Monte Carlo gPC Methods for Diffusive Kinetic Flocking Models with Uncertainties
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Abstract
In this paper we introduce and discuss numerical schemes for the approximation of kinetic equations for flocking behavior with phase transitions that incorporate uncertain quantities. This class of schemes here considered make use of a Monte Carlo approach in the phase space coupled with a stochastic Galerkin expansion in the random space. The proposed methods naturally preserve the positivity of the statistical moments of the solution and are capable to achieve high accuracy in the random space. Several tests on a kinetic alignment model with self propulsion validate the proposed methods both in the homogeneous and inhomogeneous setting, shading light on the influence of uncertainties in phase transition phenomena driven by noise such as their smoothing and confidence band.
Keywords
Uncertainty quantification Stochastic Galerkin Collective behaviorMathematics Subject Classification (2010)
35Q83 65C05 65M701 Introduction
Uncertainty quantification (UQ) for partial differential equations describing real world phenomena gained in recent years lot of momentum in various communities. Without intending to review the very huge literature on this topic we mention [3, 18, 22, 34, 40, 44, 45, 46, 48, 49] and the references therein. One of the main advantages of this approach relies in its capability to provide a sound mathematical framework to reproduce realistic experiments through the introduction of the stochastic quantities, reflecting our incomplete information on some features on the systems’ modelling. This argument is especially true for the description of emergent social structures in interacting agents’ systems in socioeconomic and life sciences. Common examples are the emergence of consensus phenomena in opinion dynamics, flocking and milling patterns in swarming of animals or humans and the formation of stable wealth distributions in economic systems, see [39]. It is worth observing how for these models we can have at most statistical information on initial conditions and on the modeling parameters, which are in practice substituted by empirical social forces embedding a huge variability, see for example [5].
As a followup question to the progress of the analytical understanding of real phenomena, the existence of phase transitions driven by noise represents a deeply fascinating issue, see [17, 19, 23, 25, 28, 29] and the references therein. The notion of phase transition has been fruitfully borrowed from thermodynamics to highlight the phase change of the system under specific stimuli. In particular, it is of interest the emergence of patterns in the collective dynamics for critical strengths of noise as in the classical Kuramoto model [8, 14, 37, 42] or in collective behavior [6, 28]. In the present paper we concentrate on a flocking model for interacting agents with selfpropulsion and diffusion where the parameters are assumed to be stochastic. This model has been recently investigated in [1, 6, 7, 10] in absence of uncertainties. The introduction of uncertain quantities points in the direction of a more realistic description of the underlying processes and helps us to compute possible deviations from the prescribed deterministic behavior.
Suitable numerical methods that preserve the positivity of the distribution function are developed and are based on the socalled of Monte Carlo generalized polynomial chaos (MCgPC) methods. The introduced class of schemes is based on a Monte Carlo approach in the phase space that is coupled with stochastic Galerkin decomposition in the random space [34, 35, 39, 46]. This method has been recently proposed in [18] in the case of zero diffusion and it will be extended for models incorporating noise in the present setting. Furthermore, beside the natural positivity preservation, the resulting methods are spectrally accurate in the random space for sufficiently regular uncertainties. Furthermore, the adoption of a spectrally accurate methods in the random space is particularly efficient if compared with other nonintrusive approaches.
The rest of this paper has the following structure: in Section 2 we introduce in detail the class of kinetic flocking models of interest, the issue of phase transition will be treated with a particular focus on the relation between selfpropulsion strength and noise intensity in the case they both depend on uncertain quantities. Section 3 focuses on the construction of Monte Carlo generalized polynomial chaos methods, a reduction in the computational complexity is here achieved through a Monte Carlo meanfield algorithm discussed in previous works. Finally, Section 4 is devoted to numerical test for the validation of the proposed schemes. Here continuous and Monte Carlo schemes will be compared to show the effectiveness of the MCgPC approach.
2 Phase Transitions in Kinetic Flocking Models with Uncertainties
Kinetic models for aggregationdiffusion dynamics encountered extensive investigation in recent years [6, 7, 11, 13, 15, 16, 24, 26]. This class of models describes the aggregate behavior of large systems of selfpropelled particles for which stable patterns emerge asymptotically. Nevertheless, at the present times, the study of the influence of ineradicable uncertainties in the modeling parameters is quite an unexplored area. We mention in this direction [3, 18, 30, 31, 44].
In the following we will treat separately the case spacehomogeneous case and then we will provide some insights on the nonlocalised inhomogeneous setting.
