Symbolic Generation of Parallel Solvers for Unconstrained Optimization

Abstract

<p>In this thesis we consider the need to generate efficient solvers for inverse imaging problems in a way that supports both quality and performance in software, as well as flexibility in the underlying mathematical models. Many problem domains involve large data sizes and rates, and changes in mathematical modelling are limited only by researcher ingenuity and driven by the value of the application. We use a problem in Magnetic Resonance Imaging to illustrate this situation, motivate the need for better software tools and test the tools we develop. The problem is the determination of velocity profiles, think blood-flow patterns, using Phase Contrast Angiography. Despite the name, this method is completely noninvasive, not requiring the injection of contrast agents, but it is too time-consuming with present imaging and computing technology.</p> <p>Our approach is to separate the specification, the mathematical model, from the implementation details required for performance, using a custom language. The Domain Specific Language (DSL) provided to scientists allows for a complete abstraction from the highly optimized generated code. The mathematical DSL is converted to an internal representation we refer to as the Coconut Expression Library. Our expression library uses the directed acyclic graphs as an underlying data structure, which lends itself nicely to our automatic simplifications, differentiation and subexpression elimination. We show how parallelization and other optimizations are encoded as rules which are applied automatically rather than schemes that need to be implemented by the programmer in the low-level implementation. Finally, we present results, both in terms of numerical results and computational performance.</p>