325 research outputs found
Mean-field optimal control and optimality conditions in the space of probability measures
We derive a framework to compute optimal controls for problems with states in
the space of probability measures. Since many optimal control problems
constrained by a system of ordinary differential equations (ODE) modelling
interacting particles converge to optimal control problems constrained by a
partial differential equation (PDE) in the mean-field limit, it is interesting
to have a calculus directly on the mesoscopic level of probability measures
which allows us to derive the corresponding first-order optimality system. In
addition to this new calculus, we provide relations for the resulting system to
the first-order optimality system derived on the particle level, and the
first-order optimality system based on -calculus under additional
regularity assumptions. We further justify the use of the -adjoint in
numerical simulations by establishing a link between the adjoint in the space
of probability measures and the adjoint corresponding to -calculus.
Moreover, we prove a convergence rate for the convergence of the optimal
controls corresponding to the particle formulation to the optimal controls of
the mean-field problem as the number of particles tends to infinity
Instantaneous control of interacting particle systems in the mean-field limit
Controlling large particle systems in collective dynamics by a few agents is
a subject of high practical importance, e.g., in evacuation dynamics. In this
paper we study an instantaneous control approach to steer an interacting
particle system into a certain spatial region by repulsive forces from a few
external agents, which might be interpreted as shepherd dogs leading sheep to
their home. We introduce an appropriate mathematical model and the
corresponding optimization problem. In particular, we are interested in the
interaction of numerous particles, which can be approximated by a mean-field
equation. Due to the high-dimensional phase space this will require a tailored
optimization strategy. The arising control problems are solved using adjoint
information to compute the descent directions. Numerical results on the
microscopic and the macroscopic level indicate the convergence of optimal
controls and optimal states in the mean-field limit,i.e., for an increasing
number of particles.Comment: arXiv admin note: substantial text overlap with arXiv:1610.0132
An analytical framework for a consensus-based global optimization method
In this paper we provide an analytical framework for investigating the
efficiency of a consensus-based model for tackling global optimization
problems. This work justifies the optimization algorithm in the mean-field
sense showing the convergence to the global minimizer for a large class of
functions. Theoretical results on consensus estimates are then illustrated by
numerical simulations where variants of the method including nonlinear
diffusion are introduced
Port-Hamiltonian structure of interacting particle systems and its mean-field limit
We derive a minimal port-Hamiltonian formulation of a general class of
interacting particle systems driven by alignment and potential-based force
dynamics which include the Cucker-Smale model with potential interaction and
the second order Kuramoto model. The port-Hamiltonian structure allows to
characterize conserved quantities such as Casimir functions as well as the
long-time behaviour using a LaSalle-type argument on the particle level. It is
then shown that the port-Hamiltonian structure is preserved in the mean-field
limit and an analogue of the LaSalle invariance principle is studied in the
space of probability measures equipped with the 2-Wasserstein-metric. The
results on the particle and mean-field limit yield a new perspective on uniform
stability of general interacting particle systems. Moreover, as the minimal
port-Hamiltonian formulation is closed we identify the ports of the subsystems
which admit generalized mass-spring-damper structure modelling the binary
interaction of two particles. Using the information of ports we discuss the
coupling of difference species in a port-Hamiltonian preserving manner
Ensemble-based gradient inference for particle methods in optimization and sampling
We propose an approach based on function evaluations and Bayesian inference
to extract higher-order differential information of objective functions {from a
given ensemble of particles}. Pointwise evaluation of some
potential in an ensemble contains implicit information about
first or higher order derivatives, which can be made explicit with little
computational effort (ensemble-based gradient inference -- EGI). We suggest to
use this information for the improvement of established ensemble-based
numerical methods for optimization and sampling such as Consensus-based
optimization and Langevin-based samplers. Numerical studies indicate that the
augmented algorithms are often superior to their gradient-free variants, in
particular the augmented methods help the ensembles to escape their initial
domain, to explore multimodal, non-Gaussian settings and to speed up the
collapse at the end of optimization dynamics.}
The code for the numerical examples in this manuscript can be found in the
paper's Github repository
(https://github.com/MercuryBench/ensemble-based-gradient.git)
Time-continuous microscopic pedestrian models: an overview
We give an overview of time-continuous pedestrian models with a focus on
data-driven modelling. Starting from pioneer, reactive force-based models we
move forward to modern, active pedestrian models with sophisticated
collision-avoidance and anticipation techniques through optimisation problems.
The overview focuses on the mathematical aspects of the models and their
different components. We include methods used for data-based calibration of
model parameters, hybrid approaches incorporating neural networks, and purely
data-based models fitted by deep learning. Some development perspectives of
modelling paradigms we expect to grow in the coming years are outlined in the
conclusion.Comment: 26 pages; chapter accepted for publication in Crowd Dynamics (vol. 4
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