how the dependence on h= 0, . . . , M is only apparent since for each times t≥0the ratio

ˆ

fh,i+1/ˆ

fh,i in (19)does not depend on the speciﬁc projection thanks to background interactions.

3.2 Positivity of statistical moments

In general positivity of the solution, or of its statistical moments, is not achievable once we apply

stochastic Galerkin methods and the solution of the system fM(θ, v, t) looses a genuine physical

meaning. In this section we provide explicit conditions to preserve nonnegativity of projections

ˆ

fh(v, t) and therefore of the statistical moments of fM(θ, v, t), that have been obtained in 2from

direct inspection of the Galerkin projections (6)-(7). In particular, we will show how in the

background interactions case we are able to provide reliable conditions, without restriction on ∆v,

for positivity preservation.

In recent works [6,7] a particle scheme has been proposed to enforce positivity of statistical

quantities for uncertainty quantiﬁcation of mean-ﬁeld models. The core idea of the approach

presented in the cited works is to approximate the expected solution of a mean-ﬁeld type model

by reformulating the problem in a Monte Carlo (MC) setting in the phase space, which is then

expanded through a SG generalized polynomial chaos (SG-gPC) method. The expected solution

is then reconstructed from expected positions and velocities of the microscopic system, which is

considered in the gPC setting. We will refer to this method as MCgPC. The solution of the

MCgPC approach is still spectrally accurate in the random space whereas in the phase space it

assumes to accuracy of the Monte Carlo method. The approach presented here for the linear case

provide high accuracy also in V.

Let us introduce the time discretization tn=n∆t, ∆t > 0 and n= 0, . . . , T and consider the

following forward Euler method for all h= 0, . . . , M

ˆ

fn+1

h,i =ˆ

fn

h,i + ∆tFn

h,i+1/2− Fn

h,i−1/2

∆v,(21)

where the ﬂux has the form introduced in (18). We can prove the following result

Theorem 4. Under the time step restriction

∆t≤∆v2

2 (M∆v+D), M = max

i|˜

Ci+1/2|, D = max

iDi+1/2

the explicit scheme (21)preserves nonnegativity, i.e. ˆ

fn+1

h,i ≥0provided ˆ

fn

h,i ≥0.

Proof. The proof of this result is analogous for all h= 0, . . . , M to the result for explicit scheme

obtained in [22].

9

We observe that no explicit dependence on the expansion degree h= 0, . . . , M appears in

the derived restriction thanks to the background-type interactions. Furthermore, the restriction

on ∆tin 4ensures nonnegativity without additional bounds on the spatial grid as for example

happen for central type schemes. The derived condition automatically holds for higher order

strong stability preserving (SSP) methods like Runge-Kutta and multistep methods since these

are convex combinations of the forward Euler integration. The the prove nonnegativity of the

scheme is extended straightforwardly to each SSP type time integration.

We highlight how the derived parabolic restriction to enforce nonnegativity of explicit schemes

can be quite heavy for practical applications. A convenient strategy to lighten this burden resorts

to the technology of semi-implicit methods, see [3] for an introduction. Indeed, we can prove

nonnegativity of the solutions {ˆ

fh}M

h=0 by considering the set of modiﬁed ﬂuxes

˜

Fn+1

h,i+1/2=˜

Cn

i+1/2h(1 −δn

i+1/2)ˆ

fn+1

h,i+1 +δn

i+1/2ˆ

fn+1

h,i i+Di+1/2

ˆ

fn+1

h,i+1 −ˆ

fn+1

h,i

∆v.(22)

The scheme is semi-implicit since we compute the background dependent ˜

Ci+1/2and weight func-

tions δi+1/2at time tn. As a consequence, it is easily seen how in the case of a ﬁxed background

the scheme is coherent with a fully implicit method.

The following result holds

Proposition 1. Let us consider a semi-implicit method for all h= 0, . . . , M

ˆ

fn+1

h,i =ˆ

fh,i + ∆t

˜

Fn+1

i+1/2−˜

Fn+1

i−1/2

∆v,

with ﬂuxes deﬁned in (22). Under the time step restriction

∆t < ∆v

2M, M = max

i|˜

Cn

i+1/2|,

the semi-implicit scheme preserves nonneagivity, i.e. ˆ

fn+1

h,i ≥0if ˆ

fn

h,i ≥0for all i= 1, . . . , N and

h= 0, . . . , M .

