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15 March 2018, Volume 36 Issue 2
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WEAK ERROR ESTIMATES FOR TRAJECTORIES OF SPDEs UNDER SPECTRAL GALERKIN DISCRETIZATION
CharlesEdouard Bréhier, Martin Hairer, Andrew M. Stuart
2018, 36(2): 159182. DOI:
10.4208/jcm.1607m20160539
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149
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We consider stochastic semilinear evolution equations which are driven by additive, spatially correlated, Wiener noise, and in particular consider problems of heat equation (analytic semigroup) and dampeddriven wave equations (bounded semigroup) type. We discretize these equations by means of a spectral Galerkin projection, and we study the approximation of the probability distribution of the trajectories:test functions are regular, but depend on the values of the process on the interval[0,
T
].
We introduce a new approach in the context of quantative weak error analysis for discretization of SPDEs. The weak error is formulated using a deterministic function (Itô map) of the stochastic convolution found when the nonlinear term is dropped. The regularity properties of the Itô map are exploited, and in particular secondorder Taylor expansions employed, to transfer the error from spectral approximation of the stochastic convolution into the weak error of interest.
We prove that the weak rate of convergence is twice the strong rate of convergence in two situations. First, we assume that the covariance operator commutes with the generator of the semigroup:the first order term in the weak error expansion cancels out thanks to an independence property. Second, we remove the commuting assumption, and extend the previous result, thanks to the analysis of a new error term depending on a commutator.
ON EFFECTIVE STOCHASTIC GALERKIN FINITE ELEMENT METHOD FOR STOCHASTIC OPTIMAL CONTROL GOVERNED BY INTEGRALDIFFERENTIAL EQUATIONS WITH RANDOM COEFFICIENTS
Wanfang Shen, Liang Ge
2018, 36(2): 183201. DOI:
10.4208/jcm.1611m20160676
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262
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In this paper, we apply stochastic Galerkin finite element methods to the optimal control problem governed by an elliptic integraldifferential PDEs with random field. The control problem has the control constraints of obstacle type. A new gradient algorithm based on the preconditioner conjugate gradient algorithm (PCG) is developed for this optimal control problem. This algorithm can transform a part of the state equation matrix and costate equation matrix into block diagonal matrix and then solve the optimal control systems iteratively. The proof of convergence for this algorithm is also discussed. Finally numerical examples of a medial size are presented to illustrate our theoretical results.
ANALYSIS OF MULTIINDEX MONTE CARLO ESTIMATORS FOR A ZAKAI SPDE
Christoph Reisinger, Zhenru Wang
2018, 36(2): 202236. DOI:
10.4208/jcm.1612m20160681
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168
)
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In this article, we propose a spacetime MultiIndex Monte Carlo (MIMC) estimator for a onedimensional parabolic stochastic partial differential equation (SPDE) of Zakai type. We compare the complexity with the Multilevel Monte Carlo (MLMC) method of Giles and Reisinger (2012), and find, by means of Fourier analysis, that the MIMC method:(i) has suboptimal complexity of
O
(
ε
^{2}
log
ε

^{3}
) for a root mean square error (RMSE)
ε
if the same spatial discretisation as in the MLMC method is used; (ii) has a better complexity of
O
(
ε
^{2}
log
ε
) if a carefully adapted discretisation is used; (iii) has to be adapted for nonsmooth functionals. Numerical tests confirm these findings empirically.
