490 research outputs found
A modified semi--implict Euler-Maruyama Scheme for finite element discretization of SPDEs with additive noise
We consider the numerical approximation of a general second order
semi--linear parabolic stochastic partial differential equation (SPDE) driven
by additive space-time noise. We introduce a new modified scheme using a linear
functional of the noise with a semi--implicit Euler--Maruyama method in time
and in space we analyse a finite element method (although extension to finite
differences or finite volumes would be possible). We prove convergence in the
root mean square norm for a diffusion reaction equation and diffusion
advection reaction equation. We present numerical results for a linear reaction
diffusion equation in two dimensions as well as a nonlinear example of
two-dimensional stochastic advection diffusion reaction equation. We see from
both the analysis and numerics that the proposed scheme has better convergence
properties than the standard semi--implicit Euler--Maruyama method
Monte Carlo versus multilevel Monte Carlo in weak error simulations of SPDE approximations
The simulation of the expectation of a stochastic quantity E[Y] by Monte
Carlo methods is known to be computationally expensive especially if the
stochastic quantity or its approximation Y_n is expensive to simulate, e.g.,
the solution of a stochastic partial differential equation. If the convergence
of Y_n to Y in terms of the error |E[Y - Y_n]| is to be simulated, this will
typically be done by a Monte Carlo method, i.e., |E[Y] - E_N[Y_n]| is computed.
In this article upper and lower bounds for the additional error caused by this
are determined and compared to those of |E_N[Y - Y_n]|, which are found to be
smaller. Furthermore, the corresponding results for multilevel Monte Carlo
estimators, for which the additional sampling error converges with the same
rate as |E[Y - Y_n]|, are presented. Simulations of a stochastic heat equation
driven by multiplicative Wiener noise and a geometric Brownian motion are
performed which confirm the theoretical results and show the consequences of
the presented theory for weak error simulations.Comment: 16 pages, 5 figures; formulated Section 2 independently of SPDEs,
shortened Section 3, added example of geometric Brownian motion in Section
Convergence rates of the truncated Euler-Maruyama method for stochastic differential equations
Influenced by Higham, Mao and Stuart [9], several numerical methods have been developed to study the strong convergence of the numerical solutions to stochastic differential equations (SDEs) under the local Lipschitz condition. These numerical methods include the tamed Euler–Maruyama (EM) method, the tamed Milstein method, the stopped EM, the backward EM, the backward forward EM, etc. Recently, we developed a new explicit method in [23], called the truncated EM method, for the nonlinear SDE dx(t) = f (x(t))dt + g(x(t))dB(t) and established the strong convergence theory under the local Lip- schitz condition plus the Khasminskii-type condition xT f (x) + p−1 |g(x)|2 ≤ K(1 + |x|2). However, due to the page limit there, we did not study the convergence rates for the method, which is the aim of this paper. We will, under some additional conditions, discuss the rates of Lq -convergence of the truncated EM method for 2 ≤ q < p and show that the order of Lq -convergence can be arbitrarily close to q/2
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