Proof. The proof of this result is analogous for all h= 0, . . . , M to the result for semi-implicit

scheme obtained in [22].

Extensions to higher order semi-implicit schemes have been obtained in [3].

3.3 Entropy dissipation

We concentrate on the case of ﬁxed background. In Section 2.1 we have seen how the Fokker-

Planck problems of interest can be rewritten in Landau form (14). In particular, it can be proven

how the numerical ﬂux for this reformulation is given by the following equivalent form

Fh,i+1/2=Di+1/2

∆v¯

f∞

h,i+1/2 ˆ

fh,i+1

ˆ

f∞

h,i+1

−ˆ

fh,i

ˆ

f∞

h,i !,(23)

with

¯

f∞

h,i+1/2=

ˆ

f∞

h,i+1 ˆ

f∞

h,i

ˆ

f∞

h,i+1 −ˆ

f∞

h,i

log ˆ

f∞

h,i+1

ˆ

f∞

h,i !,

since for all h= 0, . . . , M we have λi+1/2= log ˆ

f∞

h,i −log f∞

h,i+1 and the weight functions are

rewritten as

δi+1/2=1

log ˆ

f∞

h,i −log f∞

h,i+1

+

ˆ

f∞

h,i+1

ˆ

f∞

h,i+1 −f∞

h,i

.

We can prove the following result

10

Theorem 5. Let us consider the conservative discretization (17)for all t≥0and h= 0 ...,M.

The numerical ﬂux (18)satisﬁes the discrete entropy dissipation

d

dtH∆v(f, f ∞) = −I∆(f, f ∞),

where

H∆v(ˆ

fh,ˆ

f∞)=∆v

N

X

i=0

ˆ

fi,h log ˆ

fh,i

ˆ

f∞

h,i !,

and

I∆v(ˆ

fh,i,ˆ

f∞

h,i) =

N

X

i=0 "log ˆ

fh,i+1

ˆ

f∞

h,i+1 !− ˆ

fh,i

ˆ

f∞

h,i !#· ˆ

fh,i+1

ˆ

f∞

h,i+1

−ˆ

fh,i

ˆ

f∞

h,i !¯

f∞

h,i+1/2Di+1/2≥0.

Proof. From the deﬁnition of relative entropy for all h= 0, . . . , M we have

d

dtH(ˆ

fh,ˆ

f∞

h)=∆v

N

X

i=0

dˆ

fh

dt log ˆ

fh,i

ˆ

f∞

h,i !+ 1!

=

N

X

i=0 log ˆ

fh,i

ˆ

f∞

h,i !+ 1!Fh,i+1/2− Fh,i−1/2

After summation by parts we have

d

dtH(ˆ

fh,ˆ

f∞

h) = −

N

X

i=0 "log ˆ

fh,i+1

ˆ

f∞

h,i+1 !−log ˆ

fh,i

ˆ

f∞

h,i !#Fh,i+1/2,

and from the reformulation of the ﬂux in (23) we may conclude since (x−y) log(x/y)≥0 for all

x, y ≥0.

4 Numerical tests

In the present section present several tests for Fokker-Planck equations with background interac-

tions and uncertain initial distribution. We adopt the introduced structure preserving stochastic

Galerkin method here discussed. In particular we will consider ﬁxed and evolving backgrounds

both. As discussed in Section 3the essential aspect for the accurate computation of the large time

distribution of the problem (1) lies in the numerical approximation of the integral

λi+1/2=Zvi+1

vi

1

D(v)(B[g](v, t) + D0(v))dv,

which deﬁnes the quasi-stationary states of each projection. In general a high order quadrature

method is needed. In the following numerical examples we will consider open Newton-Cotes

quadrature methods up to the 6th order and the Gauss-Legendre quadrature. Through the text

we will refer to these methods as SPk,k= 2,4,6, G, where the index kindicates the order of

the adopted quadrature method with Greferring to the Gauss-Legendre case. To highlight the

advantages of this approach a nonconstant diﬀusion function is considered for bounded domains. In

all the tests we considered suitable restrictions on the time discretization to guarantee positivity of

the expected solution of the problems both in the explicit and semi-implicit integration. Extension

to the multidimensional case is considered at the end of this section.