A FIRSTORDER NUMERICAL SCHEME FOR FORWARDBACKWARD STOCHASTIC DIFFERENTIAL EQUATIONS IN BOUNDED DOMAINS
Jie Yang, Guannan Zhang, Weidong Zhao
2018, 36(2): 237258. DOI:
10.4208/jcm.1612m20160582
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184
)
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We propose a novel numerical scheme for decoupled forwardbackward stochastic differential equations (FBSDEs) in bounded domains, which corresponds to a class of nonlinear parabolic partial differential equations with Dirichlet boundary conditions. The key idea is to exploit the regularity of the solution (
Y
_{t}
,
Z
_{t}
) with respect to
X
_{t}
to avoid direct approximation of the involved random exit time. Especially, in the onedimensional case, we prove that the probability of
X
_{t}
exiting the domain within △
t
is on the order of
O
((△
t
)
^{ε}
exp(1/(△
t
)
^{2ε}
)), if the distance between the start point
X
_{0}
and the boundary is at least on the order of
O
((△
t
)
^{(1)/2ε}
) for any fixed
ε
> 0. Hence, in spatial discretization, we set the mesh size △
x
~
O
((△
t
)
^{(1)/2ε}
), so that all the interior grid points are sufficiently far from the boundary, which makes the error caused by the exit time decay
subexponentially
with respect to △
t
. The accuracy of the approximate solution near the boundary can be guaranteed by means of highorder piecewise polynomial interpolation. Our method is developed using the implicit Euler scheme and cubic polynomial interpolation, which leads to an overall firstorder convergence rate with respect to △
t
.
A FAST STOCHASTIC GALERKIN METHOD FOR A CONSTRAINED OPTIMAL CONTROL PROBLEM GOVERNED BY A RANDOM FRACTIONAL DIFFUSION EQUATION
Ning Du, Wanfang Shen
2018, 36(2): 259275. DOI:
10.4208/jcm.1612m20160696
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129
)
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We develop a fast stochastic Galerkin method for an optimal control problem governed by a random spacefractional diffusion equation with deterministic constrained control. Optimal control problems governed by a fractional diffusion equation tends to provide a better description for transport or conduction processes in heterogeneous media. However, the fractional control problem introduces significant computation complexity due to the nonlocal nature of fractional differential operators, and this is further worsen by the large number of random space dimensions to discretize the probability space. We approximate the optimality system by a gradient algorithm combined with the stochastic Galerkin method through the discretization with respect to both the spatial space and the probability space. The resulting linear system can be decoupled for the random and spatial variable, and thus solved separately. A fast preconditioned BiConjugate Gradient Stabilized method is developed to efficiently solve the decoupled systems derived from the fractional diffusion operators in the spatial space. Numerical experiments show the utility of the method.
EXPONENTIAL INTEGRATORS FOR STOCHASTIC SCHRÖDINGER EQUATIONS DRIVEN BY ITÔ NOISE
Rikard Anton, David Cohen
2018, 36(2): 276309. DOI:
10.4208/jcm.1701m20160525
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154
)
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We study an explicit exponential scheme for the time discretisation of stochastic Schrödinger Equations Driven by additive or Multiplicative Itô Noise. The numerical scheme is shown to converge with strong order 1 if the noise is additive and with strong order 1/2 for multiplicative noise. In addition, if the noise is additive, we show that the exact solutions of the linear stochastic Schrödinger equations satisfy trace formulas for the expected mass, energy, and momentum (i. e., linear drifts in these quantities). Furthermore, we inspect the behaviour of the numerical solutions with respect to these trace formulas. Several numerical simulations are presented and confirm our theoretical results.
A SPARSE GRID STOCHASTIC COLLOCATION AND FINITE VOLUME ELEMENT METHOD FOR CONSTRAINED OPTIMAL CONTROL PROBLEM GOVERNED BY RANDOM ELLIPTIC EQUATIONS
Liang Ge, Tongjun Sun
2018, 36(2): 310330. DOI:
10.4208/jcm.1703m20160692
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187
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In this paper, a hybird approximation scheme for an optimal control problem governed by an elliptic equation with random field in its coefficients is considered. The random coefficients are smooth in the physical space and depend on a large number of random variables in the probability space. The necessary and sufficient optimality conditions for the optimal control problem are obtained. The scheme is established to approximate the optimality system through the discretization by using finite volume element method for the spatial space and a sparse grid stochastic collocation method based on the Smolyak approximation for the probability space, respectively. This scheme naturally leads to the discrete solutions of an uncoupled deterministic problem. The existence and uniqueness of the discrete solutions are proved. A priori error estimates are derived for the state, the costate and the control variables. Numerical examples are presented to illustrate our theoretical results.
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