Approximating Rough Stochastic PDEs

Hairer, Martin; Maas, Jan; Weber, Hendrik

doi:10.1002/cpa.21495

Cited by 32 publications

(47 citation statements)

References 43 publications

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“…Studying this issue in general is an extremely difficult task and goes beyond the scope of this paper. Similar questions have been addressed in the past in the context of the Allen-Cahn or the stochastic Ginzburg-Landau equations in relation to stochastic quantization [31,32], with very appealing new results in larger dimensions [33,34]. Therefore, with this in mind, we consider the simplest turbulent systems that exhibit random transitions between multiple coexisting attractors.…”

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confidence: 74%

Computation of rare transitions in the barotropic quasi-geostrophic equations

Laurie

Bouchet

2015

New J. Phys.

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We investigate the theoretical and numerical computation of rare transitions in simple geophysical turbulent models. We consider the barotropic quasi-geostrophic and two-dimensional Navier-Stokes equations in regimes where bistability between two coexisting large-scale attractors exist. By means of large deviations and instanton theory with the use of an Onsager-Machlup path integral formalism for the transition probability, we show how one can directly compute the most probable transition path between two coexisting attractors analytically in an equilibrium (Langevin) framework and numerically otherwise. We adapt a class of numerical optimization algorithms known as minimum action methods to simple geophysical turbulent models. We show that by numerically minimizing an appropriate action functional in a large deviation limit, one can predict the most likely transition path for a rare transition between two states. By considering examples where theoretical predictions can be made, we show that the minimum action method successfully predicts the most likely transition path. Finally, we discuss the application and extension of such numerical optimization schemes to the computation of rare transitions observed in direct numerical simulations and experiments and to other, more complex, turbulent systems. models. Stochastic resonance, however, remains a very interesting possibility. To address such issues one should study the attractors and the dynamics of the rare transition between attractors in a hierarchy of models from the simplest to more complex ones used by the climate community. For these complex climate models, which genuinely reproduce the turbulent nature of the Earthʼs atmosphere and ocean dynamics, such a task is currently inconceivable and seems unreachable in the foreseeable future using direct numerical simulations. The reasons are the rarity of the transitions and the computational complexity of these models. The main aim of this paper is to make a step in the direction of this challenge by studying bistability and the associated transitions in turbulent dynamics using tools that will allow one to compute transitions in more complex systems in the near future.These rare transitions are essential phenomena because they correspond to drastic changes in complex system behavior. Moreover, they cannot be studied using conventional tools. They contain dynamics occurring on multiple and extremely different timescales, usually with no spectral gap. This prevents the use of classical tools from dynamical system theory. The theoretical understanding of these transitions is an extremely difficult problem due to the complexity, the large number of degrees of freedom, and the non-equilibrium nature of many of these flows. Up to now, there have been an extremely limited number of theoretical results, where analysis has been limited to analogies with models of very few degrees of freedom [27] or to specific classes of systems that can be directly related to equilibrium Langevin dynamics [28]. For this reason, the u...

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confidence: 74%

Computation of rare transitions in the barotropic quasi-geostrophic equations

Laurie

Bouchet

2015

New J. Phys.

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“…The statement given here is actually slightly stronger than the bound stated in [1] because the norm appearing on the left hand side of (2) is bounded uniformly in t instead of allowing a blow up near 0. In [1] we had to introduce this blowup due to a slightly modified definition of the Gaussian process X : the process used in [1] does not start at 0, but with stationary initial condition, which was convenient for other reasons.…”

Section: R θ (T; X Y) := δ˜ θ (T; X Y) −θ(T X) δX (T; X Y)mentioning

confidence: 71%

“…In [1] we had to introduce this blowup due to a slightly modified definition of the Gaussian process X : the process used in [1] does not start at 0, but with stationary initial condition, which was convenient for other reasons. When going through the proof given in [1], one realises that when considering the process with zero initial condition, one can apply bound (3.74) for all times t and there is no need to use (3.75) for small times. Based on this version of Lemma 1, it is then straightforward to use the a priori information on the time regularity of R θ , combined with the fact that the "tilde" processes coincide with the "non-tilde" processes before time τ , to obtain the bound E R …”

Section: R θ (T; X Y) := δ˜ θ (T; X Y) −θ(T X) δX (T; X Y)mentioning

confidence: 99%

Erratum to: Rough Burgers-like equations with multiplicative noise

Hairer

Weber

2013

Probab. Theory Relat. Fields

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“…The reasons for not doing so are that the integral is well-defined without it and that non-geometric situations can arise naturally in the context of numerical approximations, see for example [HM10,HMW12].…”

Section: Remark 34mentioning

confidence: 99%

Solving the KPZ Equation

Hairer

2013

XVIIth International Congress on Mathematical Physics

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We introduce a new concept of solution to the KPZ equation which is shown to extend the classical Cole-Hopf solution. This notion provides a factorisation of the Cole-Hopf solution map into a "universal" measurable map from the probability space into an explicitly described auxiliary metric space, composed with a new solution map that has very good continuity properties. The advantage of such a formulation is that it essentially provides a pathwise notion of a solution, together with a very detailed approximation theory. In particular, our construction completely bypasses the Cole-Hopf transform, thus laying the groundwork for proving that the KPZ equation describes the fluctuations of systems in the KPZ universality class.As a corollary of our construction, we obtain very detailed new regularity results about the solution, as well as its derivative with respect to the initial condition. Other byproducts of the proof include an explicit approximation to the stationary solution of the KPZ equation, a well-posedness result for the FokkerPlanck equation associated to a particle diffusing in a rough space-time dependent potential, and a new periodic homogenisation result for the heat equation with a space-time periodic potential. One ingredient in our construction is an example of a non-Gaussian rough path such that the area process of its natural approximations needs to be renormalised by a diverging term for the approximations to converge.

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Approximating Rough Stochastic PDEs

Cited by 32 publications

References 43 publications

Computation of rare transitions in the barotropic quasi-geostrophic equations

Computation of rare transitions in the barotropic quasi-geostrophic equations

Erratum to: Rough Burgers-like equations with multiplicative noise

Solving the KPZ Equation

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