Variational methods

International audienceThis contribution presents derivative-based methods for local sensitivity analysis, called Variational Sensitivity Analysis (VSA). If one defines an output called the response function, its sensitivity to inputs variations around a nominal value can be studied using derivative (gradient) information. The main issue of VSA is then to provide an efficient way of computing gradients. This contribution first presents the theoretical grounds of VSA: framework and problem statement, tangent and adjoint methods. Then it covers pratical means to compute derivatives, from naive to more sophisticated approaches, discussing their various 2 merits. Finally, applications of VSA are reviewed and some examples are presented, covering various applications fields: oceanography, glaciology, meteorology

A comparison of approximation techniques for variance-based sensitivity analysis of biochemical reaction systems

<p>Abstract</p> <p>Background</p> <p>Sensitivity analysis is an indispensable tool for the analysis of complex systems. In a recent paper, we have introduced a thermodynamically consistent variance-based sensitivity analysis approach for studying the robustness and fragility properties of biochemical reaction systems under uncertainty in the standard chemical potentials of the activated complexes of the reactions and the standard chemical potentials of the molecular species. In that approach, key sensitivity indices were estimated by Monte Carlo sampling, which is computationally very demanding and impractical for large biochemical reaction systems. Computationally efficient algorithms are needed to make variance-based sensitivity analysis applicable to realistic cellular networks, modeled by biochemical reaction systems that consist of a large number of reactions and molecular species.</p> <p>Results</p> <p>We present four techniques, derivative approximation (DA), polynomial approximation (PA), Gauss-Hermite integration (GHI), and orthonormal Hermite approximation (OHA), for <it>analytically </it>approximating the variance-based sensitivity indices associated with a biochemical reaction system. By using a well-known model of the mitogen-activated protein kinase signaling cascade as a case study, we numerically compare the approximation quality of these techniques against traditional Monte Carlo sampling. Our results indicate that, although DA is computationally the most attractive technique, special care should be exercised when using it for sensitivity analysis, since it may only be accurate at low levels of uncertainty. On the other hand, PA, GHI, and OHA are computationally more demanding than DA but can work well at high levels of uncertainty. GHI results in a slightly better accuracy than PA, but it is more difficult to implement. OHA produces the most accurate approximation results and can be implemented in a straightforward manner. It turns out that the computational cost of the four approximation techniques considered in this paper is orders of magnitude smaller than traditional Monte Carlo estimation. Software, coded in MATLAB^®, which implements all sensitivity analysis techniques discussed in this paper, is available free of charge.</p> <p>Conclusions</p> <p>Estimating variance-based sensitivity indices of a large biochemical reaction system is a computationally challenging task that can only be addressed via approximations. Among the methods presented in this paper, a technique based on orthonormal Hermite polynomials seems to be an acceptable candidate for the job, producing very good approximation results for a wide range of uncertainty levels in a fraction of the time required by traditional Monte Carlo sampling.</p

Multiplexing information flow through dynamic signalling systems

We consider how a signalling system can act as an information hub by multiplexing information arising from multiple signals. We formally define multiplexing, mathematically characterise which systems can multiplex and how well they can do it. While the results of this paper are theoretical, to motivate the idea of multiplexing, we provide experimental evidence that tentatively suggests that the NF-κB transcription factor can multiplex information about changes in multiple signals. We believe that our theoretical results may resolve the apparent paradox of how a system like NF-κB that regulates cell fate and inflammatory signalling in response to diverse stimuli can appear to have the low information carrying capacity suggested by recent studies on scalar signals. In carrying out our study, we introduce new methods for the analysis of large, nonlinear stochastic dynamic models, and develop computational algorithms that facilitate the calculation of fundamental constructs of information theory such as Kullback–Leibler divergences and sensitivity matrices, and link these methods to a new theory about multiplexing information. We show that many current models such as those of the NF-κB system cannot multiplex effectively and provide models that overcome this limitation using post-transcriptional modifications

Optimised Adjoint Sensitivity Analysis Using Adjoint Guided Mesh Adaptivity Applied to Neutron Detector Response Calculations

This article presents a new approach for the efficient calculation of sensitivities in radiation dose estimates, subject to imprecisely known nuclear material cross-section data. The method is a combined application of adjoint-based models to perform, simultaneously, both the sensitivity calculation together with optimal adaptive mesh refinement. Adjoint-based sensitivity methods are known for their efficiency since they enable sensitivities of all parameters to be formed through only two solutions to the problem. However, the efficient solutions can also be obtained by their computation on optimal meshes, here guided by goal-based adjoint approaches. It is shown that both mesh adaptivity and sensitivity can be computed by the same adjoint solution, meaning both can be performed without additional costs. A simple demonstration is presented based on the Maynard fixed-source problem where uncertainties, with respect to material cross-sections, of doses received in local regions are examined. It is shown that the method is able to calculate sensitivities with reduced computational costs in terms of memory and potentially computational time through reduced mesh size when using adaptive resolution in comparison to uniform resolution. In particular, it is the local spatial contributions to the sensitivity that are resolved more effectively due to the adaptive meshes concentrating resolution in those areas contributing the most to its value