126 research outputs found
A Data-Driven Approach for Discovering Stochastic Dynamical Systems with Non-Gaussian Levy Noise
With the rapid increase of valuable observational, experimental and
simulating data for complex systems, great efforts are being devoted to
discovering governing laws underlying the evolution of these systems. However,
the existing techniques are limited to extract governing laws from data as
either deterministic differential equations or stochastic differential
equations with Gaussian noise. In the present work, we develop a new
data-driven approach to extract stochastic dynamical systems with non-Gaussian
symmetric L\'evy noise, as well as Gaussian noise. First, we establish a
feasible theoretical framework, by expressing the drift coefficient, diffusion
coefficient and jump measure (i.e., anomalous diffusion) for the underlying
stochastic dynamical system in terms of sample paths data. We then design a
numerical algorithm to compute the drift, diffusion coefficient and jump
measure, and thus extract a governing stochastic differential equation with
Gaussian and non-Gaussian noise. Finally, we demonstrate the efficacy and
accuracy of our approach by applying to several prototypical one-, two- and
three-dimensional systems. This new approach will become a tool in discovering
governing dynamical laws from noisy data sets, from observing or simulating
complex phenomena, such as rare events triggered by random fluctuations with
heavy as well as light tail statistical features.Comment: 36 page
Exponential stability of non-autonomous stochastic partial differential equations with finite memory
The exponential stability, in both mean square and almost sure senses, for
energy solutions to a class of nonlinear and non-autonomous stochastic PDEs
with finite memory is investigated. Various criteria for stability are
obtained. An example is presented to demonstrate the main results
Modeling nonlinear random vibration: Implication of the energy conservation law
Nonlinear random vibration under excitations of both Gaussian and Poisson
white noises is considered. The model is based on stochastic differential
equations, and the corresponding stochastic integrals are defined in such a way
that the energy conservation law is satisfied. It is shown that Stratonovich
integral and Di Paola-Falsone integral should be used for excitations of
Gaussian and Poisson white noises, respectively, in order for the model to
satisfy the underlining physical laws (e.g., energy conservation). Numerical
examples are presented to illustrate the theoretical results.Comment: 7 figure
Compactly Generated Shape Index Theory and its Application to a Retarded Nonautonomous Parabolic Equation
We establish the compactly generated shape (H-shape) index theory for local
semiflows on complete metric spaces via more general shape index pairs, and
define the H-shape cohomology index to develop the Morse equations. The main
advantages are that the quotient space is not necessarily metrizable for
the shape index pair and N\sm E need not to be a neighborhood of the
compact invariant set. Moreover, in this new theory, the phase space is not
required to be separable. We apply H-shape index theory to an abstract retarded
nonautonomous parabolic equation to obtain the existence of bounded full
solutions
An alternative expression of Di Paola and Falson's formula for stochastic dynamics
Di Paola and Falsone's formula is widely used in studying stochastic dynamics
of nonlinear systems under Poisson white noise. In this short communication, an
alternative expression is presented. Compared to Di Paola and Falsone's
original expression, the alternative one is applicable under more general
condition, and shows significantly improved performance in numerical
implementation. The alternative expression turns out to be a special case of
the Marcus integrals.Comment: 12 pages, 2 figure
Fokker-Planck Equations for Stochastic Dynamical Systems with Symmetric L\'evy Motions
The Fokker-Planck equations for stochastic dynamical systems, with
non-Gaussian stable symmetric L\'evy motions, have a nonlocal or
fractional Laplacian term. This nonlocality is the manifestation of the effect
of non-Gaussian fluctuations. Taking advantage of the Toeplitz matrix structure
of the time-space discretization, a fast and accurate numerical algorithm is
proposed to simulate the nonlocal Fokker-Planck equations, under either
absorbing or natural conditions. The scheme is shown to satisfy a discrete
maximum principle and to be convergent. It is validated against a known exact
solution and the numerical solutions obtained by using other methods. The
numerical results for two prototypical stochastic systems, the
Ornstein-Uhlenbeck system and the double-well system are shown
A regularity result for the nonlocal Fokker-Planck equation with Ornstein-Uhlenbeck drift
Despite there are numerous theoretical studies of stochastic differential
equations with a symmetric -stable L\'evy noise, very few regularity
results exist in the case of . In this paper, we study the
fractional Fokker-Planck equation with Ornstein-Uhlenbeck drift, and prove that
there exists a unique solution, which is in space for when
.Comment: 6 page
State estimation under non-Gaussian Levy noise: A modified Kalman filtering method
The Kalman filter is extensively used for state estimation for linear systems
under Gaussian noise. When non-Gaussian L\'evy noise is present, the
conventional Kalman filter may fail to be effective due to the fact that the
non-Gaussian L\'evy noise may have infinite variance. A modified Kalman filter
for linear systems with non-Gaussian L\'evy noise is devised. It works
effectively with reasonable computational cost. Simulation results are
presented to illustrate this non-Gaussian filtering method
Random data Cauchy problem for the wave equation on compact manifold
Inspired by the work of Burq and Tzvetkov (Invent. math. 173(2008),
449-475.), firstly, we construct the local strong solution to the cubic
nonlinear wave equation with random data for a large set of initial data in
with , where M is a three dimensional compact
manifold with boundary, moreover, our result improves the result of Theorem 2
in (Invent. math. 173(2008), 449-475.); secondly, we construct the local strong
solution to the quintic nonlinear wave equation with random data for a large
set of initial data in with , where M is a two
dimensional compact boundaryless manifold; finally, we construct the local
strong solution to the quintic nonlinear wave equation with random data for a
large set of initial data in with , where M is
a two dimensional compact manifold with boundary.Comment: We correct some misprint
Probabilistic pointwise convergence problem of some dispersive equations
In this paper, we investigate the almost surely pointwise convergence problem
of free KdV equation, free wave equation, free elliptic and non-elliptic
Schr\"odinger equation respectively. We firstly establish some estimates
related to the Wiener decomposition of frequency spaces which are just Lemmas
2.1-2.6 in this paper. Secondly, by using Lemmas 2.1-2.6, 3.1, we establish the
probabilistic estimates of some random series which are just Lemmas 3.2-3.11 in
this paper. Finally, combining the density theorem in L with Lemmas
3.2-3.11, we obtain almost surely pointwise convergence of the solutions to
corresponding equations with randomized initial data in , which require
much less regularity of the initial data than the rough data case. At the same
time, we present the probabilistic density theorem, which is Lemma 3.11 in this
paper
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