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 ).
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