60,140 research outputs found
Dimension reduction for systems with slow relaxation
We develop reduced, stochastic models for high dimensional, dissipative
dynamical systems that relax very slowly to equilibrium and can encode long
term memory. We present a variety of empirical and first principles approaches
for model reduction, and build a mathematical framework for analyzing the
reduced models. We introduce the notions of universal and asymptotic filters to
characterize `optimal' model reductions for sloppy linear models. We illustrate
our methods by applying them to the practically important problem of modeling
evaporation in oil spills.Comment: 48 Pages, 13 figures. Paper dedicated to the memory of Leo Kadanof
Discrete-time multi-scale systems
We introduce multi-scale filtering by the way of certain double convolution
systems. We prove stability theorems for these systems and make connections
with function theory in the poly-disc. Finally, we compare the framework
developed here with the white noise space framework, within which a similar
class of double convolution systems has been defined earlier
Convergence of invariant densities in the small-noise limit
This paper presents a systematic numerical study of the effects of noise on
the invariant probability densities of dynamical systems with varying degrees
of hyperbolicity. It is found that the rate of convergence of invariant
densities in the small-noise limit is frequently governed by power laws. In
addition, a simple heuristic is proposed and found to correctly predict the
power law exponent in exponentially mixing systems. In systems which are not
exponentially mixing, the heuristic provides only an upper bound on the power
law exponent. As this numerical study requires the computation of invariant
densities across more than 2 decades of noise amplitudes, it also provides an
opportunity to discuss and compare standard numerical methods for computing
invariant probability densities.Comment: 27 pages, 19 figures, revised with minor correction
Continuous Tensor Network States for Quantum Fields
We introduce a new class of states for bosonic quantum fields which extend
tensor network states to the continuum and generalize continuous matrix product
states (cMPS) to spatial dimensions . By construction, they are
Euclidean invariant, and are genuine continuum limits of discrete tensor
network states. Admitting both a functional integral and an operator
representation, they share the important properties of their discrete
counterparts: expressiveness, invariance under gauge transformations, simple
rescaling flow, and compact expressions for the -point functions of local
observables. While we discuss mostly the continuous tensor network states
extending Projected Entangled Pair States (PEPS), we propose a generalization
bearing similarities with the continuum Multi-scale Entanglement
Renormalization Ansatz (cMERA).Comment: 16 pages, 5 figures, close to published versio
Sampling from a system-theoretic viewpoint: Part I - Concepts and tools
This paper is first in a series of papers studying a system-theoretic approach to the problem of reconstructing an analog signal from its samples. The idea, borrowed from earlier treatments in the control literature, is to address the problem as a hybrid model-matching problem in which performance is measured by system norms. In this paper we present the paradigm and revise underlying technical tools, such as the lifting technique and some topics of the operator theory. This material facilitates a systematic and unified treatment of a wide range of sampling and reconstruction problems, recovering many hitherto considered different solutions and leading to new results. Some of these applications are discussed in the second part
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