In this paper we present a variational technique that handles coarse-graining and passing to a limit in a unified manner. The technique is based on a duality structure, which is present in many gradient flows and other variational evolutions, and which often arises from a large-deviations principle. It has three main features: (a) a natural interaction between the duality structure and the coarse-graining, (b) application to systems with nondissipative effects, and (c) application to coarse-graining of approximate solutions which solve the equation only to some error. As examples, we use this technique to solve three limit problems, the overdamped limit of the Vlasov-Fokker-Planck equation and the smallnoise limit of randomly perturbed Hamiltonian systems with one and with many degrees of freedom.
Mathematics Subject Classification

Enhancement of multiple-scale elongated structures in noisy image data is relevant for many biomedical applications but commonly used PDE-based enhancement techniques often fail at crossings in an image. To get an overview of how an image is composed of local multiple-scale elongated structures we construct a continuous wavelet transform on the similitude group, SIM (2). Our unitary transform maps the space of images onto a reproducing kernel space defined on SIM (2), allowing us to robustly relate Euclidean (and scaling) invariant operators on images to leftinvariant operators on the corresponding continuous wavelet transform. Rather than often used wavelet (soft-)thresholding techniques, we employ the group structure in the wavelet domain to arrive at left-invariant evolutions and flows (diffusion), for contextual crossing preserving enhancement of multiple scale elongated structures in noisy images. We present experiments that display benefits of our work compared to recent PDE techniques acting directly on the images and to our previous work on left-invariant diffusions on Coherent state transforms defined on Euclidean motion group.

In molecular dynamics and sampling of high dimensional Gibbs measures coarse-graining is an important technique to reduce the dimensionality of the problem. We will study and quantify the coarse-graining error between the coarse-grained dynamics and an effective dynamics. The effective dynamics is a Markov process on the coarse-grained state space obtained by a closure procedure from the coarse-grained coefficients. We obtain error estimates both in relative entropy and Wasserstein distance, for both Langevin and overdamped Langevin dynamics. The approach allows for vectorial coarse-graining maps. Hereby, the quality of the chosen coarse-graining is measured by certain functional inequalities encoding the scale separation of the Gibbs measure. The method is based on error estimates between solutions of (kinetic) Fokker-Planck equations in terms of large-deviation rate functionals.Here X t ∈ d is the state of the system at time t, V is a potential, β = 1/(k B T a ) is the inverse temperature, W d t is a d-dimensional Brownian motion and X 0 is the initial state of the system.

In this paper we introduce a new generalisation of the relative Fisher Information for Markov jump processes on a finite or countable state space, and prove an inequality which connects this object with the relative entropy and a large deviation rate functional. In addition to possessing various favourable properties, we show that this generalised Fisher Information converges to the classical Fisher Information in an appropriate limit. We then use this generalised Fisher Information and the aforementioned inequality to qualitatively study coarse-graining problems for jump processes on discrete spaces.Remark 1.1. The space P(X ) is a subset of 1 (X ), and the weak measure topology on P(X ) coincides with the σ( 1 , ∞ )-topology on 1 (X ). Recall that by Schur's theorem, weak and strong convergence on 1 (X ) are the same, even though the weak and strong topologies may be different; therefore functions f : [0, T ] → 1 (X ) are strongly continuous if and only they are weakly continuous. Since 'weak measure convergence' in P(X ) is the same as the σ( 1 , ∞ )-convergence in 1 (X ), we will omit the term 'weak' in our discussion and notation, and simply talk about 'continuous' functions from [0, T ] to P(X ) or to 1 (X ).

We study the convergence to equilibrium of an underdamped Langevin equation that is controlled by a linear feedback force. Specifically, we are interested in sampling the possibly multimodal invariant probability distribution of a Langevin system at small noise (or low temperature), for which the dynamics can easily get trapped inside metastable subsets of the phase space. We follow Chen et al. [J. Math. Phys. 56, 113302 (2015)] and consider a Langevin equation that is simulated at a high temperature, with the control playing the role of a friction that balances the additional noise so as to restore the original invariant measure at a lower temperature. We discuss different limits as the temperature ratio goes to infinity and prove convergence to a limit dynamics. It turns out that, depending on whether the lower (“target”) or the higher (“simulation”) temperature is fixed, the controlled dynamics converges either to the overdamped Langevin equation or to a deterministic gradient flow. This implies that (a) the ergodic limit and the large temperature separation limit do not commute in general and that (b) it is not possible to accelerate the speed of convergence to the ergodic limit by making the temperature separation larger and larger. We discuss the implications of these observations from the perspective of stochastic optimization algorithms and enhanced sampling schemes in molecular dynamics.

Coarse-graining is central to reducing dimensionality in molecular dynamics, and is typically characterized by a mapping which projects the full state of the system to a smaller class of variables. While extensive literature has been devoted to coarse-graining starting from reversible systems, not much is known in the non-reversible setting. In this article, starting with a non-reversible dynamics, we introduce and study an effective dynamics which approximates the (non-closed) projected dynamics. Under fairly weak conditions on the system, we prove error bounds on the trajectorial error between the projected and the effective dynamics. In addition to extending existing results to the non-reversible setting, our error estimates also indicate that the notion of mean force motivated by this effective dynamics is a good one.

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