66,103 research outputs found
Uncertainty Quantification for Linear Hyperbolic Equations with Stochastic Process or Random Field Coefficients
In this paper hyperbolic partial differential equations with random
coefficients are discussed. Such random partial differential equations appear
for instance in traffic flow problems as well as in many physical processes in
random media. Two types of models are presented: The first has a time-dependent
coefficient modeled by the Ornstein--Uhlenbeck process. The second has a random
field coefficient with a given covariance in space. For the former a formula
for the exact solution in terms of moments is derived. In both cases stable
numerical schemes are introduced to solve these random partial differential
equations. Simulation results including convergence studies conclude the
theoretical findings
Statistics of non-linear stochastic dynamical systems under L\'evy noises by a convolution quadrature approach
This paper describes a novel numerical approach to find the statistics of the
non-stationary response of scalar non-linear systems excited by L\'evy white
noises. The proposed numerical procedure relies on the introduction of an
integral transform of Wiener-Hopf type into the equation governing the
characteristic function. Once this equation is rewritten as partial
integro-differential equation, it is then solved by applying the method of
convolution quadrature originally proposed by Lubich, here extended to deal
with this particular integral transform. The proposed approach is relevant for
two reasons: 1) Statistics of systems with several different drift terms can be
handled in an efficient way, independently from the kind of white noise; 2) The
particular form of Wiener-Hopf integral transform and its numerical evaluation,
both introduced in this study, are generalizations of fractional
integro-differential operators of potential type and Gr\"unwald-Letnikov
fractional derivatives, respectively.Comment: 20 pages, 5 figure
Universal Nonlinear Filtering Using Feynman Path Integrals II: The Continuous-Continuous Model with Additive Noise
In this paper, the Feynman path integral formulation of the
continuous-continuous filtering problem, a fundamental problem of applied
science, is investigated for the case when the noise in the signal and
measurement model is additive. It is shown that it leads to an independent and
self-contained analysis and solution of the problem. A consequence of this
analysis is Feynman path integral formula for the conditional probability
density that manifests the underlying physics of the problem. A corollary of
the path integral formula is the Yau algorithm that has been shown to be
superior to all other known algorithms. The Feynman path integral formulation
is shown to lead to practical and implementable algorithms. In particular, the
solution of the Yau PDE is reduced to one of function computation and
integration.Comment: Interdisciplinary, 41 pages, 5 figures, JHEP3 class; added more
discussion and reference
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