## B03 - Multilevel coarse graining of multiscale problems

**Head(s):** Prof. Dr. Beate Koksch (FU Berlin), Prof. Dr. Roland Netz (FU Berlin), Prof. Dr. Christof Schütte (FU Berlin)

**Project member(s):** Cihan Ayaz, Dr. Andreas Bittracher, Laura Lavacchi, Dr. Johann Moschner, Dr. Wei Zhang

**Participating institution(s):** FU Berlin

### Project Summary

Molecular dynamics (MD) describes the dynamical behaviour of molecular systems in atomistic resolution. In many cases of biological interest one is interested in the dynamical behaviour on long timescales related to the biological function of the molecular system under consideration. Due to the cascades of time and length scales involved, extension of MD simulations to the biologically relevant timescales is infeasible in many cases, and thus coarse graining (CG) methods are required to construct lower-dimensional and easier evaluated surrogate models. The central requirement on such reduced models is that they can reproduce the interesting long timescales of the full dynamics. We are interested in CG methods using reaction coordinates (RC) or order parameters that are observables of the system but span a low-dimensional manifold allowing the characterisation of effects on long timescales by free energies or related concepts.

In this project, we are aiming at creating an integrated, mathematically verifiable multilevel CG scheme that is applicable to high-dimensional molecular systems and accurately reproduces the effective dynamics. We will do so by building on the progress made by the projects B02 and B03 in the first funding period, expanding the individually developed methods and combining them into a joint algorithmic framework. The basis of our efforts will be the theory of transition manifolds, recently developed in project B03, that places RC-based CG approaches onto a solid mathematical footing. Its core object is the transition manifold, the “dynamical backbone” of the effective dynamics, that characterises good RCs as parametrisations of this backbone. Using this theory, we developed a CG algorithm requiring only local information about the full system, that was demonstrated to be very accurate in first applications to molecular systems with some dominant long timescales. One primary task for the coming funding period is the extension of the theory of transition manifolds to systems with cascades of scales. We expect this will lead to an adaptive multilevel method for the computation of RCs and accurate simulation of the effective dynamics. One of the achievements of project B02, on the other hand, was the formulation of generalised Langevin dynamics for arbitrary one-dimensional RCs, and the data-based estimation of the corresponding drift-, diffusion- and memory terms. Further steps towards our goal are thus the extension of this model to multi-dimensional RCs, and integration into the aforementioned multilevel approach. The performance of the resulting joint multilevel algorithm will be demonstrated in application to different realistic molecular systems.

### Project publications

Koltai, P. and Wu, H. and Noé, F. and Schütte, Ch.
(2018)
*Optimal data-driven estimation of generalized Markov state models for non-equilibrium dynamics.*
Computation, 6(1)
(22).
ISSN 2079-3197 (online)

Kappler, J. and Daldrop, J.O. and Brünig, F.N. and Boehle, M.D. and Netz, R.R.
(2018)
*Memory-induced acceleration and slowdown of barrier crossing.*
J. Chem. Phys., 148
(1).
014903.

Klus, S. and Nüske, F. and Koltai, P. and Wu, H. and Kevrekidis, I. and Schütte, Ch. and Noé, F.
(2018)
*Data-driven model reduction and transfer operator approximation.*
Journal of Nonlinear Science, 28
(1).
pp. 1-26.

Polthier, L.
(2017)
*Algebraic Multilevel Methods for Markov Chains.*
SFB 1114 Preprint in arXiv:1711.04332
.
pp. 1-19.
(Unpublished)

Banisch, R. and Trstanova, Z. and Bittracher, A. and Klus, S. and Koltai, P.
(2017)
*Diffusion maps tailored to arbitrary non-degenerate Ito processes.*
SFB 1114 Preprint in arXiv:1710.03484
.
pp. 1-24.
(Unpublished)

Bittracher, A. and Koltai, P. and Klus, S. and Banisch, R. and Dellnitz, M. and Schütte, Ch.
(2017)
*Transition manifolds of complex metastable systems: Theory and data-driven computation of effective dynamics.*
Journal of Nonlinear Science
.
pp. 1-42.
ISSN 1432-1467 (online)

Klus, S. and Schütte, Ch.
(2016)
*Towards tensor-based methods for the numerical approximation of the Perron-Frobenius and Koopman operator.*
Journal of Computational Dynamics
.
ISSN 2158-2491

Klus, S. and Koltai, P. and Schütte, Ch.
(2016)
*On the numerical approximation of the Perron-Frobenius and Koopman operator.*
Journal of Computational Dynamics, 3
(1).
pp. 51-79.
ISSN 2158-2491

Klus, S. and Gelß, P. and Peitz, S. and Schütte, Ch.
(2016)
*Tensor-based dynamic mode decomposition.*
SIAM Journal on Scientific Computing
.
ISSN ISSN 1064-8275 (print); 1095-7197 (electronic)
(Submitted)

Kornhuber, R. and Yserentant, H.
(2016)
*Numerical Homogenization of Elliptic Multiscale
Problems by Subspace Decomposition.*
Multiscale Model. Simul., 14
(3).
pp. 1017-1036.
ISSN print: 1540-3459; online: 1540-3467

Enciso, M. and Schütte, Ch. and Delle Site, L.
(2015)
*Influence of pH and sequence in peptide aggregation via
molecular simulation.*
Journal of Chemical Physics, 143
(24).
p. 243130.
ISSN 0021-9606

Schuster, I. and Strathmann, H. and Paige, B. and Sejdinovic, Dino
(2015)
*Kernel Sequential Monte Carlo.*
SFB 1114 Preprint in arXiv:1510.03105
.
(Submitted)