Quadratic and rate-independent limits for a large-deviations functional

Bonaschi, GA Giovanni; Peletier, MA Mark

doi:10.1007/s00161-015-0470-1

“…In [BP16,Thm. 4.2] it was then proved that that the functionals J Ψn,E := 1 nJ Ψn,E converge in the sense of Mosco, with respect to the weak-strict topology in BV([0, T ]; R d ), to the functional J Ψ0,p,E : BV([0, T ]; R d ) → with Ψ 0 (v) = A|v|, p given by (1.4) and the associated total variation functional Var Ψ0,p,E defined in (3.20) ahead, and with I K * denoting the indicator function of the set K * = [−A, A].…”

Section: Introductionmentioning

confidence: 99%

“…Since the (null-)minimizers of J Ψ0,p,E are Balanced Viscosity solutions of the rate-independent system driven by Ψ 0 and E (cf. Proposition 3.6 ahead), [BP16,Thm. 4.2] ultimately establishes a connection between the jump process X h and the latter rate-independent system, understood in a Balanced Viscosity sense.…”

Section: Introductionmentioning

confidence: 99%

“…The aim of this paper is twofold. First of all, we intend to extend the 'stochastic generation' of Balanced Viscosity solutions investigated in [BP16], to the multi-dimensional rate-independent system (1.1), where now…”

Section: Introductionmentioning

confidence: 99%

“…[Mie16], to a limiting rate-independent system, understood in the sense of Balanced Viscosity solutions. In particular, our analysis encompasses both the vanishing-viscosity approximation of rate-independent systems from [MRS12a, MRS16], and their stochastic derivation developed in [BP16].where ∂Ψ 0 : R d ⇒ R d is the subdifferential of Ψ 0 in the sense of convex analysis, whereas DE is the differential of the map u → E(t, u). Due to the 0-homogeneity of ∂Ψ 0 , (1.1) is invariant for time rescalings, i.e.…”

mentioning

confidence: 99%

“…[Mie16], to a limiting rate-independent system, understood in the sense of Balanced Viscosity solutions. In particular, our analysis encompasses both the vanishing-viscosity approximation of rate-independent systems from [MRS12a, MRS16], and their stochastic derivation developed in [BP16].…”

mentioning

confidence: 99%

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Generation of balanced viscosity solutions to rate-independent systems via variational convergence

Bonaschi

¹

,

Rossi

²

2019

Annali di Matematica

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In this paper we investigate the origin of the Balanced Viscosity solution concept for rate-independent evolution in the setting of a finite-dimensional space. Namely, given a family of dissipation potentials (Ψn)n with superlinear growth at infinity and a smooth energy functional E, we enucleate sufficient conditions on them ensuring that the associated gradient systems (Ψn, E) Evolutionary cf. [Mie16], to a limiting rate-independent system, understood in the sense of Balanced Viscosity solutions. In particular, our analysis encompasses both the vanishing-viscosity approximation of rate-independent systems from [MRS12a, MRS16], and their stochastic derivation developed in [BP16].where ∂Ψ 0 : R d ⇒ R d is the subdifferential of Ψ 0 in the sense of convex analysis, whereas DE is the differential of the map u → E(t, u). Due to the 0-homogeneity of ∂Ψ 0 , (1.1) is invariant for time rescalings, i.e. it is rateindependent. Now, it is well known that, even in the case of a smooth energy E, if u → E(t, u) fails to be strictly convex, then absolutely continuous solutions to (1.1) need not exist. In the last two decades, this has motivated the development of various weak solvability concepts for (1.1) and, in general, for rate-independent systems in infinite-dimensional Banach spaces, or even topological spaces. The analysis of these solution notions has posed several challenges. Date: October 09, 2017. G.A.B. kindly acknowledges support from the Nederlandse Oxrganisatie voor Wetenschappelijk Onderzoek (NWO), VICI Grant 639.033.008. R.R. has been partially supported by GNAMPA (Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni) of INdAM (Istituto Nazionale di Alta Matematica). 1 2 GIOVANNI A. BONASCHI AND RICCARDA ROSSI Energetic and Balanced Viscosity solutions to rate-independent systems. While referring to [Mie11] and [MR15] for a survey of all weak notions of rate-independent evolution, we may recall here the concept of Energetic solution, first proposed in [MT99] (cf. also [DMT02] for the concept of quasistatic evolution in fracture), and fully analyzed in [MT04]. It consists of the global stability conditionand of the (Ψ 0 , E)-energy balanceinvolving the dissipated energy Var Ψ0 (u; [0, t]) (where Var Ψ0 denotes the notion of total variation induced by Ψ 0 ), the stored energy E(t, u(t)) at the process time t, the initial energy E(0, u(0)), and the work of the external forces. Since the energetic formulation (S)-(E Ψ0,E ) only features the (assumedly smooth) power of the external forces ∂ t E, and no other derivatives, it is particularly suited to solutions with discontinuities in time. It is also considerably flexible and can be indeed given for rate-independent processes in general topological spaces, cf.[MM05]. That is why, it has been exploited in a great variety of applicative contexts, cf. [Mie05, MR15]. Nonetheless, over the years it has become apparent that, in the very case of a nonconvex dependence u → E(t, u), the global stability (S) fails provide a truthful description o...