11

4.1 Test 1: Stationary background distribution

Let us consider the evolution of a distribution function f(θ, v, t) in the presence of uncertainty that

follows (1), with v∈[−1,1], and interacting with a given background distribution g(v, t) = g(v)

for all t≥0 of the form

g(v) = βexp −(w−ug)2

2σ2

g, ug∈(−1,1), σ2

g= 0.01,(24)

with β > 0 a normalizing constant such that R1

−1g(v)dv = 1. We consider a nonconstant diﬀusion

D(v) = σ2

2(1 −v2)2with given σ2that will be speciﬁed later on. Furthermore, the nonlocal

operator in (2) is deﬁned in terms of the interaction function

P(v, v∗) = χ(|v−v∗| ≤ ∆),(25)

where ∆ >0 is a constant measuring the maximal distance under which interactions may occur.

The introduced function P(·,·) is usually deﬁned as bounded conﬁdence function. This model

has been proposed in the literature to describe the evolution of the distribution of agents having

opinion vat time t≥0, see [19,25]. In particular the presence of background interactions is

generally considered to take into account the inﬂuence of external actors in opinion dynamics like

the case of media [4] or the action of possible control strategies [2]. Extensions to the case of

uncertain interactions have been proposed in [26].

In this ﬁrst test we consider as initial distribution

f(θ, v, 0) = C(θ)exp −(v−u1(θ))2

2σ2

0+ exp −(v−u2(θ))2

2σ2

0,(26)

with C(θ) such that RVf(θ, v, 0) = ρ(θ)>0 for all θ∈IΘand ui(θ), i= 1,2 given by

u1(θ) = ¯u+κ θ, u2(θ) = −¯u+κ θ,

being θ∼ U ([−1,1]). In the case ∆ = 2 it follows that P≡1 and we can compute the explicit

stationary distribution

f∞(θ, v) = C(θ)

(1 −v2)21 + v

1−vug/(2σ2)

exp −1−ugv

σ2(1 −v2).

The stochastic Galerkin decomposition of the resulting problem can be performed by consid-

ering a Legendre polynomial basis {Φh}M

h=0 being Ψ(θ) = 1

2χ(θ∈[−1,1]). The resulting system

of equations have the form (5) whose asymptotic solution for all h= 0, . . . , M reads

ˆ

f∞

h(v) = Ch

(1 −v2)21 + v

1−vug/(2σ2)

exp (−1−ugv

σ2

f(1 −v2)).(27)

being Ch=1

2RIΘC(θ)Φh(θ)dθ. In Figure 1we present the evolution of the L1relative error

computed with respect to the exact stationary state for the SPk,k= 2,4,6, G, schemes for

various quadrature methods. To exemplify the advantages we consider the two projections h= 0

(left), and h= 1 (right). In particular, for each SPkwe considered N= 41 gridpoints for the

discretization of the state variable. We can observe how we achieve diﬀerent accuracy in terms of

the steady states of the problem in relation to the considered quadrature rules for both h= 0,1.

Further, with low order quadrature we approach to the numerical steady state of the method faster

than with high order rules. We observe that with a Gauss-Legendre method we essentially reach

machine precision in ﬁnite time for each projection. In the same ﬁgure we show the dissipation

of the relative entropy functional H(ˆ

fh,ˆ

f∞

h) discussed in Section 2.1 with h= 0,1 obtained with

the structure preserving method. We present the case of two coarse grids obtained with N= 11,

N= 21 gridpoints compare with the exact dissipation of the relative entropy.

12

0 5 10 15 20 25

10-15

10-10

10-5

100

0 5 10 15 20 25

10-15

10-10

10-5

100

0 0.5 1 1.5 2

0

0.2

0.4

0.6

0.8

0 0.5 1 1.5 2

0

0.05

0.1

Figure 1: Example 1. Top row: evolution of the L1relative error with respect to the stationary

solution (27) for the SPkscheme with diﬀerent quadrature methods. We considered the initial

uncertain distribution f(θ, v, 0) in (26) with ¯u= 0.25, ρ(θ) = 1 + 0.5θ,θ∼ U ([−1,1]), and

σ2= 2 ·10−1. For all hthe solution has been computed for N= 41 gridpoints over the time

interval [0,25], ∆t= ∆v2/(2σ2). Bottom row: dissipation of the numerical entropies H(ˆ

f0,ˆ

f∞

0),

H(ˆ

f1,ˆ

f∞

1) for the SPkscheme with Gaussian quadrature for two coarse grids with N= 11 and

N= 21 gridpoints.

