A Hybrid Automaton Framework for Modeling Cyber-Physical Systems with Partial Differential Equation Dynamics
This paper presents a novel hybrid automaton framework called Partial Differential Hybrid Automata (PDHA) that can model cyber-physical systems with continuous dynamics described by partial differential equations.
Convergence Rates for Approximating Random Evolution Equations with Finite-Dimensional Noise
This paper presents an abstract framework to obtain convergence rates for the approximation of random evolution equations corresponding to a random family of forms determined by finite-dimensional noise. The full discretisation error in space, time, and randomness is considered, where polynomial chaos expansion (PCE) is used for the semi-discretisation in randomness.
Rigorous Enclosure of Self-Similar Solutions to Semilinear PDEs on Unbounded Domains
The authors develop computer-assisted tools to rigorously enclose self-similar solutions of semilinear partial differential equations on unbounded domains, using a spectral approach based on an eigenbasis of the operator L = -Δ - x⋅∇.
Geometric Structure and Discretization of Adjoint Systems for Evolutionary Partial Differential Equations
The adjoint system associated with an evolutionary partial differential equation has an infinite-dimensional Hamiltonian structure, which is useful for connecting the fully continuous, semi-discrete, and fully discrete levels. The discretize-then-optimize and optimize-then-discretize methods can be uniquely characterized in terms of an adjoint-variational quadratic conservation law.
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