7,193 research outputs found
Discontinuities handled with events in Assimulo
Often integrating ordinary differential equations or differential algebraic equations (DAE) do not constitute the problem alone. A common complement is finding the root of an algebraic function (an event function) that depends on the states of the problem. This formulation of a model enables the possibility of including discontinuities, an important part of the Functional Mock-up Interface standard which allows hybrid models of differential algebraic equations. The problem of root-finding during integration is however difficult. Both in a theoretical aspect and as a software problem. An implementation of software for root-finding is done in Assimulo, a Python/Cython wrapper for integrators. The implementation takes the Functional Mock-up Interface standard into consideration. The implementation is made usable for a wide variety of integration algorithms and is also verified and benchmarked with advanced industrial models, showing good indications of being robust and scaling well for large systems
Connecting the Dots: Towards Continuous Time Hamiltonian Monte Carlo
Continuous time Hamiltonian Monte Carlo is introduced, as a powerful
alternative to Markov chain Monte Carlo methods for continuous target
distributions. The method is constructed in two steps: First Hamiltonian
dynamics are chosen as the deterministic dynamics in a continuous time
piecewise deterministic Markov process. Under very mild restrictions, such a
process will have the desired target distribution as an invariant distribution.
Secondly, the numerical implementation of such processes, based on adaptive
numerical integration of second order ordinary differential equations is
considered. The numerical implementation yields an approximate, yet highly
robust algorithm that, unlike conventional Hamiltonian Monte Carlo, enables the
exploitation of the complete Hamiltonian trajectories (hence the title). The
proposed algorithm may yield large speedups and improvements in stability
relative to relevant benchmarks, while incurring numerical errors that are
negligible relative to the overall Monte Carlo errors
Engineering and programming manual: Two-dimensional kinetic reference computer program (TDK)
The Two Dimensional Kinetics (TDK) computer program is a primary tool in applying the JANNAF liquid rocket thrust chamber performance prediction methodology. The development of a methodology that includes all aspects of rocket engine performance from analytical calculation to test measurements, that is physically accurate and consistent, and that serves as an industry and government reference is presented. Recent interest in rocket engines that operate at high expansion ratio, such as most Orbit Transfer Vehicle (OTV) engine designs, has required an extension of the analytical methods used by the TDK computer program. Thus, the version of TDK that is described in this manual is in many respects different from the 1973 version of the program. This new material reflects the new capabilities of the TDK computer program, the most important of which are described
Perturbations of slowly rotating black holes: massive vector fields in the Kerr metric
We discuss a general method to study linear perturbations of slowly rotating
black holes which is valid for any perturbation field, and particularly
advantageous when the field equations are not separable. As an illustration of
the method we investigate massive vector (Proca) perturbations in the Kerr
metric, which do not appear to be separable in the standard Teukolsky
formalism. Working in a perturbative scheme, we discuss two important effects
induced by rotation: a Zeeman-like shift of nonaxisymmetric quasinormal modes
and bound states with different azimuthal number m, and the coupling between
axial and polar modes with different multipolar index l. We explicitly compute
the perturbation equations up to second order in rotation, but in principle the
method can be extended to any order. Working at first order in rotation we show
that polar and axial Proca modes can be computed by solving two decoupled sets
of equations, and we derive a single master equation describing axial
perturbations of spin s=0 and s=+-1. By extending the calculation to second
order we can study the superradiant regime of Proca perturbations in a
self-consistent way. For the first time we show that Proca fields around Kerr
black holes exhibit a superradiant instability, which is significantly stronger
than for massive scalar fields. Because of this instability, astrophysical
observations of spinning black holes provide the tightest upper limit on the
mass of the photon: mv<4x10^-20 eV under our most conservative assumptions.
Spin measurements for the largest black holes could reduce this bound to
mv<10^-22 eV or lower.Comment: v1: 29 pages, 9 figures, 3 appendices. v2: References added and
improved discussion. Matches the version to appear in Physical Review D.
Mathematica notebooks available here http://blackholes.ist.utl.pt/?page=Files
and http://www.phy.olemiss.edu/~berti/qnms.htm
Direct event location techniques based on Adams multistep methods for discontinuous ODEs
In this paper we consider numerical techniques to locate the event points of the differential system x′=f(x), where f is a discontinuous vector field along an event surface splitting the state space into two different regions R1 and R2 and f(x)=fi(x) when x∈Ri, for i=1,2 while f1(x)≠f2(x) when x∈Σ. Methods based on Adams multistep schemes which approach the event surface Σ from one side only and in a finite number of steps are proposed. Particularly, these techniques do not require the evaluation of the vector field f1 (respectively, f2) in the region R2 (respectively R1) and are based on the computation–at each step– of a new time ste
Birth/birth-death processes and their computable transition probabilities with biological applications
Birth-death processes track the size of a univariate population, but many
biological systems involve interaction between populations, necessitating
models for two or more populations simultaneously. A lack of efficient methods
for evaluating finite-time transition probabilities of bivariate processes,
however, has restricted statistical inference in these models. Researchers rely
on computationally expensive methods such as matrix exponentiation or Monte
Carlo approximation, restricting likelihood-based inference to small systems,
or indirect methods such as approximate Bayesian computation. In this paper, we
introduce the birth(death)/birth-death process, a tractable bivariate extension
of the birth-death process. We develop an efficient and robust algorithm to
calculate the transition probabilities of birth(death)/birth-death processes
using a continued fraction representation of their Laplace transforms. Next, we
identify several exemplary models arising in molecular epidemiology,
macro-parasite evolution, and infectious disease modeling that fall within this
class, and demonstrate advantages of our proposed method over existing
approaches to inference in these models. Notably, the ubiquitous stochastic
susceptible-infectious-removed (SIR) model falls within this class, and we
emphasize that computable transition probabilities newly enable direct
inference of parameters in the SIR model. We also propose a very fast method
for approximating the transition probabilities under the SIR model via a novel
branching process simplification, and compare it to the continued fraction
representation method with application to the 17th century plague in Eyam.
Although the two methods produce similar maximum a posteriori estimates, the
branching process approximation fails to capture the correlation structure in
the joint posterior distribution
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