# Active-dissipative nonlinear spatially extended media: Complexity, coarse-graining, multiscale analysis and numerical methods

Lead Research Organisation:
Imperial College London

Department Name: Department of Chemical Engineering

### Abstract

Spatially extended systems (SES), i.e. infinite dimensional dynamical systems described through partial differential equations deterministic or stochastic in large or unbounded domains, are typically characterized by the presence of a wide range of characteristic length and time scales which often leads to complex spatio-temporal behavior. SES arise frequently as mathematical models of a large variety of natural phenomena and technological applications. The complexity of SES and their dynamics is such that it is very difficult, if not impossible to analyze them directly, either mathematically or, in several cases, numerically. It is imperative, therefore, to seek a low-dimensional description of SES, i.e. to produce coarse grained models that capture most, if not all of the essential dynamic features of the particular applications and which are much easier to study analytically and numerically. The primary aim of the proposed research is the development of state-of-the-art efficient methods for mode reduction and coarse-graining of SES, both deterministic and stochastic.

### Planned Impact

The immediate impact will be academic and felt across a spectrum of disciplines, such as applied mathematics, applied physics, numerical analysis, fluid dynamics and stochastic modelling. This stems from the nature of the proposed research which is a combination of applied mathematics, applied nonlinear dynamics, numerical methods and stochastic processes. The impact on the non-academic sector will be felt towards the end of the project and once several successful trials of simulations of complex interfacial flows have been completed. These should be of great interest to workers in the industrial sector with interests in the development of predictive computational models for technological processes that involve complex thin-film flows and more general to industries that exploit such flows, e.g. in the rapidly growing area of micro-/nano-technology.

### Publications

Duncan A
(2015)

*A Multiscale Analysis of Diffusions on Rapidly Varying Surfaces*in Journal of Nonlinear Science
Duncan AB
(2016)

*Noise-induced transitions in rugged energy landscapes.*in Physical review. E
Goddard B
(2012)

*The Overdamped Limit of Dynamic Density Functional Theory: Rigorous Results*in Multiscale Modeling & Simulation
Goddard BD
(2012)

*General dynamical density functional theory for classical fluids.*in Physical review letters
Goddard BD
(2013)

*Unification of dynamic density functional theory for colloidal fluids to include inertia and hydrodynamic interactions: derivation and numerical experiments.*in Journal of physics. Condensed matter : an Institute of Physics journal
Gomes S
(2017)

*Controlling roughening processes in the stochastic Kuramoto-Sivashinsky equation*in Physica D: Nonlinear Phenomena
Gomes SN
(2015)

*Controlling spatiotemporal chaos in active dissipative-dispersive nonlinear systems.*in Physical review. E, Statistical, nonlinear, and soft matter physics
Joubaud R
(2014)

*Langevin Dynamics with Space-Time Periodic Nonequilibrium Forcing*in Journal of Statistical Physics
Kalliadasis S
(2015)

*A new framework for extracting coarse-grained models from time series with multiscale structure*in Journal of Computational Physics
Krumscheid S
(2013)

