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Keywords
(11)
Burgers Equation
Mathematical Analysis
Multiplicative Noise
Numerical Analysis
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Ordinary Differential Equation
Random Variable
reactiondiffusion equation
Stochastic Heat Equation
Stochastic Integral
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A Break of the Complexity of the Numerical Approximation of Nonlinear SPDEs with Multiplicative Noise
A Break of the Complexity of the Numerical Approximation of Nonlinear SPDEs with Multiplicative Noise,Arnulf Jentzen,Michael Roeckner
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A Break of the Complexity of the Numerical Approximation of Nonlinear SPDEs with Multiplicative Noise
(
Citations: 2
)
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Arnulf Jentzen
,
Michael Roeckner
A new algorithm for simulating stochastic partial differential equations (SPDEs) of evolutionary type, which is in some sense an infinite dimensional analog of Milstein's scheme for finite dimensional stochastic ordinary differential equations (SODEs), is introduced and analyzed in this article. The Milstein scheme is known to be impressively efficient for scalar onedimensional SODEs but only for some special multidimensional SODEs due to difficult simulations of iterated stochastic integrals in the general multidimensional SODE case. It is a key observation of this article that, in contrast to what one may expect, its infinite dimensional counterpart introduced here is very easy to simulate and this, therefore, leads to a break of the complexity (number of computational operations and random variables needed to compute the scheme) in comparison to previously considered algorithms for simulating nonlinear SPDEs with multiplicative trace class noise. The analysis is supported by numerical results for a stochastic heat equation, stochastic
reaction diffusion
equations and a stochastic
Burgers equation
showing significant computational savings.
Published in 2010.
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Citation Context
(2)
...[
5
], the mixed terms of the stochastic increments are missing, as has been pointed out in Refs [
3
,
4
]...
Annika Lang
,
et al.
Erratum: Almost sure convergence of a semidiscrete Milstein scheme fo...
...[
5
], the mixed terms of the stochastic increments are missing, as has been pointed out in Refs [
3
,
4
]...
Annika Lang
,
et al.
Erratum: Almost sure convergence of a semidiscrete Milstein scheme fo...
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(
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Published in 1996.
GALERKIN FINITE ELEMENT APPROXIMATIONS OF STOCHASTIC ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS
(
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IVO BABUSKA
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,
GEORGIOS E. ZOURARIS
Published in 2002.
Weak order for the discretization of the stochastic heat equation
(
Citations: 13
)
Arnaud Debussche
,
Jacques Printems
Journal:
Mathematics of Computation  Math. Comput.
, vol. 78, no. 266, pp. 845863, 2009
Numerical Approximation of Some Linear Stochastic Partial Differential Equations Driven by Special Additive Noises
(
Citations: 25
)
Qiang Du
,
Tianyu Zhang
Journal:
Siam Journal on Numerical Analysis  SIAM J NUMER ANAL
, vol. 40, no. 4, pp. 14211445, 2002
Asymptotic Theory for the Probability Density Functions in Burgers Turbulence
(
Citations: 14
)
Weinan E
,
Eric Vanden Eijnden
Journal:
Physical Review Letters  PHYS REV LETT
, vol. 83, no. 13, pp. 25722575, 1999
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Citations
(2)
Erratum: Almost sure convergence of a semidiscrete Milstein scheme for SPDEs of Zakai type
Annika Lang
,
PaoLiu Chow
,
Jürgen Potthoff
Journal:
Stochastics An International Journal of Probability and Stochastic Processes
, vol. 84, no. 4, pp. 561561, 2012
Erratum: Almost sure convergence of a semidiscrete Milstein scheme for SPDEs of Zakai type
Annika Lang
,
PaoLiu Chow
,
Jürgen Potthoff
Journal:
Stochastics An International Journal of Probability and Stochastic Processes
, vol. aheadofp, no. aheadofp, pp. 11, 2011