A Hybrid Galerkin–Monte-Carlo Approach to Higher-Dimensional Population Balances in Polymerization Kinetics

Population balance models describing not only the chain-length distribution of a polymer but also additional properties like branching or composition are still difficult to solve numerically. For simulation of such systems two essentially different approaches are discussed in the literature: deterministic solvers based on rate equations and stochastic Monte-Carlo (MC) strategies based on chemical master equations. The paper presents a novel hybrid approach to polymer reaction kinetics that combines the best of these two worlds. We discuss the theoretical conditions of the algorithm, describe its numerical realization, and show that, if applicable, it is more efficient than full-scale MC approaches and leads to more detailed information in additional property indices than deterministic solvers

Fully-Automatic Multiresolution Idealization for Filtered Ion Channel Recordings: Flickering Event Detection

We propose a new model-free segmentation method, JULES, which combines recent statistical multiresolution techniques with local deconvolution for idealization of ion channel recordings. The multiresolution criterion takes into account scales down to the sampling rate enabling the detection of flickering events, i.e., events on small temporal scales, even below the filter frequency. For such small scales the deconvolution step allows for a precise determination of dwell times and, in particular, of amplitude levels, a task which is not possible with common thresholding methods. This is confirmed theoretically and in a comprehensive simulation study. In addition, JULES can be applied as a preprocessing method for a refined hidden Markov analysis. Our new methodolodgy allows us to show that gramicidin A flickering events have the same amplitude as the slow gating events. JULES is available as an R function jules in the package clampSeg

Observation Uncertainty in Reversible Markov Chains

In many applications one is interested in finding a simplified model which captures the essential dynamical behavior of a real life process. If the essential dynamics can be assumed to be (approximately) memoryless then a reasonable choice for a model is a Markov model whose parameters are estimated by means of Bayesian inference from an observed time series. We propose an efficient Monte Carlo Markov Chain framework to assess the uncertainty of the Markov model and related observables. The derived Gibbs sampler allows for sampling distributions of transition matrices subject to reversibility and/or sparsity constraints. The performance of the suggested sampling scheme is demonstrated and discussed for a variety of model examples. The uncertainty analysis of functions of the Markov model under investigation is discussed in application to the identification of conformations of the trialanine molecule via Robust Perron Cluster Analysis (PCCA+)

Optimal Fuzzy Aggregation of Networks

This paper is concerned with the problem of fuzzy aggregation of a network with non-negative weights on its edges into a small number of clusters. Specifically we want to optimally define a probability of affiliation of each of the n nodes of the network to each of m < n clusters or aggregates. We take a dynamical perspective on this problem by analyzing the discrete-time Markov chain associated with the network and mapping it onto a Markov chain describing transitions between the clusters. We show that every such aggregated Markov chain and affiliation function can be lifted again onto the full network to define the so-called lifted transition matrix between the nodes of the network. The optimal aggregated Markov chain and affiliation function can then be determined by minimizing some appropriately defined distance between the lifted transition matrix and the transition matrix of the original chain. In general, the resulting constrained nonlinear minimization problem comes out to have continuous level sets of minimizers. We exploit this fact to devise an algorithm for identification of the optimal cluster number by choosing specific minimizers from the level sets. Numerical minimization is performed by some appropriately adapted version of restricted line search using projected gradient descent. The resulting algorithmic scheme is shown to perform well on several test examples

Importance sampling in path space for diffusion processes with slow-fast variables

Importance sampling is a widely used technique to reduce the variance of a Monte Carlo estimator by an appropriate change of measure. In this work, we study importance sampling in the framework of diffusion process and consider the change of measure which is realized by adding a control force to the original dynamics. For certain exponential type expectation, the corresponding control force of the optimal change of measure leads to a zero-variance estimator and is related to the solution of a Hamilton-Jacobi-Bellmann equation. We focus on certain diffusions with both slow and fast variables, and the main result is that we obtain an upper bound of the relative error for the importance sampling estimators with control obtained from the limiting dynamics. We demonstrate our approximation strategy with a simple numerical example