Stability Results for Bounded Stationary Solutions of Reaction-Diffusion-ODE Systems

Chris, Kowall,; Marciniak-Czochra, Anna; Münnich, Finn

doi:10.48550/arxiv.2201.12748

Cited by 1 publication

(1 citation statement)

References 41 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…To leverage different data features, we use different distances of the spatial patterns. This is motivated by applicability of different functional setting in nonlinear stability analysis of reactiondiffusion equations [34]. Specifically, we use discrete equivalents of the following norms…”

Section: Multi-feature Correlation Integral Likelihoodmentioning

confidence: 99%

A Bayesian Approach to Modelling Biological Pattern Formation with Limited Data

Kazarnikov¹,

Scheichl²,

Haario³

et al. 2022

Preprint

Self Cite

View full text Add to dashboard Cite

Pattern formation plays an important role in the development of living organisms. Since the classical work of Alan Turing, a pre-eminent way of modelling has been through reactiondiffusion mechanisms. Alternative models have been proposed, that link dynamics of diffusing molecular signals with tissue mechanics. Model validation is complicated as in many experimental situations only the limiting, stationary regime of the pattern formation process is observable, without any knowledge of the transient behaviour or the initial state. To overcome this problem, the initial state of the model can be randomised. But then fixed values of the model parameters correspond to a family of patterns rather than a fixed stationary solution, and standard estimation approaches, such the least squares, are not suitable. Instead, statistical characteristics of the patterns should be compared, which is difficult given the typically limited amount of available data in practical applications. To deal with this problem, we extend the recently developed statistical approach (the Correlation Integral Likelihood method) for parameter identification by pattern data. We introduce modifications that allow increasing the accuracy of the identification process without resizing the data set. The proposed approach is tested using different classes of pattern formation models and severely limited data sets. For all considered equations, parallel, GPU-based implementations of the numerical solvers with efficient time stepping schemes are provided.

show abstract

Section: Multi-feature Correlation Integral Likelihoodmentioning

confidence: 99%