-1 -0.5 0 0.5 1

0

0.5

1

1.5

-0.05 0 0.05

-1 -0.5 0 0.5 1

0

0.005

0.01

0.015

-0.05 0 0.05

Figure 2: Example 1. Large time behavior of expectation (left) and variance (right) of f(θ, v, t)

obtained with SPkschemes and k= 2, G and an uncertain initial distribution of the form (26).

We can observe how the high accuracy of the proposed scheme reﬂects in an arbitrary accurate

numerical description of the large time statistical moments of the solution of the problem. t∈[0, T ]

with T= 15 and N= 41, ∆t= ∆v2/(2σ2).

13

E[f]SPk

Time 2 4 6 G

1 2.0785 1.9989 2.0025 2.0026

5 1.9949 4.2572 2.2868 2.3361

10 1.9953 3.9141 6.4698 7.3367

Var(f)SPk

Time 2 4 6 G

1 2.0870 2.0001 2.0030 2.0031

5 1.9978 4.4192 2.2398 2.2789

10 1.9982 3.9309 6.6929 7.3405

Table 1: Example 1. Estimation of the order of convergence toward the reference stationary

state for SP–CC scheme with RK4 method. Rates have been computed using N= 21,41,81,

σ2/2=0.1, ∆t= ∆w2/σ2.

The high accuracy of the scheme in the description of the large time behavior in each polynomial

space reﬂects in a high accuracy in the approximation of statistical moments of the solution of

the problem, see Figure 2. Here we considered the schemes SP2and SPG, that is the structure

preserving schemes with approximation of λi+1/2with a midpoint and Gauss-Legendre method

respectively. We highlight how the expectation is positive thanks to the properties of the scheme.

In Table 1we estimate the order of convergence of the structure preserving method in terms of

accuracy of the expected quantities E[f], and Var(f) in their stochastic Galerkin approximation

(6)-(7). It is easily observed how for the approximation of the variance it is required the solutions

of the whole set of projections h= 0, . . . , M , we will consider M= 10. Here we used N= 21,41,81

and the order of convergence of the explicit structure preserving schemes is measured as log2

e1(t)

e2(t),

where e1(t) is the relative error at time t≥0 of the expected solution and its variance computed

with N= 21 gridpoints with respect to that computed with N= 41 gridpoints and, likewise,

e2(t) is the relative error at time t≥0 computed with N= 41 with respect to that computed

with N= 81 gridpoints. The time integration has been performed with RK4 at each time step

chosen in such a way that the restriction for positivity of the scheme in Theorem 4is satisﬁed, i.e.

∆t=O(∆v2). We can observe how the SPkschemes are second order accurate in the transient

regimes and assume the order of the quadrature method near the expected steady state and its

related variance.

In the more general case P(v, v∗)6= 1 we have no analytical insight on the large time solution

ˆ

f∞

h(v) in each polynomial space. In Figure 3we consider the case of bounded conﬁdence type

interactions (25) with ∆ = 1.0 and a ﬁxed background distribution g(v) of the form

g(v) = βexp −(v−ug)2

2σ2

g+ exp −(v+ug)2

2σ2

g,

with ug=1

2and σ2

g= 10−2. We considered the uncertain initial density (26) with deterministic

initial mass ρ(θ) = 1 and uncertainty in u1(θ), u2(θ) so that

u1=1

2+1

4θ, u2=−1

2+1

4θ,

with θ∼ U([−1,1]). The integral B[g](v) has been evaluated through a trapezoidal rule. As

observed in Section 2.1 the large time solution for all h= 0, . . . , M does not depend on the

uncertainties of the initial distribution. Indeed, the variance annihilates as we can observe in 3(d)

and the asymptotic state coincides with E[f].

14

-1 -0.5 0 0.5 1

0

0.5

1

1.5

(a) t= 0

-1 -0.5 0 0.5 1

0

0.5

1

(b) t= 10

(c) E(f) (d) Var(f)

Figure 3: Example 1. Top row: initial distribution and solution at time T= 10 in the case of

bounded conﬁdence interactions and ∆ = 1 obtained with SPG,N= 41 gridpoints and M= 5

projections. Bottom row: evolution of the expected solution (left) and its variance (right) in the

interval [0,10].

