130 research outputs found
Testing quantum mechanics: a statistical approach
As experiments continue to push the quantum-classical boundary using
increasingly complex dynamical systems, the interpretation of experimental data
becomes more and more challenging: when the observations are noisy, indirect,
and limited, how can we be sure that we are observing quantum behavior? This
tutorial highlights some of the difficulties in such experimental tests of
quantum mechanics, using optomechanics as the central example, and discusses
how the issues can be resolved using techniques from statistics and insights
from quantum information theory.Comment: v1: 2 pages; v2: invited tutorial for Quantum Measurements and
Quantum Metrology, substantial expansion of v1, 19 pages; v3: accepted; v4:
corrected some errors, publishe
Dynamic Models and Nonlinear Filtering of Wave Propagation in Random Fields
In this paper, a general model of wireless channels is established based on
the physics of wave propagation. Then the problems of inverse scattering and
channel prediction are formulated as nonlinear filtering problems. The
solutions to the nonlinear filtering problems are given in the form of dynamic
evolution equations of the estimated quantities. Finally, examples are provided
to illustrate the practical applications of the proposed theory.Comment: 12 pages, 1 figur
Duality for nonlinear filtering
This thesis is concerned with the stochastic filtering problem for a hidden
Markov model (HMM) with the white noise observation model. For this filtering
problem, we make three types of original contributions: (1) dual
controllability characterization of stochastic observability, (2) dual minimum
variance optimal control formulation of the stochastic filtering problem, and
(3) filter stability analysis using the dual optimal control formulation.
For the first contribution of this thesis, a backward stochastic differential
equation (BSDE) is proposed as the dual control system. The observability
(detectability) of the HMM is shown to be equivalent to the controllability
(stabilizability) of the dual control system. For the linear-Gaussian model,
the dual relationship reduces to classical duality in linear systems theory.
The second contribution is to transform the minimum variance estimation
problem into an optimal control problem. The constraint is given by the dual
control system. The optimal solution is obtained via two approaches: (1) by an
application of maximum principle and (2) by the martingale characterization of
the optimal value. The optimal solution is used to derive the nonlinear filter.
The third contribution is to carry out filter stability analysis by studying
the dual optimal control problem. Two approaches are presented through Chapters
7 and 8. In Chapter 7, conditional Poincar\'e inequality (PI) is introduced.
Based on conditional PI, various convergence rates are obtained and related to
literature. In Chapter 8, the stabilizability of the dual control system is
shown to be a necessary and sufficient condition for filter stability on
certain finite state space model.Comment: Ph.D. Thesis of the autho
Solution properties of the incompressible Euler system with rough path advection
The present paper aims to establish the local well-posedness of Euler's fluid equations on geometric rough paths. In particular, we consider the Euler equations for the incompressible flow of an ideal fluid whose Lagrangian transport velocity possesses an additional rough-in-time, divergence-free vector field. In recent work, we have demonstrated that this system can be derived from Clebsch and Hamilton-Pontryagin variational principles that possess a perturbative geometric rough path Lie-advection constraint. In this paper, we prove the local well-posedness of the system in -Sobolev spaces with integer regularity and establish a Beale-Kato-Majda (BKM) blow-up criterion in terms of the -norm of the vorticity. In dimension two, we show that the -norms of the vorticity are conserved, which yields global well-posedness and a Wong-Zakai approximation theorem for the stochastic version of the equation.publishedVersionPaid open acces
Sequential Bayesian inference for static parameters in dynamic state space models
A method for sequential Bayesian inference of the static parameters of a
dynamic state space model is proposed. The method is based on the observation
that many dynamic state space models have a relatively small number of static
parameters (or hyper-parameters), so that in principle the posterior can be
computed and stored on a discrete grid of practical size which can be tracked
dynamically. Further to this, this approach is able to use any existing
methodology which computes the filtering and prediction distributions of the
state process. Kalman filter and its extensions to non-linear/non-Gaussian
situations have been used in this paper. This is illustrated using several
applications: linear Gaussian model, Binomial model, stochastic volatility
model and the extremely non-linear univariate non-stationary growth model.
Performance has been compared to both existing on-line method and off-line
methods
Predictability of extreme events in a branching diffusion model
We propose a framework for studying predictability of extreme events in
complex systems. Major conceptual elements -- hierarchical structure, spatial
dynamics, and external driving -- are combined in a classical branching
diffusion with immigration. New elements -- observation space and observed
events -- are introduced in order to formulate a prediction problem patterned
after the geophysical and environmental applications. The problem consists of
estimating the likelihood of occurrence of an extreme event given the
observations of smaller events while the complete internal dynamics of the
system is unknown. We look for premonitory patterns that emerge as an extreme
event approaches; those patterns are deviations from the long-term system's
averages. We have found a single control parameter that governs multiple
spatio-temporal premonitory patterns. For that purpose, we derive i) complete
analytic description of time- and space-dependent size distribution of
particles generated by a single immigrant; ii) the steady-state moments that
correspond to multiple immigrants; and iii) size- and space-based asymptotic
for the particle size distribution. Our results suggest a mechanism for
universal premonitory patterns and provide a natural framework for their
theoretical and empirical study
- …