18,838 research outputs found
Multiple barrier-crossings of an Ornstein-Uhlenbeck diffusion in consecutive periods
We investigate the joint distribution and the multivariate survival functions
for the maxima of an Ornstein-Uhlenbeck (OU) process in consecutive
time-intervals. A PDE method, alongside an eigenfunction expansion is adopted,
with which we first calculate the distribution and the survival functions for
the maximum of a homogeneous OU-process in a single interval. By a
deterministic time-change and a parameter translation, this result can be
extended to an inhomogeneous OU-process. Next, we derive a general formula for
the joint distribution and the survival functions for the maxima of a
continuous Markov process in consecutive periods. With these results, one can
obtain semi-analytical expressions for the joint distribution and the
multivariate survival functions for the maxima of an OU-process, with piecewise
constant parameter functions, in consecutive time periods. The joint
distribution and the survival functions can be evaluated numerically by an
iterated quadrature scheme, which can be implemented efficiently by matrix
multiplications. Moreover, we show that the computation can be further
simplified to the product of single quadratures if the filtration is enlarged.
Such results may be used for the modelling of heatwaves and related risk
management challenges.Comment: 38 pages, 10 figures, 2 table
A Constrained Approach to Multiscale Stochastic Simulation of\ud Chemically Reacting Systems
Stochastic simulation of coupled chemical reactions is often computationally intensive, especially if a chemical system contains reactions occurring on different time scales. In this paper we introduce a multiscale methodology suitable to address this problem. It is based on the Conditional Stochastic Simulation Algorithm (CSSA) which samples from the conditional distribution of the suitably defined fast variables, given values for the slow variables. In the Constrained Multiscale Algorithm (CMA) a single realization of the CSSA is then used for each value of the slow variable to approximate the effective drift and diffusion terms, in a similar manner to the constrained mean-force computations in other applications such as molecular dynamics. We then show how using the ensuing Stochastic Differential Equation (SDE) approximation, we can in turn approximate average switching times in stochastic chemical systems
Language-based Abstractions for Dynamical Systems
Ordinary differential equations (ODEs) are the primary means to modelling
dynamical systems in many natural and engineering sciences. The number of
equations required to describe a system with high heterogeneity limits our
capability of effectively performing analyses. This has motivated a large body
of research, across many disciplines, into abstraction techniques that provide
smaller ODE systems while preserving the original dynamics in some appropriate
sense. In this paper we give an overview of a recently proposed
computer-science perspective to this problem, where ODE reduction is recast to
finding an appropriate equivalence relation over ODE variables, akin to
classical models of computation based on labelled transition systems.Comment: In Proceedings QAPL 2017, arXiv:1707.0366
A fast and well-conditioned spectral method for singular integral equations
We develop a spectral method for solving univariate singular integral
equations over unions of intervals by utilizing Chebyshev and ultraspherical
polynomials to reformulate the equations as almost-banded infinite-dimensional
systems. This is accomplished by utilizing low rank approximations for sparse
representations of the bivariate kernels. The resulting system can be solved in
operations using an adaptive QR factorization, where is
the bandwidth and is the optimal number of unknowns needed to resolve the
true solution. The complexity is reduced to operations by
pre-caching the QR factorization when the same operator is used for multiple
right-hand sides. Stability is proved by showing that the resulting linear
operator can be diagonally preconditioned to be a compact perturbation of the
identity. Applications considered include the Faraday cage, and acoustic
scattering for the Helmholtz and gravity Helmholtz equations, including
spectrally accurate numerical evaluation of the far- and near-field solution.
The Julia software package SingularIntegralEquations.jl implements our method
with a convenient, user-friendly interface
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