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Stochastic Parameterizing Manifolds and Non-Markovian Reduced Equations - Stochastic Manifolds for Nonlinear SPDEs II (Paperback, 2015 ed.)
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Stochastic Parameterizing Manifolds and Non-Markovian Reduced Equations - Stochastic Manifolds for Nonlinear SPDEs II (Paperback, 2015 ed.)
Series: SpringerBriefs in Mathematics
Expected to ship within 10 - 15 working days
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In this second volume, a general approach is developed to provide
approximate parameterizations of the "small" scales by the "large"
ones for a broad class of stochastic partial differential equations
(SPDEs). This is accomplished via the concept of parameterizing
manifolds (PMs), which are stochastic manifolds that improve, for a
given realization of the noise, in mean square error the partial
knowledge of the full SPDE solution when compared to its projection
onto some resolved modes. Backward-forward systems are designed to
give access to such PMs in practice. The key idea consists of
representing the modes with high wave numbers as a pullback limit
depending on the time-history of the modes with low wave numbers.
Non-Markovian stochastic reduced systems are then derived based on
such a PM approach. The reduced systems take the form of stochastic
differential equations involving random coefficients that convey
memory effects. The theory is illustrated on a stochastic
Burgers-type equation.
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