15

0

10

0

-1 020

1

2

4

(a) g(v, t)

-1 -0.5 0 0.5 1

0

0.5

1

1.5

2

(b) t= 20

(c) E(f) (d) Var(f)

Figure 4: Example 2. Top row: (left): evolution of the background distribution according to the

linear advection equation (28) with α= 0.05, (right) expected solution of the (1) and bounded

conﬁdence interactions with ∆ = 1.0 obtained with the stochastic Galerkin SPGscheme and semi-

implicit time integration for h= 0,...,5, in red we represent the estimated conﬁdence bands.

Bottom row: evolution over the time interval [0,20] of the expected solution and of its variance.

We considered N= 41 and ∆t= CFL∆v, CFL = 0.5 so that the solution of the scheme advection

equation is stable.

4.2 Example 2: Evolving background distribution

In this section we test the performance of the introduced structure preserving stochastic Galerkin

scheme in the case of an evolving background distribution. To exemplify a dynamic background

distribution we consider the solution of a linear advection equation

∂tg(v, t) + α∂vg(v, t) = 0, α > 0,(28)

which is coupled to the original stochastic Fokker-Planck equation in (1) through the operator

B[g](v, t). The initial background is considered of the form (24), with ug=−1

2, we consider

periodic boundary conditions for (28) and α= 0.05. The advection equation is solved numerically

with a Lax-Wendroﬀ scheme for each time t≥0. In the following we consider as uncertain initial

distribution (26) with ¯u= 0.5, κ= 0.25, and the mass is ρ(θ) = 1 + 1

2θwith uniform perturbation

θ∼ U([−1,1]).

In Table 2we estimate the order of convergence of the stochastic Galerkin structure preserving

scheme in terms of the expected quantities E(f), Var(f) in the case of a background distribution

evolving as (28). The numerical integration is a second-order semi-implicit method, see [3,22].

The approximation of the variance we considered M= 5 projections. We can observe that the

16

E[f]SPk

Time 2 4 6 G

1 1.8976 2.0834 2.0972 2.0975

5 2.4162 4.7225 4.7940 4.7955

10 2.6446 4.5139 4.5082 4.5082

20 2.0685 2.5829 2.6271 2.6273

Var(f)SPk

Time 2 4 6 G

1 1.8834 2.1088 1.8565 1.8568

5 2.4162 4.3191 4.3937 4.3953

10 2.5793 4.5869 4.5809 4.5809

20 2.2616 2.4154 2.4678 2.4679

Table 2: Example 2. Estimation of the order of convergence of the scheme in the case of dy-

namic background g(v, t) for second order semi-implicit method. The evolution of the background

distribution follows an advection equation with α= 0.05. The rates have been computed using

N= 21,41,81, σ2/2=0.1, ∆t= CFL∆vwith CFL = 0.5.

dynamic background distribution prevents the formation of steady state solution of the original

problem. Indeed, for each SPk,k= 2,4,6, G the scheme initially increases its order according to

the quadrature method and for large times it is reduced to the initial second-order.

In Figure 4we can observe the behavior of (1) in the bounded domain v∈[−1,1] and inter-

acting through a bounded conﬁdence type P(v, v∗) with ∆ = 1, where the evolving background

follows the advection (28).

4.3 Example 3: 2D model of swarming

We consider a kinetic swarming model with self-propulsion and diﬀusion with uncertain initial

distribution. In the deterministic framework we refer to [5] for the nonlinear case of the model

which leads to a provable sharp phase transition that discriminate the minimal amount of noise

needed to obtain symmetric distribution with zero mean. The study of possible uncertain quantit-

ies in the dynamics is here of paramount importance since coeﬃcients like the noise intensity and

the self-propulsion strength are commonly based on ﬁeld observations and empirical evidence. We

refer to [6] for a more detailed analysis of the inﬂuence of uncertain quantities in problems with

phase transition. In the following, we consider the case of uncertain initial distribution.

