1,351 research outputs found
Lower Bounds on Complexity of Lyapunov Functions for Switched Linear Systems
We show that for any positive integer , there are families of switched
linear systems---in fixed dimension and defined by two matrices only---that are
stable under arbitrary switching but do not admit (i) a polynomial Lyapunov
function of degree , or (ii) a polytopic Lyapunov function with facets, or (iii) a piecewise quadratic Lyapunov function with
pieces. This implies that there cannot be an upper bound on the size of the
linear and semidefinite programs that search for such stability certificates.
Several constructive and non-constructive arguments are presented which connect
our problem to known (and rather classical) results in the literature regarding
the finiteness conjecture, undecidability, and non-algebraicity of the joint
spectral radius. In particular, we show that existence of an extremal piecewise
algebraic Lyapunov function implies the finiteness property of the optimal
product, generalizing a result of Lagarias and Wang. As a corollary, we prove
that the finiteness property holds for sets of matrices with an extremal
Lyapunov function belonging to some of the most popular function classes in
controls
Active Self-Assembly of Algorithmic Shapes and Patterns in Polylogarithmic Time
We describe a computational model for studying the complexity of
self-assembled structures with active molecular components. Our model captures
notions of growth and movement ubiquitous in biological systems. The model is
inspired by biology's fantastic ability to assemble biomolecules that form
systems with complicated structure and dynamics, from molecular motors that
walk on rigid tracks and proteins that dynamically alter the structure of the
cell during mitosis, to embryonic development where large-scale complicated
organisms efficiently grow from a single cell. Using this active self-assembly
model, we show how to efficiently self-assemble shapes and patterns from simple
monomers. For example, we show how to grow a line of monomers in time and
number of monomer states that is merely logarithmic in the length of the line.
Our main results show how to grow arbitrary connected two-dimensional
geometric shapes and patterns in expected time that is polylogarithmic in the
size of the shape, plus roughly the time required to run a Turing machine
deciding whether or not a given pixel is in the shape. We do this while keeping
the number of monomer types logarithmic in shape size, plus those monomers
required by the Kolmogorov complexity of the shape or pattern. This work thus
highlights the efficiency advantages of active self-assembly over passive
self-assembly and motivates experimental effort to construct general-purpose
active molecular self-assembly systems
Robust-to-Dynamics Optimization
A robust-to-dynamics optimization (RDO) problem is an optimization problem
specified by two pieces of input: (i) a mathematical program (an objective
function and a feasible set
), and (ii) a dynamical system (a map
). Its goal is to minimize over the
set of initial conditions that forever remain in
under . The focus of this paper is on the case where the
mathematical program is a linear program and the dynamical system is either a
known linear map, or an uncertain linear map that can change over time. In both
cases, we study a converging sequence of polyhedral outer approximations and
(lifted) spectrahedral inner approximations to . Our inner
approximations are optimized with respect to the objective function and
their semidefinite characterization---which has a semidefinite constraint of
fixed size---is obtained by applying polar duality to convex sets that are
invariant under (multiple) linear maps. We characterize three barriers that can
stop convergence of the outer approximations from being finite. We prove that
once these barriers are removed, our inner and outer approximating procedures
find an optimal solution and a certificate of optimality for the RDO problem in
a finite number of steps. Moreover, in the case where the dynamics are linear,
we show that this phenomenon occurs in a number of steps that can be computed
in time polynomial in the bit size of the input data. Our analysis also leads
to a polynomial-time algorithm for RDO instances where the spectral radius of
the linear map is bounded above by any constant less than one. Finally, in our
concluding section, we propose a broader research agenda for studying
optimization problems with dynamical systems constraints, of which RDO is a
special case
Error analysis of coarse-grained kinetic Monte Carlo method
In this paper we investigate the approximation properties of the
coarse-graining procedure applied to kinetic Monte Carlo simulations of lattice
stochastic dynamics. We provide both analytical and numerical evidence that the
hierarchy of the coarse models is built in a systematic way that allows for
error control in both transient and long-time simulations. We demonstrate that
the numerical accuracy of the CGMC algorithm as an approximation of stochastic
lattice spin flip dynamics is of order two in terms of the coarse-graining
ratio and that the natural small parameter is the coarse-graining ratio over
the range of particle/particle interactions. The error estimate is shown to
hold in the weak convergence sense. We employ the derived analytical results to
guide CGMC algorithms and we demonstrate a CPU speed-up in demanding
computational regimes that involve nucleation, phase transitions and
metastability.Comment: 30 page
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