7,245 research outputs found
Discrete Lie Advection of Differential Forms
In this paper, we present a numerical technique for performing Lie advection
of arbitrary differential forms. Leveraging advances in high-resolution finite
volume methods for scalar hyperbolic conservation laws, we first discretize the
interior product (also called contraction) through integrals over Eulerian
approximations of extrusions. This, along with Cartan's homotopy formula and a
discrete exterior derivative, can then be used to derive a discrete Lie
derivative. The usefulness of this operator is demonstrated through the
numerical advection of scalar fields and 1-forms on regular grids.Comment: Accepted version; to be published in J. FoC
Geometry of the ergodic quotient reveals coherent structures in flows
Dynamical systems that exhibit diverse behaviors can rarely be completely
understood using a single approach. However, by identifying coherent structures
in their state spaces, i.e., regions of uniform and simpler behavior, we could
hope to study each of the structures separately and then form the understanding
of the system as a whole. The method we present in this paper uses trajectory
averages of scalar functions on the state space to: (a) identify invariant sets
in the state space, (b) form coherent structures by aggregating invariant sets
that are similar across multiple spatial scales. First, we construct the
ergodic quotient, the object obtained by mapping trajectories to the space of
trajectory averages of a function basis on the state space. Second, we endow
the ergodic quotient with a metric structure that successfully captures how
similar the invariant sets are in the state space. Finally, we parametrize the
ergodic quotient using intrinsic diffusion modes on it. By segmenting the
ergodic quotient based on the diffusion modes, we extract coherent features in
the state space of the dynamical system. The algorithm is validated by
analyzing the Arnold-Beltrami-Childress flow, which was the test-bed for
alternative approaches: the Ulam's approximation of the transfer operator and
the computation of Lagrangian Coherent Structures. Furthermore, we explain how
the method extends the Poincar\'e map analysis for periodic flows. As a
demonstration, we apply the method to a periodically-driven three-dimensional
Hill's vortex flow, discovering unknown coherent structures in its state space.
In the end, we discuss differences between the ergodic quotient and
alternatives, propose a generalization to analysis of (quasi-)periodic
structures, and lay out future research directions.Comment: Submitted to Elsevier Physica D: Nonlinear Phenomen
The inefficiency of re-weighted sampling and the curse of system size in high order path integration
Computing averages over a target probability density by statistical
re-weighting of a set of samples with a different distribution is a strategy
which is commonly adopted in fields as diverse as atomistic simulation and
finance. Here we present a very general analysis of the accuracy and efficiency
of this approach, highlighting some of its weaknesses. We then give an example
of how our results can be used, specifically to assess the feasibility of
high-order path integral methods. We demonstrate that the most promising of
these techniques -- which is based on re-weighted sampling -- is bound to fail
as the size of the system is increased, because of the exponential growth of
the statistical uncertainty in the re-weighted average
An evaluation of the accuracy of some radar wind profiling techniques
Major advances in Doppler radar measurement in optically clear air have made it feasible to monitor radial velocities in the troposphere and lower stratosphere. For most applications the three dimensional wind vector is monitored rather than the radial velocity. Measurement of the wind vector with a single radar can be made assuming a spatially linear, time invariant wind field. The components and derivatives of the wind are estimated by the parameters of a linear regression of the radial velocities on functions of their spatial locations. The accuracy of the wind measurement thus depends on the locations of the radial velocities. The suitability is evaluated of some of the common retrieval techniques for simultaneous measurement of both the vertical and horizontal wind components. The techniques considered for study are fixed beam, azimuthal scanning (VAD) and elevation scanning (VED)
Group Importance Sampling for Particle Filtering and MCMC
Bayesian methods and their implementations by means of sophisticated Monte
Carlo techniques have become very popular in signal processing over the last
years. Importance Sampling (IS) is a well-known Monte Carlo technique that
approximates integrals involving a posterior distribution by means of weighted
samples. In this work, we study the assignation of a single weighted sample
which compresses the information contained in a population of weighted samples.
Part of the theory that we present as Group Importance Sampling (GIS) has been
employed implicitly in different works in the literature. The provided analysis
yields several theoretical and practical consequences. For instance, we discuss
the application of GIS into the Sequential Importance Resampling framework and
show that Independent Multiple Try Metropolis schemes can be interpreted as a
standard Metropolis-Hastings algorithm, following the GIS approach. We also
introduce two novel Markov Chain Monte Carlo (MCMC) techniques based on GIS.
The first one, named Group Metropolis Sampling method, produces a Markov chain
of sets of weighted samples. All these sets are then employed for obtaining a
unique global estimator. The second one is the Distributed Particle
Metropolis-Hastings technique, where different parallel particle filters are
jointly used to drive an MCMC algorithm. Different resampled trajectories are
compared and then tested with a proper acceptance probability. The novel
schemes are tested in different numerical experiments such as learning the
hyperparameters of Gaussian Processes, two localization problems in a wireless
sensor network (with synthetic and real data) and the tracking of vegetation
parameters given satellite observations, where they are compared with several
benchmark Monte Carlo techniques. Three illustrative Matlab demos are also
provided.Comment: To appear in Digital Signal Processing. Related Matlab demos are
provided at https://github.com/lukafree/GIS.gi
Approximate Bayesian Computation in State Space Models
A new approach to inference in state space models is proposed, based on
approximate Bayesian computation (ABC). ABC avoids evaluation of the likelihood
function by matching observed summary statistics with statistics computed from
data simulated from the true process; exact inference being feasible only if
the statistics are sufficient. With finite sample sufficiency unattainable in
the state space setting, we seek asymptotic sufficiency via the maximum
likelihood estimator (MLE) of the parameters of an auxiliary model. We prove
that this auxiliary model-based approach achieves Bayesian consistency, and
that - in a precise limiting sense - the proximity to (asymptotic) sufficiency
yielded by the MLE is replicated by the score. In multiple parameter settings a
separate treatment of scalar parameters, based on integrated likelihood
techniques, is advocated as a way of avoiding the curse of dimensionality. Some
attention is given to a structure in which the state variable is driven by a
continuous time process, with exact inference typically infeasible in this case
as a result of intractable transitions. The ABC method is demonstrated using
the unscented Kalman filter as a fast and simple way of producing an
approximation in this setting, with a stochastic volatility model for financial
returns used for illustration
Discrete Geometric Structures in Homogenization and Inverse Homogenization with application to EIT
We introduce a new geometric approach for the homogenization and inverse
homogenization of the divergence form elliptic operator with rough conductivity
coefficients in dimension two. We show that conductivity
coefficients are in one-to-one correspondence with divergence-free matrices and
convex functions over the domain . Although homogenization is a
non-linear and non-injective operator when applied directly to conductivity
coefficients, homogenization becomes a linear interpolation operator over
triangulations of when re-expressed using convex functions, and is a
volume averaging operator when re-expressed with divergence-free matrices.
Using optimal weighted Delaunay triangulations for linearly interpolating
convex functions, we obtain an optimally robust homogenization algorithm for
arbitrary rough coefficients. Next, we consider inverse homogenization and show
how to decompose it into a linear ill-posed problem and a well-posed non-linear
problem. We apply this new geometric approach to Electrical Impedance
Tomography (EIT). It is known that the EIT problem admits at most one isotropic
solution. If an isotropic solution exists, we show how to compute it from any
conductivity having the same boundary Dirichlet-to-Neumann map. It is known
that the EIT problem admits a unique (stable with respect to -convergence)
solution in the space of divergence-free matrices. As such we suggest that the
space of convex functions is the natural space in which to parameterize
solutions of the EIT problem
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