DynamicBounds.jl: Validated Bounds and Relaxations of Differential Equations

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This package provides An abstraction layer for constructing valid relaxations and bounds on the solution sets (and numerical solutions of) parametric ODEs, and DAEs. Evaluations of the parametric ODES/DAEs are made available by linking the specific integrator to an equivalent evaluation using solvers present in DifferentialEquations.jl.

The abstraction layer for DynamicBounds is divided into three major parts. First, there are problems (<:) which hold all the information required to define a well-posed parametric differential equation problem. Second, there are integrators (<:) which hold all the information require to compute relaxations of the problem or integrate the parametric differential equation problem at a particular parameter value.

• DifferentialInequality: Provides valid bounds and relaxations of the numerical solutions of systems of ODEs via second-order implicit methods.
• DiscretizeRelax: Provides valid bounds and relaxations of the numerical solutions of systems of ODEs via second-order implicit methods.
  • Wilhelm2019: Provides valid bounds and relaxations of the numerical solutions of systems of ODEs via second-order implicit methods.
  • Wilhelm2019: Provides valid bounds and relaxations of the numerical solutions of systems of ODEs via second-order implicit methods.

• Define the new integrator structure and extend make.
  • The integrator should be an abstract subtype of AbstractODERelaxIntegator for parametric systems of ODEs.
  • The integrator should be an abstract subtype of AbstractDAERelaxIntegator for parametric systems of DAEs.
  • Otherwise, it should be a subtype of abstract subtype of AbstractDERelaxIntegator associated with a specific problem form.
• Extend relax!
• Extend integrate!
• Extend the support/set/get/get!/getall! for all supported attributes
  • Fully extending these functions for each attribute is desirable but also can be a burden. We recommend the following as a minimal extension:
    • For interval-only methods:
      • get!(integrator, Value{Nominal}(), value)
      • get!(integrator, Gradient{Nominal}(), value)
      • get!(integrator, Gradient{Nominal}(), value)

    • For relaxation methods:

    • set!(integator, Gradient())
    • set!(integrator, Value{Nominal}())
• Define new problems by adding a structure <: AbstractDERelaxProblem
• Add a corresponding structure <: AbstractRelaxationAttribute to hold the solution information.
• The extend the support and get functions for all supported attributes

• ReachabilityAnalysis.jl: A well-developed package in Julia for novel reachability approaches. There is some overlap between the functionality enclosed in this package and DynamicBounds.jl. Both packages provide methods for performing reachability analysis. However, DynamicBounds.jl is meant to provide a specialized interface to additionally query dynamic relaxations and supply optimization problems with a structured abstraction layer such that optimizers may be written in a reachability (or relaxation) algorithm agnostic fashion.