Direct Matrix Method for Computing Jacobians of Discretized Nonlinear Integro-Differential Operators
Computation of the Jacobian for discretized nonlinear integro-differential equations is an important step for many numerical algorithms. The direct matrix (or operator) method for analytically computing the Jacobian are relatively well-known in the scientific computing and numerical analysis communities and are typically organized as a collection of "rules of thumb" gained through experience. What is not often realized, however, is that these heuristic rules can be formally organized into a collection of simple rules reminiscent of the rules for calculating derivatives in single-variable calculus.
In this work, I have used a MATLAB-based notation to formalize several differentiation rules for computing analytical Jacobians for discretized nonlinear integro-differential equations (Chu, 2009). Using these rules can dramatically reduce
- the effort required to calculate analytical Jacobians and
- improve the computational performance of numerical algorithms that require the Jacobian.
Examples
- Electrochemical thin-films (source code)
- Double layer charging of metal colloid spheres at high applied electric fields (source code)
- Nanostructure shape (source code)
References
- Chu, K. T. (2009). A direct matrix method for computing analytical Jacobians of discretized nonlinear integro-differential equations. Journal of Computational Physics, 228(15), 5526–5538. https://doi.org/10.1016/j.jcp.2009.04.031 [pdf]