SciMLOperators.jl: The SciML Operators Interface

Many functions, from linear solvers to differential equations, require the use of matrix-free operators in order to achieve maximum performance in many scenarios. SciMLOperators.jl defines the abstract interface for how operators in the SciML ecosystem are supposed to be defined. It gives the common set of functions and traits which solvers can rely on for properly performing their tasks. Along with that, SciMLOperators.jl provides definitions for the basic standard operators which are used in building blocks for most tasks, both simplifying the use of operators while also demonstrating to users how such operators can be built and used in practice.

Why SciMLOperators?

SciMLOperators.jl has the design that is required in order to be used in all scenarios of equation solvers. For example, Magnus integrators for differential equations require defining an operator $u' = A(t)u$, while Munthe-Kaas methods require defining operators of the form $u' = A(u)u$. Thus the operators need some form of time and state dependence which the solvers can update and query when they are non-constant (update_coefficients!). Additionally, the operators need the ability to act like "normal" functions for equation solvers. For example, if A(u,p,t) has the same operation as update_coefficients(A,u,p,t); A*u, then A can be used in any place where a differential equation definition f(u,p,t) is used without requring the user or solver to do any extra work.

Thus while previous good efforts for matrix-free operators have existed in the Julia ecosystem, such as LinearMaps.jl, those operator interfaces lack these aspects in order to actually be fully seamless with downstream equation solvers. This necessitates the definition and use of an extended operator interface with all of these properties, hence the AbstractSciMLOperator interface.