Approximation de Gauss | EPFL Graph Search

L'approximation de Gauss nommée d'après le physicien allemand Carl Friedrich Gauss, est l'approximation linéaire de l'optique géométrique obtenue dans certaines conditions appelées conditions de Gauss. Cette approximation, souvent applicable en pratique, permet de simplifier les relations mathématiques de l'optique géométrique. On obtient dans ces conditions un stigmatisme approché. Les écarts à cette approximation rencontrés dans les instruments d'optique sont appelés aberrations géométriques. Conditions de Gauss Les conditions dans lesquelles on peut appliquer l'approximation de Gauss sont les suivantes :

les angles d'incidence des rayons par rapport à l'axe optique de l'élément sont faibles ;
le point d'incidence est proche de l'axe optique : on dit alors que l'on travaille avec des rayons paraxiaux.

Lorsque ces conditions sont respectées, on peut considérer le système optique comme approximativement stigmatique. Pour réaliser ces conditions, on peut utiliser des

Reduction techniques for PDEs built upon Reduced Basis and Domain Decomposition Methods with applications to hemodynamics

Luca Pegolotti

This thesis focuses on the development and validation of a reduced order technique for cardiovascular simulations. The method is based on the combined use of the Reduced Basis method and a Domain Decomposition approach and can be seen as a particular implementation of the Reduced Basis Element method. Our contributions include the application to the unsteady three-dimensional Navier--Stokes equations, the introduction of a reduced coupling between subdomains, and the reconstruction of arteries with deformed elementary building blocks. The technique is divided into two main stages: the offline and the online phases. In the offline phase, we define a library of reference building blocks (e.g., tubes and bifurcations) and associate with each of these a set of Reduced Basis functions for velocity and pressure. The set of Reduced Basis functions is obtained by Proper Orthogonal Decomposition of a large number of flow solutions called snapshots; this step is expensive in terms of computational time. In the online phase, the artery of interest is geometrically approximated as a composition of subdomains, which are obtained from the parametrized deformation of the aforementioned building blocks. The local solution in each subdomain is then found as a linear combination of the Reduced Basis functions defined in the corresponding building block. The strategy to couple the local solutions is of utmost importance. In this thesis, we devise a nonconforming method for the coupling of Partial Differential Equations that takes advantage of the definition of a small number of Lagrange multiplier basis functions on the interfaces. We show that this strategy allows us to preserve the h-convergence properties of the discretization method of choice for the primal variable even when a small number of Lagrange multiplier basis functions is employed. Moreover, we test the flexibility of the approach in scenarios in which different discretization algorithms are employed in the subdomains, and we also use it in a fluid-structure interaction benchmark. The introduction of the Lagrange multipliers, however, is associated with stability problems deriving from the saddle-point structure of the global system. In our Reduced Order Model, the stability is recovered by means of supremizers enrichment. In our numerical simulations, we specifically focus on the effects of the Reduced Basis and geometrical approximations on the quality of the results. We show that the Reduced Order Model performs similarly to the corresponding high-fidelity one in terms of accuracy. Compared to other popular models for cardiovascular simulations (namely 1D models), it also allows us to compute a local reconstruction of the Wall-Shear Stress on the vessel wall. The speedup with respect to the Finite Element method is substantial (at least one order of magnitude), although the current implementation presents bottlenecks that are addressed in depth throughout the thesis.

EPFL2021

Fast adaptive uniformisation of the chemical master equation

Maria-Emanuela-Canini Mateescu, Verena Wolf

Within systems biology there is an increasing interest in the stochastic behaviour of biochemical reaction networks. An appropriate stochastic description is provided by the chemical master equation, which represents a continuous-time Markov chain (CTMC). The uniformisation technique is an efficient method to compute probability distributions of a CTMC if the number of states is manageable. However, the size of a CTMC that represents a biochemical reaction network is usually far beyond what is feasible. In this study, the authors present an on-the-fly variant of uniformisation, where they improve the original algorithm at the cost of a small approximation error. By means of several examples, the authors show that their approach is particularly well-suited for biochemical reaction networks.

2010

Fast Adaptive Uniformization for the Chemical Master Equation

Maria-Emanuela-Canini Mateescu, Verena Wolf

Within systems biology there is an increasing interest in the stochastic behavior of biochemical reaction networks. An appropriate stochastic description is provided by the chemical master equation, which represents a continuous-time Markov chain (CTMC). Standard Uniformization (SU) is an efficient method for the transient analysis of CTMCs. For systems with very different time scales, such as biochemical reaction networks, SU is computationally expensive. In these cases, a variant of SU, called adaptive uniformization (AU), is known to reduce the large number of iterations needed by SU. The additional difficulty of AU is that it requires the solution of a birth process. In this paper we present an on-the-fly variant of AU, where we improve the original algorithm for AU at the cost of a small approximation error. By means of several examples, we show that our approach is particularly well-suited for biochemical reaction networks.

Ieee Computer Soc Press, Customer Service Center, Po Box 3014, 10662 Los Vaqueros Circle, Los Alamitos, Ca 90720-1264 Usa2009

Reduction techniques for PDEs built upon Reduced Basis and Domain Decomposition Methods with applications to hemodynamics

Luca Pegolotti

EPFL2021