2.1 Space Homogeneous Problem
In [6] the authors investigated the space homogeneous version of (3) in the deterministic setting. It has been proven that a phase transition between the socalled polarized and unpolarized motion takes place as the noise intensity D increases and for a specific range of the values of the selfpropulsion strength α. At a difference with the cited results, the present formulation of the model embeds from the very beginning the presence of uncertain quantities.
Theorem 1

For small enough diffusion\(D(\theta )\rightarrow 0\)for all 𝜃 ∈ I_{𝜃}there is a function u = u(D(𝜃)) with\(\lim _{D\rightarrow 0}u(D(\theta )) = 1\), such that\(f^{\infty }(\theta ,v)\)with\(u=(u(D(\theta )),0,\dots ,0)\)is a stationary solution of the original problem.

For large enough diffusion coefficient Dthe only stationary solution is the symmetric distribution given by (4) with u_{f} ≡ 0.
This behavior is reminiscent of the one observed for the Vicsek model [6, 23], where agents move with a constant speed and interact with their neighbors via a local alignment force and are subject to noise action. In particular, a critical noise intensity has been discovered whose value determines a phase transition between the ordered states and a chaotic state characterized by a null asymptotic average velocity of the system of agents. The critical noise value is theoretically demonstrated in [23] for the Vicsek model while for the present model there is a strong numerical evidence for its existence [6].
2.2 Space Inhomogeneous Case
In the following we will present a numerical insight for the case of a VFP model with Cucker–Smale type interaction forces (5) in dimension d_{x} = 1 and d_{v} = 1, selfpropulsion forces and uncertain diffusion coefficient. In particular, numerical results obtained with accurate numerical schemes highlight that a phase transition can occur also in this regime, shedding light on the deep interplay between alignment forces and noise strength in more general settings with not localized alignment [20, 38].
3 Stochastic Galerkin Methods for Kinetic Equations
We introduce Stochastic Galerkin (SG) numerical methods with applications to the nonlinear Vlasov–Fokker–Planck (VFP) equation (1). We discuss the class of stochastic Galerkin (SG) methods and, in particular, we concentrate on the generalized Polynomial Chaos (gPC) decomposition [22, 46, 47, 49]. These methods gained increased popularity in recent years since they guarantees spectral accuracy in the random space under suitable regularity conditions.
In our schemes, we exploit Monte Carlo methods for the approximation of the numerical solution of the (VFP) in the phase space taking advantage of particle based reformulation of the problem that converges in distribution to the solution for an increasing number of agents. The core idea then is to apply SGgPC techniques for the efficient approximation of the random field in the resulting MC approximation. We highlight that the combination of the two approaches leads to a positive approximation of statistical moments of the solution even in the nonlinear case.
In the following we will derive a SGgPC scheme for the continuous problem and we will consider specific formulation for the socalled Monte Carlo gPC methods (MCgPC), see [18].
3.1 Preliminaries on SGgPC Expansion
Theorem 2
Proof
To tackle efficiently the general fully nonlinear case in the following we will introduce a novel scheme that exploits the spectral convergence in the random space of SGgCP methods and that naturally preserves the positivity of the numerical distribution.
3.2 Particle Based SGgPC Formulation of Kinetic Equations
As for classical spectral methods, the solution of the coupled SG system (6) for f^{M} looses its positivity and then a clear physical meaning. In order to overcome the difficulty recently the socalled Monte Carlo gPC (MCgPC) scheme has been proposed. These methods combine the advantages of a Monte Carlo approach for the approximation of f in the phase space and they conserve spectral accuracy in the random space. We refer to [18] for the study a wide range of meanfield equations describing the emergence of patterns and ordered behavior in large interacting systems in the zero diffusion limit.
The Monte Carlo (MC) method is a probabilistic particle method that describe the evolution of density functions by resorting to the computation of interactions in a finite set of randomly chosen particles of which it is known a priori the dynamics, see [21, 39]. In MC methods the updated distribution function in the phase space is typically reconstructed from particles as postprocessing. Several approaches are possible for the reconstruction step, which can estimated through a histogram, weighted integration rule methods or by a convolution of the empirical particle distribution with a suitable mollifier, see [33]. All the mentioned approximations preserve the positivity of the obtained numerical distribution function. Concerning the order of accuracy for MC methods, we have a convergence rate of the order \(\mathcal O(\frac {1}{\sqrt {N}})\) where N is the number of considered samples, see [12].
Theorem 3
In the next section we derive a SGgPC decomposition of (10) so that we will preserve the exponential convergence in the random space with respect to all the uncertain quantities.
