Scientific Computing Seminar
Spring 2008

Kevin Long
Department of Mathematics & Statistics

Tuesday, 3/4/2008
4:00 -- 5:00 p.m.
Experimental Sciences Building, Room 120

"Unifying Intrusive Parallel Finite Element Computations through Software-Based Fréchet Differentiation"


Use of computational simulation in scientific research or engineering analysis usually requires not only the solution of a forward problem, but simulation-based optimization as well, followed by estimation of uncertainties due to modeling errors, computational errors, and uncertain measurements. The most efficient algorithms for large-scale simulation-based optimization, sensitivity analysis, and uncertainty quantification are intrusive, meaning that implementation of the algorithm requires modification of the simulation code to provide certain operations not needed by a forward simulation, and thus not provided by most simulation codes. Furthermore, intrusive algorithms are becoming increasingly important even for forward problems, because physics-based preconditioners usually require non-standard operations. The development time required for implementation of these operations and their integration into large, complex simulation codes has become a significant barrier to transfer of new algorithms from the research lab to production environments.

This work develops a mathematical framework in which intrusive and non-intrusive PDE simulation methodologies are unified. This conceptual unification is of practical importance because in this framework the core mathematical operation involved in taking a PDE to a discrete system of equations can be automated, allowing an efficient simulation code to assemble itself given a high-level symbolic description of the weak form of the PDE and of its associated geometry, boundary conditions, and discretization scheme. If requested, this process can also automatically provide the additional operations needed by intrusive algorithms. What is perhaps surprising is that simulation code implemented using this general-purpose method can actually be more efficient than hand-tuned special-purpose simulators.

In this talk I will briefly survey intrusive algorithms for PDE-based computation, then go on to develop the mathematical foundations for their unification. The central lemmas are simply stated but of subtle importance; I will show how they provide a foothold for automation of simulation development, automation of intrusive modifications, and automation of performance optimizations. I will then present quantitative performance results and examples of application to problems in PDE-constrained optimization.

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