Abstract
There is an interplay between models, specified by variables and equations, and their connections to one another. This dichotomy should be reflected in the abstract as well. Without referring to the models directly – only that a model consists of spaces and maps between them – the most readily apparent feature of a multi-model system is its topology. We propose that this topology should be modeled first, and then the spaces and maps of the individual models be specified in accordance with the topology. Axiomatically, this construction leads to sheaves. Sheaf theory provides a toolbox for constructing predictive models described by systems of equations. Sheaves are mathematical objects that manage the combination of bits of local information into a consistent whole. The power of this approach is that complex models can be assembled from smaller, easier-to-construct models. The models discussed in this chapter span the study of continuous dynamical systems, partial differential equations, probabilistic graphical models, and discrete approximations of these models.
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Which is in general not the direct sum, since P may be infinite!
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A poset is locally finite if for every pair x, y ∈ P, the set {z ∈ P: x ≤ z ≤ y} is finite.
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A hypergraph is literally a set of sets of vertices. Each element of a hypergraph is called a hyperedge. A hypergraph is given a direction by specifying the order of vertices in each hyperedge.
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We take signed measures rather than probability measures for algebraic convenience. Throughout, if we start with probability measures, they remain so. Thus nothing is lost by this perspective.
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This partial order is the 1-skeleton of the nerve of \(\mathcal{U}\).
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Acknowledgements
The author would like to thank the anonymous referees for the thoughtful suggestions that have improved this chapter considerably. This work was partially supported under the DARPA SIMPLEX program through SPAWAR, Federal contract N66001-15-C-4040.
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Robinson, M. (2017). Sheaf and Duality Methods for Analyzing Multi-Model Systems. In: Pesenson, I., Le Gia, Q., Mayeli, A., Mhaskar, H., Zhou, DX. (eds) Recent Applications of Harmonic Analysis to Function Spaces, Differential Equations, and Data Science. Applied and Numerical Harmonic Analysis. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-55556-0_8
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