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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 external distribution called the background. The proposed methods are capable to preserve physical properties in the approximation of statistical moments of the problem like nonnegativity, 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.
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 IΘR, and with known probability distribution
function Ψ(θ) : IΘR+. 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), vVRdv,θIΘ
and t0 is the time. The introduced distribution represents the proportion of particles/agents
in [v, v +dv] at time t0 and for given value of uncertainty θIΘ. 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 vVRdvand B[·] is the operator
B[g](v, t) = ZV
P(v, v)(vv)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 θIΘ. 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 σalgebra and Pa probability measure, and let us defined a
random variable
θ: (Ω, F )(IΘ,BR),
with IΘRand BRis the Borel set. We focus on real-valued distributions of the form f(θ, v, t) :
×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
Eh(θk(θ)] = ZIΘ
Φh(θk(θ)Ψ(θ)=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 fL2(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, tk(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 , tk(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)]2E[(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, t2
k(θ)+2
M
X
k=0
k1
X
h=0
ˆ
fk(v, t)ˆ
fh(v, tk(θh(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)E2
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, th(θ)!2
dvΨ(θ)dθ.
We can reformulate the problem (5) in a more compact form as follows
tˆ
f=vhBˆ
f+Dvˆ
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 kvB[g](v, t)kLCB, with CB>0, and if DCDwe have
kˆ
f(v, t)k2
L2et(CB+2CD)kˆ
f(v, 0)k2
L2
Proof. We multiply (5) by ˆ
fh(v, t) and integrate over VR
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)=ZV2
vˆ
fh(v, t)D(v)ˆ
fh(v, t)dv
≤ −CDZVvˆ
fh(v, t)2.
Finally, after summation h= 0, . . . , M of (8) we obtained
1
2kˆ
f(v, t)k2
L2CB
2kˆ
f(v, t)k2
L2− kvˆ
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, vh(θ)Ψ(θ)=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 θIΘ. 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 θIΘ.
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)vV = 0.
The following result holds
5
Theorem 2. Let the smooth function Φ(x),xR+be convex. Then, if F(θ, v, t)is the solution
to (13)in VRand F(θ, v, t)is bounded for all θIΘthe 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, v00 (F(θ, v , t)) |vF(θ, v, t)|2dv.
Proof. The proof of this result follows the strategy adopted in [13] for all θIΘ.
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 θIΘby
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, tk(θ) log PM
k=0 ˆ
fk(v, tk(θ)
PM
k=0 ˆ
f
k(v, tk(θ)dv
=ZV
D(v)
M
X
k=0
ˆ
fk(v, tk(θ) vlog PM
k=0 ˆ
fk(v, tk
PM
k=0 ˆ
f
k(v, tk(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),
Ihk =ZIΘvlog fM(θ, v, t)vlog fM ,(θ, v)2Φh(θ).
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),xR+be convex. Then, if Fh(v, t)is the solution
to (15)in VRand 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(v00(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 viV, 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,i1/2(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
1exp(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 t0the 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=nt, ∆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,i1/2
v,(21)
where the flux has the form introduced in (18). We can prove the following result
Theorem 4. Under the time step restriction
tv2
2 (Mv+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
i1/2
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 t0and h= 0 ...,M.
The numerical flux (18)satisfies the discrete entropy dissipation
d
dtHv(f, f ) = −I(f, f ),
where
Hv(ˆ
fh,ˆ
f)=∆v
N
X
i=0
ˆ
fi,h log ˆ
fh,i
ˆ
f
h,i !,
and
Iv(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+1/20.
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,i1/2
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 (xy) 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 t0 of the form
g(v) = βexp (wug)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(|vv| ≤ ∆),(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 t0, 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 (vu1(θ))2
2σ2
0+ exp (vu2(θ))2
2σ2
0,(26)
with C(θ) such that RVf(θ, v, 0) = ρ(θ)>0 for all θIΘand 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 P1 and we can compute the explicit
stationary distribution
f(θ, v) = C(θ)
(1 v2)21 + v
1vug/(2σ2)
exp 1ugv
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
1vug/(2σ2)
exp (1ugv
σ2
f(1 v2)).(27)
being Ch=1
2RIΘC(θh(θ). 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 ·101. 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= ∆w22.
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 t0 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 t0 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 (vug)2
2σ2
g+ exp (v+ug)2
2σ2
g,
with ug=1
2and σ2
g= 102. 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 t0. 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
vR2at time t0 and uncertain initial condition f(θ, v, 0) having mass ρ(θ). In details the
model reads
tf(θ, v, t) = v·αv(|v|21)f(θ, v, t)+(vug)f(θ, v, t) + Dvf(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
2ug·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(vxµx)2
σ2
x
+(vyµy)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|21) ˆ
fh(v, t)+(vug)ˆ
fh(v, t) + Dvˆ
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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