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Post-edited  An introduction to molecular dynamics
Stoltz, Gabriel (Auteur de la Conférence) | CIRM (Editeur )

The aim of this two-hour lecture is to present the mathematical underpinnings of some common numerical approaches to compute average properties as predicted by statistical physics. The first part provides an overview of the most important concepts of statistical physics (in particular thermodynamic ensembles). The aim of the second part is to provide an introduction to the practical computation of averages with respect to the Boltzmann-Gibbs measure using appropriate stochastic dynamics of Langevin type. Rigorous ergodicity results as well as elements on the estimation of numerical errors are provided. The last part is devoted to the computation of transport coefficients such as the mobility or autodiffusion in fluids, relying either on integrated equilibrium correlations à la Green-Kubo, or on the linear response of nonequilibrium dynamics in their steady-states. The aim of this two-hour lecture is to present the mathematical underpinnings of some common numerical approaches to compute average properties as predicted by statistical physics. The first part provides an overview of the most important concepts of statistical physics (in particular thermodynamic ensembles). The aim of the second part is to provide an introduction to the practical computation of averages with respect to the Boltzmann-Gibbs ...

82B31 ; 82B80 ; 65C30 ; 82C31 ; 82C70 ; 60H10

In this lecture, we shall discuss the key steps involved in the use of least squares regression for approximating the solution to BSDEs. This includes how to obtain explicit error estimates, and how these error estimates can be used to tune the parameters of the numerical scheme based on complexity considerations.
The algorithms are based on a two stage approximation process. Firstly, a suitable discrete time process is chosen to approximate the of the continuous time solution of the BSDE. The nodes of the discrete time processes can be expressed as conditional expectations. As we shall demonstrate, the choice of discrete time process is very important, as its properties will impact the performance of the overall numerical scheme. In the second stage, the conditional expectation is approximated in functional form using least squares regression on synthetically generated data n Monte Carlo simulations drawn from a suitable probability distribution. A key feature of the regression step is that the explanatory variables are built on a user chosen finite dimensional linear space of functions, which the user specifies by setting basis functions. The choice of basis functions is made on the hypothesis that it contains the solution, so regularity and boundedness assumptions are used in its construction. The impact of the choice of the basis functions is exposed in error estimates.
In addition to the choice of discrete time approximation and the basis functions, the Markovian structure of the problem gives significant additional freedom with regards to the Monte Carlo simulations. We demonstrate how to use this additional freedom to develop generic stratified sampling approaches that are independent of the underlying transition density function. Moreover, we demonstrate how to leverage the stratification method to develop a HPC algorithm for implementation on GPUs.
Thanks to the Feynmann-Kac relation between the the solution of a BSDE and its associated semilinear PDE, the approximation of the BSDE can be directly used to approximate the solution of the PDE. Moreover, the smoothness properties of the PDE play a crucial role in the selection of the hypothesis space of regressions functions, so this relationship is vitally important for the numerical scheme.
We conclude with some draw backs of the regression approach, notably the curse of dimensionality.
In this lecture, we shall discuss the key steps involved in the use of least squares regression for approximating the solution to BSDEs. This includes how to obtain explicit error estimates, and how these error estimates can be used to tune the parameters of the numerical scheme based on complexity considerations.
The algorithms are based on a two stage approximation process. Firstly, a suitable discrete time process is chosen to approximate the ...

65C05 ; 65C30 ; 93E24 ; 60H35 ; 60H10

Multi angle  Cubature methods and applications
Crisan, Dan (Auteur de la Conférence) | CIRM (Editeur )

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

Multi angle  An introduction to BSDE
Imkeller, Peter (Auteur de la Conférence) | CIRM (Editeur )

Backward stochastic differential equations have been a very successful and active tool for stochastic finance and insurance for some decades. More generally they serve as a central method in applications of control theory in many areas. We introduce BSDE by looking at a simple utility optimization problem in financial stochastics. We shall derive an important class of BSDE by applying the martingale optimality principle to solve an optimal investment problem for a financial agent whose income is partly affected by market external risk. We then present the basics of existence and uniqueness theory for solutions to BSDE the coefficients of which satisfy global Lipschitz conditions. Backward stochastic differential equations have been a very successful and active tool for stochastic finance and insurance for some decades. More generally they serve as a central method in applications of control theory in many areas. We introduce BSDE by looking at a simple utility optimization problem in financial stochastics. We shall derive an important class of BSDE by applying the martingale optimality principle to solve an optimal ...

91B24 ; 60H15 ; 60H10 ; 91G80

We introduce a new strategy for the solution of Mean Field Games in the presence of major and minor players. This approach is based on a formulation of the fixed point step in spaces of controls. We use it to highlight the differences between open and closed loop problems. We illustrate the implementation of this approach for linear quadratic and finite state space games, and we provide numerical results motivated by applications in biology and cyber-security. We introduce a new strategy for the solution of Mean Field Games in the presence of major and minor players. This approach is based on a formulation of the fixed point step in spaces of controls. We use it to highlight the differences between open and closed loop problems. We illustrate the implementation of this approach for linear quadratic and finite state space games, and we provide numerical results motivated by applications in biology and ...

93E20 ; 60H10 ; 60K35 ; 49K45

In this talk, I will introduce the stochastic downscaling method (SDM) that borrows techniques from small scale turbulence (S.B. Pope) for the simulation of wind flows thanks to hybrid methods (deterministic-stochastic). I will present the downscaling method used to refine a wind forecast at a sufficiently small scale, and the way wind turbines are implemented in the model. Comparisons with traditional numerical methods (LES) and validation w.r.t. experimental data will also be provided. In this talk, I will introduce the stochastic downscaling method (SDM) that borrows techniques from small scale turbulence (S.B. Pope) for the simulation of wind flows thanks to hybrid methods (deterministic-stochastic). I will present the downscaling method used to refine a wind forecast at a sufficiently small scale, and the way wind turbines are implemented in the model. Comparisons with traditional numerical methods (LES) and validation ...

60H10 ; 86A10 ; 86-08 ; 76F55 ; 76M35

Multi angle  Some applications of irreversibility
Rey-Bellet, Luc (Auteur de la Conférence) | CIRM (Editeur )

Multi angle  Fast slow systems with chaotic noise
Kelly, David (Auteur de la Conférence) | CIRM (Editeur )

It has long been observed that multi-scale systems, particularly those in climatology, exhibit behavior typical of stochastic models, most notably in the unpredictability and statistical variability of events. This is often in spite of the fact that the underlying physical model is completely deterministic. One possible explanation for this stochastic behavior is deterministic chaotic effects. In fact, it has been well established that the statistical properties of chaotic systems can be well approximated by stochastic differential equations. In this talk, we focus on fast-slow ODEs, where the fast, chaotic variables are fed into the slow variables to yield a diffusion approximation. In particular we focus on the case where the chaotic noise is multidimensional and multiplicative. The tools from rough path theory prove useful in this difficult setting. It has long been observed that multi-scale systems, particularly those in climatology, exhibit behavior typical of stochastic models, most notably in the unpredictability and statistical variability of events. This is often in spite of the fact that the underlying physical model is completely deterministic. One possible explanation for this stochastic behavior is deterministic chaotic effects. In fact, it has been well established that the ...

60H10 ; 37D20 ; 37D25 ; 37A50

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