We consider a model for the evolution of the density of individuals f=f(θ, v, t) having velocity

v∈R2at time t≥0 and uncertain initial condition f(θ, v, 0) having mass ρ(θ). In details the

model reads

∂tf(θ, v, t) = ∇v·αv(|v|2−1)f(θ, v, t)+(v−ug)f(θ, v, t) + D∇vf(θ, v, t),(29)

being α > 0 the self-propulsion strength and D > 0 a constant noise intensity. At a diﬀerence with

the original nonlinear case here the agents interact with a background distribution g(v) through

its mean velocity ug=RVvg(v)dv. It may be shown that a free energy functional is deﬁned which

dissipates along solutions. Further, stationary solutions have the form

f∞(θ, v) = C(θ) exp −1

Dα|v|4

4+ (1 −α)|v|2

2−ug·v,

with C(θ)>0 a normalization constant. In particular, we focus on the 2D case and we consider

the ﬁxed background distribution

g(v) = 1

2πσxσy

exp −1

2(vx−µx)2

σ2

x

+(vy−µy)2

σ2

y, v = (vx, vy).(30)

17

(a) D= 0.8, µx=µy= 0 (b) D= 0.2, µx=µy= 0

(c) D= 0.8, µx=µy= 1 (d) D= 0.2, µx=µy= 1

(e) D= 0.8, µx= 0, µy= 1 (f) D= 0.2, µx= 0, µy= 1

Figure 5: Example 3. Expected solutions at time T= 100 of the 2D swarming model (29) with

initial uncertain distribution of mass ρ(θ) = 1 + 0.5θ,θ∼ U ([−1,1]) and several background

distributions (30) obtained through the structure preserving stochastic Galerkin method h=

0,...,10. Uniform grid for the velocity domain [−4,4] ×[−4,4] with N= 51 gridpoints in both

directions, time integration with second order semi-implicit method ∆t=O(∆v). We visualize

the upper conﬁdence band through the red mesh.

18

The extension of the presented structure preserving methods to the multidimensional case has

been established in [22,23]. The idea is to apply a structure preserving scheme to each dimension

of the stochastic Galerkin projections

∂tˆ

fh(v, t) = ∇v·hαv(|v|2−1) ˆ

fh(v, t)+(v−ug)ˆ

fh(v, t) + D∇vˆ

fh(v, t)i, h = 0, . . . , M (31)

with initial distribution ˆ

fh(v, 0) = E[f(θ, v , 0)Φh(θ)]. In Figure 5we present the large time

distributions for the choices of diﬀusion D= 0.2, D= 0.8 and three conﬁguration of the back-

ground distribution. We applied the SPGscheme to (31) for h= 0,...,5 to obtain the approx-

imation of expected distribution and of the variance. The initial distribution is here such that

RVf(θ, v, 0)dv = 1 + 1

2θ,θ∼ U([−1,1]).

A uniform grid over [−4,4] ×[−4,4] with N= 51 gridpoints in each direction has been con-

sidered. The time integration over the time interval [0,100] has been performed taking advantage

of the second order semi-implicit method with ∆t=O(∆v). The surfaces represent the expected

solution whereas the red grids represent the upper conﬁdence band that may be computed as usual

as E[f] + pVar(f).

We can clearly observe the inﬂuence of the background in shaping the large time distribution

of the problem, which is steered towards the background mean. The computed conﬁdence bands,

furthermore, make clear how the behavior is stable under the action of initial uncertainties.

Conclusion

We studied the application of structure preserving type schemes to the stochastic Galerkin ap-

proximation of Fokker-Planck equations with uncertain initial distribution and background inter-

actions. The developed methods are capable to preserve the stationary state of the problem with

arbitrary accuracy and deﬁne nonnegative expected solutions under suitable time step restrictions.

Both explicit and semi-implicit type time integrations have been taken into account. Furthermore,

we have proven discrete relative entropy dissipation property for the derived scheme for each pro-

jection of the original model. Several applications to prototype problems in socio-economic and life

sciences have been proposed both in case of ﬁxed and evolving background distribution together

with the extension of the method to the multidimensional case. Extensions of the scheme to the

case of vanishing diﬀusion and for more general anisotropic diﬀusion functions are under study

both in the deterministic and uncertain setting.

Acknowledgments

This work has been written within the activities of GNFM (Gruppo Nazionale per la Fisica Mate-

matica) of INdAM (Istituto Nazionale di Alta Matematica), Italy. The author acknowledges partial

support from the Excellence Project CUP: E11G18000350001 of the Department of Mathematical

Sciences “G. L. Lagrange”, Politecnico di Torino, Italy.

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