##
**Stochastic parameterizing manifolds and non-Markovian reduced equations. Stochastic manifolds for nonlinear SPDEs II.**
*(English)*
Zbl 1331.37009

SpringerBriefs in Mathematics. Cham: Springer (ISBN 978-3-319-12519-0/pbk; 978-3-319-12520-6/ebook). xvii, 129 p. (2015).

This volume is the second part of a two-volume set of monographs related to the concept of stochastic parameterizing manifold for nonlinear SPDEs (for the first volume see [the authors, Approximation of stochastic invariant manifolds. Stochastic manifolds for nonlinear SPDEs I. Cham: Springer (2015; Zbl 1319.60002)]. The monograph, continues the important tradition of the previous 3 decades on inertial or approximate inertial manifolds (and related concepts) which has already been applied to random and stochastic systems, and tries to offer generalizations that will apply to an important type of SPDEs arising in applications, and with a view towards the development of numerical algorithms for the computation of these entities. The proposed manifolds are pathwise global objects, thus enabling the reductions obtained to work well even away from the critical values of certain “bifurcation” parameters of the system. The monograph contributes to the important discussion concerning the development of reduced models from mathematically complicated infinite-dimensional stochastic models which may be consistent with the physical principles but nevertheless are intractable with up-to-date computing capacities.

The fundamental idea throughout the monograph, is the decomposition of the state space \({\mathcal H}\) where the random fields generated by the SPDE reside into two parts \({\mathcal H}^{c}\) (finite-dimensional) and \({\mathcal H}^{s}\) which are invariant under the action of the linear part of the system. Each of these parts contains contributions of different qualitative nature, the low and the high modes respectively, and the assumption is that the information carried by the high modes, is dissipated by the action of the linear dynamics fast enough so that in the long run the low modes carry most of the important information concerning the random field. Clearly, nonlinearity leads to a coupling between the two classes of modes and the question arising is whether we may describe the transient evolution of the high modes in terms of the evolution of the low modes. In other terms, it is interesting to find out whether an \(m\)-dimensional (where \(m=\mathrm{dim}({\mathcal H}^{c})\)) local stochastic invariant manifold \({\mathfrak M}\) is admitted by the nonlinear random dynamical system of the form \({\mathfrak M}=\{ \xi + \hat{h}(\xi, \omega) \,\, : \,\, \xi \in {\mathfrak B} \subset {\mathcal H}^{c}\}\) where \({\mathfrak B}\) is a deterministic neighbourghood of the origin and \(\hat{h} : {\mathfrak B} \times \Omega \to {\mathcal H}^{s}\) is a function “parameterizing” the manifold. Once the function \(\hat{h}\) is approximated, reduced descriptions of the full system in terms of the low modes are possible. Such structures are allowed to depend on certain “bifurcation” parameters of the system. The questions concerning the existence of such invariant manifolds have been treated in detail in Volume I, while Volume II focuses on approximation procedures for \(\hat{h}\).

After a brief introduction in Chapter 1, the next two chapters briefly review fundamental concepts, treated in detail in Volume I, related to important definitions from the theory of random dynamical systems, information related to the general class of systems for which the proposed theory is applicable, notions from spectral theory and basic results from the theory of approximation of stochastic invariant manifolds. This makes Volume II independent and accessible to readers who are more interested in the application side and the computational aspect of the theory (leaving theoretical issues to Volume I). Chapter 4 deals with a general framework for the approximation of \(\hat{h}\). The starting point is a coupled forward-backward system for the spatio-temporal evolution of the low and the high contents of the random field, \(u_{c}\) and \(u_{s}\) respectively, and a characterization of \(\hat{h}\) in terms of the pullback limit of the process \(u_{s}\). This characterization is theoretical, as \(u_{s}\) is in practice not available, so an alternative way of characterizing \(\hat{h}\) must be obtained. This leads to the notion of stochastic parameterizing manifold which is motivated by a least square minimization problem associated with the problem of parameterization of the high modes in terms of the low modes for sufficiently large time scales. The actual approximation is provided by the solution of properly constructed forward-backward schemes. There are various options, when constructing such approximations, a theme which is explored in the remaining part of the monograph. In Chapter 5, the problem of reducing the dynamics of the full system through the utilization of the idea of stochastic parameterizing manifold is considered, using a method akin to the nonlinear Galerkin method but with the approximate inertial manifolds replaced with the stochastic parameterizing manifolds. This construction leads to a closure scheme, which allows for the derivation of a lower-dimensional random coefficient system, for the evolution of the amplitude of the lower modes which carry most of the important information concerning the dynamics, which is non-Markovian as it involves the past of the noise path, with appropriately exponentially decaying terms. This reduction procedure is illustrated in detail in Chapter 6, for the stochastic Burgers equation, driven by a multiplicative white noise process and the performance of the proposed reduction scheme is assessed through numerical experiments. Finally, in Chapter 7, an alternative approximation scheme is proposed, which is better suited for numerical approximation, the performance of which is once more tested on the stochastic Burgers equation with very satisfactory results concernign the statistics of the random field generated by the SPDE.

