21,713 research outputs found
Joint Spectral Radius and Path-Complete Graph Lyapunov Functions
We introduce the framework of path-complete graph Lyapunov functions for
approximation of the joint spectral radius. The approach is based on the
analysis of the underlying switched system via inequalities imposed among
multiple Lyapunov functions associated to a labeled directed graph. Inspired by
concepts in automata theory and symbolic dynamics, we define a class of graphs
called path-complete graphs, and show that any such graph gives rise to a
method for proving stability of the switched system. This enables us to derive
several asymptotically tight hierarchies of semidefinite programming
relaxations that unify and generalize many existing techniques such as common
quadratic, common sum of squares, and maximum/minimum-of-quadratics Lyapunov
functions. We compare the quality of approximation obtained by certain classes
of path-complete graphs including a family of dual graphs and all path-complete
graphs with two nodes on an alphabet of two matrices. We provide approximation
guarantees for several families of path-complete graphs, such as the De Bruijn
graphs, establishing as a byproduct a constructive converse Lyapunov theorem
for maximum/minimum-of-quadratics Lyapunov functions.Comment: To appear in SIAM Journal on Control and Optimization. Version 2 has
gone through two major rounds of revision. In particular, a section on the
performance of our algorithm on application-motivated problems has been added
and a more comprehensive literature review is presente
Symbolic-Numeric Algorithms for Computer Analysis of Spheroidal Quantum Dot Models
A computation scheme for solving elliptic boundary value problems with
axially symmetric confining potentials using different sets of one-parameter
basis functions is presented. The efficiency of the proposed symbolic-numerical
algorithms implemented in Maple is shown by examples of spheroidal quantum dot
models, for which energy spectra and eigenfunctions versus the spheroid aspect
ratio were calculated within the conventional effective mass approximation.
Critical values of the aspect ratio, at which the discrete spectrum of models
with finite-wall potentials is transformed into a continuous one in strong
dimensional quantization regime, were revealed using the exact and adiabatic
classifications.Comment: 6 figures, Submitted to Proc. of The 12th International Workshop on
Computer Algebra in Scientific Computing (CASC 2010) Tsakhkadzor, Armenia,
September 5 - 12, 201
Surrogate Accelerated Bayesian Inversion for the Determination of the Thermal Diffusivity of a Material
Determination of the thermal properties of a material is an important task in
many scientific and engineering applications. How a material behaves when
subjected to high or fluctuating temperatures can be critical to the safety and
longevity of a system's essential components. The laser flash experiment is a
well-established technique for indirectly measuring the thermal diffusivity,
and hence the thermal conductivity, of a material. In previous works,
optimization schemes have been used to find estimates of the thermal
conductivity and other quantities of interest which best fit a given model to
experimental data. Adopting a Bayesian approach allows for prior beliefs about
uncertain model inputs to be conditioned on experimental data to determine a
posterior distribution, but probing this distribution using sampling techniques
such as Markov chain Monte Carlo methods can be incredibly computationally
intensive. This difficulty is especially true for forward models consisting of
time-dependent partial differential equations. We pose the problem of
determining the thermal conductivity of a material via the laser flash
experiment as a Bayesian inverse problem in which the laser intensity is also
treated as uncertain. We introduce a parametric surrogate model that takes the
form of a stochastic Galerkin finite element approximation, also known as a
generalized polynomial chaos expansion, and show how it can be used to sample
efficiently from the approximate posterior distribution. This approach gives
access not only to the sought-after estimate of the thermal conductivity but
also important information about its relationship to the laser intensity, and
information for uncertainty quantification. We also investigate the effects of
the spatial profile of the laser on the estimated posterior distribution for
the thermal conductivity
Sparse Randomized Kaczmarz for Support Recovery of Jointly Sparse Corrupted Multiple Measurement Vectors
While single measurement vector (SMV) models have been widely studied in
signal processing, there is a surging interest in addressing the multiple
measurement vectors (MMV) problem. In the MMV setting, more than one
measurement vector is available and the multiple signals to be recovered share
some commonalities such as a common support. Applications in which MMV is a
naturally occurring phenomenon include online streaming, medical imaging, and
video recovery. This work presents a stochastic iterative algorithm for the
support recovery of jointly sparse corrupted MMV. We present a variant of the
Sparse Randomized Kaczmarz algorithm for corrupted MMV and compare our proposed
method with an existing Kaczmarz type algorithm for MMV problems. We also
showcase the usefulness of our approach in the online (streaming) setting and
provide empirical evidence that suggests the robustness of the proposed method
to the distribution of the corruption and the number of corruptions occurring.Comment: 13 pages, 6 figure
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