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In various scientific and industrial fields, stochastic simulations are taking on a new importance. This is due to the increasing power of computers and practitioners’ aim to simulate more and more complex systems, and thus use random parameters as well as random noises to model the parametric uncertainties and the lack of knowledge on the physics of these systems. The error analysis of these computations is a highly complex mathematical undertaking. Approaching these issues, the authors present stochastic numerical methods and prove accurate convergence rate estimates in terms of their numerical parameters (number of simulations, time discretization steps). As a result, the book is a self-contained and rigorous study of the numerical methods within a theoretical framework. After briefly reviewing the basics, the authors first introduce fundamental notions in stochastic calculus and continuous-time martingale theory, then develop the analysis of pure-jump Markov processes, Poisson processes, and stochastic differential equations. In particular, they review the essential properties of Itô integrals and prove fundamental results on the probabilistic analysis of parabolic partial differential equations. These results in turn provide the basis for developing stochastic numerical methods, both from an algorithmic and theoretical point of view. The book combines advanced mathematical tools, theoretical analysis of stochastic numerical methods, and practical issues at a high level, so as to provide optimal results on the accuracy of Monte Carlo simulations of stochastic processes. It is intended for master and Ph.D. students in the field of stochastic processes and their numerical applications, as well as for physicists, biologists, economists and other professionals working with stochastic simulations, who will benefit from the ability to reliably estimate and control the accuracy of their simulations.
Parameter estimation in stochastic differential equations and stochastic partial differential equations is the science, art and technology of modeling complex phenomena. The subject has attracted researchers from several areas of mathematics. This volume presents the estimation of the unknown parameters in the corresponding continuous models based on continuous and discrete observations and examines extensively maximum likelihood, minimum contrast and Bayesian methods.
With this hands-on introduction readers will learn what SDEs are all about and how they should use them in practice.
We introduce computationally efficient Monte Carlo methods for studying the statistics of stochastic differential equations in two distinct settings. In the first, we derive importance sampling methods for data assimilation when the noise in the model and observations are small. The methods are formulated in discrete time, where the "posterior" distribution we want to sample from can be analyzed in an accessible small noise expansion. We show that a "symmetrization" procedure akin to antithetic coupling can improve the order of accuracy of the sampling methods, which is illustrated with numerical examples. In the second setting, we develop "stochastic continuation" methods to estimate level sets for statistics of stochastic differential equations with respect to their parameters. We adapt Keller's Pseudo-Arclength continuation method to this setting using stochastic approximation, and generalized least squares regression. Furthermore, we show that the methods can be improved through the use of coupling methods to reduce the variance of the derivative estimates that are involved.
No detailed description available for "Parametric Estimates by the Monte Carlo Method".
This book presents the texts of seminars presented during the years 1995 and 1996 at the Université Paris VI and is the first attempt to present a survey on this subject. Starting from the classical conditions for existence and unicity of a solution in the most simple case-which requires more than basic stochartic calculus-several refinements on the hypotheses are introduced to obtain more general results.
A central problem in numerous applications is estimating the unknown parameters of a system of ordinary differential equations (ODEs) from noisy measurements of a function of some of the states at discrete times. Formulating this dynamic inverse problem in a Bayesian statistical framework, state and parameter estimation can be performed using sequential Monte Carlo (SMC) methods, such as particle filters (PFs) and ensemble Kalman filters (EnKFs).Addressing the issue of particle retention in PF-SMC, we propose to solve ODE systems within a PF framework with higher order numerical integrators which can handle stiffness and to base the choice of the innovation variance on estimates of discretization errors. Using linear multistep method (LMM) numerical solvers in this context gives a handle on the stability and accuracy of propagation, and provides a natural and systematic way to rigorously estimate the innovation variance via well-known local error estimates.We explore computationally efficient implementations of LMM PF-SMC by considering parallelized and vectorized formulations. While PF algorithms are known to be amenable to parallelization due to the independent propagation of each particle, by formulating the problem in a vectorized fashion, it is possible to arrive at an implementation of the method which takes full advantage of multiple processors.We employ a variation of LMM PF-SMC in estimating unknown parameters of a tracer kinetics model from sequences of real positron emission tomography scan data. A combination of optimization and statistical inference is utilized: nonlinear least squares finds optimal starting values, which then act as hyperparameters in the Bayesian framework. The LMM PF-SMC algorithm is modified to allow variable time steps to accommodate the increase in time interval length between data measurements from beginning to end of the procedure, keeping the time step the same for each particle.We also apply the idea of linking innovation variance with numerical integration error estimates to EnKFs by employing a stochastic interpretation of the discretization error in numerical integrators, extending the technique to deterministic, large-scale nonlinear evolution models. The resulting algorithm, which introduces LMM time integrators into the EnKF framework, proves especially effective in predicting unmeasured system components.
This text is used by for the resolution of partial differential equations, trasnport equations, the Boltzmann equation and the parabolic equations of diffusion.
A unified Bayesian treatment of the state-of-the-art filtering, smoothing, and parameter estimation algorithms for non-linear state space models.