The monograph is well written and contains novel and important results for researchers in the field of analytical or numerical random dynamical systems and SPDEs. The clarity of presentation as well as the detailed list of references, makes it also appealing to research students as well as to newcomers to the field.

The fundamental idea throughout the monograph, is the decomposition of the state space \({\mathcal H}\) where the random fields generated by the SPDE reside into two parts \({\mathcal H}^{c}\) (finite-dimensional) and \({\mathcal H}^{s}\) which are invariant under the action of the linear part of the system. Each of these parts contains contributions of different qualitative nature, the low and the high modes respectively, and the assumption is that the information carried by the high modes, is dissipated by the action of the linear dynamics fast enough so that in the long run the low modes carry most of the important information concerning the random field. Clearly, nonlinearity leads to a coupling between the two classes of modes and the question arising is whether we may describe the transient evolution of the high modes in terms of the evolution of the low modes. In other terms, it is interesting to find out whether an \(m\)-dimensional (where \(m=\mathrm{dim}({\mathcal H}^{c})\)) local stochastic invariant manifold \({\mathfrak M}\) is admitted by the nonlinear random dynamical system of the form \({\mathfrak M}=\{ \xi + \hat{h}(\xi, \omega) \,\, : \,\, \xi \in {\mathfrak B} \subset {\mathcal H}^{c}\}\) where \({\mathfrak B}\) is a deterministic neighbourghood of the origin and \(\hat{h} : {\mathfrak B} \times \Omega \to {\mathcal H}^{s}\) is a function “parameterizing” the manifold. Once the function \(\hat{h}\) is approximated, reduced descriptions of the full system in terms of the low modes are possible. Such structures are allowed to depend on certain “bifurcation” parameters of the system. The questions concerning the existence of such invariant manifolds have been treated in detail in Volume I, while Volume II focuses on approximation procedures for \(\hat{h}\).

After a brief introduction in Chapter 1, the next two chapters briefly review fundamental concepts, treated in detail in Volume I, related to important definitions from the theory of random dynamical systems, information related to the general class of systems for which the proposed theory is applicable, notions from spectral theory and basic results from the theory of approximation of stochastic invariant manifolds. This makes Volume II independent and accessible to readers who are more interested in the application side and the computational aspect of the theory (leaving theoretical issues to Volume I). Chapter 4 deals with a general framework for the approximation of \(\hat{h}\). The starting point is a coupled forward-backward system for the spatio-temporal evolution of the low and the high contents of the random field, \(u_{c}\) and \(u_{s}\) respectively, and a characterization of \(\hat{h}\) in terms of the pullback limit of the process \(u_{s}\). This characterization is theoretical, as \(u_{s}\) is in practice not available, so an alternative way of characterizing \(\hat{h}\) must be obtained. This leads to the notion of stochastic parameterizing manifold which is motivated by a least square minimization problem associated with the problem of parameterization of the high modes in terms of the low modes for sufficiently large time scales. The actual approximation is provided by the solution of properly constructed forward-backward schemes. There are various options, when constructing such approximations, a theme which is explored in the remaining part of the monograph. In Chapter 5, the problem of reducing the dynamics of the full system through the utilization of the idea of stochastic parameterizing manifold is considered, using a method akin to the nonlinear Galerkin method but with the approximate inertial manifolds replaced with the stochastic parameterizing manifolds. This construction leads to a closure scheme, which allows for the derivation of a lower-dimensional random coefficient system, for the evolution of the amplitude of the lower modes which carry most of the important information concerning the dynamics, which is non-Markovian as it involves the past of the noise path, with appropriately exponentially decaying terms. This reduction procedure is illustrated in detail in Chapter 6, for the stochastic Burgers equation, driven by a multiplicative white noise process and the performance of the proposed reduction scheme is assessed through numerical experiments. Finally, in Chapter 7, an alternative approximation scheme is proposed, which is better suited for numerical approximation, the performance of which is once more tested on the stochastic Burgers equation with very satisfactory results concernign the statistics of the random field generated by the SPDE.

The monograph is well written and contains novel and important results for researchers in the field of analytical or numerical random dynamical systems and SPDEs. The clarity of presentation as well as the detailed list of references, makes it also appealing to research students as well as to newcomers to the field.

Reviewer: Athanasios Yannacopoulos (Athens)

### MSC:

37H10 | Generation, random and stochastic difference and differential equations |

60H30 | Applications of stochastic analysis (to PDEs, etc.) |

35R60 | PDEs with randomness, stochastic partial differential equations |