Sparse Automatic Differentiation for Complex Networks of Differential-Algebraic Equations Using Abstract Elementary Algebra
This work addresses the limitation of built-in data types in mainstream languages for complex mathematical modeling, offering a more flexible approach for numerical solver developers.
The authors propose using abstract elementary algebra in numerical libraries to match the flexibility of domain-specific languages, demonstrating a sparse automatic differentiation method for complex networks of differential-algebraic equations that scales linearly with system size.
Most numerical solvers and libraries nowadays are implemented to use mathematical models created with language-specific built-in data types (e.g. real in Fortran or double in C) and their respective elementary algebra implementations. However, the built-in elementary algebra typically has limited functionality and often restricts the flexibility of mathematical models and the analysis types that can be applied to those models. To overcome this limitation, a number of domain-specific languages such as gPROMS or Modelica with more feature-rich built-in data types have been proposed. In this paper, we argue that if numerical libraries and solvers are designed to use abstract elementary algebra rather than the language-specific built-in algebra, modern mainstream languages can be as effective as any domain-specific language. We illustrate our ideas using the example of sparse Jacobian matrix computation. We implement an automatic differentiation method that takes advantage of sparse system structures and is straightforward to parallelize in a distributed memory setting. Furthermore, we show that the computational cost scales linearly with the size of the system.