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Global sensitivity analysis in stochastic systems Le Maître, Olivier | CIRM H

Multi angle

y

Research schools

Stochastic models are used in many scientific fields, including mechanics, physics, life sciences, queues and social-network studies, chemistry. Stochastic modeling is necessary when deterministic ones cannot capture features of the dynamics, for instance, to represent effects of unresolved small-scale fluctuations, or when systems are subjected to important inherent noise. Often, stochastic models are not completely known and involve some calibrated parameters that should be considered as uncertain. In this case, it is critical to assess the impact of the uncertain model parameters on the stochastic model predictions. This is usually achieved by performing a sensitivity analysis (SA) which characterizes changes in a model output when the uncertain parameters are varied. In the case of a stochastic model, one classically applies the SA to statistical moments of the prediction, estimating, for instance, the derivatives with respect to the uncertain parameters of the output mean and variance. In this presentation, we introduce new approaches of SA in a stochastic system based on variance decomposition methods (ANOVA, Sobol). Compared to previous methods, our SA methods are global, with respect to both the parameters and stochasticity, and decompose the variance into stochastic, parametric and mixed contributions.
We consider first the case of uncertain Stochastic Differential Equations (SDE), that is systems with external noisy forcing and uncertain parameters. A polynomial chaos (PC) analysis with stochastic expansion coefficients is proposed to approximate the SDE solution. We first use a Galerkin formalism to determine the expansion coefficients, leading to a hierarchy of SDEs. Under the mild assumption that the noise and uncertain parameters are independent, the Galerkin formalism naturally separates parametric uncertainty and stochastic forcing dependencies, enabling an orthogonal decomposition of the variance, and consequently identify contributions arising
from the uncertainty in parameters, the stochastic forcing, and a coupled term. Non-intrusive approaches are subsequently considered for application to more complex systems hardly amenable to Galerkin projection. We also discuss parallel implementations and application to derived quantity of interest, in particular, a novel sampling strategy for non-smooth quantities of interest but smooth SDE solution. Numerical examples are provided to illustrate the output of the SA and the computational complexity of the method.
Second, we consider the case of stochastic simulators governed by a set of reaction channels with stochastic dynamics. Reformulating the system dynamics in terms of independent standardized Poisson processes permits the identification of individual realizations of each reaction channel dynamic and a quantitative characterization of the inherent stochasticity sources. By judiciously exploiting the inherent stochasticity of the system, we can then compute the global sensitivities associated with individual reaction channels, as well as the importance of channel interactions. This approach is subsequently extended to account for the effects of uncertain parameters and we propose dedicated algorithms to perform the Sobols decomposition of the variance into contributions from an arbitrary subset of uncertain parameters and stochastic reaction channels. The algorithms are illustrated in simplified systems, including the birth-death, Schlgl, and Michaelis-Menten models. The sensitivity analysis output is also contrasted with a local derivative-based sensitivity analysis method.
Stochastic models are used in many scientific fields, including mechanics, physics, life sciences, queues and social-network studies, chemistry. Stochastic modeling is necessary when deterministic ones cannot capture features of the dynamics, for instance, to represent effects of unresolved small-scale fluctuations, or when systems are subjected to important inherent noise. Often, stochastic models are not completely known and involve some ...

60H35 ; 65C30 ; 65D15

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Cubature methods and applications Crisan, Dan | CIRM H

Multi angle

y

Research schools

The talk will have two parts: In the first part, I will go over some of the basic feature of cubature methods for approximating solutions of classical SDEs and how they can be adapted to solve Backward SDEs. In the second part, I will introduce some recent results on the use of cubature method for approximating solutions of McKean-Vlasov SDEs.

65C30 ; 60H10 ; 34F05 ; 60H35 ; 91G60

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Random differential equations in scientific computing Neckel, Tobias ; Rupp, Florian | Versita 2013

Ouvrage

V

- xxii; 624 p.
ISBN 978-83-7656-025-0

Localisation : Ouvrage RdC (NECK)

équation différentielle aléatoire # variable aléatoire # calcul stochastique # équation différentielle ordinaire # transformée de Fourier # système dynamique aléatoire

60-02 ; 60H35 ; 65C30

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Numerical approximations of stochastic differential equations with non-globally Lipschitz continuous coefficients Hutzenthaler, Martin ; Jentzen, Arnulf | American Mathematical Society 2014

Ouvrage

V

- v; 99 p.
ISBN 978-1-4704-0984-5

Memoirs of the american mathematical society , 1112

Localisation : Collection 1er étage

équation différentielle stochastique # événement rare # convergence forte # approximation numérique # condition de Lipschitz locale # condition de Lyapunov

60H35 ; 65C05 ; 65C30

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Monte-Carlo methods and stochastic processes:
from linear to non-linear
Gobet, Emmanuel | CRC Press 2016

Ouvrage

y

- xxv; 309 p.
ISBN 978-1-4987-4622-9

Localisation : Ouvrage RdC (GOBE)

méthode de Monte-Carlo # processus stochastique # méthode de simulation # convergence d'algorithmes # équation différentielle stochastique # dynamique non linéaire

65C30 ; 65C35 ; 82C80 ; 65C60 ; 65C10 ; 65C05 ; 65-02 ; 65C50 ; 90C15 ; 90C27 ; 60H15 ; 35R60 ; 60F05 ; 60H35 ; 60H10 ; 34F05 ; 62J02 ; 62F25

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Multi angle

y

Research schools

We will first recall, for a general audience, the use of Monte Carlo and Multi-level Monte Carlo methods in the context of Uncertainty Quantification. Then we will discuss the recently developed Adaptive Multilevel Monte Carlo (MLMC) Methods for (i) It Stochastic Differential Equations, (ii) Stochastic Reaction Networks modeled by Pure Jump Markov Processes and (iii) Partial Differential Equations with random inputs. In this context, the notion of adaptivity includes several aspects such as mesh refinements based on either a priori or a posteriori error estimates, the local choice of different time stepping methods and the selection of the total number of levels and the number of samples at different levels. Our Adaptive MLMC estimator uses a hierarchy of adaptively refined, non-uniform time discretizations, and, as such, it may be considered a generalization of the uniform discretization MLMC method introduced independently by M. Giles and S. Heinrich. In particular, we show that our adaptive MLMC algorithms are asymptotically accurate and have the correct complexity with an improved control of the multiplicative constant factor in the asymptotic analysis. In this context, we developed novel techniques for estimation of parameters needed in our MLMC algorithms, such as the variance of the difference between consecutive approximations. These techniques take particular care of the deepest levels, where for efficiency reasons only few realizations are available to produce essential estimates. Moreover, we show the asymptotic normality of the statistical error in the MLMC estimator, justifying in this way our error estimate that allows prescribing both the required accuracy and confidence level in the final result. We present several examples to illustrate the above results and the corresponding computational savings. We will first recall, for a general audience, the use of Monte Carlo and Multi-level Monte Carlo methods in the context of Uncertainty Quantification. Then we will discuss the recently developed Adaptive Multilevel Monte Carlo (MLMC) Methods for (i) It Stochastic Differential Equations, (ii) Stochastic Reaction Networks modeled by Pure Jump Markov Processes and (iii) Partial Differential Equations with random inputs. In this context, the notion ...

65C30 ; 65C05 ; 60H15 ; 60H35 ; 35R60

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