3.2.1 Stochastic Galerkin Scheme for the Particle System
The convergence of the SGgPC expansion (6) for sufficiently regular function follows from standard results in polynomial approximation theory, we recall for example [27]. In general, thanks to the property of the introduced polynomial basis we have
Theorem 4
Theorem 5
Proof
In the last section we will give numerical evidence of this result.
3.2.2 Monte Carlo gPC Scheme
We now approximate the limiting stochastic kinetic equation taking advantage of the particle reformulation of the problem. In fact, since the solution of the system of SDEs (10) converges in distribution to the solution of the original problem (1) for \(N\rightarrow +\infty \), we can approximate the original dynamics by means of a Monte Carlo (MC) method in the phase space. The main drawback of this approach lies in the computational cost \(\mathcal {O}(M^{2}N^{2})\), since at each time step and for each gPC projection each agent modifies its velocity in a genuine nonlinear way.
For the reconstruction of expected quantities, in the present manuscript we consider the histogram of position and velocity of the set of particles in the phase space, we point the reader to [33] for possible alternatives. Thanks to the MC approach the resulting method preserves the positivity of the expected distribution function.
Remark 1
4 Numerical Tests
In this section we present several numerical examples based on (3) both in the homogeneous and inhomogeneous cases. We test the effectiveness of the MCgPC scheme through several tests based on VFP equations. In all test the integration of the system of stochastic differential equations (10) is performed through a standard Euler–Maruyama method whereas the solution of the system of PDEs derived from the SG procedure is solved through a standard central scheme coupled with a fourth order Runge–Kutta integration. In the whole section we will consider a uniform noise, therefore Gauss–Legendre polynomial basis are chosen in the gPC setting. Numerical investigations on the influence of uncertainties in phase transition phenomena are presented through the section. Finally, we explore the non localized inhomogeneous model with Cucker–Smale type interactions.
4.1 Test 1: Space Homogeneous Case
We can easily observe the regions of maximal sensitivity with respect to the presence of uncertainties. In particular, for high diffusion values the variability of the expected average velocity vanishes and we may argue that the phase transition predicted in [6] is actually a quite stable pattern in the space homogeneous regime. Nevertheless, the averaging of uncertain quantities acts as a smoothing factor of the phase transition as we can clearly observe in Fig. 4b and d. For a vanishing influence of 𝜃 ∈ I_{𝜃} given by \(\lambda \rightarrow 0\) in (16) the transition becomes sharper coherently with the deterministic case, see Fig. 4d. Summarizing the main effect of the uncertainties is the smoothing of the transition point making it less sharper and abrupt than in the deterministic cases.
4.2 Test 2. Space Inhomogeneous Case
In this section we focus on inhomogeneous models. First we consider the localised case for which a phase transition is expected to happen similarly to the homogeneous case, see [6], even if this has not yet been proved. To explore other possible alternatives we will also consider the case of Cucker–Smale type interactions for which no analogous theoretical results regarding phase transitions currently exist. In all the tests of the present section we consider a sample of N = 10^{5} particles for the MCgPC scheme.
4.2.1 Test 2A. Localized Interaction Case
4.2.2 Test 2B. Cucker–Smale Type Interactions
5 Conclusion
The introduction of uncertainties in swarming dynamics is of paramount importance for the largescale description of realistic phenomena. We concentrated in this manuscript on the case of Vlasov–Fokker–Planck equations for emerging collective behavior with phase transitions depending on the strength of selfpropelling and noise terms. We extended to these models the construction of nonnegative gPC schemes for kinetictype equations. The proposed approach takes advantage of a Monte Carlo approximation of the kinetic equation in phase space which is coupled with a stochastic Galerkin gPC expansion. Several numerical tests were proposed both in the homogeneous and inhomogeneous settings, proving the effectiveness of the approach and shedding light on the action of uncertainties in terms of the stability of the phase transitions, We generically observe that they lead to a smoothing of the sharp transition values. Several extensions of the present work are possible from both the analytical and numerical viewpoints and in connection with more general kernels in the collision/interaction dynamics.
Notes
Acknowledgements
JAC was partially supported by the EPSRC grant number EP/P031587/1. MZ is member of the National Group on Mathematical Physics (GNFM) of INdAM (Istituto Italiano di Alta Matematica), Italy. This work has been written within the activities of the Excellence Project CUP: E11G18000350001 of the Department of Mathematical Sciences ”G. L. Lagrange” of Politecnico di Torino funded by MIUR (Italian Ministry for Education, University and Research).
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