*Semiparametric Drift and Diffusion Estimation for Multiscale Diffusions*in Multiscale Modeling & SimulationDescription | Spatially extended systems (SES), i.e. infinite-dimensional dynamical systems described through partial differential equations (PDEs) deterministic or stochastic (SPDEs) in large or unbounded domains, are typically characterized by the presence of a wide range of characteristic length and time scales which often leads to complex spatiotemporal behavior. They arise frequently as mathematical models of a large variety of natural phenomena and technological applications. From fluid and solid mechanics, reaction-advection-diffusion processes, materials science and glaciology, atmosphere/ocean and climate processes, to biological cell and population dynamics, DNA - protein structure/interaction and epidemiology. SES also appear routinely in engineering applications, such as the rapidly growing field of micro-electromechanical devices and nanotechnology, microreactors, large-scale chemical plants and electric power grids. Their study is therefore crucial for the understanding of several physical and biological systems, as well as for the analysis, control and optimization of technological processes. Of particular interest are the identification and understanding of different regimes in the parameter space, the emergence of underlying scaling laws and the description of the developed complex spatiotemporal pattern formation dynamics. The primary aim of the proposed research was the development of state-of-the-art efficient methods for low-dimensional representation (mode reduction and coarse-graining) of SES, both deterministic and stochastic. The project led to the development of a generic theoretical/numerical framework for the accurate and efficient analysis of nonlinear dissipative PDEs, both deterministic and stochastic. Human resources: It should be noted that this grant also supported a talented young researcher, Sebastian Krumscheid, who worked under the joint supervision of the PI and Prof. G.A. Pavliotis from the Mathematics Department of IC. Hence Sebastian worked in a highly cross-disciplinary environment which also enlarged his knwoledge base flexibility in using knowledge between different subject. He now has a high-profile research prosition at the Mathematics Institute of Computational Science and Engineering, Department of Mathematics, Ecole Polytechnique Federale de Lausane, Lausanne, Switzerland. |

Exploitation Route | The results of this project will have significant impact on a wide variety of applications where the interaction and transfer of information and energy between different spatiotemporal scales dominates the dynamics. In particular, the impact of our work on model noisy weakly nonlinear SPDEs, might appear less obvious, but most physical systems are subject to external or internal random fluctuations with examples ranging from biology and climate modeling to technological applications while synchronization phenomena are abundant in science, nature, engineering, and even social life. Hence, the precise characterization of the influence of noise and synchronization on the long-time dynamics is crucial for the understanding and description of the emerging complex dynamics in several physical, biological and technological settings. Many of these settings can be described by model SES, like the ones we worked with in this project. Harnessing noise and synchronization effects can also facilitate the control and optimisation of technological processes, such as the process of nanostructuring of solid surfaces by ion-beam erosion crucial in applied condensed matter research where the "holy grail" is smart surfaces for advanced materials. |

Sectors | Chemicals,Energy,Environment,Manufacturing, including Industrial Biotechology |

URL | http://www.imperial.ac.uk/complex-multiscale-systems |

Description | In many physical, technological, social, and economic applications, one is commonly faced with the task of estimating statistical properties, such as mean first passage times of a temporal continuous process, from empirical data (experimental observations). Typically, however, an accurate and reliable estimation of such properties directly from the data alone is not possible as the time series is often too short, or the particular phenomenon of interest is only rarely observed. We propose here a theoretical-computational framework which provides us with a systematic and rational estimation of statistical quantities of a given temporal process, such as waiting times between subsequent bursts of activity in intermittent signals. Our framework is illustrated with applications from real-world data sets, ranging from marine biology to paleoclimatic data. |

First Year Of Impact | 2015 |

Sector | Chemicals,Energy,Environment,Manufacturing, including Industrial Biotechology |

Description | Platform Grant |

Amount | £2,000,000 (GBP) |

Funding ID | EP/L020564/1 |

Organisation | Engineering and Physical Sciences Research Council (EPSRC) |

Sector | Academic/University |

Country | United Kingdom |

Start | 06/2014 |

End | 06/2019 |

Description | Responsive Mode-Standard Grants |

Amount | £500,000 (GBP) |

Funding ID | EP/L025159/1 |

Organisation | Engineering and Physical Sciences Research Council (EPSRC) |

Sector | Academic/University |

Country | United Kingdom |

Start | 11/2014 |

End | 11/2017 |

Title | New course graining procedures for spatially extended systems (SES) |

Description | --The development of a systematic methodology for deriving low-dimensional stochastic models that describe accurately the dynamics of SES. --The development of efficient methods for the accurate numerical solution of PDEs and SPDEs in large and unbounded domains. --The rigorous understanding and accurate and efficient modelling of the emergence of stochastic effects in SES due to spatiotemporal chaos. --The application of the analytical and numerical techniques that we will develop to the generalised Kuramoto-Sivashinsky equation, a simple prototype that retains the basic elements of any nonlinear process involving wave evolution. |