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Sharp asymptotic estimates for expectations, probabilities, and mean first passage times in stochastic systems with small noise

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Schäfer,

Vanden‐Eijnden

2023

Comm Pure Appl Math

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0

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Freidlin‐Wentzell theory of large deviations can be used to compute the likelihood of extreme or rare events in stochastic dynamical systems via the solution of an optimization problem. The approach gives exponential estimates that often need to be refined via calculation of a prefactor. Here it is shown how to perform these computations in practice. Specifically, sharp asymptotic estimates are derived for expectations, probabilities, and mean first passage times in a form that is geared towards numerical purposes: they require solving well‐posed matrix Riccati equations involving the minimizer of the Freidlin‐Wentzell action as input, either forward or backward in time with appropriate initial or final conditions tailored to the estimate at hand. The usefulness of our approach is illustrated on several examples. In particular, invariant measure probabilities and mean first passage times are calculated in models involving stochastic partial differential equations of reaction‐advection‐diffusion type.

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Exploring families of energy-dissipation landscapes via tilting: three types of EDP convergence

Mielke

¹

,

Montefusco

²

,

Peletier

³

2021

Continuum Mech. Thermodyn.

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We introduce two new concepts of convergence of gradient systems $$({\mathbf{Q}}, {{\mathcal {E}}}_\varepsilon ,{{\mathcal {R}}}_\varepsilon )$$ ( Q , E ε , R ε ) to a limiting gradient system $$({\mathbf{Q}},{{\mathcal {E}}}_0,{{\mathcal {R}}}_0)$$ ( Q , E 0 , R 0 ) . These new concepts are called ‘EDP convergence with tilting’ and ‘contact–EDP convergence with tilting.’ Both are based on the energy-dissipation-principle (EDP) formulation of solutions of gradient systems and can be seen as refinements of the Gamma-convergence for gradient flows first introduced by Sandier and Serfaty. The two new concepts are constructed in order to avoid the ‘unnatural’ limiting gradient structures that sometimes arise as limits in EDP convergence. EDP convergence with tilting is a strengthening of EDP convergence by requiring EDP convergence for a full family of ‘tilted’ copies of $$({\mathbf{Q}}, {{\mathcal {E}}}_\varepsilon ,{{\mathcal {R}}}_\varepsilon )$$ ( Q , E ε , R ε ) . It avoids unnatural limiting gradient structures, but many interesting systems are non-convergent according to this concept. Contact–EDP convergence with tilting is a relaxation of EDP convergence with tilting and still avoids unnatural limits but applies to a broader class of sequences $$({\mathbf{Q}}, {{\mathcal {E}}}_\varepsilon ,{{\mathcal {R}}}_\varepsilon )$$ ( Q , E ε , R ε ) . In this paper, we define these concepts, study their properties, and connect them with classical EDP convergence. We illustrate the different concepts on a number of test problems.

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Quadratic and rate-independent limits for a large-deviations functional

Cited by 18 publications

References 43 publications

Generation of balanced viscosity solutions to rate-independent systems via variational convergence

Generation of balanced viscosity solutions to rate-independent systems via variational convergence

Sharp asymptotic estimates for expectations, probabilities, and mean first passage times in stochastic systems with small noise

Exploring families of energy-dissipation landscapes via tilting: three types of EDP convergence

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