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Register a new userEfficient numerical approximation of a non-regular Fokker--Planck equation associated with first-passage time distributions
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Added by
Gregor Gantner
Publication date
2021
fields
Informatics Engineering
and research's language is
English
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In neuroscience, the distribution of a decision time is modelled by means of a one-dimensional Fokker--Planck equation with time-dependent boundaries and space-time-dependent drift. Efficient approximation of the solution to this equation is required, e.g., for model evaluation and parameter fitting. However, the prescribed boundary conditions lead to a strong singularity and thus to slow convergence of numerical approximations. In this article we demonstrate that the solution can be related to the solution of a parabolic PDE on a rectangular space-time domain with homogeneous initial and boundary conditions by transformation and subtraction of a known function. We verify that the solution of the new PDE is indeed more regular than the solution of the original PDE and proceed to discretize the new PDE using a space-time minimal residual method. We also demonstrate that the solution depends analytically on the parameters determining the boundaries as well as the drift. This justifies the use of a sparse tensor product interpolation method to approximate the PDE solution for various parameter ranges. The predicted convergence rates of the minimal residual method and that of the interpolation method are supported by numerical simulations.
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Fractional Fokker-Planck equation plays an important role in describing anomalous dynamics. To the best of our knowledge, the existing discussions mainly focus on this kind of equation involving one diffusion operator. In this paper, we first derive the fractional Fokker-Planck equation with two-scale diffusion from the Levy process framework, and then the fully discrete scheme is built by using the $L_{1}$ scheme for time discretization and finite element method for space. With the help of the sharp regularity estimate of the solution, we optimally get the spatial and temporal error estimates. Finally, we validate the effectiveness of the provided algorithm by extensive numerical experiments.
We consider the problem of computing first-passage time distributions for reaction processes modelled by master equations. We show that this generally intractable class of problems is equivalent to a sequential Bayesian inference problem for an auxiliary observation process. The solution can be approximated efficiently by solving a closed set of coupled ordinary differential equations (for the low-order moments of the process) whose size scales with the number of species. We apply it to an epidemic model and a trimerisation process, and show good agreement with stochastic simulations.
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In this article, we develop and analyze a full discretization, based on the spatial spectral Galerkin method and the temporal drift implicit Euler scheme, for the stochastic Cahn--Hilliard equation driven by multiplicative space-time white noise. By introducing an appropriate decomposition of the numerical approximation, we first use the factorization method to deduce the a priori estimate and regularity estimate of the proposed full discretization. With the help of the variation approach, we then obtain the sharp spatial and temporal convergence rate in negative Sobolev space in mean square sense. Furthermore, the sharp mean square convergence rates in both time and space are derived via the Sobolev interpolation inequality and semigroup theory. To the best of our knowledge, this is the first result on the convergence rate of temporally and fully discrete numerical methods for the stochastic Cahn--Hilliard equation driven by multiplicative space-time white noise.
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