The Experts below are selected from a list of 10716 Experts worldwide ranked by ideXlab platform
Fausto Gozzi - One of the best experts on this subject based on the ideXlab platform.
-
Kolmogorov Equation ASSOCIATED TO A STOCHASTIC NAVIER-STOKES Equation
Journal of Functional Analysis, 1998Co-Authors: Franco Flandoli, Fausto GozziAbstract:Abstract A direct solution of the Kolmogorov Equation associated to a stochastic Navier–Stokes Equation is given, with restriction to two space dimensions and periodic boundary conditions. The existence of a variational solution is proved, using a special property of the nonlinear operator.
Stephen S.-t. Yau - One of the best experts on this subject based on the ideXlab platform.
-
hermite spectral method to 1 d forward Kolmogorov Equation and its application to nonlinear filtering problems
IEEE Transactions on Automatic Control, 2013Co-Authors: Xue Luo, Stephen S.-t. YauAbstract:In this paper, we investigate the Hermite spectral method (HSM) to numerically solve the forward Kolmogorov Equation (FKE). A useful guideline of choosing the scaling factor of the generalized Hermite functions is given in this paper. It greatly improves the resolution of HSM. The convergence rate of HSM to FKE is analyzed in the suitable function space and has been verified by the numerical simulation. As an important application and our primary motivation to study the HSM to FKE, we work on the implementation of the nonlinear filtering (NLF) problems with a real-time algorithm developed by S.-T. Yau and the second author in 2008. The HSM to FKE is served as the off-line computation in this algorithm. The translating factor of the generalized Hermite functions and the moving-window technique are introduced to deal with the drifting of the posterior conditional density function of the states in the on-line experiments. Two numerical experiments of NLF problems are carried out to illustrate the feasibility of our algorithm. Moreover, our algorithm surpasses the particle filters as a real-time solver to NLF.
-
hermite spectral method to 1d forward Kolmogorov Equation and its application to nonlinear filtering problems
arXiv: Optimization and Control, 2013Co-Authors: Xue Luo, Stephen S.-t. YauAbstract:In this paper, we investigate the Hermite spectral method (HSM) to numerically solve the forward Kolmogorov Equation (FKE). A useful guideline of choosing the scaling factor of the generalized Hermite functions is given in this paper. It greatly improves the resolution of HSM. The convergence rate of HSM to FKE is analyzed in the suitable function space and has been verified by the numerical simulation. As an important application and our primary motivation to study the HSM to FKE, we work on the implementation of the nonlinear filtering (NLF) problem with a real-time algorithm developed in [17]. The HSM to FKE is served as the off-line computation in this algorithm. The translating factor of the generalized Hermite functions and the moving-window technique are introduced to deal with the drifting of the posterior conditional density function of the states in the on-line experiments. Two numerical experiments of NLF problems are carried out to illustrate the feasibility of our algorithm. Moreover, our algorithm surpasses the particle filter as a real-time solver to NLF.
-
Wavelet-Galerkin method for the Kolmogorov Equation
Mathematical and Computer Modelling, 2004Co-Authors: Zhigang Liang, Stephen S.-t. YauAbstract:It is well known that the Kolmogorov Equation plays an important role in applied science. For example, the nonlinear filtering problem, which plays a key role in modern technologies, was solved by Yau and Yau [1] by reducing it to the off-line computation of the Kolmogorov Equation. In this paper, we develop a theorical foundation of using the wavelet-Galerkin method to solve linear parabolic P.D.E. We apply our theory to the Kolmogorov Equation. We give a rigorous proof that the solution of the Kolmogorov Equation can be approximated very well in any finite domain by our wavelet-Galerkin method. An example is provided by using Daubechies D"4 scaling functions.
-
finite dimensional filters with nonlinear drift xi explicit solution of the generalized Kolmogorov Equation in brockett mitter program
Advances in Mathematics, 1998Co-Authors: Shingtung Yau, Stephen S.-t. YauAbstract:Abstract Ever since the technique of the Kalman–Bucy filter was popularized, there has been an intense interest in finding new classes of finite-dimensional recursive filters. In the late 1970s the concept of the estimation algebra of a filtering system was introduced. Brockett, Clark, and Mitter proposed to use the Wei–Norman approach to solve the nonlinear filtering problem. In 1990, Tam, Wong, and Yau presented a rigorous proof of the Brocket–Mitter program which allows one to construct finite-dimensional recursive filters from finite–dimensional estimation algebras. Later Yau wrote down explicitly a system of ordinary differential Equations and generalized Kolmogorov Equation to which the robust Duncan–Mortenser– Zakai Equation can be reduced. Thus there remains three fundamental problems in Brockett–Mitter program. The first is the problem of finding explicit solution to the generalized Kolmogorov Equation. The second is the problem of finding real-time solution of a system of ODEs. The third is the Brockett's problem of classification of finite–dimensional estimation algebras. In this paper, we solve the first problem.
