21 research outputs found
Linear Parabolic Problems in Random Moving Domains
We consider linear parabolic equations on a random non-cylindrical domain.
Utilizing the domain mapping method, we write the problem as a partial
differential equation with random coefficients on a cylindrical deterministic
domain. Exploiting the deterministic results concerning equations on
non-cylindrical domains, we state the necessary assumptions about the velocity
filed and in addition, about the flow transformation that this field generates.
In this paper we consider both cases, the uniformly bounded with respect to the
sample and log-normal type transformation. In addition, we give an explicit
example of a log-normal type transformation and prove that it does not satisfy
the uniformly bounded condition. We define a general framework for considering
linear parabolic problems on random non-cylindrical domains. As the first
example, we consider the heat equation on a random tube domain and prove its
well-posedness. Moreover, as the other example we consider the parabolic Stokes
equation which illustrates the case when it is not enough just to study the
plain-back transformation of the function, but instead to consider for example
the Piola type transformation, in order to keep the divergence free property
Feedback Loops in Opinion Dynamics of Agent-Based Models with Multiplicative Noise
We introduce an agent-based model for co-evolving opinions and social dynamics, under the influence of multiplicative noise. In this model, every agent is characterized by a position in a social space and a continuous opinion state variable. Agents’ movements are governed by the positions and opinions of other agents and similarly, the opinion dynamics are influenced by agents’ spatial proximity and their opinion similarity. Using numerical simulations and formal analyses, we study this feedback loop between opinion dynamics and the mobility of agents in a social space. We investigate the behaviour of this ABM in different regimes and explore the influence of various factors on the appearance of emerging phenomena such as group formation and opinion consensus. We study the empirical distribution, and, in the limit of infinite number of agents, we derive a corresponding reduced model given by a partial differential equation (PDE). Finally, using numerical examples, we show that a resulting PDE model is a good approximation of the original ABM
Feedback Loops in Opinion Dynamics of Agent-Based Models with Multiplicative Noise
We introduce an agent-based model for co-evolving opinion and social
dynamics, under the influence of multiplicative noise. In this model, every
agent is characterized by a position in a social space and a continuous opinion
state variable. Agents' movements are governed by positions and opinions of
other agents and similarly, the opinion dynamics is influenced by agents'
spatial proximity and their opinion similarity. Using numerical simulations and
formal analysis, we study this feedback loop between opinion dynamics and
mobility of agents in a social space. We investigate the behavior of this ABM
in different regimes and explore the influence of various factors on appearance
of emerging phenomena such as group formation and opinion consensus. We study
the empirical distribution and in the limit of infinite number of agents we
derive a corresponding reduced model given by a partial differential equation
(PDE). Finally, using numerical examples we show that a resulting PDE model is
a good approximation of the original ABM
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Feedback Loops in Opinion Dynamics of Agent-Based Models with Multiplicative Noise
We introduce an agent-based model for co-evolving opinions and social dynamics, under the influence of multiplicative noise. In this model, every agent is characterized by a position in a social space and a continuous opinion state variable. Agents’ movements are governed by the positions and opinions of other agents and similarly, the opinion dynamics are influenced by agents’ spatial proximity and their opinion similarity. Using numerical simulations and formal analyses, we study this feedback loop between opinion dynamics and the mobility of agents in a social space. We investigate the behaviour of this ABM in different regimes and explore the influence of various factors on the appearance of emerging phenomena such as group formation and opinion consensus. We study the empirical distribution, and, in the limit of infinite number of agents, we derive a corresponding reduced model given by a partial differential equation (PDE). Finally, using numerical examples, we show that a resulting PDE model is a good approximation of the original ABM
Multilevel Representations of Isotropic Gaussian Random Fields on the Sphere
Series expansions of isotropic Gaussian random fields on with
independent Gaussian coefficients and localized basis functions are
constructed. Such representations provide an alternative to the standard
Karhunen-Lo\`eve expansions of isotropic random fields in terms of spherical
harmonics. Their multilevel localized structure of basis functions is
especially useful in adaptive algorithms. The basis functions are obtained by
applying the square root of the covariance operator to spherical needlets.
Localization of the resulting covariance-dependent multilevel basis is shown
under decay conditions on the angular power spectrum of the random field. In
addition, numerical illustrations are given and an application to random
elliptic PDEs on the sphere is analyzed
Exponential stability of the flow for a generalised Burgers equation on a circle
The paper deals with the problem of stability for the flow of the 1D Burgers
equation on a circle. Using some ideas from the theory of positivity preserving
semigroups, we establish the strong contraction in the norm. As a
consequence, it is proved that the equation with a bounded external force
possesses a unique bounded solution on , which is exponentially
stable in as . In the case of a random external force, we
show that the difference between two trajectories goes to zero with probability
.Comment: 13 page
Random partial differential equations on evolving hypersurfaces
Partial differential equations with random coefficients (random PDEs) is a
very developed and popular field. The variety of applications, especially in biology, motivate us to consider the random PDEs on curved moving domains. We introduce and analyse the advection-diffusion equations with random coefficients on moving hypersurfaces. We consider both cases, uniform and log-normal distributions of coefficients. Furthermore, we will introduce and analyse a surface finite element discretisation of the equation. We show unique solvability of the resulting semi-discrete problem and prove optimal error bounds for the semi-discrete solution and Monte Carlo samplings of its expectation.
Our theoretical findings are illustrated by numerical experiments. In the end
we present an outlook for the case when the velocity of a hypersurface is
an uniformly bounded random field and the domain is flat
From interacting agents to density-based modeling with stochastic PDEs
Many real-world processes can naturally be modeled as systems of interacting
agents. However, the long-term simulation of such agent-based models is often
intractable when the system becomes too large. In this paper, starting from a
stochastic spatio-temporal agent-based model (ABM), we present a reduced model
in terms of stochastic PDEs that describes the evolution of agent number
densities for large populations. We discuss the algorithmic details of both
approaches; regarding the SPDE model, we apply Finite Element discretization in
space which not only ensures efficient simulation but also serves as a
regularization of the SPDE. Illustrative examples for the spreading of an
innovation among agents are given and used for comparing ABM and SPDE models
An evolving space framework for Oseen equations on a moving domain
This article considers non-stationary incompressible linear fluid equations in a moving domain. We demonstrate the existence and uniqueness of an appropriate weak formulation of the problem by making use of the theory of time-dependent Bochner spaces. It is not possible to directly apply established evolving Hilbert space theory due to the incompressibility constraint. After we have established the well-posedness, we derive and analyse a first order time discretisation of the system
Non-stationary incompressible linear fluid equations in a moving domain
This article considers non-stationary incompressible linear fluid equations
in a moving domain. We demonstrate the existence and uniqueness of an
appropriate weak formulation of the problem by making use of the theory of
time-dependent Bochner spaces. It is not possible to directly apply established
evolving Hilbert space theory due to the incompressibility constraint. After we
have established the well-posedness, we derive and analyse a time
discretisation of the system