arXiv subjects

D. T. Crommelin

Publications and source records attributed to D. T. Crommelin.

4 recordsLinked to original sources

arXiv2025

Asymptotic behavior for a general class of spreading models

Growing literatures on epidemic and rumor dynamics show that infection and information coevolve. We present a unified framework for modeling the spread of infection and information: a general class of interaction-driven fluid-limit models expressed as coupled ODEs. The class includes the SIR epidemic model, the Daley-Kendall rumor model, and many extensions. For this general class, we derive theoretical results: under explicit graph-theoretic conditions, we obtain a classification of asymptotic behavior and motivate a conjecture of exponential decay for vanishing states. When these conditions are violated, the classification can fail, and decay may become non-exponential (e.g., algebraic). In deriving the main result, we establish asymptotic stability and $L^1$-integrability properties for state variables. Alongside these results, we introduce the dependency graph that captures outflow dependencies and offers a new angle on the structure of this model class. Finally, we illustrate the results with several examples, including a heterogeneous rumor model and a rumor-dependent SIR model, showing how small changes to the dependency graph can flip asymptotic behavior and reshape epidemic trajectories.

q-bio.PE↗

arXiv2019

Efficient estimation of divergence-based sensitivity indices with Gaussian process surrogates

We consider the estimation of sensitivity indices based on divergence measures such as Hellinger distance. For sensitivity analysis of complex models, these divergence-based indices can be estimated by Monte-Carlo sampling (MCS) in combination with kernel density estimation (KDE). In a direct approach, the complex model must be evaluated at every input point generated by MCS, resulting in samples in the input-output space that can be used for density estimation. However, if the computational cost of the complex model strongly limits the number of model evaluations, this direct method gives large errors. We propose to use Gaussian process (GP) surrogates to increase the number of samples in the combined input-output space. By enlarging this sample set, the KDE becomes more accurate, leading to improved estimates. To compare the GP surrogates, we use a surrogate constructed by samples obtained with stochastic collocation, combined with Lagrange interpolation. Furthermore, we propose a new estimation method for these sensitivity indices based on minimum spanning trees. Finally, we also propose a new type of sensitivity indices based on divergence measures, namely direct sensitivity indices. These are useful when the input data is dependent.

math.ST↗

arXiv2019

Clustering-based collocation for uncertainty propagation with multivariate dependent inputs

In this article, we propose the use of partitioning and clustering methods as an alternative to Gaussian quadrature for stochastic collocation. The key idea is to use cluster centers as the nodes for collocation. In this way, we can extend the use of collocation methods to uncertainty propagation with multivariate, dependent input, in which the output approximation is piecewise constant on the clusters. The approach is particularly useful in situations where the probability distribution of the input is unknown, and only a sample from the input distribution is available. We examine several clustering methods and assess the convergence of collocation based on these methods both theoretically and numerically. We demonstrate good performance of the proposed methods, most notably for the challenging case of nonlinearly dependent inputs in higher dimensions. Numerical tests with input dimension up to 16 are included, using as benchmarks the Genz test functions and a test case from computational fluid dynamics (lid-driven cavity flow).

math.NA↗

arXiv2018

Quantifying dependencies for sensitivity analysis with multivariate input sample data

We present a novel method for quantifying dependencies in multivariate datasets, based on estimating the Rényi entropy by minimum spanning trees (MSTs). The length of the MSTs can be used to order pairs of variables from strongly to weakly dependent, making it a useful tool for sensitivity analysis with dependent input variables. It is well-suited for cases where the input distribution is unknown and only a sample of the inputs is available. We introduce an estimator to quantify dependency based on the MST length, and investigate its properties with several numerical examples. To reduce the computational cost of constructing the exact MST for large datasets, we explore methods to compute approximations to the exact MST, and find the multilevel approach introduced recently by Zhong et al. (2015) to be the most accurate. We apply our proposed method to an artificial testcase based on the Ishigami function, as well as to a real-world testcase involving sediment transport in the North Sea. The results are consistent with prior knowledge and heuristic understanding, as well as with variance-based analysis using Sobol indices in the case where these indices can be computed.

math.ST↗