-
Explicit solution of a Kolmogorov Equation
Applied Mathematics and Optimization, 1996Co-Authors: S.-t. Yau, Stephen S.-t. YauAbstract:Ever since the technique of the Kalman-Bucy filter was popularized, there has been an intense interest in finding new classes of finite-dimensional recursive filters. In the late seventies, the concept of the estimation algebra of a filtering system was introduced. It has been the major tool in studying the Duncan-Mortensen-Zakai Equation. Recently the second author has constructed general finite-dimensional filters which contain both Kalman-Bucy filters and Benes filter as special cases. In this paper we consider a filtering system with arbitrary nonlinear drift f ( x ) which satisfies some regularity assumption at infinity. This is a natural assumption in view of Theorem 10 of [DTWY] in a special case. Under the assumption on the observation h( x )=constant, we propose writing down the solution of the Duncan-Mortensen-Zakai Equation explicitly.
Luciano Tubaro - One of the best experts on this subject based on the ideXlab platform.
-
Kolmogorov Equation associated to the stochastic reflection problem on a smooth convex set of a hilbert space
arXiv: Probability, 2009Co-Authors: Viorel Barbu, Giuseppe Da Prato, Luciano TubaroAbstract:We consider the stochastic reflection problem associated with a self-adjoint operator $A$ and a cylindrical Wiener process on a convex set $K$ with nonempty interior and regular boundary $\Sigma$ in a Hilbert space $H$. We prove the existence and uniqueness of a smooth solution for the corresponding elliptic infinite-dimensional Kolmogorov Equation with Neumann boundary condition on $\Sigma$.
-
Kolmogorov Equation associated to the stochastic reflection problem on a smooth convex set of a Hilbert space
Annals of Probability, 2009Co-Authors: Viorel Barbu, Giuseppe Da Prato, Luciano TubaroAbstract:We consider the stochastic reflection problem associated with a selfadjoint operator A and a cylindrical Wiener process on a convex set K with nonempty interior and regular boundary E in a Hilbert space H. We prove the existence and uniqueness of a smooth solution for the corresponding elliptic infinite-dimensional Kolmogorov Equation with Neumann boundary condition on E.
Franco Flandoli - One of the best experts on this subject based on the ideXlab platform.
-
Kolmogorov Equations Associated to the Stochastic Two Dimensional Euler Equations
SIAM Journal on Mathematical Analysis, 2019Co-Authors: Franco Flandoli, Dejun LuoAbstract:The Kolmogorov Equation associated to a stochastic two dimensional Euler Equation with transport type noise and random initial conditions is studied in the weak sense by a direct approach, based on...
-
A spectral-based numerical method for Kolmogorov Equations in Hilbert spaces
Infinite Dimensional Analysis Quantum Probability and Related Topics, 2016Co-Authors: Francisco J. Delgado-vences, Franco FlandoliAbstract:We propose a numerical solution for the solution of the Fokker–Planck–Kolmogorov (FPK) Equations associated with stochastic partial differential Equations in Hilbert spaces. The method is based on the spectral decomposition of the Ornstein–Uhlenbeck semigroup associated to the Kolmogorov Equation. This allows us to write the solution of the Kolmogorov Equation as a deterministic version of the Wiener–Chaos Expansion. By using this expansion we reformulate the Kolmogorov Equation as an infinite system of ordinary differential Equations, and by truncating it we set a linear finite system of differential Equations. The solution of such system allow us to build an approximation to the solution of the Kolmogorov Equations. We test the numerical method with the Kolmogorov Equations associated with a stochastic diffusion Equation, a Fisher–KPP stochastic Equation and a stochastic Burgers Equation in dimension 1.
-
Kolmogorov Equation ASSOCIATED TO A STOCHASTIC NAVIER-STOKES Equation
Journal of Functional Analysis, 1998Co-Authors: Franco Flandoli, Fausto GozziAbstract:Abstract A direct solution of the Kolmogorov Equation associated to a stochastic Navier–Stokes Equation is given, with restriction to two space dimensions and periodic boundary conditions. The existence of a variational solution is proved, using a special property of the nonlinear operator.
Viorel Barbu - One of the best experts on this subject based on the ideXlab platform.
-
Kolmogorov Equation associated to the stochastic reflection problem on a smooth convex set of a hilbert space
arXiv: Probability, 2009Co-Authors: Viorel Barbu, Giuseppe Da Prato, Luciano TubaroAbstract:We consider the stochastic reflection problem associated with a self-adjoint operator $A$ and a cylindrical Wiener process on a convex set $K$ with nonempty interior and regular boundary $\Sigma$ in a Hilbert space $H$. We prove the existence and uniqueness of a smooth solution for the corresponding elliptic infinite-dimensional Kolmogorov Equation with Neumann boundary condition on $\Sigma$.
-
Kolmogorov Equation associated to the stochastic reflection problem on a smooth convex set of a Hilbert space
Annals of Probability, 2009Co-Authors: Viorel Barbu, Giuseppe Da Prato, Luciano TubaroAbstract:We consider the stochastic reflection problem associated with a selfadjoint operator A and a cylindrical Wiener process on a convex set K with nonempty interior and regular boundary E in a Hilbert space H. We prove the existence and uniqueness of a smooth solution for the corresponding elliptic infinite-dimensional Kolmogorov Equation with Neumann boundary condition on E.
