204 research outputs found
Computational Issues for Optimal Shape Design in Hemodynamics
A Fluid-Structure Interaction model is studied for aortic flow, based on Koiter's shell model for the structure, Navier-Stokes equation for the fluid and transpiration for the coupling. It accounts for wall deformation while yet working on a fixed geometry. The model is established first. Then a numerical approximation is proposed and some tests are given. The model is also used for optimal design of a stent and possible recovery of the arterial wall elastic coefficients by inverse methods
A reduced basis for option pricing
We introduce a reduced basis method for the efficient numerical solution of partial integro-differential equations which arise in option pricing theory. Our method uses a basis of functions constructed from a sequence of Black-Scholes solutions with different volatilities. We show that this choice of basis leads to a sparse representation of option pricing functions, yielding an approximation whose precision is exponential in the number of basis functions. A Galerkin method using this basis for solving the pricing PDE is presented. Numerical tests based on the CEV diffusion model and the Merton jump diffusion model show that the method has better numerical performance relative to commonly used finite-difference and finite-element methods. We also compare our method with a numerical Proper Orthogonal Decomposition (POD). Finally, we show that this approach may be used advantageously for the calibration of local volatility functions.
Dynamic Programming for Mean-field type Control
International audienceFor mean-field type control problems, stochastic dynamic programming requires adaptation. We propose to reformulate the problem as a distributed control problem by assuming that the PDF of the stochastic process exists. Then we show that Bellman's principle applies to the dynamic programming value function where the dependency on is functional as in P.L. Lions' analysis of mean-filed games (2007). We derive HJB equations and apply them to two examples, a portfolio optimization and a systemic risk model
Simulation of the 3D Radiative Transfer with Anisotropic Scattering for Convective Trails
The integro-differential formulation of the RTE and its solution by
iterations on the source has been extended here to handle anisotropic
scattering. The iterative part of the method is O(N ln N ), thanks to an
efficient use of H-matrices. The precision is good enough to evaluate the
effect of sensitive parameters for the study of contrails. Most of the time the
stratified 1D approximation should suffice, but in complex cases with high
relief the 3D formulation is needed
Reflective Conditions for Radiative Transfer in Integral Form with H-Matrices
In a recent article the authors showed that the radiative Transfer equations
with multiple frequencies and scattering can be formulated as a nonlinear
integral system. In the present article, the formulation is extended to handle
reflective boundary conditions. The fixed point method to solve the system is
shown to be monotone. The discretization is done with a Finite Element
Method. The convolution integrals are precomputed at every vertices of the mesh
and stored in compressed hierarchical matrices, using Partially Pivoted
Adaptive Cross-Approximation. Then the fixed point iterations involve only
matrix vector products. The method is , with respect to
the number of vertices, when everything is smooth. A numerical implementation
is proposed and tested on two examples. As there are some analogies with ray
tracing the programming is complex
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