22,153 research outputs found
A class of robust numerical methods for solving dynamical systems with multiple time scales
In this paper, we develop a class of robust numerical methods for solving dynamical systems with multiple time scales. We first represent the solution of a multiscale dynamical system as a transformation of a slowly varying solution. Then, under the scale separation assumption, we provide a systematic way to construct the transformation map and derive the dynamic equation for the slowly varying solution. We also provide the convergence analysis of the proposed method. Finally, we present several numerical examples, including ODE system with three and four separated time scales to demonstrate the accuracy and efficiency of the proposed method. Numerical results verify that our method is robust in solving ODE systems with multiple time scale, where the time step does not depend on the multiscale parameters
A class of robust numerical methods for solving dynamical systems with multiple time scales
In this paper, we develop a class of robust numerical methods for solving dynamical systems with multiple time scales. We first represent the solution of a multiscale dynamical system as a transformation of a slowly varying solution. Then, under the scale separation assumption, we provide a systematic way to construct the transformation map and derive the dynamic equation for the slowly varying solution. We also provide the convergence analysis of the proposed method. Finally, we present several numerical examples, including ODE system with three and four separated time scales to demonstrate the accuracy and efficiency of the proposed method. Numerical results verify that our method is robust in solving ODE systems with multiple time scale, where the time step does not depend on the multiscale parameters
A new framework for extracting coarse-grained models from time series with multiscale structure
In many applications it is desirable to infer coarse-grained models from
observational data. The observed process often corresponds only to a few
selected degrees of freedom of a high-dimensional dynamical system with
multiple time scales. In this work we consider the inference problem of
identifying an appropriate coarse-grained model from a single time series of a
multiscale system. It is known that estimators such as the maximum likelihood
estimator or the quadratic variation of the path estimator can be strongly
biased in this setting. Here we present a novel parametric inference
methodology for problems with linear parameter dependency that does not suffer
from this drawback. Furthermore, we demonstrate through a wide spectrum of
examples that our methodology can be used to derive appropriate coarse-grained
models from time series of partial observations of a multiscale system in an
effective and systematic fashion
Pulses and Snakes in Ginzburg--Landau Equation
Using a variational formulation for partial differential equations (PDEs)
combined with numerical simulations on ordinary differential equations (ODEs),
we find two categories (pulses and snakes) of dissipative solitons, and analyze
the dependence of both their shape and stability on the physical parameters of
the cubic-quintic Ginzburg-Landau equation (CGLE). In contrast to the regular
solitary waves investigated in numerous integrable and non-integrable systems
over the last three decades, these dissipative solitons are not stationary in
time. Rather, they are spatially confined pulse-type structures whose envelopes
exhibit complicated temporal dynamics. Numerical simulations reveal very
interesting bifurcations sequences as the parameters of the CGLE are varied.
Our predictions on the variation of the soliton amplitude, width, position,
speed and phase of the solutions using the variational formulation agree with
simulation results.Comment: 30 pages, 14 figure
Universal Convexification via Risk-Aversion
We develop a framework for convexifying a fairly general class of
optimization problems. Under additional assumptions, we analyze the
suboptimality of the solution to the convexified problem relative to the
original nonconvex problem and prove additive approximation guarantees. We then
develop algorithms based on stochastic gradient methods to solve the resulting
optimization problems and show bounds on convergence rates. %We show a simple
application of this framework to supervised learning, where one can perform
integration explicitly and can use standard (non-stochastic) optimization
algorithms with better convergence guarantees. We then extend this framework to
apply to a general class of discrete-time dynamical systems. In this context,
our convexification approach falls under the well-studied paradigm of
risk-sensitive Markov Decision Processes. We derive the first known model-based
and model-free policy gradient optimization algorithms with guaranteed
convergence to the optimal solution. Finally, we present numerical results
validating our formulation in different applications
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