اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدRandom walk approximation for irreversible drift-diffusion process on manifold: ergodicity, unconditional stability and convergence
64
0
0.0
(
0
)
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
Irreversible drift-diffusion processes are very common in biochemical reactions. They have a non-equilibrium stationary state (invariant measure) which does not satisfy detailed balance. For the corresponding Fokker-Planck equation on a closed manifold, via Voronoi tessellation, we propose two upwind finite volume schemes with or without the information of the invariant measure. Both two schemes enjoy stochastic $Q$-matrix structures and can be decomposed as a gradient flow part and a Hamiltonian flow part, which enable us to prove unconditional stability, ergodicity and error estimates. Based on two upwind schemes, several numerical examples - including sampling accelerated by a mixture flow, image transformations and simulations for stochastic model of chaotic system - are conducted. These two structure-preserving schemes also give a natural random walk approximation for a generic irreversible drift-diffusion process on a manifold. Thus they can be adapted to manifold-related computations induced from high dimensional molecular dynamics.
قيم البحث
اقرأ أيضاً
We aim at the development and analysis of the numerical schemes for approximately solving the backward diffusion-wave problem, which involves a fractional derivative in time with order $alphain(1,2)$. From terminal observations at two time levels, i.
e., $u(T_1)$ and $u(T_2)$, we simultaneously recover two initial data $u(0)$ and $u_t(0)$ and hence the solution $u(t)$ for all $t > 0$. First of all, existence, uniqueness and Lipschitz stability of the backward diffusion-wave problem were established under some conditions about $T_1$ and $T_2$. Moreover, for noisy data, we propose a quasi-boundary value scheme to regularize the mildly ill-posed problem, and show the convergence of the regularized solution. Next, to numerically solve the regularized problem, a fully discrete scheme is proposed by applying finite element method in space and convolution quadrature in time. We establish error bounds of the discrete solution in both cases of smooth and nonsmooth data. The error estimate is very useful in practice since it indicates the way to choose discretization parameters and regularization parameter, according to the noise level. The theoretical results are supported by numerical experiments.
In this article, we discuss the numerical solution of diffusion equations on random surfaces within the isogeometric framework. Complex computational geometries, given only by surface triangulations, are recast into the isogeometric context by transf
orming them into quadrangulations and a subsequent interpolation procedure. Moreover, we describe in detail, how diffusion problems on random surfaces can be modelled and how quantities of interest may be derived. In particular, we propose a low rank approximation algorithm for the high-dimensional space-time correlation of the random solution. Extensive numerical studies are provided to quantify and validate the approach.
We present a novel class of high-order space-time finite element schemes for the Poisson-Nernst-Planck (PNP) equations. We prove that our schemes are mass conservative, positivity preserving, and unconditionally energy stable for any order of approxi
mation. To the best of our knowledge, this is the first class of (arbitrarily) high-order accurate schemes for the PNP equations that simultaneously achieve all these three properties. This is accomplished via (1) using finite elements to directly approximate the so-called entropy variable instead of the density variable, and (2) using a discontinuous Galerkin (DG) discretization in time. The entropy variable formulation, which was originally developed by Metti et al. [17] under the name of a log-density formulation, guarantees both positivity of densities and a continuous-in-time energy stability result. The DG in time discretization further ensures an unconditional energy stability in the fully discrete level for any approximation order, where the lowest order case is exactly the backward Euler discretization and in this case we recover the method of Metti et al. [17].
We consider stochastic differential equations driven by a general Levy processes (SDEs) with infinite activity and the related, via the Feynman-Kac formula, Dirichlet problem for parabolic integro-differential equation (PIDE). We approximate the solu
tion of PIDE using a numerical method for the SDEs. The method is based on three ingredients: (i) we approximate small jumps by a diffusion; (ii) we use restricted jump-adaptive time-stepping; and (iii) between the jumps we exploit a weak Euler approximation. We prove weak convergence of the considered algorithm and present an in-depth analysis of how its error and computational cost depend on the jump activity level. Results of some numerical experiments, including pricing of barrier basket currency options, are presented.
A study is presented on the convergence of the computation of coupled advection-diffusion-reaction equations. In the computation, the equations with different coefficients and even types are assigned in two subdomains, and Schwarz iteration is made b
etween the equations when marching from a time level to the next one. The analysis starts with the linear systems resulting from the full discretization of the equations by explicit schemes. Conditions for convergence are derived, and its speedup and the effects of difference in the equations are discussed. Then, it proceeds to an implicit scheme, and a recursive expression for convergence speed is derived. An optimal interface condition for the Schwarz iteration is obtained, and it leads to perfect convergence, that is, convergence within two times of iteration. Furthermore, the methods and analyses are extended to the coupling of the viscous Burgers equations. Numerical experiments indicate that the conclusions, such as the perfect convergence, drawn in the linear situations may remain in the Burgers equations computation.
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
