Two Steps Towards Automating Efficient Solution of Inverse Problems

We have combined symbolic mathematics with code generation to create a user–friendly environment for specifying finite element methods. The motivation behind this approach is to automate the boring and error-prone task of implementing variational forms of PDEs, which is a cornerstone when developing finite element simulators. By employing a symbolic engine in a high–level language we allow the user to specify the weak form of the PDE in an abstract and user–friendly format. Furthermore, the symbolic framework allows us to do certain calculations like differentiation automatically. In addition to the symbolic framework, we use a high–level library for expressing linear algebra algorithms in terms of block matrices and block preconditioners. Our efforts have resulted in the open source software package SyFi [3], which is part of the FEniCS project [9]. SyFi stands for Symbolic Finite elements and is implemented in C++ and Python. SyFi is built on top of the symbolic C++ library GiNaC [5] and uses its Python interface Swiginac [17]. In addition to SyFi, we employ the in-house code PyCC [16] for block matrices and preconditioners. SyFi is largely divided in two parts, a kernel implementing finite elements etc. and a form compiler. The SyFi form compiler (SFC) takes as input variational forms and functionals on the following form (see also the FEniCS Form Compiler (FFC) [10, 11, 12, 13] and UFC (Unified Form-assembly Code) [1, 2]),