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Institute
VoronoiGraph.jl
(2023)
This Package implements a variation of the Voronoi Graph Traversal algorithm by Polianskii and Pokorny [1]. It constructs a Voronoi Diagram from a set of points by performing a random walk on the graph of the vertices of the diagram. Unlike many other Voronoi implementations this algorithm is not limited to 2 or 3 dimensions and promises good performance even in higher dimensions.
This article addresses the problem of estimating the Koopman generator of a Markov process. The direct computation of the infinitesimal generator is not easy because of the discretization of the state space, in particular because of the trade-off inherent in the choice of the best lag time to study the process. Short lag times implies a strong discretization of the state space and a consequent loss of Markovianity. Large lag times bypass events on fast timescales. We propose a method to approximate the generator with the computation of the Newton polynomial extrapolation. This technique is a multistep approach which uses as its input Koopman transfer operators evaluated for a series of lag times. Thus, the estimated infinitesimal generator combines information from different time resolutions and does not bias only fast- or slow-decaying dynamics. We show that the multi-scale Newton method can improve the estimation of the generator in comparison to the computation using finite difference or matrix logarithm methods.
The Robust Perron Cluster Analysis (PCCA+) has become a popular spectral clustering algorithm for coarse-graining transition matrices of nearly decomposable Markov chains with transition states. Originally developed for reversible Markov chains, the algorithm only worked for transition matrices with real eigenvalues. In this paper, we therefore extend the theoretical framework of PCCA+ to Markov chains with a complex eigen-decomposition. We show that by replacing a complex conjugate pair of eigenvectors by their real and imaginary components, a real representation of the same subspace is obtained, which is suitable for the cluster analysis. We show that our approach leads to the same results as the generalized PCCA+ (GPCCA), which replaces the complex eigen-decomposition by a conceptually more difficult real Schur decomposition. We apply the method on non-reversible Markov chains, including circular chains, and demonstrate its efficiency compared to GPCCA. The experiments are performed in the Matlab programming language and codes are provided.
cmdtools
(2021)
Estimating the rate of rare conformational changes in molecular systems is one of the goals of molecular dynamics simulations. In the past few decades, a lot of progress has been done in data-based approaches toward this problem. In contrast, model-based methods, such as the Square Root Approximation (SqRA), directly derive these quantities from the potential energy functions. In this article, we demonstrate how the SqRA formalism naturally blends with the tensor structure obtained by coupling multiple systems, resulting in the tensor-based Square Root Approximation (tSqRA). It enables efficient treatment of high-dimensional systems using the SqRA and provides an algebraic expression of the impact of coupling energies between molecular subsystems. Based on the tSqRA, we also develop the projected rate estimation, a hybrid data-model-based algorithm that efficiently estimates the slowest rates for coupled systems. In addition, we investigate the possibility of integrating low-rank approximations within this framework to maximize the potential of the tSqRA.
The dominant eigenfunctions of the Koopman operator characterize the metastabilities and slow-timescale dynamics of stochastic diffusion processes. In the context of molecular dynamics and Markov state modeling, they allow for a description of the location and frequencies of rare transitions, which are hard to obtain by direct simulation alone. In this article, we reformulate the eigenproblem in terms of the ISOKANN framework, an iterative algorithm that learns the eigenfunctions by alternating between short burst simulations and a mixture of machine learning and classical numerics, which naturally leads to a proof of convergence. We furthermore show how the intermediate iterates can be used to reduce the sampling variance by importance sampling and optimal control (enhanced sampling), as well as to select locations for further training (adaptive sampling). We demonstrate the usage of our proposed method in experiments, increasing the approximation accuracy by several orders of magnitude.
Voronoi Graph - Improved raycasting and integration schemes for high dimensional Voronoi diagrams
(2024)
The computation of Voronoi Diagrams, or their dual Delauney triangulations is difficult in high dimensions. In a recent publication Polianskii and Pokorny propose an iterative randomized algorithm facilitating the approximation of Voronoi tesselations in high dimensions. In this paper, we provide an improved vertex search method that is not only exact but even faster than the bisection method that was previously recommended. Building on this we also provide a depth-first graph-traversal algorithm which allows us to compute the entire Voronoi diagram. This enables us to compare the outcomes with those of classical algorithms like qHull, which we either match or marginally beat in terms of computation time. We furthermore show how the raycasting algorithm naturally lends to a Monte Carlo approximation for the volume and boundary integrals of the Voronoi cells, both of which are of importance for finite Volume methods. We compare the Monte-Carlo methods to the exact polygonal integration, as well as a hybrid approximation scheme.
Markov processes serve as foundational models in many scientific disciplines,
such as molecular dynamics, and their simulation forms a common basis for
analysis. While simulations produce useful trajectories, obtaining macroscopic
information directly from microstate data presents significant challenges. This
paper addresses this gap by introducing the concept of membership functions
being the macrostates themselves. We derive equations for the holding times of
these macrostates and demonstrate their consistency with the classical definition.
Furthermore, we discuss the application of the ISOKANN method for learning
these quantities from simulation data. In addition, we present a novel method
for extracting transition paths based on the ISOKANN results and demonstrate
its efficacy by applying it to simulations of the 𝜇-opioid receptor. With this
approach we provide a new perspective on analyzing the macroscopic behaviour
of Markov systems.
ISOKANN.jl
(2024)