Type Of Material | Improvements to research infrastructure |

Provided To Others? | No |

Impact | In many physical, technological, social, and economic applications, one is commonly faced with the task of estimating statistical properties, such as mean first passage times of a temporal continuous process, from empirical data (experimental observations). Typically, however, an accurate and reliable estimation of such properties directly from the data alone is not possible as the time series is often too short, or the particular phenomenon of interest is only rarely observed. We propose here a theoretical-computational framework which provides us with a systematic and rational estimation of statistical quantities of a given temporal process, such as waiting times between subsequent bursts of activity in intermittent signals. Our framework is illustrated with applications from real-world data sets, ranging from marine biology to paleoclimatic data. |

URL | http://www.imperial.ac.uk/complex-multiscale-systems |

Title | Mode reduction and coarse graining |

Description | Spatially extended systems (SES), i.e. infinite-dimensional dynamical systems described through partial differential equations (PDEs) deterministic or stochastic (SPDEs) in large or unbounded domains, are typically characterized by the presence of a wide range of characteristic length and time scales which often leads to complex spatiotemporal behaviour. They arise frequently as mathematical models of a large variety of natural phenomena and technological applications. From fluid and solid mechanics, reaction-advection-diffusion processes, materials science and glaciology, atmosphere/ocean and climate processes, to biological cell and population dynamics, DNA - protein structure/interaction and epidemiology. SES also appear routinely in engineering applications, such as the rapidly growing field of micro-electromechanical devices and nanotechnology, microreactors, large-scale chemical plants and electric power grids. Their study is therefore crucial for the understanding of several physical and biological systems, as well as for the analysis, control and optimization of technological processes. Of particular interest are the identification and understanding of different regimes in the parameter space, the emergence of underlying scaling laws and the description of the developed complex spatiotemporal pattern formation dynamics. The primary aim here was the development of state-of-the-art efficient methods for mode reduction and coarse-graining of spatially extended systems, both deterministic and stochastic. |

Type Of Material | Computer model/algorithm |

Provided To Others? | No |

Impact | There have been different methodologies proposed over the last decades, both deterministic (e.g. adiabatic elimination and center-manifold projection techniques, they have various assumptions) and stochastic (e.g. optimal prediction and the Mori-Zwanzig formalism in statistical mechanics). It is important to emphasize however that the stochastic ones have been developed mainly for Hamiltonian systems. For such systems the invariant measure is known and hence one can prescribe the statistical properties of the unresolved degrees of freedom from the beginning. Here we developed a novel mode reduction strategy for stochastic dissipative systems. It was exemplified successfully with a model prototype that retains the basic ingredient of a wide spectrum of nonlinear processes, namely instability, dissipation, dispersion and nonlinearity. This paves the way to much more involved systems, e.g. the Euler equations, or even full Navier-Stokes, with huge computational savings. As part of this project we also developed a theoretical-computational framework which provides us with a systematic and rational estimation of statistical quantities of a given temporal process, such as waiting times between subsequent bursts of activity in intermittent signals. Our framework is illustrated with applications from real-world data sets, ranging from marine biology to climate change. |

URL | http://www.imperial.ac.uk/complex-multiscale-systems |

Description | Collaboration with Prof. G.A. Pavliotis |

Organisation | Imperial College London |

Country | United Kingdom |

Sector | Academic/University |

PI Contribution | This grant was instrumental in cementing and enhancing my collaboration with Prof. G.A. Pavliotis from the Mathematics Department of IC. In particular, it paved the way for a follow up grant, EP/L025159, but also a much bigger and highly interdisciplinary activity, currently supported by platform grant EP/L020564. The contributions by myself and my team were related to elements from physics of fluids, multiscale fluid dynamics, statistical mechanics of classical fluids and general numerical techniques. |

Collaborator Contribution | The contributions by Prof. Pavliotis were related to elements of stochastic processes, homogenization theory and multiscale analysis of stochastic partial differential equations. |

Impact | Several publications in high profile-high impact factor journals listed under "Publications". |

Start Year | 2014 |