51,294 research outputs found
Time-vectorized numerical integration for systems of ODEs
Stiff systems of ordinary differential equations (ODEs) and sparse training
data are common in scientific problems. This paper describes efficient,
implicit, vectorized methods for integrating stiff systems of ordinary
differential equations through time and calculating parameter gradients with
the adjoint method. The main innovation is to vectorize the problem both over
the number of independent times series and over a batch or "chunk" of
sequential time steps, effectively vectorizing the assembly of the implicit
system of ODEs. The block-bidiagonal structure of the linearized implicit
system for the backward Euler method allows for further vectorization using
parallel cyclic reduction (PCR). Vectorizing over both axes of the input data
provides a higher bandwidth of calculations to the computing device, allowing
even problems with comparatively sparse data to fully utilize modern GPUs and
achieving speed ups of greater than 100x, compared to standard, sequential time
integration. We demonstrate the advantages of implicit, vectorized time
integration with several example problems, drawn from both analytical stiff and
non-stiff ODE models as well as neural ODE models. We also describe and provide
a freely available open-source implementation of the methods developed here
Solution of Ordinary Differential Equations in Gradient-Based Multidisciplinary Design Optimization
A gradient-based approach to multidisciplinary design optimization enables efficient scalability to large numbers of design variables. However, the need for derivatives causes difficulties when integrating ordinary differential equations (ODEs) in models. To simplify this, we propose the use of the general linear methods framework, which unifies all Runge-Kutta and linear multistep methods. This approach enables rapid implementation of integration methods without the need to differentiate each one, even in a gradient-based optimization context. We also develop a new parallel time integration algorithm that enables vectorization across time steps. We present a set of benchmarking results using a stiff ODE, a non-stiff nonlinear ODE, and an orbital dynamics ODE, and compare integration methods. In a modular gradient-based multidisciplinary design optimization context, we find that the new parallel time integration algorithm with high-order implicit methods, especially Gauss-Legendre collocation, is the best choice for a broad range of problems
The Escape Problem for Irreversible Systems
The problem of noise-induced escape from a metastable state arises in
physics, chemistry, biology, systems engineering, and other areas. The problem
is well understood when the underlying dynamics of the system obey detailed
balance. When this assumption fails many of the results of classical
transition-rate theory no longer apply, and no general method exists for
computing the weak-noise asymptotics of fundamental quantities such as the mean
escape time. In this paper we present a general technique for analysing the
weak-noise limit of a wide range of stochastically perturbed continuous-time
nonlinear dynamical systems. We simplify the original problem, which involves
solving a partial differential equation, into one in which only ordinary
differential equations need be solved. This allows us to resolve some old
issues for the case when detailed balance holds. When it does not hold, we show
how the formula for the mean escape time asymptotics depends on the dynamics of
the system along the most probable escape path. We also present new results on
short-time behavior and discuss the possibility of focusing along the escape
path.Comment: 24 pages, APS revtex macros (version 2.1) now available from PBB via
`get oldrevtex.sty
Inference in complex biological systems with Gaussian processes and parallel tempering
Parameter inference in mathematical models of complex biological
systems, expressed as coupled ordinary differential equations (ODEs), is a challenging problem. These depend on kinetic parameters, which cannot all be measured and have to be ascertained a different way. However, the computational
costs associated with repeatedly solving the ODEs are often staggering, making
many techniques impractical. Therefore, aimed at reducing this cost, new concepts using gradient matching have been proposed. This paper combines current
adaptive gradient matching approaches, using Gaussian processes, with a parallel tempering scheme, in order to compare 2 different paradigms using the same
nonlinear regression method. We use 2 ODE systems to assess our technique,
showing an improvement over the recent method in Calderhead et al. (2008)
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