Molecular dynamics and related computational methods enable the description of biological systems with all-atom detail. However, these approaches are limited regarding simulation times and system sizes. A systematic way to bridge the micro-macro scale range between molecular dynamics and experiments is to apply coarse-graining (CG) techniques. The basic idea of (CG) is to replace the high-dimensional all-atom description of the system by a reduced representation that preserves a suf.cient accuracy of the properties of interest. Obviously, numerical evaluation of the coarse-grained system would require less resources so that an increase of orders of magnitude in the simulated time and length scales can be achieved in this way. Many different CG approaches have been introduced over the years. Most of them are validated by means of numerical experiments only, while reliable theoretical insight into their approximation properties is missing. The ambitious goal of this project is to put CG approximations on a solid mathematical footing.

To this end, we will start with the three main challenges every approach to CG is facing: (1) How to identify a suitable reduced state space, i.e., a lower-dimensional subspace of the all-atom state space, onto which the full-atom dynamics can be projected without destroying its essential properties (by essential properties of the dynamics we mean their longest timescales and the transport and kinetic properties associated with them), (2) how to find a closed representation of the projected dynamics in terms of the resulting coarse graining coordinates (CGC), and (3) how to provide an ef.cient numerical realization. Our approach to (1) will rely on novel CG error estimators exploiting ideas from dimension-adaptive sparse quadrature. CG is then understood as a projection of the transfer operator, re.ecting the full-atom dynamics, onto the reduced state space. Concerning (2) we will investigate existing, prevalent methods, including very recent approaches to the extraction of CG dynamics for deterministic, discrete dynamical systems, with respect to their applicability in the context of projected transfer operators. Basic multigrid ideas of scale separation and localization will be used in (3) to derive and analyze a multiscale discretization of the transfer operator together with a multilevel strategy integrating the adaptive concepts from (1) and (2). In order to validate our approach and compare it with existing, heuristic CG strategies, we will .rst apply this multilevel strategy to systems with fast-slow scale separation, where analytical results are available. We will also investigate whether existing strategies can be justi.ed by our abstract results. The theoretical considerations will be complemented by applications to small but realistic molecular systems. The long-term goal of the project is a multilevel coarse grained description of supramolecular aggregation processes related to neurodegenerative diseases, with simulations complemented and validated by experiments.

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