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Register a new userLarge time behaviors of upwind schemes by jump processes
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We revisit the traditional upwind schemes for linear conservation laws in the viewpoint of jump processes, allowing studying upwind schemes using probabilistic tools. In particular, for Fokker-Planck equations on $mathbb{R}$, in the case of weak confinement, we show that the solution of upwind scheme converges to a stationary solution. In the case of strong confinement, using a discrete Poincare inequality, we prove that the $O(h)$ numeric error under $ell^1$ norm is uniform in time, and establish the uniform exponential convergence to the steady states. Compared with the traditional results of exponential convergence of upwind schemes, our result is in the whole space without boundary. We also establish similar results on torus for which the stationary solution of the scheme does not have detailed balance. This work shows an interesting connection between standard numerical methods and time continuous Markov chains, and could motivate better understanding of numerical analysis for conservation laws.
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This work introduces and studies a new family of velocity jump Markov processes directly amenable to exact simulation with the following two properties: i) trajectories converge in law when a time-step parameter vanishes towards a given Langevin or Hamil-tonian dynamics; ii) the stationary distribution of the process is always exactly given by the product of a Gaussian (for velocities) by any target log-density whose gradient is pointwise computabe together with some additional explicit appropriate upper bound. The process does not exhibit any velocity reflections (jump sizes can be controlled) and is suitable for the factorization method. We provide a rigorous mathematical proof of: i) the small time-step convergence towards Hamiltonian/Langevin dynamics, as well as ii) the exponentially fast convergence towards the target distribution when suitable noise on velocity is present. Numerical implementation is detailed and illustrated.
This paper describes the structure of solutions to Kolmogorovs equations for nonhomogeneous jump Markov processes and applications of these results to control of jump stochastic systems. These equations were studied by Feller (1940), who clarified in 1945 in the errata to that paper that some of its results covered only nonexplosive Markov processes. We present the results for possibly explosive Markov processes. The paper is based on the invited talk presented by the authors at the International Conference dedicated to the 200th anniversary of the birth of P. L.~Chebyshev.
In this paper we introduce non-decreasing jump processes with independent and time non-homogeneous increments. Although they are not Levy processes, they somehow generalize subordinators in the sense that their Laplace exponents are possibly different Bernv{s}tein functions for each time $t$. By means of these processes, a generalization of subordinate semigroups in the sense of Bochner is proposed. Because of time-inhomogeneity, two-parameter semigroups (propagators) arise and we provide a Phillips formula which leads to time dependent generators. The inverse processes are also investigated and the corresponding governing equations obtained in the form of generalized variable order fractional equations. An application to a generalized subordinate Brownian motion is also examined.
This paper addresses the question whether there are numerical schemes for constant-coefficient advection problems that can yield convergent solutions for an infinite time horizon. The motivation is that such methods may serve as building blocks for long-time accurate solutions in more complex advection-dominated problems. After establishing a new notion of convergence in an infinite time limit of numerical methods, we first show that linear methods cannot meet this convergence criterion. Then we present a new numerical methodology, based on a nonlinear jet scheme framework. We show that these methods do satisfy the new convergence criterion, thus establishing that numerical methods exist that converge on an infinite time horizon, and demonstrate the long-time accuracy gains incurred by this property.
This paper introduces a novel methodology for the identification of switching dynamics for switched autoregressive linear models. Switching behavior is assumed to follow a Markov model. The systems outputs are contaminated by possibly large values of measurement noise. Although the procedure provided can handle other noise distributions, for simplicity, it is assumed that the distribution is Normal with unknown variance. Given noisy input-output data, we aim at identifying switched system coefficients, parameters of the noise distribution, dynamics of switching and probability transition matrix of Markovian model. System dynamics are estimated using previous results which exploit algebraic constraints that system trajectories have to satisfy. Switching dynamics are computed with solving a maximum likelihood estimation problem. The efficiency of proposed approach is shown with several academic examples. Although the noise to output ratio can be high, the method is shown to be extremely effective in the situations where a large number of measurements is available.
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