arXiv subjects

Fabian Michel

Publications and source records attributed to Fabian Michel.

8 recordsLinked to original sources

arXiv2025

Wasserstein error bounds for aggregations of continuous-time Markov chains

We study the approximation of a (finite) continuous-time Markov chain by a Markov chain on a reduced state space, and we provide formal error bounds for the approximated transient distributions in the Wasserstein distance. These bounds extend previous work on error bounds in the total variation distance, and are the first step towards a generalization to continuous-time Markov processes with continuous state spaces. A Wasserstein matrix norm is used to bound the error caused by the lower-dimensional approximation of the dynamics. In order to control the propagation of the accumulated error, we rely on the concept of coarse Ricci curvature of a Markov chain. The practical applicability of the presented bounds depends strongly on the curvature of the chain. Examples for CTMCs taken from the literature (where we added a metric on the state space) show that a negative curvature results in exponentially exploding bounds. On the other hand, certain CTMCs which we call translation-invariant always have non-negative curvature. When measuring the error in the total variation distance (a special case of the Wasserstein distance with the discrete metric), the curvature is also always non-negative. If it is strictly positive, the bounds presented in this paper are an improvement over previous work.

math.PR↗

arXiv2025

The Arnoldi Aggregation for Approximate Transient Distributions of Markov Chains

The paper proposes a new aggregation method, based on the Arnoldi iteration, for computing approximate transient distributions of Markov chains. This aggregation is not partition-based, which means that an aggregate state may represent any portion of any original state, leading to a reduced system which is not a Markov chain. Results on exactness (in case the algorithm finds an invariant Krylov subspace) and minimality of the size of the Arnoldi aggregation are proven. For practical use, a heuristic is proposed for deciding when to stop expanding the state space once a certain accuracy has been reached. Apart from the theory, the paper also includes an extensive empirical section where the new aggregation algorithm is tested on several models and compared to a lumping-based state space reduction scheme.

math.PR↗

arXiv2025

Formal Approximations of the Transient Distributions of the M/G/1 Workload Process

This paper calculates transient distributions of a special class of Markov processes with continuous state space and in continuous time, up to an explicit error bound. We approximate specific queues on R with one-sided Lévy input, such as the M/G/1 workload process, with a finite-state Markov chain. The transient distribution of the original process is approximated by a distribution with a density which is piecewise constant on the state space. Easy-to-calculate error bounds for the difference between the approximated and actual transient distributions are provided in the Wasserstein distance. Our method is fast: to achieve a practically useful error bound, it usually requires only a few seconds or at most minutes of computation time.

math.PR↗

arXiv2024

Formal Error Bounds for the State Space Reduction of Markov Chains

We study the approximation of a Markov chain on a reduced state space, for both discrete- and continuous-time Markov chains. In this context, we extend the existing theory of formal error bounds for the approximated transient distributions. As a special case, we consider aggregated (or lumped) Markov chains, where the state space reduction is achieved by partitioning the state space into macro states. In the discrete-time setting, we bound the stepwise increment of the error, and in the continuous-time setting, we bound the rate at which the error grows. In addition, the same error bounds can also be applied to bound how far an approximated stationary distribution is from stationarity. Subsequently, we compare these error bounds with relevant concepts from the literature, such as exact and ordinary lumpability, as well as deflatability and aggregatability. These concepts define stricter than necessary conditions to identify settings in which the aggregation error is zero. We also consider possible algorithms for finding suitable aggregations for which the formal error bounds are low, and we analyse first experiments with these algorithms on a range of different models.

math.PR↗

arXiv2023

Interacting Edge-Reinforced Random Walks

We consider the edge-reinforced random walk with multiple (but finitely many) walkers which influence the edge weights together. The walker which moves at a given time step is chosen uniformly at random, or according to a fixed order. First, we consider 2 walkers with linear reinforcement on a line graph comprising three nodes. We show that the edge weights evolve similarly to the setting with a single walker which corresponds to a Pólya urn. In particular, the left edge weight proportion is a martingale at certain stopping times, showing that a (random) limiting proportion exists. We then look at an arbitrary number of walkers on Z with very general reinforcement. We show that in this case, the behaviour is also the same as for a single walker: either all walkers are recurrent or all walkers have finite range. In the particular case of reinforcements of "sequence type", we give a criterion for recurrence.

math.PR↗

arXiv2023

Variations on Reinforced Random Walks

This thesis examines edge-reinforced random walks with some modifications to the standard definition. An overview of known results relating to the standard model is given and the proof of recurrence for the standard linearly edge-reinforced random walk on bounded degree graphs with small initial edge weights is repeated. Then, the edge-reinforced random walk with multiple walkers influencing each other is considered. The following new results are shown: on a segment of three nodes, the edge weights resemble a Pólya urn and the fraction of the edge weights divided by the total weight forms a converging martingale. On Z, the behavior is the same as for a single walker - either all walkers have finite range or all walkers are recurrent. Finally, edge-reinforced random walks with a bias in a certain direction are analysed, in particular on Z. It is shown that the bias can introduce a phase transition between recurrence and transience, depending on the strength of the bias, thus fundamentally altering the behavior in comparison to the standard linearly reinforced random walk.

math.PR↗

arXiv2023

Linearly Edge-Reinforced Random Walks

This thesis examines linearly edge-reinforced random walks on infinite trees. In particular, recurrence and transience of such random walks on general (fixed) trees as well as on Galton-Watson trees (i.e. random trees) is characterized, and shown to be related to the branching number of these trees and a so-called reinforcement parameter. A phase transition from transience to recurrence takes place at a critical parameter value. As a tool, random walks in random environment are introduced and known results are repeated, together with detailed proofs. A result on quasi-independent percolation is proved as a by-product. Finally, for the edge-reinforced random walk on Z, the existence of a kind of stationary / limiting distribution with finite moments is shown.

math.PR↗

arXiv2020

Surface Diffusion Control Enables Tailored Aspect Ratio Nanostructures in Area-Selective Atomic Layer Deposition

Area-selective atomic layer deposition is a key technology for modern microelectronics as it eliminates alignment errors inherent to conventional approaches by enabling material deposition only in specific areas. Typically, the selectivity originates from surface modifications of the substrate that allow or block precursor adsorption. The control of the deposition process currently remains a major challenge as the selectivity of the no-growth areas is lost quickly. Here, we show that surface modifications of the substrate strongly manipulate the surface diffusion. The selective deposition of TiO$_2$ on poly (methyl methacrylate) and SiO$_2$ yields localized nanostructures with tailored aspect ratios. Controlling the surface diffusion allows to tune such nanostructures as it boosts the growth rate at the interface of the growth and no-growth areas. Kinetic Monte-Carlo calculations reveal that species move from high to low diffusion areas. Further, we identify the catalytic activity of TiCl$_4$ during the formation of carboxylic acid on poly (methyl methacrylate) as the reaction mechanism responsible for the loss of selectivity, and show that process optimization leads to higher selectivity. Our work enables the precise control of area-selective atomic layer deposition on the nanoscale, and offers new strategies in area-selective deposition processes by exploiting surface diffusion effects.

cond-mat.mtrl-sci↗