54,129 research outputs found
Approximate parameter inference in systems biology using gradient matching: a comparative evaluation
Background: A challenging problem in current systems biology is that of
parameter inference in biological pathways expressed as coupled ordinary
differential equations (ODEs). Conventional methods that repeatedly numerically
solve the ODEs have large associated computational costs. Aimed at reducing this
cost, new concepts using gradient matching have been proposed, which bypass
the need for numerical integration. This paper presents a recently established
adaptive gradient matching approach, using Gaussian processes, combined with a
parallel tempering scheme, and conducts a comparative evaluation with current
state of the art methods used for parameter inference in ODEs. Among these
contemporary methods is a technique based on reproducing kernel Hilbert spaces
(RKHS). This has previously shown promising results for parameter estimation,
but under lax experimental settings. We look at a range of scenarios to test the
robustness of this method. We also change the approach of inferring the penalty
parameter from AIC to cross validation to improve the stability of the method.
Methodology: Methodology for the recently proposed adaptive gradient
matching method using Gaussian processes, upon which we build our new
method, is provided. Details of a competing method using reproducing kernel
Hilbert spaces are also described here.
Results: We conduct a comparative analysis for the methods described in this
paper, using two benchmark ODE systems. The analyses are repeated under
different experimental settings, to observe the sensitivity of the techniques.
Conclusions: Our study reveals that for known noise variance, our proposed
method based on Gaussian processes and parallel tempering achieves overall the
best performance. When the noise variance is unknown, the RKHS method
proves to be more robust
Fifty Years of Stiffness
The notion of stiffness, which originated in several applications of a
different nature, has dominated the activities related to the numerical
treatment of differential problems for the last fifty years. Contrary to what
usually happens in Mathematics, its definition has been, for a long time, not
formally precise (actually, there are too many of them). Again, the needs of
applications, especially those arising in the construction of robust and
general purpose codes, require nowadays a formally precise definition. In this
paper, we review the evolution of such a notion and we also provide a precise
definition which encompasses all the previous ones.Comment: 24 pages, 11 figure
Differential-Algebraic Equations and Beyond: From Smooth to Nonsmooth Constrained Dynamical Systems
The present article presents a summarizing view at differential-algebraic
equations (DAEs) and analyzes how new application fields and corresponding
mathematical models lead to innovations both in theory and in numerical
analysis for this problem class. Recent numerical methods for nonsmooth
dynamical systems subject to unilateral contact and friction illustrate the
topicality of this development.Comment: Preprint of Book Chapte
Extrapolation-based implicit-explicit general linear methods
For many systems of differential equations modeling problems in science and
engineering, there are natural splittings of the right hand side into two
parts, one non-stiff or mildly stiff, and the other one stiff. For such systems
implicit-explicit (IMEX) integration combines an explicit scheme for the
non-stiff part with an implicit scheme for the stiff part.
In a recent series of papers two of the authors (Sandu and Zhang) have
developed IMEX GLMs, a family of implicit-explicit schemes based on general
linear methods. It has been shown that, due to their high stage order, IMEX
GLMs require no additional coupling order conditions, and are not marred by
order reduction.
This work develops a new extrapolation-based approach to construct practical
IMEX GLM pairs of high order. We look for methods with large absolute stability
region, assuming that the implicit part of the method is A- or L-stable. We
provide examples of IMEX GLMs with optimal stability properties. Their
application to a two dimensional test problem confirms the theoretical
findings
Travelling waves in a tissue interaction model for skin pattern formation
Tissue interaction plays a major role in many morphogenetic processes, particularly those associated with skin organ primordia. We examine travelling wave solutions in a tissue interaction model for skin pattern formation which is firmly based on the known biology. From a phase space analysis we conjecture the existence of travelling waves with specific wave speeds. Subsequently, analytical approximations to the wave profiles are derived using perturbation methods. We then show numerically that such travelling wave solutions do exist and that they are in good agreement with our analytical results. Finally, the biological implications of our analysis are discussed
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