Automation of Computational Mathematical Modeling

Doctoral thesis, 2004

This thesis concerns the Automation of Computational Mathematical Modeling (CMM), involving the key steps of automation of (i) discretization, (ii) discrete solution, (iii) error control, (iv) modeling, and (v) optimization. The automation of (i)--(ii) is realized through multi-adaptive Galerkin methods on general discretizations of space-time with time steps variable in space. The automation of (iii) is realized through a posteriori error estimation based on residuals of computed solutions and stability factors or stability weights obtained by the solution of associated dual problems. The automation of (iv) is studied in a model case, while that of (v) is reduced to an application of (i)--(iv). An open-source software system for the Automation of Computational Mathematical Modeling is developed and implemented, based on general order multi-adaptive Galerkin methods, including automatic evaluation of variational forms, automatic discrete solution, automatic generation and solution of dual problems, and automatic model reduction.